Control and Experimental Evaluation of Speed-variable Switched Differential Pump Concept

Transcription

1 Master's Thesis Control and Experimental Evaluation of Speed-variable Switched Differential Pump Concept MCE4-121 Morten Grønkjær Henrik Rahn Aalborg University Board of Studies of Energy June 9 th 215

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3 Title: Control and Experimental Evaluation of Speed-variable Switched Differential Pump Concept Semester: 1 th Semester theme: Master s Thesis in Mechatronic Control Engineering Project period: 2/ /6-215 ECTS: 3 Supervisor: Lasse Schmidt, Torben Ole Andersen Project group: MCE4-121 SYNOPSIS: Morten Grønkjær Henrik Rahn Copies: 5 Pages, total: 15 Appendices: A-C & DVD-ROM The Speed-variable Switched Differential Pump Concept (SvSDP) is a continuation of the energy efficient hydraulic Speed-variable Differential Pump (SvDP) concept original proposed by Bosch Rexroth A/S. The SvSDP consist of three gear pumps driven by a single electric motor connected directly to the inlet and outlet ports of differential cylinders. A SvSDP system is designed and dimensioned. To allow for general evaluation of the concept, a multi-purpose test bench for hydraulic systems is also designed and constructed. A detailed, non-linear dynamic model of the system is developed, and the parameterization of the model is conducted based on experimental results. The model is linearized and Relative Gain Array (RGA) analysis is used to identify the amount of cross couplings in the system. It is found that decentralized control of the system is not immediately applicable, why decoupling strategies are investigated. No experimental verification is made of the designed closed loop control systems, but from simulation results it is found, that the decoupled control strategy gives acceptable and consistent performance of the closed loop system. The simulated closed loop position tracking shows maximum tracking errors of less that 3 mm even for nonrealizable ramp trajectories. By signing this document, each member of the group confirms that all participated in the project work and thereby all members are collectively liable for the content of the report. Furthermore, all group members confirm that the report does not include plagiarism.

4 II Control and Experimental Evaluation of SvSDP Concept

5 Control and Experimental Evaluation of SvSDP Concept III Summary To provide a more energy efficient alternative to typical valve controlled hydraulic actuation systems, the Speed-variable Differential Pump (SvDP) concept has been proposed by Bosch Rexroth A/S. The basic idea of the SvDP concept is to be able to connect regular fixed-displacement gear pumps directly to the inlet and outlet ports of differential cylinders. The pumps are driven by the shaft of a single electrical motor, where the rotational direction of the pumps are opposite, such that one pump supplies flow to the inlet side of the cylinder, while the other is driven by the corresponding outlet flow. The magnitude of the flow is varied by controlling the speed of the electric motor, which removes the necessity of valves in the main flow. The experimental evaluation of the concept has been the topic of different student projects, but it has been found that the dynamic performance of the system was insufficient for closed loop position tracking applications. An improved concept, denoted as a Speed-variable Switched Differential Pump (SvSDP) system, utilizing an additional pump mounted on the same motor shaft, was proposed in the Fall of 214. The basis of this project is the experimental evaluation and verification of this solution proposal. The main advantage of the SvSDP system over the original SvDP system, is that it may be dimensioned such that an excess of flow will be available for pressure build-up, independent of the movement direction of the cylinder. This allows for operation of the hydraulic system at high enough pressure to ensure sufficient stiffness of the oil. Proportional valves on each side of the cylinder allow for control of the pressure level through throttling of excess flow, where the utilization of the proportional valves influences the overall energy efficiency of the system, depending on the operating conditions. An SvSDP system, which utilizes the main components available from the original SvDP system, is designed and dimensioned. To allow for general evaluation of the concept, a multi-purpose test bench for hydraulic systems is designed and constructed. The test bench consists of two similar hydraulic cylinders, both connected to a slide with variable inertia. One of the cylinders is connected to the designed SvSDP system, while the opposite cylinder may be controlled through use of a servo-valve to emulate various load cases, including both opposing and overrunning loads. A centralized data acquisition and control system for the test bench is also set up. A detailed, non-linear dynamic model of the system is developed, and the parameterization and verification of the model is conducted based on experimental results. To ensure that the developed model properly describes the physics of the system, specifically targeted test sequences are defined, which allows for isolated evaluation and verification of parameters relating to friction, oil bulk modulus and pump displacement and leakage. To investigate suitable application-independent control strategies for the SvSDP concept, the model is linearized and Relative Gain Array (RGA) analysis is used to identify the amount of cross couplings in the system. It is found that decentralized control of the system is not immediately applicable, why decoupling strategies are investigated. The decoupling strategies are developed with the goal of allowing free choice of valve utilization strategy, without otherwise influencing the performance of the control system. No experimental verification is made of the designed closed loop control systems, but from simulation results it is found, that the decoupled control strategy gives acceptable and consistent performance of the closed loop system, independent on the choice of valve utilization strategy. The simulated closed loop position tracking shows maximum tracking errors of less that 3 mm even for non-realizable ramp trajectories. It is furthermore found that the valve utilization scheme may be changed during operation, without introducing discontinuities in the control signals, as long as both proportional valves are closed during the scheme change.

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7 Preface V Preface This Master s Thesis titled "Control and Experimental Evaluation of Speed-variable Switched Differential Pump Concept" is the documentation for the work carried out by student group MCE4-121 at Aalborg University, Board of Studies of Energy. The project has been written in the spring semester of 215 with the semester theme Masters s Thesis in Mechatronic Control Engineering. The project has concerned the Speed-variable Switched Differential Pump (SvSDP) concept, which is an improvement proposal for the original Speed-variable Differential Pump (SvDP) concept, initially proposed by Bosch Rexroth A/S. The purpose of the project has been two-fold: 1) Experimentally evaluate and verify the performance of the SvSDP concept, especially with respect to dynamic performance. 2) Develop application-independent control strategies for the SvSDP concept. Due to the extent of the project, as well as various practical issues, it has, unfortunately, not been possible to obtain all the intended experiment results prior to the project submission. Persistent effort will be made to acquire the remaining data before the project evaluation. Acknowledgement Several industrial partners and suppliers have been contacted during the project period. The authors would particularly like to thank Bosch Rexroth A/S for providing personal aid and financial fonding of a large part of project. Additional thanks go to Fjero A/S for providing hydraulic cylinders for the project at a particularly favorable price. Reader s guide As this project is a direction extension of the work presented in the paper "System Optimization of New Speed-varible Differential Pump Concept" and corresponding Appendix Report, a significant amount of references are made to these sources. This project report has however, to the greatest possible extent, been written as a self-contained document, why prior knowledge of the SvDP concept should not be necessary. Still, the mentioned paper and corresponding appendix report may be found on the enclosed DVD for easy reference. The remaining information used in the report, has been found from reports, paper, web pages, literature and information from the supervisor. References for these sources use Harvard-style notation. The references are marked with company name or last name of the author as well as year of publication, [Name, Year]. All sources are found in the bibliography, sorted alphabetically by author name. Figures, tables and equations has number/letter notations, with the first number representing the appendix/supplement and the second number the order in the chapter, e.g. Fig Unless otherwise specified, all units are in SI-units. Enclosed to this report is a DVD containing the following: PDF version of the project report. PDF version of the paper "System Optimization of New Speed-varible Differential Pump Concept" and corresponding Appendix Report from Fall 214. Constructed Simulink models and MATLAB script. Developed LabVIEW software and SolidWorks CAD models. References, such as digital literature, web pages and data-sheets.

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9 Contents VII Contents 1 Introduction 1 2 Project Objective 9 I Design, Dimensioning and Setup of General Purpose Test Bench 11 3 Requirement Specification and Topology Selection Specification and Component Limitations Component Reuse and Limitations Test Bench Topology Presentation Topology Selection Design and Dimensioning of Test Bench and Components General Considerations Dimensioning of Hydraulic Components of Main System Model and Operation Map of the Load System Dimensioning of the Mechanical Frame and Components Implementation of Data Acquisition and Control System Software Functionality of the DAC Alternative DAC Proposal II Modeling and Development of Control Strategies for SvSDP System 41 6 Modeling and Parameterization of the SvSDP System Test Sequences for Parameter Determination Mechanical Model Hydraulic Model Control Specification and Model Linearization Purpose, Considerations and Limitations of Control System Model Linearization Verification of Linear Model Cross-Coupling Analysis Discussion of Control Strategies System Decoupling Methods Output Transformation Influence of Unidirectional Valve Flow Input Transformation Method Input Transformation Method Input Transformation Method Input Transformation Method Input Transformation Method Discussion of the Results of the Decoupling Analysis

10 VIII Control and Experimental Evaluation of SvSDP Concept 9 Design and Evaluation of SvSDP Control Strategies Design of the Pressure Level Control System Design of Motion Controller Energy Efficient Valve Utilization Conclusion 15 Appendices 17 A Dimensioning and Mounting Design of Cylinders 19 A.1 Cylinder Buckling A.2 Cylinder Mounting A.3 Stresses due to Misalignment B Strength Analysis of Mechanical Components 121 B.1 Strength of Bolted Friction Joints B.2 Stresses in the Frame Profile B.3 Stresses in Trunnion Pins, Bearings and Eyes B.4 Stresses in Base Plate, Double Eye and Pin Bolt B.5 End-stop Mechanism B.6 Sliding Rail Unit and Topology C Specification of Check Valve Manifold Block 135 Bibliography 137

11 Chapter 1 - Introduction 1 1 Introduction Hydraulic actuation systems are used in numerous industrial applications. Such a system may typically be comprised of a pump unit and a hydraulic cylinder, which is controlled through the use of a proportional valve as illustrated schematically in Fig M M Figure 1.1: Hydraulic cylinder controlled by a proportional valve. By varying the position of the valve spool, the pressure difference across the valve gives rise to a flow. To avoid excessive losses, the pressurized oil may be supplied through the use of a variable displacement pump driven by an electric motor at a fixed speed. The displacement of the pump is then controlled such that the desired pressure is maintained without throttling of excess flow. This allows for a typical energy efficiency of the pump unit of approximately 87% [Rasmussen et al., 1996, p. 171], however as the flow to and from the actuator has to pass through the proportional valve at a certain pressure drop, energy losses due to throttling are unavoidable. The necessary pressure drop for a given flow depends on the opening area of the valve, why a larger valve may be used to reduce the pressure drop, at the expense of an increased sensitivity to inaccuracies in the positioning of the valve spool. Additionally, for applications with high demands to the dynamic performance, it is advantageous to maintain an elevated minimum pressure in order to maintain sufficient stiffness of the hydraulic fluid and thereby increase the natural frequency of the system [Andersen and Hansen, 23, Sec. 3.5]. This, again, increases the pressure drop across the valve and thereby reduces the energy efficiency of the system. As an alternative to the typical proportional valve control of hydraulic cylinders, the Speed-variable Differential Pump (SvDP) Concept, illustrated in Fig. 1.2a, has been proposed by Bosch Rexroth A/S. In the original SvDP concept, two external gear pumps are mounted on the shaft of a single servo-motor, with the outlets directly connected to the cylinder chambers. The pumps are oppositely oriented such that for a given rotational direction, one of the pumps will pump oil into one cylinder chamber, while the other pump will remove oil from the opposite chamber. The nominal displacements of the pumps should therefore be chosen such that the desired piston velocity may be realized, and such that the ratio between the displacement of the two pumps is equal to the ratio between the piston and rod side areas of the cylinder. Due to the mechanical link between the two pumps any mismatch between the displacement ratio and cylinder area ratio will result in either an excess or deficiency of flow into the cylinder, i.e. both chamber pressures will either increase or decrease.

12 2 Control and Experimental Evaluation of SvSDP Concept M M Pressure relief valves Proportional valves M M Passive antisystem cavitation (a) Original SvDP concept. (b) Augmented SvDP concept proposed in [Madsen and Bertelsen, 213]. Figure 1.2: Speed-variable Differentiable Pump Concepts (SvDP). The pump and motor symbols corresponds to the configuration of the pumps for positive shaft velocities. To maintain bounded pressures and avoid cavitation, the original SvDP concept has been augmented with the additional valves shown in Fig. 1.2b, as proposed in [Madsen and Bertelsen, 213]. The combination of check valves constitutes a passive anti-cavitation system, while the proportional valves allow for active throttling of the flow to maintain acceptable pressure levels. This modified version of the SvDP concept has been evaluated experimentally in [Madsen and Bertelsen, 213] and [Nielsen and Jensen, 214], which has shown improved energy efficiency compared to a valve operated system, however at the expense of a reduced dynamic tracking performance, when used for closed loop piston position tracking. To improve the dynamic performance of the SvDP concept, the root cause of this deficiency has been analyzed in [Grønkjær and Rahn, 214a], and a solution has been proposed. To allow for this project report to be read without prior knowledge of the system or any previous work, the remainder of this chapter is a resume of the main results and findings presented in [Grønkjær and Rahn, 214a]. The model of the original SvDP system in [Grønkjær and Rahn, 214a] has been verified based on experimental results obtained by [Nielsen and Jensen, 214], such that the simulation method could be used for evaluation of the proposed solution concept. The reduction in dynamic performance is found to be most profound when one of the anticavitation systems is active, i.e. during low pressure operation. This is partly due to the reduced oil bulk modulus (stiffness), which depends on the oil pressure as shown in Fig The open loop bandwidth of the system is approximately halved as the pressures decreases, why the purpose of the solution concept is to allow for operation at heightened pressures in the entire operation area.

13 Chapter 1 - Introduction 3 Effective Bulk Modulus [bar] V %,air,atm =.5% V %,air,atm = 1.% V %,air,atm = 1.5% V %,air,atm = 2.% Pressure [bar] Figure 1.3: Bulk modulus (oil stiffness) as function of pressure, see Eq for formula, [Andersen and Hansen, 23]. From Fig. 1.3 it can be seen, that the bulk modulus flattens around bar, why this pressure level is a good compromise between fluid stiffness and minimum chamber pressures. Increasing the minimum pressure much above this level have only minimal influence on the bulk modulus and thereby the dynamic performance of the system. However, increasing the minimum chamber pressure influences the energy efficiency of the system, as the losses associated with throttling from the corresponding chamber are increased. The target minimum chamber pressure is denoted the set pressure and it has been chosen to be p set = 25 bar. As an aid in the solution concept generation and evaluation, the Match Ratio χ has been proposed. The match ratio describes the possibility of operating the system at heightened pressures. It is defined in Eq. 1.1, with Q in and Q out being the effective flow into and out of the cylinder respectively. As the pumps are non-ideal, the volumetric efficiency of the pumps change with shaft rotational velocity ω m and the pressure drop across the respective pump. The areas A in and A out corresponds to the cylinder piston and rod-side areas depending on which is associated with the inlet port and which is associated with the outlet port. χ = Q / in(ω m, p in ) Ain (1.1) Q out (ω m, p out ) A out The match ratio may be interpreted as follows: χ = 1 corresponds to the perfectly matched system, i.e. the idea of the original concept. χ < 1 corresponds to a system where the effective motor displacement is greater than the pump displacement, why the chamber pressures will decrease. χ > 1 corresponds to a system where the effective pump displacement is greater than the motor displacement, why the chamber pressures will increase. The proposed solution, denoted as the Speed-variable Switched Differential Pump concept (SvSDP) is illustrated in Fig. 1.4a.

14 4 Control and Experimental Evaluation of SvSDP Concept p A A p p B Ar M 1.3 M P1 P2 P3 Match Ratio [ ] p A, p B = 25 bar p A, p B = 5 bar p A, p B = 75 bar p A, p B = 1 bar Shaft Velocity [RPM] (a) Hydraulic diagram of the SvSDP concept. (b) Match ratio χ for the SvSDP. Figure 1.4: Speed-variable Switched Displacement Pump (SvSDP) concept. From comparison with Fig. 1.2b, the modification is seen to consist of the implementation of an additional pump, mounted on same shaft as the remaining pumps, as well as two additional check valves. The purpose of the additional pump is to ensure a surplus of flow into the cylinder regardless of the shaft rotational direction. The combination of the two pumps, P1 and P2, constitutes a single equivalent unit denoted as a Switched-Displacement Pump (SDP). The characteristic of the SDP is that the pumping displacement is greater than the motoring displacement. The resulting match ratio as a function of shaft velocity is sketched in Fig. 1.4b for different chamber pressures, assuming no utilization of the proportional valves. The velocity region in which χ < 1 is denoted the critical range and as seen in Fig. 1.4b, this range is pressure dependent. By purposefully mismatching the pump displacement ratio to the cylinder area ratio even more, the width of the critical velocity range can be reduced at the cost of increased throttling losses. Selecting suitable pumps is therefore a trade off between narrow critical range and acceptable energy efficiency. Inside the critical range, the desired minimum chamber pressure cannot be maintained and the dynamic performance of the system will degrade due to the lowered oil stiffness. When the system operates outside the critical velocity range, i.e. where χ 1, the chamber pressures must be actively controlled to avoid utilization of the pressure relief valves and to ensure acceptable overall system energy efficiency. The amount of excess flow is directly linked to the shaft rotational velocity, which in turn is a function of the velocity reference. This excess flow should therefore actively be throttled at low pressure drop to ensure good system energy efficiency. Since the nominal displacement of the three pumps may be chosen somewhat arbitrarily, an optimization procedure is used to determine these. The optimization problem is based on the system operating in steady state conditions, where it is assumed that the chamber pressures are controlled as Fig The constraints of this optimization problem is that the desired piston velocity may be realized (including a margin) within the shaft speed range of the servo motor, and that the match ratio χ > 1 for both positive and negative shaft speeds. The objective function is a normalized weighted sum of both the size of the critical velocity range and the throttling losses associated with the purposeful mismatch between pump displacements and the cylinder area ratio. The optimization algorithm is run at different load pressures, to account for different

15 Chapter 1 - Introduction 5 load conditions, which converges to approximately the same design. See [Grønkjær and Rahn, 214b, p.39-41] for further details. Which chamber pressure that should be controlled depends on the load acting on the cylinder. The theoretical cylinder force is directly proportional to the load pressure p L. The load pressure is a weighted difference between the chamber pressures as seen in Eq. 1.2, with the cylinder area ratio denoted α. p L = p A α p B α = A r A p (1.2) As seen from Eq. 1.2, different chamber pressure combinations may result in the same load pressure, why controlling the motion using the load pressure alone does not guarantee a certain minimum chamber pressure. If the minimum chamber pressure can be fixed to the set pressure, p set = min (p A, p B ), by utilizing the proportional valves, the chamber pressures can be described directly as functions of the load pressure. As neither of the chamber pressures may decrease below the set pressure in normal operation (operation outside the critical velocity range), a switching condition exists where both chamber pressures are equal to the set pressure. This is shown in Eq. 1.3, where the set pressure is inserted into the definition for the load pressure, Eq p A = p B = p set p L,sw = p set α p set = p set (1 α) (1.3) For load pressures above and below the switching pressure p L,sw, the chamber pressures can be written as Eq. 1.4 and Eq p L p set (1 α) : p A = p L + α p set p B = p set (1.4) p L < p set (1 α) : p A = p set p B = p set p L α (1.5) To better visualize the chamber pressure, the pressure mappings in Eq. 1.4 and Eq. 1.5 are shown in Fig Chamber Pressure [bar] Chamber A Chamber B Load Pressure [bar] Figure 1.5: Steady-state relation between load pressure and chamber pressure. In [Grønkjær and Rahn, 214a] decentralized control is used to control the pressure level and to control the piston motion. The controllers have been developed in order to determine the dynamic performance of the SvSDP compared with both the original SvDP concept (B-SvDP), a system identical to Fig. 1.2b, and a valve operated reference system (VoRS), a system identical to Fig In the B-SvDP system the pump sizes of the two pumps are selected, such that the pump displacement ratio most closely match the cylinder area ratio. In the VoRS the dynamic model of the proportional valve is modeled based on a Bosch Rexroth 4WREE1 proportional valve, where it is assumed that the flow characteristics of the valve is matched to the asymmetric cylinder. The variable displacement pump in the VoRS is modeled to be ideal, both in respect

16 6 Control and Experimental Evaluation of SvSDP Concept to energy efficiency and dynamic performance, such that the reference system is given the most favorable conditions. In this way the VoRS emulates an upper limit for what might be achievable with a valve operated system using this specific valve. The simulation results of a piston position trajectory emulating a real life application with sufficiently smooth trajectory reference are shown in Fig VoRS B SvDP SvSDP (con) SvSDP (agg) Position Error [mm].5.5 Normalized Control Signal [ ].5.5 Piston Side Pressure [bar] Rod Side Pressure [bar] Time [s] Figure 1.6: Simulation results of a trajectory emulating a real life trajectory [Grønkjær and Rahn, 214b, p. 55]. VoRS = Valve operated Reference System [Fig. 1.1], B-SvDP = Base line SVDP [Fig. 1.2b], SvSDP = SvDP with Switched-Displacement Pump [Fig. 1.4a]. (con) = conservatively tuned controller, (agg) = aggressively tuned controller. The simulation of the SvSDP is carried out with a proportional valve utilization method, which does not corresponds to the most energy efficient valve utilization method. A method to achieve the most energy efficient valve utilization dynamically had not been fully developed prior to completion of the work in [Grønkjær and Rahn, 214b], why it is further investigated in this project. As seen from Fig. 1.6 the SvSDP (agg) have a position error comparable to that of the reference system VoRS. Two other trajectories, a sine trajectory and an up-down ramp, have also been simulated, and the results are summarized in Fig VoRS B SvDP SvSDP (con) SvSDP (agg) Normalized RMS Error Real life Sine Ramp Normalized Max Error Real life Sine Ramp Normalized Input Energy Real life Sine Ramp Figure 1.7: Normalized RMS and maximum tracking error, together with normalized required input energy for all three trajectories [Grønkjær and Rahn, 214b, p ].

17 Chapter 1 - Introduction 7 The results in Fig. 1.7 shows that using the SvSDP concept, comparable dynamic performance and superior energy efficient may be achieved compared to a valve operated reference system. The improved dynamic performance of the SvSDP concept compared with the B-SvDP, is achieved at the cost of a decreased energy efficiency, which however is still better than the VoRS. Previously obtained experimental results from [Nielsen and Jensen, 214] suggest an additional practical problem not implemented in the simulations shown in this section (see [Grønkjær and Rahn, 214b,p ] for details). A delay of 22 ms have been observed between the zero crossing of the shaft velocity and the change in chamber pressure, which resulted in a large overshoot in shaft velocity reference and poor dynamic performance in general. This delay is believed to be due to slow check valve dynamics in the original SvDP system. All the check valves used in the original SvDP system is very large, with a maximum flow rate of 45 L/min, it is expected that the poppet inside the valve is of non-neglectable mass. In this project the check valve dynamics are thus modeled and the findings are used as a basis for the selection the check valve sizes for the experimental evaluation of the system. As the results obtained in [Grønkjær and Rahn, 214a] are based purely on simulation, the main purpose of this project is the experimental validation of these results.

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19 Chapter 2 - Project Objective 9 2 Project Objective This project is intended as a direct continuation to the work and results presented in [Grønkjær and Rahn, 214a] and [Grønkjær and Rahn, 214b], which are summarized in Chapter 1. The main purpose of this project is therefore to experimental evaluate the SvSDP concept, to determine if the simulated results may be verified from experimental results. At the time of this work, reconstruction of the laboratory facilities is carried out, which means that it has not been possible to obtain an overview of the available test equipment. As a consequence of this situation, a new complete test bench is designed, ordered and assembled by the authors, making it possible to evaluate the dynamic performance of the SvSDP concept. Based on the identified improvement areas, the work of this project is thus divided into two parts: Design, Dimensioning and Setup of General Purpose Test Bench Modeling and Development of Control Strategies for SvSDP Systemt Design, Dimensioning and Setup of General Purpose Test Bench This part presents the work related to the development of a test bench for the SvSDP system, with flexible loading possibilities. The topics presented in this part are: Investigation, evaluation and selection of test bench topology Detailed mechanical design and dimensioning of the test bench components Selection of hydraulic components, and evaluation of the possible operating range Implementation of data acquisition and control system for the test bench and the SvSDP system Modeling and Development of Control Strategies for SvSDP System This part present both the development and verification of the dynamic model of the SvSDP system, as well as the considerations and investigation of control strategies for the SvSDP system. The focus of the work is to design a suitable application-independent control structure for the SvSDP system, which makes it possible to track a position reference in the experimental evaluation of the concept. A partial focus of the control design is to develop a control structure which makes it possible to utilize the proportional valves in the most energy efficient way. Development and verification of dynamic model for the SvSDP, including modeling of check valve dynamics Specification of goals and limitations for control system Model linearization and analysis for determination of suitable control strategies Development, parameterization and evaluation of control system, with special focus on energy efficient utilization of the proportional valves

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21 Part I Design, Dimensioning and Setup of General Purpose Test Bench 11

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23 Chapter 3 - Requirement Specification and Topology Selection 13 3 Requirement Specification and Topology Selection This part presents the considerations and results for the design, dimensioning and modeling of a test bench for evaluation of both the dynamic performance and energy efficiency of the SvSDP concept. The content of this part is not considered as the main focus of this project, but nonetheless represents a large part of the work carried out within the project period. As a consequence, this part is structured as a brief review of mainly the considerations and choices made during the course of the test bench design, from requirement specification to topology selection, dimensioning, assembly and implementation of control and data acquisition software and hardware. Much of the support material may therefore be found in the appendix, where specific references to this are made throughout the part. In this chapter, the general desirable traits of the test bench are listed, along with the overall system specifications. These are primarily based on the reuse of components from the original SvDP system developed by [Madsen and Bertelsen, 213]. The considered test bench topologies, that supports the desirable traits are presented in Section 3.3. The final design of the test bench is presented in Chapter 4 including the considerations regarding the detailed mechanical design, as well as the discussion and selection of topologies for various subsystems. The considerations relates to both flexibility, size, ease of assembly, cost and complexity. Suitable hydraulic components are additionally selected. In Chapter 5, the structure and functionality of the implemented Data Acquisition and Control System is briefly described. 3.1 Specification and Component Limitations To experimentally evaluate the performance of the SvSDP concept, a cylinder, a test bench and the corresponding hydraulic component must be selected. To allow for testing under various operating conditions, different topologies that allow for a degree of loading flexibility, both with respect to inertia and load force, are presented in this chapter. To evaluate the different topologies, simple calculations are made based on the desired specifications. The free variables in the design are considered the stroke length of the cylinder and the nominal piston velocity. Nominal, in this case, denotes the achievable piston velocity at a shaft speed 15 2% below the limitation of the utilized pumps which is 3 RPM [Bosch Rexroth AG, 212a]. To allow for reasonable piston velocities and sufficiently long trajectories, these free parameters are chosen as: Cylinder stroke length: l stroke = 7 mm Nominal shaft speed: n nom = 25 RPM Nominal piston velocity: ẋ nom = 25 mm/s Desirable traits of the topologies are: Changeable load inertia Changeable load force, preferably independent of the inertia Moving parts should be close to ground for increased safety and accessibility

24 14 Control and Experimental Evaluation of SvSDP Concept 3.2 Component Reuse and Limitations As it has been attempted to reuse as many components as possible from the original SvDP system developed by [Madsen and Bertelsen, 213], the basis for the dimensioning of the system is chosen based on the limitations of these components. From the original SvDP setup, it has been possible to reuse the main components shown in Tab Component QTY Vendor Specification Motor 1 Rexroth Synchronous Servo Motor IndraDyn S - MSK71E-3-NN-S1-UG-NNNN Servo-drive 1 Rexroth INDRADRIVE HCS2 composed of Converter 1 Rexroth HCS2.1E-W7-A-3-NNNN Control Unit 1 Rexroth CSH1.1C-ET-ENS-NNN-MA1-NN-S-NN-FW AZPF Pump 1 Rexroth AZPF-22-14LHO3KB-S9997 Proportional Valves 2 Rexroth KKDSR1NB/HCG24NK4V Pressure Relief Valves 2 Rexroth DBDS 2 G1X/2 Table 3.1: List of components reused from the original SvDP system developed by [Madsen and Bertelsen, 213]. Since the development of this project has coincided with a larger renovation of the laboratory facilities, it has not been possible to achieve lists of other available components in advance. The remaining part of the test bench has therefore been designed and specified, without relying on other components being readily available Component Limitations As the mechanical design is based on the expected load forces, the component limitations of both the motor, the converter and the pumps listed in Tab. 3.1 are investigated in this subsection. Simple power calculations translates the most restrictive of these limitations into an approximation of the expectable continuous load force in the end of this subsection. Torque / Speed / Power limitations of Servo-drive The torque limitations of the motor due to heating issues are given in the data sheet [Bosch Rexroth AG, 213, p ] and shown by the black line in Fig. 3.1.

25 Chapter 3 - Requirement Specification and Topology Selection 15 Shaft Torque [Nm] Shaft Speed [RPM] Continuous Motor Limit Continuous Brake Limit (Int) Continuous Brake Limit (Ext) Absolute Motor Limit Figure 3.1: Torque limitations of the motor. (Int) = Internal brake resistor [existing], (Ext) = External brake resistor [acquisition]. The existing converter has a built-in brake resistor rated to a continuous power of.5 kw. As seen in Fig. 3.1, 4 quadrant operation is therefore severely limited, why an external brake resistor (ext), with a rated continuous power of 5.5 kw, has been supplied by Bosch Rexroth A/S. With this augmentation, the continuous torque limitation of the servo drive is limited mainly by the motor and not by the brake resistor [Bosch Rexroth AG, 21b, p. 16]. It has additionally been verified, that the continuous motor torque is within the power limitations of the converter. Speed / Pressure limitations of pumps The external gear pump limitations are listed in Tab Specification Value Comment Max Continuous Pressure 25 bar Max Intermittent Pressure 28 bar Duty time up to 2 s Max Peak Pressure 3 bar Max Speed 3 RPM At intermittent pressure Table 3.2: External gear pump series F (AZPF) [Bosch Rexroth AG, 212a, p. 16] Torque limitations of drive shaft Since all three pumps are connected to the same drive shaft, it is investigated whether the drive shaft couplings limits the maximum torque. The maximum transferable torque in the couplings between the pumps are 65 Nm, whereas the maximum torque on the tapered key shaft is 16 Nm [Bosch Rexroth AG, 212a, p. 19]. As motor torque is only required for 2 pumps at the same time (the third pump will either help drive the remaining pumps or contribute with torque), the shaft couplings are found not to be the limiting factor. The most strict requirement follows from the torque limitations of the electric motor. Assuming a design such that the nominal piston velocity ẋ nom = 25 mm/s is realized at a nominal shaft speed of ω m,nom = 25 RPM, at the max continuous torque limit of T mot,cont = 26.4 Nm and a loss free hydraulic system, the power equality gives the load force in Eq F L = ω m T ẋ = 27.6 kn 25 kn (3.1) This load force of 25 kn is thus used in the evaluation of the presented topologies.

26 16 Control and Experimental Evaluation of SvSDP Concept 3.3 Test Bench Topology Presentation In this section, three different test bench topologies are presented. The presentation consist of a sketch and a functional description, followed by simple load calculations and a list of pros/cons. The load calculation is carried out to roughly estimate the physical dimensions of the test bench needed to realize the required load force. Due to the purpose of the test bench, as well as the time limitation imposed in order to realize the test bench within the project period, only a brief topology evaluation is made. More focus is instead put into the detailed mechanical design and verification of sufficient component strength Topology 1 The functional principle of Topology 1 is illustrated in Fig m θ Figure 3.2: Sketch of test bench topology 1, consisting of a cylinder driving a sliding mass linearly on an inclined plane. The load conditions may be changed by varying the mass m and the inclination angle θ. To ease the evaluation of the hydraulic system, a linear load characteristic is advantageous. This may be achieved by having a stack of mass plates slide linearly along a platform, where the inertia may be changed in discrete steps by varying the number of utilized mass plates. The load force may be changed, by allowing the platform to be inclined, either in discrete steps using e.g. a mechanical lock pin or continuously through the use of an additional actuator as shown in Fig Load calculations: The load inertia m x as seen from the cylinder is constant and equal to the mass of the slide, and the load component F L,x due to gravity additionally depends on the inclination angle as given in Eq m x = m F L,x = m g sin(θ) (3.2) Assuming the inclination angle to vary between ±45, the mass required to realize a load force of 25kN is the improper value m 36 kg.

27 Chapter 3 - Requirement Specification and Topology Selection 17 Pros: Position independent load profile, easing system analysis. Independent variation of inertia and load profile. Small frame length ( 2 x stroke length). Cons: Requires either two cylinders or a mechanical locking system for variation of the inclination angle. Requirement to stiff support of moving mass independent of inclination angle. The platform must be elevated sufficiently above the ground level if pulling loads (θ < ) are to be realized. This increases the requirements to the mechanical frame and increases the necessary safety considerations and features, due to operation in the height. Improper large mass required to realize the nominal load forces. Inertia limited by the mass of the slide Topology 2 The functional principle of Topology 2 is illustrated in Fig a b θ m 1 m 2 Figure 3.3: Sketch of test bench topology 2, consisting of a cylinder driving an arm about a central pivot point. The load conditions may be changed by individual adjustment of the masses m 1 and m 2. By using a rotating load, a high inertia and load force may be achieved using lower mass, due to the gearing ratio between the movement of the cylinder and the rotation of the arm. By using masses in each end of an arm, which is rotating about a central pivot point, the gravitational force may be changed independent of the inertia. The moment of inertia may be kept constant as long as m 1 + m 2 = const. Having m 1 > m 2 allows for pushing loads, while m 1 < m 2 allows for pulling loads. As seen from the cylinder, both the inertia and load force will have a prescribed non-linear position dependency, independent of the distribution of mass. Load calculations: The topology is simplistically evaluated by considering the rotating arm to be in the horizontal position, i.e. θ =. Furthermore, the cylinder is considered to be mounted such that at this position, the cylinder is perpendicular to the arm. As seen from the cylinder, both the inertia m x

28 18 Control and Experimental Evaluation of SvSDP Concept and load force F L,x varies with the rotation of the arm, but may for small rotations around the horizontal position be described by Eq m x = (a + b)2 a 2 (m 1 + m 2 ) F L,x = a + b g (m 1 m 2 ) (3.3) a The possible stroke length of the cylinder depends on the length a and the possible rotation angle θ max. Assuming that the arm may rotate in the interval θ [ θ max ; θ max ], and furthermore that the cylinder orientation is approximately vertical at both end positions, the possible stroke length may then be approximated by Eq s 2 a sin(θ max ) (3.4) In order to realize the desired stroke length of 7mm for a maximum rotation angle of θ max = 6, the required minimum distance from the pivot point to the attack point of the cylinder must be a 4 mm. Assuming the mass to be positioned entirely on one end of the arm, at a very large arm length of (a + b) = 2 m, the mass required to realize a load force of 25 kn is still m 1 5 kg. The corresponding inertia in this configuration is m x 125 kg. Pros: Independent variation of inertia and load force. Possible to realize high inertia using only low mass, due to the mechanical gearing. Simple mechanical construction. Low friction, since ball or roller bearings can be used in the pivot joint. Cons: Nonlinear position dependent load profile with respect to both inertia and load force. For a given piston velocity, the velocity of the moving masses may become high due to the gearing ratio. The moving mass is elevated from the ground level increasing the necessary safety considerations and features. High stiffness requirements to arm profile, to avoid low frequency mechanical modes. Large construction required to realize both a stroke length of 7 mm and a load force of 25 kn Topology 3 The functional principle of Topology 3 is illustrated in Fig m Figure 3.4: Sketch of test bench topology 3, consisting of a main and a load cylinder driving a slide of mass plates. The inertia may be changed by adjustment of the mass. Various load forces may be realized through control of the load cylinder.

29 Chapter 3 - Requirement Specification and Topology Selection 19 A general deficiency of Topology 1 and 2 are the large masses and dimensions. Higher load forces may be realized without using additional mass, by utilizing a second cylinder to control the load force. The load force may thus both be controlled as a constant to ease system analysis, or it may be made position/velocity dependent to emulate various load cases. Pros: Flexible load force emulation. Moving mass close to ground level, reducing the safety features and the requirements to the mechanical construction. Ideally, the load force is decoupled from the inertia and can be controlled independently. Cons: Limited maximum inertia, due to the lack of mechanical gearing. Additional hydraulic components required to drive the load system. Large length of test bench ( 3 x stroke length). The dynamic performance of the load system must be sufficiently high, to allow for separation of the dynamics of the main system from that of the load system. Frame must be strong enough to withstand the opposing cylinder forces. 3.4 Topology Selection Based on the presented topologies and the listed pros and cons. Topology 3 is chosen due to its flexibility in load emulation. Topology 1 is considered inferior to Topology 3 due to the inability to realize high load forces using reasonable masses. Additionally, elevating the moving mass greatly increases the risks associated with operation of the system. Topology 2 is attractive due to its high inertia possibilities. A combination of topology 2 and 3, where an additional cylinder is placed between the ground and the opposing arm, allows for high inertia and high external force, but still with increased safety requirements and requirements to the stiffness of the arm. Furthermore, this combined topology will still have a non-linear load characteristic.

30

31 Chapter 4 - Design and Dimensioning of Test Bench and Components 4 21 Design and Dimensioning of Test Bench and Components This chapter presents the final design of the constructed test bench, which is shown in Fig Figure 4.1: The constructed test bench, including the hydraulics of the SvSDP system. As may be seen from Fig. 4.1, the test bench mainly consist of a frame, made of a combination of square tubes and U-beam profiles, two opposing cylinders and a moving slide. The main cylinder is driven by the designed SvSDP system, while the opposite cylinder may be used for emulation of both opposing and overrunning loads. The main elements of the test bench have been highlighted in the drawing in Fig. 4.2, to give an overview of the different parts of the test bench. Mass plate Sliding element U-beam Upper square tube Rubber end stop Rails Load pin Load cylinder Main cylinder Plate for rails Minimess for pressure sensor Position sensor Cable tray Lower square tube Foot E-chain Base plate Figure 4.2: Perspective view of the designed mechanical frame of the test bench.

32 22 Control and Experimental Evaluation of SvSDP Concept The design and construction process for the test setup has included the following work by the authors: Overall design, detailed hydraulic and mechanical dimensioning, CAD implementation of the test bench design, preparation of working drawings and specification of machining tolerances, component specification, quotation of prices, budgeting, contact with suppliers, coordination and follow-up of component orders, as well as assembly of the mechanic frame. To document this work, the content of this chapter therefore consists of the considerations regarding the design details, the dimensioning of the hydraulic main and load systems as well as the strength requirements for the test bench frame. A review of the considerations behind both the general design, as well as the choice of various components is presented initially in Section 4.1. Following this, the dimensioning and selection of hydraulic components for both the SvSDP system and the load system is presented in Section 4.2 and Section 4.3 respectively. The model of hydraulic load system is furthermore presented in Section 4.3, whereas the SvSDP system model is presented in Chapter 6. The dimensioning of each hydraulic system includes evaluation of the possible operation map of the corresponding system. These operation maps form the basis for the dimensioning of the mechanical components, where the general approach for strength evaluation is described in Section 4.4, along with a list of the investigated topics. 4.1 General Considerations In this section the general ideas and considerations behind the design of the test bench are reviewed. The test bench is generally designed for long term use and flexibility, such that it may be used in future projects. These projects may both relate to further investigation of the SvSDP concept, as well as to other hydraulic applications. To minimize the required machining and the overall cost of the mechanical system, the frame is made using standard steel profiles of S355J2 material. It is chosen to use the same square tube profiles for the upper and lower beam, as well as for the feet of the frame, even though the lower beam is significantly more stressed than the other components. The same plate thickness and material is used for all but the middle plate of the slide, such that all the plates can be flame cut out of a single larger plate, and thereby reduce the required machining and associated costs. The mass plates are made from identical steel plates and they can be stacked on top of each other, where the location of the handles may be alternated by flipping the plates. This again reduces the machining of the mass plates, as the handles are integrated in the layout of the plates. The mass of the mass plates are kept at 5 kg each, such that they can be handled with relative ease by two persons and/or by the use of a pallet jack. This makes it possible to change the inertia load for different operation scenarios. The mass plates are fixed to the base plate of the slide using four M36 threaded rods with associated nuts. The nuts on the threaded rods are tightened to help secure the mass plates to the base plate, such that the slide behaves as a simple rigid element during acceleration. Width of the mass plates is preferred compared to plate thickness, as wide mass plates lower the center of gravity of the inertia load. The feet of the frame are designed with very long oblong holes, which makes it possible to install the assembly on either of the available floors and in any direction in the Heavy Lab. at the Department of Energy Technology at Aalborg University. The frame is bolted together using friction joints, such that fine adjustment of the alignment of the cylinders to the rails may be made during assembly. This furthermore makes it possible to assemble the test bench, without requiring the aid of welding trained personnel. The pillow blocks connecting the base plate to the rails are mounted on the base plate using

33 Chapter 4 - Design and Dimensioning of Test Bench and Components 23 oblong holes. In this way, the parallel (horizontal) alignment of the base plate to the cylinders can be adjusted through the positioning of the bolts within the oblong holes. The selected pillow blocks are Igus Drylin R-series, based on the brief review of linear guide topologies presented in Appendix B.6. This pillow block type is selected, since the sliding element is a solid composite material, which is both inexpensive and highly tolerant to shock loads. A disadvantage with the solid composite material is a higher friction coefficient compared to a ball bearing type. But as the cylinder force is measured directly using the load pin, this friction is just part of the cylinder load. For high accuracy position and velocity measurements, it is chosen to install position sensors in both cylinders. A single position sensor is required as the minimum for use in the feedback control, but it is chosen to have the possibility to have the load system operate completely independently of the main cylinder, why two sensors are selected. This also makes it possible to use the cylinders in other applications at a later date. Identical position sensors of the type Balluff BTL7-V5E series are selected, as they use EtherCAT fieldbus communication, which is the only possible digital communication method that may be compatible with the available servo-drive control unit [Balluff GmbH, 215b, p. 74], [Bosch Rexroth AG, 21b, p. 44]. The location of the sensors in the rear end of the cylinders allows for easy access to the sensor cables. The mechanical frame is designed such that there is easy access to both the cylinder ports. Minimess test point connections for pressure transducers are mounted directly in separate welded connectors on the cylinders, to reduce the distance from the cylinder chambers to the measurement points. T-union fittings are therefore avoided at the cylinder ports. Different cylinder mounting topologies have been considered, where the full review is presented in Appendix A.2. The chosen topology utilizes trunnions on the cylinder tubes and spherical plain bearings on the piston rods. This cylinder mounting topology is selected as it makes it possible to have the mountings points close together and thus increase the buckling limitation, compared to other topologies which have the mounting points further away. The trunnion mount is chosen over a gimbal mount, due to its simplicity and therefore lower cost. The mounting type furthermore allows for compensation of misalignment in the vertical direction. Only horizontal misalignments cannot be compensated for and the effect of such misalignments are investigated in Appendix A.3. It is found that the misalignments when the cylinders are fully retracted should be less than 2 mm to avoid steel against steel contact in the cylinder gland. For the given geometry such an misalignment of 2 mm would correspond to a very coarse tolerance, why fine tolerances according to ISO are used for the affected dimensions. Due to the bolted design, it is possible to assemble the frame, the slider element and mount the cylinders independently. The cylinder piston rods are attached to the sliding element through eyes screwed onto the rods. The interface between the cylinders and the sliding element can be seen in Fig. 4.3.

34 24 Control and Experimental Evaluation of SvSDP Concept Base plate Middle plate Cable clamp Eye Lock plate Piston rod Spherical plain bearing Groove for load pin Bushing Double eye Pin bolt Rail Screwed joint Figure 4.3: Zoom view of the connection between the main cylinder piston rod and the sliding element. The spherical plain bearings allow for rotation to avoid stresses due to constrained movement. The base plate for the slide is designed, such that the load sensor pin can be mounted in either the eye for the load cylinder or the main cylinder, with a simple steel pin bolt in the opposite eye. By measuring the cylinder force directly, the real load force is obtained instead of only the theoretical cylinder force, which might otherwise be obtained by measuring the chamber pressures. In this way the cylinder friction may be compensated for in the load cylinder control system, such that the actual load on the main cylinder might be determined more accurately. A steel pin bolt inserted in the eye is shown in Fig. 4.3, where the groove for the load pin is also seen. The load pin is a Force Measurement Bolt Haehne KMB-AC35-4k-S2 [Haehne GmbH, 215] capable of measuring both the compression and tension. The load pin is originally designed for use in commercially available standard fork heads, which is the reason for the required groove to comply with the physical dimensions of the sensor. As the geometry of the load pin comply with the design of standard fork heads, it is both less expensive and it has shorter delivery time compared with custom built sensor, which were initially considered. The symmetry of the base plate allow for installation of an additional load pin in either eye, if it is desired to install two sensors at a later date. The cable for the load pin is fixed to the base plate using cable clamps on the underside of the base plate, where it connects with the e-chain. The e-chain forms a flexible cable tray for the sensor cable and the purpose of it is to avoid damage to the sensor cable during operation. The sensor cable used is a Igus Chainflex R shielded cable (CF1.INI-P5C-M12-BG-7), which is designed to withstand the continuous bending in the e-chain, without breaking the shield. A fixed cable tray is mounted on the frame feet, to secure all the cables for the pressure sensors and the position sensors. End stop buffers are installed in the frame to act as safety mechanism in the case of loss of control, such that the cylinders are protected against shock loads. The end stop buffers are selected and implemented in the frame design in such a way, that it is not possible for the cylinders to reach their full stroke even under a worst case scenario. See Appendix B.5 for details. 4.2 Dimensioning of Hydraulic Components of Main System In this section the selection of hydraulic components for the SvSDP system is presented. The hydraulic diagram of the SvSDP system is shown in Fig. 4.4.

35 Chapter 4 - Design and Dimensioning of Test Bench and Components 25 D p A d p B M PRA PRB PVA CVAR CVAP1 CVAP21 PVB CVBP3 CVBR CVAS CVAP2 CVBS M P1 P2 P Pump Dimensioning Figure 4.4: Hydraulic diagram of the SvSDP (main) system. As described in Section 3.2, the basis for the dimensioning of the system is chosen based on the component used for evaluation of the original SvDP system developed by [Madsen and Bertelsen, 213]. From this setup, it is initially attempted to reuse the available 14 cm 3 /rev pump as the B-side pump (Pump P3). Based on the displacement of the B-side pump, and the desired nominal piston and shaft velocity, it is possible to determine the approximate size of the rod side area. The outer rod and inner tube diameter are chosen from typically available material sizes [Fjero A/S, 215], in order to achieve this area while maintaining sufficient resistance towards buckling (see Appendix A.1): Tube diameter: D = 63 mm Max push pressure: 161 bar Max push force: 5.2 kn Rod diameter: d = 35 mm Max pull pressure: 25 bar Max pull force: 53.9 kn Although the individual rod and tube sizes are standard sizes, the individual components of the cylinders have been custom designed for this specific application. The authors would again like to thank Fjero A/S for their financial aid in the acquisition of the cylinders. For simplicity, both the main cylinder and the load cylinder are identical. The remaining pump sizes are then chosen based on the optimization problem presented in [Grønkjær and Rahn, 214a], by attempting to achieve a match ratio of χ 1.2 for both shaft rotational directions. This gives the pump sizes shown below, which yields the match ratio graph shown in Fig. 4.5.

36 26 Control and Experimental Evaluation of SvSDP Concept Nominal displacement of P1 pump: 16 cm 3 /rev Nominal displacement of P2 pump: 11 cm 3 /rev Nominal displacement of P3 pump: 14 cm 3 /rev Match Ratio [ ] p L = 75 bar p L = 25 bar p L = 125 bar Shaft Velocity [RPM] Figure 4.5: Match ratio χ for the selected pump sizes as function of load pressure. Minimum chamber pressure p set = 25 bar. Although it had initially been attempted to reuse the existing 14cm 3 /rev pump, three new pumps have kindly been supplied by Bosch Rexroth A/S for the final test setup. The flow/shaft speed relation at pressure differential is shown for the three selected pumps in Fig. 4.6a, to illustrate the flows expected during operation. The resulting relation between steady-state shaft speed and piston velocity for various load pressures is shown in Fig. 4.6b. 5 3 Pump Flow [L/min] Pump P1 Pump P2 Pump P Shaft Speed [RPM] Shaft Speed [RPM] p L = 5 bar p L = bar p L = 5 bar Piston Velocity [mm/s] (a) Pump flow as a function of shaft speed at pressure differential. (b) Shaft speed as a function of the piston velocity for the designed SvSDP system. Figure 4.6: Mapping of the steady state motor shaft speed to pump flow/piston velocity. For simplicity, the mapping in Fig. 4.6b is based on throttling of the excess flow using the proportional valve connected to the chamber with the lowest chamber pressure. A different valve utilization method would result in a different mapping, due to a change in the amount of inlet and outlet flow from the pumps. The plots are therefore only used to give an illustration of the relation between the piston velocity and the shaft speed Operation Map With the selected pumps, the operating limitations discussed in Section 3.2 and shown in Fig. 3.1 may be translated into the operation map shown in Fig The shaft torque limitations has here been changed into a corresponding cylinder load pressure. The motor torque is mapped into load pressures using the torque model, which is shown in Eq. 6.2, and by assuming that the chamber pressures are controlled as shown in Fig. 1.5.

37 Chapter 4 - Design and Dimensioning of Test Bench and Components Load Pressure [bar] Continuous Motor Limit Continuous Brake Limit Absolute Motor Limit Shaft Speed [RPM] Check Valve Figure 4.7: Operation map of the designed SvSDP system. The 8 required check valves are dimensioned based on a compromise between flow rate and pressure drop. The considerations for each check valve is listed in this section, whereas the exact selection criteria are listed in Tab. C.2. The valves are dimensioned for cavity mounting within a hydraulic manifold, to minimize the required tubing to connect the different valves. The check valves are not chosen to give minimum pressure drop, due to the possibility of non-neglectable dynamics of the large flow rate check valves (see [Grønkjær and Rahn, 214b, p ]). An investigation of the check valve dynamics is carried out in Section to support the selection. The annotation for the different check valve names is shown in Fig. 4.4, which may be used for reference. Design of Check Valve Manifold Block A manifold block is selected to house the check valves, such that the amount of fittings and pipes required to link the different check valves together is reduced, while also keeping the overall assembly compact. The check valve manifold block has been manufactured and supplied by Bosch Rexroth based on the specifications listed in Appendix C. The block is designed such that it can be mounted in parallel to the pump module and connected to it using only 6 pipes (1 suction and 1 pressure line for each of the 3 pumps) as shown in Fig This block design thus requires only 9 elbow fittings and straight hydraulic tubes to connect to the pump module. The final pump module is however not identical to the configuration shown in Fig. 4.8, as the sequence of pumps is determined by their displacement sizes (from largest to smallest). This requirement has been discovered late in the process, which has resulted in additional tubing and has rendered the original design impractical, due to crossing of the pipes.

38 28 Control and Experimental Evaluation of SvSDP Concept B-side A-side Manifold block Motor bracket Tank suction Tank suction Pump module P3 P2 P1 Figure 4.8: Illustration of the idea behind the design of the check valve manifold block. For simplicity, only part of the tubing is shown. This is not the final configuration of the setup, as the actual sequence of the pumps is P1-P3-P2 (not P1-P2-P3). Considerations regarding the check valve selection: CVAS & CVBS: These valves must supply all the flow from the tank to the suction side of the pumps. To avoid cavitation, these valves should thus have minimal pressure drop at full pump flow. The dynamic performance of these valves is less critical compared to the pump bypass check valves, as the dynamic performance only determines how quickly the magnitude of the virtual tank pressure changes. CVAR & CVBR: These valves must return all the flow from the suction side of the pumps to the tank. To increase the virtual tank pressure, a pressure drop of up to 1.5 bar is allowed based on specification of the maximum suction side pressure of the pumps. Similar to the suction check valves, the dynamic performance of these valves is less critical, as it only determines how quickly the magnitude of the virtual tank pressure changes. CVAP1, CVAP2 & CVBP3: The pump bypass check valves across the P1 and P3 pumps are only active during operation within the critical velocity range. For steady state operation with a piston velocity ẋ, the flow through the valves only supply the additional flow required to avoid cavitation. The maximum steady state flow through the valves thus correspond to the flow necessary to supply a pump in motoring mode when attempting to hold a load. In such a scenario, elevated chamber pressure is required in one of the cylinder chambers to hold the load, and due to the pump leakage, a non-zero shaft speed is required. As the piston is not moving, the pump connected to the opposite chamber receives no flow from the cylinder, why it must idle through the bypass check valve. To avoid cavitation, these valves should have minimum pressure drop for flow rates corresponding to operation within the critical velocity range. Since the dynamic performance of these valves determines how quickly the pressure builds up when leaving the critical velocity range, the size of the valves is chosen as small as possible. The idling check valve of the P2 pump is active both as anti-cavitation check valve for positive velocity operation and for idling flow for negative velocity operation. To avoid cavitation, the pressure drop should be minimal at low flow rates. A pressure drop of.5 bar at full P2 flow is however acceptable during idling, since the virtual

39 Chapter 4 - Design and Dimensioning of Test Bench and Components 29 tank pressure will in this case be raised to 2 bar due to the selected CVAR and CVBR. CVAP21: The "displacement switch" check valve between the P1 and P2 pumps is active for positive velocity operation outside the critical velocity range. The valve must thus support the entire P2 pump flow. Since the valve is not active for anti-cavitation purposes, a small pressure drop across the valve is considered acceptable, which, only to a small extent, influences the energy efficiency and positive critical velocity. Since the dynamic performance of this valve is crucial for the normal operation of the system, small size and spring return is preferable Pressure Relief Valves To ensure that the specified component limitations with respect to pressure, are not exceeded, the most restrictive of the pressure limitations are used as the setting for the pressure relief valves. For the piston side chambers of both the main and load cylinder, the most restrictive pressure limitation corresponds to the rated push pressure of the cylinders, i.e. 16 bar (assuming no back pressure). For the rod side chambers, the maximum relief valve pressure of 2 bar is used, which is well below both the rated pull pressure of the cylinders and the maximum pump pressure. These settings correspond to a possible main cylinder force of 49.9 kn (push) and 43.1 kn (pull) with zero back pressures. 4.3 Model and Operation Map of the Load System The load system has not been implemented prior to the project deadline, however the dimensioning, modeling and resulting operation map of the load system are presented in this section. For simplicity, the cylinder for the load side is chosen to be similar to the main cylinder, and therefore the settings for the pressure relief valves are also similar. The load system is schematically illustrated in Fig p L,A x p L,B M A r V L,B A p V L,A Q L,PRB Q L,PRA Q L,B x sv Q L,A p L,S Figure 4.9: Hydraulic diagram of the load system using the pump and the tank connections available in the Heavy Lab. The nominal static flow requirement to the load system (Eq. 4.1) is based on the desired nominal piston velocity of ẋ nom = 25 mm/s as selected in Section 3.1. Q ss,max = A p ẋ nom = 47 L/min (4.1) Since Moog D633 valves and corresponding manifold blocks are readily available with a nominal

40 3 Control and Experimental Evaluation of SvSDP Concept flow rate of Q sv-n = 4 L/min at a pressure drop of p sv-n = 35 bar per land [Moog Inc., 29, p. 4], such a valve is selected even though a slightly larger valve may be more appropriate Hydraulic Model of the Load System The pressure dynamics within the load cylinder chambers are described by the continuity equations given in Eq. 4.2, where the bulk modulus β L is assumed constant. This assumption is valid for the load system, as the chamber pressures are sufficiently high (see the operation map in Section 4.3.2). ṗ L,A = β ( L Q V L,A V ) L,A Q L,PRA ṗ L,B = β ( L Q L,A V L,B V ) L,B Q L,PRB (4.2) L,B The load system is divided into two control volumes; CV L,A and CV L,B which includes the A and B side chambers respectively. The absolute volume and volume rate of change for these control volumes are described by Eq. 4.3 and Eq V L,A = V L,A A p x VL,A = A p ẋ (4.3) V L,B = V L,B + α A p x VL,B = α A p ẋ (4.4) The 4/3 proportional valve used for control of the load cylinder is the Moog D633 servo valve. Based on [Moog Inc., 29, p. 4], the flow across each land of the servo valve is modeled as the flow through a sharp edged orifice, along with a leakage term as given by Eq Q L,x = Q sv-n psv-n x sv p sv,x sign ( p sv,x ) Q L,x-leak (4.5) x sv is the normalized spool position, and the pressure drop across each land depends on the sign of sv, as given by Eq { pl,a p sv,a = p { x L,T sv pl,s p sv,b = p x L,B sv (4.6) p L,A p L,S x sv < p L,T p L,B x sv < Both the supply pressure p L,S and tank pressure p L,T of the load system are considered constant. The additional leakage flow between the valve ports is included in the model since the utilized servo valve is zero-lapped, why leakage flow will be significant for x sv. The leakage flow is modeled as being proportional to the pressure difference between the corresponding ports as given by Eq Q L,A-leak = K sv,leak (( pl,a p L,T ) ( pl,s p L,A )) e 4 xsv x sv,leak Q L,B-leak = K sv,leak (( pl,s p L,B ) ( pl,b p L,T )) e 4 xsv x sv,leak (4.7) The exponential term is included to degrade the influence of the leakage flow for spool positions further away from the center position. The constant x sv-leak denotes the spool position at which the leakage becomes neglectable, which is chosen as x sv-leak = 4% based on the pressure signal characteristic curve of the valve [Moog Inc., 29, p. 8]. This model has been experimentally validated in [Freiberg et al., 213, p. 5-11] to be sufficiently accurate for describing the leakage flow across such a valve, where the magnitude of the leakage coefficient K sv,leak is given in Tab Symbol Description Value Unit K sv,leak Valve leakage coefficient L/min bar x sv,leak End of leakage region.4 Table 4.1: Moog D633 leakage parameters based on [Freiberg et al., 213, p. 5-11].

41 Chapter 4 - Design and Dimensioning of Test Bench and Components 31 The dynamic response of the valve spool position is modeled as a slew rate limited, criticallydamped second order system, with parameters given by Tab The simulated step responses of the normalized valve spool position are shown in Fig Description Value Unit Eigenfrequency 1 Hz Damping ratio 1 - Slew rate limit 85.7 s 1 Table 4.2: Moog D633 dynamic parameters based on [Moog Inc., 29, p. 8]. Normalized Spool Position [ ] Step Size.2 Step Size.6 Step Size Time [ms] Figure 4.1: Simulated step response of normalized valve spool position Possible Operating Conditions of the Load System The choice of supply pressure determines the possible operation map of the load system, since this determines the chamber pressures corresponding to a specific load force and piston velocity. At steady state, the continuity equation for the load system is given by Eq Q L,B = α Q L,A Q sv-n x Q sv p sv,b = α sv-n psv-n x sv p sv,b (4.8) psv-n Eliminating the valve flow gain and the spool position, Eq. 4.8 can be written as Eq. 4.9, which describes the relation between the steady state pressure drop across each spool land p L,A = p sv,b = α 2 p sv,a (4.9) Combining the definition of p sv,a and p sv,b from Eq. 4.6, with the definition of the load pressure of the load system p L,L = p L,A α p L,B, and inserting these into Eq. 4.9, the chamber pressures of the load cylinder as a function of the load pressure are given by Eq As this analysis is carried out under steady state conditions, the sign of the piston velocity is equal to the sign of the servo valve spool position, sign (ẋ) = sign (x sv ), why Eq. 4.1 is expressed based on ẋ. α p L,S + p L,L + α 3 p L,T p 1 + α 3 ẋ L,S α 2 p L,L + α 2 p L,T 1 + α 3 ẋ α 3 p L,S + p L,L + α p L,T 1 + α 3 ẋ < p L,B = α 2 p L,S α 2 p L,L + p L,T 1 + α 3 ẋ < (4.1) Eq. 4.1 is used to determine the realizable load pressure as a function of the supply pressure. The limitations on the realizable load system load pressure is that both chamber pressures are within the interval [2 bar ; 144 bar] which respectively corresponds to the pressure at which the bulk modulus of the oil is approximately constant, and to 9% of the crack pressure of the pressure relief valves. This gives a sufficient margin towards unintentionally activating the pressure relief valves. The load pressure corresponding to these limits may be found by insertion and isolation of Eq These limitations are shown in Fig. 4.11a.

42 32 Control and Experimental Evaluation of SvSDP Concept P A = 144 bar P A = 2 bar P B = 144 bar P B = 2 bar Normalized Spool Position [ ] Load Pressure [bar] Supply Pressure [bar] (a) Mapping of the realizable static load system load pressures as a function of the utilized supply pressure. Load Pressure [bar] Piston Velocity [mm/s] (b) Spool position required to realize different load conditions at p L,S = 175 bar. Figure 4.11: Static mappings for load system. Based on Fig. 4.11a, the supply pressure is chosen to p L,S = 175 bar, why the static realizable load pressure may vary between approximately p L,L [ 3 ; 7] bar, which corresponds to a load force of F L [ 9.2 ; 21.8] kn. At the selected supply pressure, the necessary normalized spool position of the valve is evaluated in the operating range as shown in Fig. 4.11b. As seen, the nominal piston speed of 25 mm/s is realized within 8% 85% of the spool travel range, which gives a sufficient margin for the possibility of required additional control effort. As seen from Fig. 4.11a, the possible operating range of the system is limited by the load system, particularly for negative load pressures. This limitation is mainly due to the relation between the static land pressure drops given in Eq Using a 4/3 proportional valve, where the spool opening area characteristic is matched to the cylinder area ratio would significantly increase the possible operating area. Alternatively, a similar result may be obtained by using two 4/3 proportional valves, one for each cylinder chamber, and scaling the input between the valves such that the reference signal to the B-side valve x sv,b-ref = α x sv,a-ref. The operating range achievable with a single valve is however considered acceptable, where the described extension possibilities may be realized without changes to the other aspects of the setup. Depending on the load conditions, the supply pressure might be lowered to avoid extensive heat dissipation due to throttling losses. 4.4 Dimensioning of the Mechanical Frame and Components This section presents the general considerations regarding strength evaluation of the key components of the designed test bench. The investigated areas are marked by circles in Fig. 4.12, and listed below figure, along with references to specific appendix section, where the detailed calculations are carried out. The calculations are partly made by analytical approximations where it is reasonable and by use of FEM analysis in SolidWorks.

43 Chapter 4 - Design and Dimensioning of Test Bench and Components 33 (g) End-stop buffer (d) Trunnion eye (e) Base plate (c) Frame profile (f) Piston rod (a) Cylinder buckling (b) Friction joints Figure 4.12: Selected are where the strength of the test bench is evaluated. The investigated areas are analyzed in depth in Appendix A and Appendix B: (a) Cylinder buckling. Determine the push force limitations of the cylinders. See Section A.1. (b) Friction force and stresses in the bolted joints. See Section B.1. (c) Stresses in the vertically mounted U-beams of the frame profile. See Section B.2. (d) Stresses in the trunnion eye and pin of the cylinder mounting. See Section B.3. (e) Stresses in the base plate for the sliding element incl. eye and pins. See Section B.4. (f) Stresses in the cylinder piston rod and gland due to misalignment. See Section A.3. (g) Strength of end-stop buffers. See Section B.5. The result of the strength evaluation of the different parts is a safety factor, SF, towards failure, i.e. yield, fatigue or slip in friction joints. The target for the safety factor of each part is SF 3 based on [Norton, 214, p ]. When this target safety factor cannot be met, the partial coefficients described in the Eurocode standard are adhered to. The characteristic load used in the analyses is chosen as F max = 5 kn, approximately corresponding to the rated maximum cylinder push force. To estimate the possible dynamic loadings, the max achievable acceleration in the simulation is used. The von-mises stress in each part due to the characteristic load is denoted as σ max. In the case of FEM solutions the von-mises stresses are given directly, while in case of analytical solution, the von-mises stress are evaluated based on Eq. 4.11, where only two-dimensional cases are considered [Norton, 214, p. 281]. σ = σx 2 + σy 2 σ x σ y + 3 τxy 2 (4.11) The safety factor with respect to yield is evaluated as the ratio SF yield = σ max σ yeild. With respect to fatigue failure, the stress levels used in the calculations correspond to the limits of the stress range [Krex, 24, p. 391]. The load is considered to vary between ± the characteristic load, such that the stress variation σ v = 2 σ max. For the utilized steel materials (S355J2), the endurance limit is considered half the ultimate tensile strength of the material [Norton, 214, p. 348], such that the safety factor with respect to fatigue is evaluated as SF fat = σ max 4 σ ut. The assumption of load variation between ±F max gives a very conservative estimate of the actual load variation, as the characteristic load is well outside the typical operation area. Additionally, this stress variation does not correspond to the number of extension/retraction cycles of the cylinder, but to changes in load force. The number of load variations is, for the intended purpose of the test bench, considered much lower than the number of extension/retraction cycles of the cylinder.

44 34 Control and Experimental Evaluation of SvSDP Concept With respect to slip in the bolted friction joints, the safety factor is considered as the ratio SF slip = F fric, where F fric is the force that may theoretically be transfered between two adjacent F transfer surfaces, and F transfer is the force that must actually be transfered due to the characteristic load. For ease of reference, the expression for the various safety factors are given in Eq SF yield = σ max σ yield SF fat = σ max 4 σ yield SF slip = F fric F transfer (4.12) Based on the strength analysis in Appendix B it is concluded that the frame has sufficient safety margins against both static yield, as well as fatigue failure. Sufficient safety margins are also achieved for the friction joints when tightening the bolts according to the description in Appendix B. The frame design is therefore deemed sufficient for the designed use.

45 Chapter 5 - Implementation of Data Acquisition and Control System 35 5 Implementation of Data Acquisition and Control System This chapter briefly described the functionality and practical implementation of the data acquisition and control system for the designed test bench. The utilized servo-drive includes an integrated control unit on which the control system for the SvSDP system may be implemented. However, as the drive is not designed for general purpose data acquisition, it naturally has only limited electrical interfaces and data logging capabilities. The native data logging capabilities of the servo-drive consists of an Oscilloscope Function, capable of recording up to 8192 measured values of 4 variables [Bosch Rexroth AG, 29b, p. 995]. The sampling rate may be varied, with a minimum sampling time of 25 µs [Bosch Rexroth AG, 212b, p. 371]. Additionally, a Sampling Trace function is available, which makes it possible to trace up to 2 variables, with a maximum number of measurement values per variable of 5. Only a limited amount of memory is available however, why the number of measurement values per variables is derated with the number of variables. This is not considered sufficient for high resolution data logging of multiple variables during test sequences which may span several seconds. As an alternative solution, a combined Data Acquisition and Control System (DAC) is therefore designed and set up, responsible for both the acquisition of measurement data, as well as control of both the SvSDP system and the load system. The DAC is set up as an EtherCAT master/slave network, where the master unit is a LabVIEW Real-Time Desktop PC (RT-machine) and the servo-drive is considered as a slave input/output device, from which measured inputs or internal variables may be read, and output voltage signals and shaft speed references may be written. With this topology, a single laptop may be used as the Human-Machine Interface (HMI) for the entire system, easing configuration and use of the DAC. A functional connection diagram of the DAC is shown in Fig. 5.1, and this chapter briefly describes the different hardware parts and the functionality implemented in the software of the DAC. RS232 IndraDrive (Servo-drive) Converter Control Unit HMI (PC) EtherCAT TCP/IP LabVIEW Real-Time Machine Motor w/ encoder ω m Position sensors x x 3 phase Vac Encoder signal EtherCAT EtherCAT Analog out Analog out Analog in Analog in p A p B Pressure sensors Servo valve F L x sv Force sensor Analog in Analog out Analog in Analog in Analog in Analog in p L,S p L,A p L,B Analog in p P12-sm Analog in p P2-pm Pressure sensors Analog in p P3-sm Q AV Proportional valves Q BV Figure 5.1: Data communication/connection diagram of the proposed Data Acquisition and Control System (DAC).

46 36 Control and Experimental Evaluation of SvSDP Concept In the attempt of establishing the connection between the RT-machine and the servo-drive, an incompatibility between the underlying EtherCAT messaging protocols supported by the servodrive and the RT-machine was discovered. The servo-drive supports Servo Profile over EtherCAT (SoE) [Bosch Rexroth AG, 29b, p. 2], whereas LabVIEW (by default) does not support this particular protocol [National Instruments, 214]. Since EtherCAT is an emerging fieldbus system, the documentation is still rather limited, why this incompatibility was only discovered based on the error message received during the connection attempt. As a consequence, the proposed DAC topology could not be realized directly, and the actually implemented DAC is thus shown Fig This modification has only slight influence on the software part of the DAC, why the functionality described in Section 5.1 is valid for both implementations, unless marked otherwise. RS232 HMI (PC2) HMI (PC1) TCP/IP IndraDrive (Servo-drive) Converter Control Unit Analog ω m,ref Analog uav ref Analog ubv ref LabVIEW Real-Time Machine Motor w/ encoder ω m 3 phase Vac Encoder signal Analog out Analog out CANopen over EtherCAT Proportional valves Q BV Q AV Position sensor Analog in Analog in p A Analog in p B Analog in p P12-sm Analog in p P2-pm Analog in p P3-sm x F M Force sensor Pressure sensors Figure 5.2: Data communication/connection diagram of the actually implemented Data Acquisition and Control System (DAC). From research of the possibilities for circumventing the incompatible EtherCAT protocols, a possible solution has been identified, which allows for the same flexibility as the initially proposed DAC. This solution requires a significant modification of the DAC software structure, why it is considered beyond the scope of this project. However, the functional principle is described in Section 5.2 for possible future reference. 5.1 Software Functionality of the DAC The software implemented on the RT-machine is a heavily modified version of the "LabVIEW Real-Time Control (NI-DAQmx)"-sample project developed by National Instruments. Real-Time Main Loop The real-time main loop implemented on the RT-machine is clocked by the sampling trigger of the input/output channels. The sequence within the main loop is as follows, where some of the functions are described in more detail in the remainder of this section: The raw measurement samples are read from the input channels, converted to engineering units and possibly filtered. It is verified whether the converted data is within specified limits, and an error is triggered if these limits are exceed. Based on the result of the limit check, as well as the target controller mode from the HMI, the state machine of the RT-machine is updated accordingly. Depending on the actual controller mode, the output values are evaluated, either directly from the target values of the HMI, or through closed loop control. The output values are written to the corresponding output channels, but the actual update of output values is triggered by the sampling trigger, i.e.

47 Chapter 5 - Implementation of Data Acquisition and Control System 37 at the beginning of the next loop iteration. Finally, both the measurements and the target outputs are stored in a data buffer for visualization and data logging. HMI Communication The communication between the HMI and the master unit is based on a TCP/IP message queue, i.e. for each action performed on the HMI, a message is constructed consisting of a header and the data to be transferred. The message is then enqueued and transferred to the RT-machine. A low priority loop on the RT-machine processes the message by reading the header in order to determine the required action, and then loads the data in to the real-time main loop. The procedure is similar for data transferred from the RT-machine to the HMI. This approach allows for use of the HMI with a high level of data integrity and low influence on the execution of the main loop. Lossless Shared Data Buffer The process data of each loop iteration, i.e. the instantaneous measurements and applied outputs, are stored in a First-In First-Out (FIFO) buffer hosted by the RT-machine. The data in the buffer is read by, and possibly saved on, the HMI computer in chunks. This allows for lossless logging of the sampled data, as long as the HMI computer reads the data from the buffer sufficiently fast. A flag is included in the log files, indicating whether an overflow has occurred, and thereby whether data has been lost. Trajectory From File To ease the conduction of tests, it is made possible for trajectories/input sequences to be loaded from text files into a FIFO buffer on the RT-machine. The use of a buffer allows for minimization of the main loop overhead, since only a memory reference and not the associated data is transferred between iterations. At the successful loading of the data from a file, the RT-machine sends back a graphical interpretation of the trajectory to the HMI, such that the validity of the trajectory may be visually verified prior to the test. Through the HMI it is possible to enable/disable repetition of the trajectory, where disabling of the repetition function stops the trajectory at the end of the current cycle. Drive Communication The drive communication for the originally proposed DAC (see Fig. 5.1) consists of full read/write access to the parameter registers of the drive, which reduces the functionality of the drive to a slave input/output module. In this way, no additional software is required on the drive control unit, why control system implementation, parameterization, error handling etc. may be carried out exclusively through the RT-machine. As described in the beginning of this chapter, this functionality could not be readily achieved with the available hardware/software, why the structure shown in Fig. 5.2 is implemented instead. As the parameter registers cannot be directly accessed, the desired drive outputs are instead transmitted as a voltage output from the RT-machine. The voltage signals are measured through the analog input channels of the drive, and the values are translated into corresponding output register values within the drive control unit. This implementation is sufficient for the performed experiments, but the drive internal data logging functions must be utilized to trace the internal variables of the drive such as measured motor velocity, estimated motor torque, etc. An alternative DAC topology, making it possible to circumvent these limitations is briefly described in Section 5.2.

48 38 Control and Experimental Evaluation of SvSDP Concept Controller Mode/State Machine A state machine is implemented in the RT-machine, which determines the current controller mode. The desired state may be changed through the HMI, but whether the state is actually changed, depends on the current state as well as the presence of errors. The five possibles states are briefly described below. Safe State (Active) - The default state when connecting to the RT-machine and no errors are present. In this state, the outputs are fixed to the "Safe Output Values", which in this case are for all outputs. Since no errors are present when in this state, changing the desired state through the HMI directly changes the actual state of the RT-machine. Safe State (Faulted) - In the event of an error, the RT-machine will immediately change into the faulted safe state and remain there. The faulted safe state is similar to the active safe state in that outputs are fixed to the "Safe Output Values", however, changing the desired state through the HMI generally will not have any effect until the error(s) is no longer present. The state can however be changed to the fault override state, such that the error may be resolved. Manual, Fault Override - The fault override state allows for manually controlling the outputs, but only with severely limited possibilities. An example of error could be exceeding limitations on the measured variables, where this state allows the user to move the system back into allowable ranges in a controlled manner. Manual - The manual state gives direct control of each output. The values may be controlled either directly through the HMI or by loading and starting a reference file. Closed Loop - In the closed loop state, the outputs are controlled indirectly by the implemented control system. The functionality of the HMI inputs changes accordingly, such that these now correspond to references for each of the controlled variables. Similarly, reference files now correspond to desired closed loop trajectories. Error Handling Error handling within the RT-machine depends on the source of the error. In the case of an error returned by the internal functions, e.g. if the value of a measurement can not be read, the software immediately clears all outputs and terminates the software loops, as it may indicate either a hardware error or an error in the basic software functionality. Expectable errors occur when measured signals exceed the expected/allowed values, e.g. if the slide position is getting too close to the ends. The reaction to these errors is instead to change into the faulted safe state as described above, and forward a message to the HMI, such that the signal that triggered the error may be identified. This allows for running test sequences and evaluate control systems, with reduced risk of damaging the hardware of the setup.

49 Chapter 5 - Implementation of Data Acquisition and Control System Alternative DAC Proposal As an alternative to the currently implemented DAC, the solution illustrated in Fig. 5.3 is proposed for future reference. ω m RS232 IndraDrive Servo-Drive (Control Main System) Control Unit Converter 3 ph Vac Motor w/ enc Microsoft Windows PC (References and Datalog) IndraWorks Analog out Analog out Analog in Analog in Position sensors x x Non-Real-Time LabVIEW HMI Integrated Real-Time Windows EtherCAT Master Environment CoE/SoE CoE LabVIEW Real-Time Machine (Control Load System) Analog in Analog out Buffer CoE/SoE Analog in Analog in Analog in Analog in Analog in Analog in Analog in Q AV Proportional valves Q BV p A p B Pressure sensors Servo valve F L x sv Force sensor p L,S p L,A p L,B p P12-sm p P2-pm p P3-sm Figure 5.3: Proposed alternative DAC solution utilizing the LabVIEW EtherCAT Library developed by Ackermann Automation (see [Ackermann Automation GmbH, 215]) The proposal is based on the LabVIEW EtherCAT Library developed by Ackermann Automation (see [Ackermann Automation GmbH, 215]), and the enclosed Real-Time environment driver for Microsoft Windows. Similar to the original DAC topology, a single PC is still used as a LabVIEW HMI. However, using the Real-Time environment, a CPU core may be reserved on the HMI PC for an EtherCAT Master process, which utilizes the Ethernet port of the PC. This allows for buffered data exchange between the LabVIEW HMI and an EtherCAT network with cycle rates of up to 1 khz [Ackermann Automation GmbH, 215]. The HMI PC thus acts as the EtherCAT Master, which should enable communication between all the utilized devices, as the Library supports both CoE (RT-machine, Position Sensor) and SoE (Servo-drive). The RT-machine is therefore reduced to a slave device, responsible for acquisition of measurements from analog channels, as well as control of the load system. This also means, that the control system for the main system could be implemented in the control unit of the servo-drive, giving a more application oriented DAC implementation. To obtain piston position feedback to the drive, the data read by the position sensor is forwarded directly to the drive, through the EtherCAT Master interface. Finally, all relevant data from the three slave devices may be streamed to the EtherCAT interface, which should allow for buffered reading and data logging within the HMI PC.

50

51 Part II Modeling and Development of Control Strategies for SvSDP System 41

52

53 Chapter 6 - Modeling and Parameterization of the SvSDP System 43 6 Modeling and Parameterization of the SvSDP System This part presents the results of the main objective of this project, i.e. the experimental evaluation and control of the SvSDP concept. This chapter presents the development, parameterization and verification of a dynamic model of the designed SvSDP system. In Chapter 7, the developed model is linearized and analyzed. Since the SvSDP system is a Multiple Input Multiple Output (MIMO) system, the choice of control strategy is based on an investigation of the couplings between the various inputs and outputs, followed by an interpretation of the results and a discussion of proper choice of control strategy. The results of the coupling analysis indicates that decentralized control may not readily be suitable, why decoupling methods are presented in Chapter 8. Finally linear feedback controllers are designed for the decoupled system in Chapter 9, and the performance of the control system(s) is evaluated in Chapter 9. As measurement data has been unavailable at the time of the initial model development, the actual work flow has been as follows: An initial model has been developed for dimensioning and selection of components, as well as control analysis and design, to better take advantage of the lead time of the ordered components. The initial model has been based on both component data sheets, as well as experimental results of the original SvDP system [Nielsen and Jensen, 214]. As the designed system is only in part similar to the original SvDP system, slight parameter and model uncertainty has been expected. As a consequence, test sequences has been defined based on the initial model, which allows for both estimation of the unknown parameters as well as model verification. The test sequences are chosen, such that each sub-model and corresponding unknown parameters may be determined as separately as possible. In this way, the scalability of the proposed concept is improved, with minimal influence from the current system configuration. Not all model verification has been carried out prior to project submission, why the simulation model cannot be fully verified. Based on the test results, some of the governing equations of the initial model has been altered slightly, to better match the observations. To avoid confusion between the initial and the final model, only the governing equations and parameters for the final model are therefore presented where experimental results have been obtained. The model is divided into two main parts: The first part is the mechanical model linking the main SvSDP system to the load system, including the inertia and friction properties, based on the setup shown in Fig The second part is the hydraulic model of the SvSDP system, including pumps, valves and pressure dynamics of the main cylinder, which is based on the diagram shown in Fig. 6.2.

54 44 Control and Experimental Evaluation of SvSDP Concept Main cylinder Load cylinder x Figure 6.1: Cross section view of the mechanical setup. p A x p B A p A r M V A V B Q AV Q BV Q PRB Q PRA Q AP Q BP p P12-sm p P2-pm p P3-sm Q CVAR Q CVAP1 Q CVAP21 Q CVBP3 Q CVBR Q CVAS V P12-sm Q CVAP2 V P2-pm V P3-sm Q CVBS M ω m Q P1 Q P2 Q P3 P1 P2 P3 Figure 6.2: Hydraulic diagram of main system Since the majority of the test setup has been designed and specified in this project, properties such as cylinder and tubing volume, slide mass, etc. are known. The unknown parameters relates mainly to the dynamics parameters of components that has not been uniquely specified in the respective data sheets, such as friction coefficients and leakage parameters. To sequentially verify the model, the governing equations of each sub-model are presented, followed by the procedure for extracting the targeted parameter from the raw measurement data. To clarify how the measurement data of each test may be processed to identify the specific parameters, examples of the defined test sequences and corresponding measured responses are presented in Section 6.1. Section 6.2 present the mechanical model, relating the cylinder pressure forces to motion of the cylinder pistons and the sliding mass, while the pressure dynamics of both cylinders are described by the hydraulic model presented in Section Test Sequences for Parameter Determination The defined test sequences utilizes only the main system inputs, why the load cylinder is disconnected from the slide and the load pin is inserted between the main cylinder and the slide, to measure the actuation force. All test sequences are initialized by a "discharge period", where both proportional valves are fully open. This reduces both chamber pressures to tank pressure, such that all the tests are carried out with identical initial conditions. The test sequences have been

55 Chapter 6 - Modeling and Parameterization of the SvSDP System 45 designed such that the measurement data from each sequence may be utilized for determination of several parameters where possible. Additionally, the parameters are determined sequentially, i.e. it is attempted to determine the parameters that may be isolated, and then use these to determine the remaining parameters. In the following, the terms "inlet" and "return" are used to denote the expanding and contracting cylinder chambers respectively, i.e. for positive velocities, the inlet side corresponds to the A-side, while the return side corresponds to the B-side, and vice versa for negative velocities. The inputs and measurement data for all test sequences may be found on the attached CD-ROM, while only the processed data as well as examples of raw measurement data are presented in this chapter Test Sequence 1 In Test Sequence 1 the return side proportional valve is held open at all times, such that the inlet pressure rises just enough to overcome the friction. For a given velocity, this thus corresponds to the minimum possible chamber pressures, why the influence of leakage terms is minimized. The purpose of this test sequence is therefore: Determination of slide and cylinder friction parameters Verification of pump flow speed dependency In each test run the normalized shaft velocity reference (ω m,ref-norm ) is changed, such that the slide moves at a different steady state velocity for each test run. As this test is specially designed to determine the friction parameters, the shaft velocity reference is changed in small steps for low velocities, such that the stiction behavior is accurately observed. One of the input sequences, and the corresponding measurements, are shown in Fig Velocity [mm/s] Position [mm] Normalized Input [ ] ω m,ref norm u AV,ref norm u BV,ref norm ẋ x Pressure [bar] 2 1 p A p B Force [kn] F M F M,P Time [s] Figure 6.3: References and measurements for Test Sequence 1. F M,P is the theoretical cylinder force calculated based on the chamber pressures. The magnitude of ω m,ref-norm is changed for each test run.

56 46 Control and Experimental Evaluation of SvSDP Concept Test Sequence 2 In Test Sequence 2 both proportional valves are held closed while the servo drive shaft speed is changed with steps in both directions. This allows for both pressure build-up and excitation of dynamics. The purpose of this test sequence is therefore: Determination of main system bulk modulus parameters Verification of pump flow pressure dependency Based on the frequency of the induced oscillations, the bulk modulus parameters may be determined. After the shaft speed of the servo drive has been driven to zero, the chamber pressures drops due to the leakage across the pumps, why the pump flow dependency may be determined. One of these input sequences, and the corresponding measurements, are shown in Fig Velocity [mm/s] Position [mm] Normalized Input [ ] ω m,ref norm u AV,ref norm u BV,ref norm ẋ x Pressure [bar] Pressure [bar] p A p B p P2 pm p P12 sm Time [s] p P3 sm Figure 6.4: References and measurements for Test Sequence 2. The virtual tank pressures p P12-sm and p P3-sm are smoothened over 1 samples to reduce the appearance of measurement noise. The magnitude of ω m,ref-norm is changed for each test run Test Sequence 3 In Test Sequence 3 the servo drive is actuated by a step reference, while both proportional valves are held closed. For sufficiently large shaft velocity references, both chamber pressures will increase, as the system will operate in χ > 1. After a time period the inlet proportional valve is actuated with a step input, which will decrease the steady state piston velocity. The size of the step input to the inlet proportional valve is changed for each test run, such that the relation between input voltage to the proportional valves and steady state velocity can be mapped. The purpose of this test sequence is therefore:

57 Chapter 6 - Modeling and Parameterization of the SvSDP System 47 Verification of the flow characteristics of the proportional valves Verification of the dynamic response of the proportional valves One of these input sequences are shown in Fig Normalized Input [ ].5.5 ω m,ref norm u AV,ref norm u BV,ref norm Figure 6.5: References and measurements for Test Sequence 3. The magnitude of both u AV,ref-norm and u BV,ref-norm are changed for each test run Test Sequence 4 In Test Sequence 4 an increasing step sequence is given to the servo drive, while both proportional valves are held fully open. The open proportional valves should minimize the loading on the servo drive and thereby the influence of the hydraulic system on the servo drive dynamics. Determination of servo drive dynamics The test sequence is shown in Fig Shaft Velocity [RPM] Normalized Input [ ] Shaft Torque [Nm] ω m,ref norm u AV,ref norm u BV,ref norm ω m T Time [s] Figure 6.6: References and measurements for Test Sequence 4. The shown torque limitation is software controlled and it may be increased in a future evaluation of the SvSDP. In this project, however, the torque limitation is unaltered for the test sequences presented in this chapter.

58 48 Control and Experimental Evaluation of SvSDP Concept 6.2 Mechanical Model The mechanical model describes the interaction between the two hydraulic systems. The mechanical system is here considered as the sliding part (from this point denoted only as the slide) only, where the stiffness of the frame is considered to be sufficiently higher than that of the combined actuation system. Additionally, the slide is considered as a single stiff body which is limited to axial motion. The position of the slide is measured with respect to the center, and the zero position, i.e. x = is defined as the point where the center of the slide intersects with the center of the frame. The motion dynamics of the slide is described by Eq. 6.1, based on the free body diagram shown in Fig M F M,P F L,P F fric,cyl F fric,sli x F fric,cyl Figure 6.7: Free body diagram of the slide with denoted forces. As the cylinders are identical, the cylinder frictions are modeled to be identical. M ẍ = F M,P ( ) F L,P + F fric,cyl (F fric,sli + F fric,cyl) (6.1) }{{}}{{} F L F fric F M,P and F L,P are the main and load cylinder theoretical pressure forces, F L is the load cylinder force including the load cylinder friction, and F fric is friction force of the main cylinder and slide. The forces pressure forces in Eq. 6.1 are described by Eq F M,P = A p (p A α p B (1 α) p atm ) F L,P = A p ( pl,a α p L,B (1 α) p atm ) (6.2) The slide and cylinder friction is determined from Test Sequence 1. In this test sequence, the return side proportional valve is always fully open, why the chamber pressures will settle after an initial transient. The cylinder friction during steady state is then determined as the difference between the real cylinder force F M, measured by the load pin, and the theoretical cylinder force F M,P, estimated by the measured pressures, as given by Eq F fric,cyl = A p (p A α p B (1 α) p atm ) F fric,sli (6.3) }{{}}{{} F M,P F M Different shaft velocity references are applied for each test run, such that the speed dependency of the friction can be determined. As the scope of this project is not to develop a general cylinder friction model, any pressure dependent cylinder friction is neglected. The processed measurement data is shown in Fig. 6.8, where Hanning windows are used to determine a weighted mean velocity, cylinder force and chamber pressures during the steady state operation. The Hanning windows are applied to the measured signals after an approximate settling time of 5 ms after the step, such that the influence of any small oscillations still present due to dynamics are minimized. The same Hanning window is also applied to the measured velocity, where it should be noted that, due to the quality of the sensor, these measurements are virtually noise-free, with fluctuations of only ±.1 mm/s.

59 Chapter 6 - Modeling and Parameterization of the SvSDP System 49 Slide Friction [N] Measurements Model Cylinder Friction [N] Measurements Model Velocity [mm/s] Velocity [mm/s] (a) Slide friction as function of velocity. (b) Cylinder friction as function of velocity. Figure 6.8: Friction model and experimental data with all 11 mass plates mounted on the slide (M = 658 kg). The slide friction measurements corresponds to a Coulomb friction coefficient within the expected range of [Igus R GmbH, 215, p. 979]. Furthermore, no apparent stiction is present in the measurements, why the slide friction is modeled by the hyperbolic tangent function given in Eq F fric,sli = µ sli M g tanh (γ sli ẋ) + B sli ẋ (6.4) The cylinder friction measurements agree with a cylinder friction model consisting of a Stribeck, a Coulomb and a Viscous friction term [Andersson et al., 26, p. 583]. To avoid abrupt changes in the cylinder friction, the sign functions in the Stribeck and Coulomb friction terms are replaced by a continuous hyperbolic tangent function, as given in Eq ( ) F fric,cyl = F c,cyl + (F s,cyl F c,cyl ) e ẋ /v s,cyl tanh (γ cyl ẋ) + B cyl ẋ (6.5) The estimated friction parameters are presented in Tab µ sli γ sli B sli F c,cyl F s,cyl v s,cyl γ cyl B cyl [ [ ] s ] [ ] N s [ mm mm [N] [N] mm ] [ s ] [ ] N s s mm mm Hydraulic Model Table 6.1: Estimated slide and cylinder friction parameters. The hydraulic model describes both the pressure dynamics and the size of the various flows. The hydraulic diagram of the main system is shown in Fig The model for the main system is, to a large extend, an expanded version of the model presented in [Grønkjær and Rahn, 214b, p. 3-9], but for continuity and self-containment the entire model is presented in this report as well. To investigate the influence of the check valve dynamics, three models are developed, with different levels of detail regarding the check valves. Simulation results of the different models are used both to determine if the selected check valves are appropriate (see Section and Appendix C), as well as to determine whether a simplified model, where the check valves are modeled as ideal, is sufficient for analysis and control development. The ideal check valve is here considered as a valve with zero pressure drop when open and with no possibility of transient reverse flow. The governing equations for all three models are presented in this section, and it is denoted, which models they are part of.

60 5 Control and Experimental Evaluation of SvSDP Concept Pressure Dynamics The pressured dynamics within the cylinder chambers are described by the continuity equations given in Eq. 6.6, which is valid for all three models. ṗ A = β A V A ( Q AP Q AV Q PRA V A ) ṗ B = β B V B ( Q BP Q BV Q PRB V B ) (6.6) The absolute volume and the volume rate of change for these chambers are described by Eq. 6.7 and Eq V A = V A + A p x VA = A p ẋ (6.7) V B = V B α A p x VB = α A p ẋ (6.8) When considering the check valves as ideal, the flow Q AP is given by Eq. 6.9, while the flow through the anti-cavitation system, as well as the pressure relief valves, is implemented by simply limiting the pressure at the crack pressure of CVAR and PRA respectively. Similarly, the pressure within the B-side chamber is limited to the crack pressure of CVBR and PRB. Q AP = Q P1 + max (Q P2, ) (6.9) When modeling the flow through the check valves, the pressure dynamics of the chambers within the check valve manifold block is also considered. These are also described by the continuity equation as given in Eq ṗ P12-sm = β P12-sm V P12-sm ( Q P1 + Q CVAS Q CVAR Q CVAP1 ) ṗ P2-pm = β P2-pm V P2-pm (Q P2 Q CVAP21 + Q CVAP2 ) ṗ P3-sm = β P3-sm V P3-sm (Q P3 + Q CVBS Q CVBR Q CVBP3 ) (6.1) The volume of these minor control volumes are constant and based on the geometry of the check valve manifold and piping. Since the bulk modulus of the oil within each chamber is a function of the respective chamber pressures, the following derivation uses "x" as a placeholder for the A, B, P12-sm, P2-pm and P3-sm control volumes. The pressure dependency of bulk modulus is modeled by Eq. 6.11, based on [Andersen and Hansen, 23, Sec ]. 1 β x = 1 + V (6.11) x,%-air β oil β x,air β oil is the bulk modulus of the pure oil, describing the maximum value of Eq The value used for β oil is determined from experiments as described below, where the utilized value represents the limitations due to finite hose stiffness etc. β air and V x,%-air represents the bulk modulus and relative volume of air entrapped in the oil. These are described by Eq [Andersen and Hansen, 23, Sec ]. β x,air = p x κ ( ) 1 patm V x,%-air = V κ %,air,atm (6.12) p x κ 1.4 is the adiabatic constant for air, p atm 1 bar is the atmospheric pressure and V %,air,atm is the relative volume of air entrapped in the oil at p x = p atm. This value influences the slope of the bulk modulus curve (see Fig. 1.3), and is also determined from experiments. The values of β oil and V %,air,atm determines the pressure dependency of the oil bulk modulus, where β oil mainly represents the maximum value, whereas V %,air,atm determines at which pressure

61 Chapter 6 - Modeling and Parameterization of the SvSDP System 51 the maximum value is reached. A method to determine these two unknown parameters is to excite the hydraulic system at different pressure levels, using the Test Sequence 2. The natural frequency of a hydraulic cylinder is mainly influenced by the chamber volumes, the load inertia as well as the oil bulk modulus as given in Eq [Grønkjær and Rahn, 214b, p. 42]. ω n,cyl = A 2 p M ( β(pa ) V A (x) + β(p ) B) α2 V B (x) (6.13) For chamber pressures above 5 bar the amount of entrapped air in the oil has only minor influence on the effective bulk modulus. By estimating ω n,cyl from the velocity step response of experiments conducted at high enough chamber pressures, then β oil may be calculated directly from Eq by setting β(p A ) = β(p B ) = β oil as shown in Eq β oil A 2 p ω 2 n,cyl M ( 1 V A + α2 V B ) p A, p B > 5 bar (6.14) The Test Sequences 2 is run for a total of 2 times, in which the magnitude of the shaft velocity has been increased for each run. The natural frequency of system is at each run determined based on the mean frequency of four oscillation periods of the velocity response. 7 of these 2 runs fulfill the criteria of p A, p B > 5 bar, which means that they are used for the estimation of the β oil parameter as shown in Tab Test run x [mm] p A [bar] p B [bar] ω n,cyl [rad/s] Est. β oil [bar] Table 6.2: Estimated β oil based on Eq. 6.14, using the measurement data obtained from the Test Sequence 2. The natural frequency w n,cyl is estimated based on the mean frequency of four oscillation cycles. The value of β oil is found by calculating the mean of the estimated β oil in Tab. 6.2 and excluding the smallest and largest estimate. The remaining V %,air,atm parameter may be found from the test runs with lower chamber pressures, by solving the quadratic equation Eq Eq is directly obtained by combining the expressions Eq. 6.11, Eq and Eq a 2 V 2 %,air,atm + b 2 V %,air,atm + c 2 = (6.15) The coefficients for Eq are given in Eq and Eq ) a 2 = a b d b 2 = c d (a + b) b V B α 2 a V A c 2 = c (c d α 2 V A V B (6.16) a = ( ) 1 patm κ p A p A κ b = ( ) 1 patm κ p B p B κ c = 1 β oil d = V A V B M A 2 p ω 2 n,cyl (6.17) 5 of the 2 test runs in Test Sequence 2 are suitable for estimation of V %,air,atm, due to their stable and low chamber pressures at the excitation points, and they are shown in Tab. 6.3.

62 52 Control and Experimental Evaluation of SvSDP Concept Test run x [mm] p A [bar] p B [bar] ω n,cyl [rad/s] Est. V %,air,atm [%] Table 6.3: Estimated V %,air,atm based on Eq. 6.15, using the measurement data obtained from the Test Sequence 2. The natural frequency w n,cyl is estimated based on the mean frequency of four oscillation cycles. The value of V %,air,atm is found by calculating the mean of the estimated V %,air,atm in Tab. 6.3 and excluding the smallest and largest estimate. The two bulk modulus parameters are summarized in Tab β oil V %,air,atm 65 bar.38 % Table 6.4: Estimated bulk modulus parameters Pump Model Pump Flow The pump flow Q Px is modeled to be a function of the pressure differential p across the pump and the shaft speed. The flow characteristics have in the initial model been made by the scaling of experimental data available for the 14 cc and 28 cc pump of the AZPF series [Bosch Rexroth AG, 29a] and [Bosch Rexroth AG, 21a]. This has been discussed in detail in [Grønkjær and Rahn, 214b, p. 5-6], where is has been found, that the pump flow characteristic can be described by a polynomial in the form of Eq Q Px = K Pxω ω m + K Pxp p + K Pxp2 ( p) 2 (6.18) K Pxω is a coefficient that relates to the effective displacement of the pump at zero pressure differential, and the coefficients K Pxp and K Pxp2 describes the pressure dependency, i.e. the leakage characteristic. The coefficients for the utilized pumps are given in Tab Pump Name [ ] D cm 3 nom rev K Pxω [ ] L/min RPM K Pxp [ ] L/min bar K Pxp2 [ ] L/min bar 2 P P P Table 6.5: Pump flow coefficient values for the utilized pumps, including the sign changes due to the reverse mounting of the pump P3, as well as the choice of positive flow direction. The nominal displacement K Pxω of the pumps are validated by removing the influence of the pump operating in motor mode, by fully opening the outlet side proportional valve and thus operating at low pressure level. The driving pump(s) will operate at low pressure level in steady state, as

63 Chapter 6 - Modeling and Parameterization of the SvSDP System 53 only the slide and cylinder friction gives rise to an inlet chamber pressure. A step reference is given to the servo drive, and after a transient, the piston moves at constant velocity. The test sequence is carried out for both directions, to validate the nominal displacement of both pump P1+P2 and pump P3. The velocity of the slider is measured, filtered using a Hanning window, and converted to a cylinder flow using the cylinder area Q P1 + Q P2 = ẋ A p and Q P3 = ẋ A p α. The experimental results are shown in Fig P1 + P2 Pump Flow [L/min] Measurement Nominal RE 1 89 Nominal RA Shaft Velocity [RPM] (a) Pump P1 and P2 in pumping mode. Pump P3 bypassed using proportional valve BV. P3 Pump Flow [L/min] Measurement Nominal RE 1 89 Nominal RA Shaft Velocity [RPM] (b) Pump P3 in pumping mode. Pump P1 bypassed using proportional valve AV. Figure 6.9: Measured and simulated pump flow. Nominal displacement of pumps based on the new data sheet "RE 1 89/2.12" [Bosch Rexroth AG, 212a, p. 4], and based on the old data sheet "RA 197/2.6" [Rexroth, 26, p. 14]. From Fig. 6.9a it may be seen, that the flow gain of the pumps are above the displacements stated in the [Bosch Rexroth AG, 212a] data sheet. The effective displacements corresponds to pumps with displacements according to the old datasheet "RA 197/2.6", with the flow coefficients given in Tab The match ratio (χ-graph) for the delivered system is therefore slightly different compared to the specified χ-graph in Fig As the test bench is unsuitable for conducting a full pump flow map, no new χ-graph is made. The pump leakage is initially modeled to only be pressure dependent. The system is actuated using Test Sequence 2, such that both chamber pressures are increased. After the shaft velocity reference step sequence, the chamber pressures drop due to the pump leakages. By measuring the position, velocity and pressures, as well as estimating the pressure gradients, the expression for the pump leakage given in Eq may be found. ṗ A = β (p A) V A (x) ( Q P1-leak A p ẋ) ṗ B = β (p B) V B (x) (Q P3-leak + α A p ẋ) ṗ P2-pm = β (p P2-pm) V P2-pm ( Q P2-leak ) (6.19) The gradients of the pressures are estimated based on an off-line smoothing of the measured pressures, such that large spikes due to noise in the measurements are compensated for. The processed leakage coefficients are shown in Fig. 6.1 for the P2 pump.

64 54 Control and Experimental Evaluation of SvSDP Concept Leakage Coefficient [(L/min)/bar] x Test run 14 Test run 17 Model Test run 15 Test run 18 Test run 16 Test run Pressure p P2 pm [bar] Figure 6.1: Pump P2 leakage coefficient. The corresponding pressure dependent pump parameters are given Tab Pump Name [ ] D cm 3 nom rev K Pxω [ ] L/min RPM K Pxp [ ] L/min bar K Pxp2 [ ] L/min bar 2 P P P Table 6.6: Estimated pump flow leakage coefficients based on measurements on the SvSDP. Pump flow gains based on [Rexroth, 26, p. 14] The preliminary data in Tab. 6.6 suggest that the delivered pumps have much lower leakage coefficients compared with the data obtained in Tab Further testing should be carried out to more accurately determine the pressure dependency of all the pumps. Pump Torque The pump torque as a function pressure differential p across the pump and the shaft speed is modeled based on scaling of experimental data available for only the 14 cc [Bosch Rexroth AG, 29a]. Based on [Grønkjær and Rahn, 214b, p. 6-7] it has been found that the pump torque characteristic may be described by a polynomial in the form of Eq T Px = sign (ω m ) K TPxC + K TPxω ω m + sign (ω m ) K TPxL p + K TPxD p (6.2) Here T TPxC is the Coulomb friction coefficient of the pump, K TPxω is the viscous friction coefficient, K TPxL is a pressure dependent friction torque coefficient and K TPxD is the nominal displacement of the pump expressed in the units given in Tab Pump Name [ ] D cm 3 nom rev [ ] K TPxC [Nm] K Nm TPxω RPM K TPxL [ ] Nm bar K TPxD [ ] Nm bar P P P Table 6.7: Pump torque coefficient values for the utilized pumps, including the sign changes due to the reverse mounting of the pump P3, as well as the choice of positive flow direction.

65 Chapter 6 - Modeling and Parameterization of the SvSDP System Proportional Valve Model The pressure compensated 2/2 proportional valves used to throttle excess flow are of type KKDS. Due to the pressure compensation, the steady state flow through the valve depends almost exclusively on the command signal as shown in Fig [Bosch Rexroth AG, 211b, p. 6]. Flow [L/min] p = 1 bar p = 2 & 1 bar p = 3 bar p = 5 bar Command value [%] Figure 6.11: Flow characteristic of the KKDS proportional valves as a function of command signal and pressure differential across the valve. Flow direction port 2 1. [Bosch Rexroth AG, 211b, p. 6] The flow characteristic is implemented in the simulation model as a lookup table to include the slight pressure dependency and non-linear behavior near the deadband. The valve spool is actuated by a solenoid amplified by a VT-SSPA1 proportional amplifier. The amplifier is considered to be the main limitation for the large signal bandwidth of the valve due to a minimum slew-rate limitation of the command signal of u v-max [Bosch Rexroth AG, 21c, p. 5]. Based on the time response from other slew-rate limited valves [Moog Inc., 29, p. 8], [Bosch Rexroth AG, 25, p. 14], the small signal response is approximated by a critically damped second order system with a settling time, which is approximately half the ramp time from 1%. This gives a small signal bandwidth of ω n = 8 t ramp. It should be noted that this approximation is based on servovalves that utilizes spool position feedback, which is not the case for the KKDS valves. The approximation is however considered sufficient for simulation and control design purposes. The parameters are summarized in Tab ω n,xv ζ xv u v-max rad/s %/s Table 6.8: Parameters for the identical proportional valves AV and BV No direct measurement of either the flow characteristics or the dynamics of the proportional valves can be made as no flow measurement or spool position measurement are available. Instead an indirect approach is used, where Test Sequence 3 is run. A input reference shaft velocity is given for the servo drive with both proportional valves closed, such that both chamber pressures may increase. After a time period, one of the proportional valves is actuated by a step input, held open for a time period and then closed again with a step input. The test sequence is carried out multiple times, where the magnitude of the proportional valve input steps are changed for each run. Steady state pressures are required, such that the valve flows may be described directly by the piston velocity and pump flow. It is required that the pump model is valid, such that the pump flows can be estimated based on shaft velocity and pressures as given in Eq and Eq ) ( ) Q AV = Q P1 (ω m, (p A p P12-sm ) + Q P1 ω m, (p P2-pm p P12-sm ) A p ẋ (6.21) ) Q BV = α A p ẋ Q P3 (ω m, (p B p P3-sm ) (6.22)

66 56 Control and Experimental Evaluation of SvSDP Concept Due to small variations in the measured piston velocity and pressures, the measured quantities are averaged using an exponential window, such that the velocity and pressures in the end of the step time period is weighted highest. As no external load can be applied to the slider, the pressure dependency of the valve flows can only be tested for small command values, due to the limited amount of friction force. The processed valve flow characteristics are shown in Fig for a simulation run. 4 3 Model Measurement 4 3 Model Measurement Flow [L/min] 2 1 Flow [L/min] Command value [%] (a) Estimated A-side proportional valve flow Command value [%] (b) Estimated B-side proportional valve flow. Figure 6.12: Illustration of the validation method, which will be used to estimated the proportional valve flow. Simulation result Servo Drive To determine the dynamic behavior of the servo drive, a step sequence is given to the drive with both proportional valves fully opened. The shaft velocity is logged at the maximum sampling frequency of 2 khz using the built-in oscilloscope in IndraLogic. The results are shown in Fig Step 3 RPM Step 9 RPM Step 15 RPM Step 21 RPM Step 27 RPM Simulation Step 6 RPM Step 12 RPM Step 15 RPM Step 18 RPM Step 24 RPM Simulation 3 Shaft Velocity [RPM] Shaft Velocity [RPM] Time [ms] Time [ms] (a) Positive shaft velocity steps. (b) Negative shaft velocity steps. Figure 6.13: Measured and simulated step-responces of the servo drive, with both proportional valves fully opened. Based on Fig. 6.13, the servo drive is modeled as a slew rate limited, under-damped second order system with unity DC-gain and parameters given in Tab. 6.9.

67 Chapter 6 - Modeling and Parameterization of the SvSDP System 57 Name Description Value Unit ω n,drive Small-signal bandwidth 12 Hz ζ drive Small-signal damping coefficient.5 - ω m,max Slew rate limit 95 krpm/s Table 6.9: Estimated servo drive parameters. It has initially been assumed that the drive is only limited by the slew rate limitation, but based on experiments it has been found, the slew rate limit is likely due to a torque limitation. This means that the slew rate limit is expected to vary slightly depending on the operating conditions, however this has not been included in the model Check Valve Model The check valve model is based on the flow characteristic from the data sheet of the utilized M-SR check valves [Bosch Rexroth AG, 211a]. Two different models are made, a quasi-static model based on steady state flow characteristic, and a dynamic model based on forces acting on the valve poppet. The dynamic model is made to investigate the effect of the valve dynamics on the system performance, while the quasi-static model is made for improved simulation performance. The dynamic model is based on physical insight and therefore utilizes estimated values for various valve parameters. This may in turn introduce significant modeling inaccuracies and simulation issues compared to the quasi-static model which may be based entirely on the available flowpressure data. The quasi-static model however, does not take into account the dynamics of the valve poppet, which should preferably be neglectable. Quasi-static Model In this model, an orifice equation is used to describe the relation between the pressure drop p cv across the valve and the flow Q cv through the valve. This relation is given in Eq Q cv = Q cv-n pcv-n x cv,norm p cv sign ( p cv ) (6.23) Q cv-n and p cv-n are flow and pressure values respectively corresponding to a specific point on the Flow-Pressure graph of the check valve data sheet [Bosch Rexroth AG, 211a, p. 4]. x cv,norm is the instantaneous position of the valve poppet, normalized with respect to the maximum position. The instantaneous normalized poppet position is modeled as given in Eq p cv < p cv-cr p cv p cv-cr x cv,norm = p cv-cr p cv < p cv-end (6.24) p cv-end p cv-cr 1 p cv-end p cv p cv-cr is the crack pressure of the specific valve and p cv-end is the pressure at which the valve is fully open, such that the Flow-Pressure relation above this value is equal to that of a non-preloaded valve. Dynamic Model For the dynamic model, the pressure flow relation is also described by Eq. 6.23, while the instantaneous poppet position is described by the differential equation of Eq m cv ẍ cv = A cv ( p cv p cv-cr ) m cv g cos(θ cv ) F cv-fl F cv-d (6.25)

68 58 Control and Experimental Evaluation of SvSDP Concept g is the gravitational acceleration and θ cv is the orientation angle of the valve, where θ cv = in case the valve is mounted vertically and such that gravity will tend to close the valve. The flow force is described by Eq [Andersen and Hansen, 27, Sec. 3.2]. F cv-fl = ( ) β 2 ρ cos p cv sign ( p cv ) (6.26) 2 β is the angle of the poppet as shown in Fig. 6.14a. The poppet mass, the stroke length, poppet area and angle are estimated based on the 3D model of the outer contour of the check valve and the 2D illustration in the data sheet, as a 3D model of the poppet itself is not available. The angle β is fixed to a single value, but the effective angle may change based on the poppet position [Andersen and Hansen, 27, Sec. 3.2]. The check valves contains a damper orifice, separating the spring chamber from the surrounding channels. The flow through the orifice D ori may for turbulent flow be described with the orifice equation. The pressure gradient ṗ sc is described by the continuity equation as shown in Eq Q ori = C d A ori 2 ρ p ori sign (ẋ cv ) ṗ sc = β V sc ( Q ori + A cv ẋ cv ) (6.27) From simulation, the poppet velocity may be found to approximately be in the range of ẋ cv [2 ; 5] mm/s, which for the given geometry and viscosity of the fluid gives Reynold s numbers of Re [97.8 ; 224.5]. These Reynold s numbers are well above the required minimum of 6 1, for valid use of the orifice equation for sharp edges orifices [Rasmussen et al., 1996, p. 59]. As the spring chamber volume V sc is tiny, the spring chamber pressure gradient ṗ sc will tend to become very large, why the pressure p sc can be approximate to be a direct function of the poppet velocity ẋ cv, given by the steady state continuity equation Eq C d A ori 2 ρ p cv sign (ẋ cv ) = A cv ẋ cv p ori sign (ẋ cv ) = A2 cv ρ 2 C 2 d A2 ori (6.28) Multiplying the pressure difference p ori with the poppet area A cv and inserting the expression for the areas, gives the damping force Eq Note that the damping force is on the same form as a drag force. F cv-d = C 2 d ρ π D6 cv D 4 ori ẋ 2 cv sign (ẋ cv ) (6.29) To validate the models, the steady state flow-pressure relation for the two models is shown in Fig. 6.14b along with the curve given in the data sheet. A slow ramp in pressure differential is given in the dynamic model to sweep the entire pressure-flow range.

69 Chapter 6 - Modeling and Parameterization of the SvSDP System 59 Out β p sc V sc D ori x cv Pressure Differential [bar] Data Sheet KE Data Sheet KE2 Data Sheet KE5 Quasi Static KE Quasi Static KE2 Quasi Static KE5 Dynamic Model KE Dynamic Model KE2 Dynamic Model KE5 In D cv Flow [L/min] (a) Geometry of valve poppet. (b) Steady state flow-pressure relation of the constructed models, with the data sheet data. Figure 6.14: Check valve models of the M-SR 15 KExx [Bosch Rexroth AG, 26]. The slight deviations are considered acceptable in modeling the valve characteristics. Specifically for the dynamic model, the match between characteristic indicates a sufficient accuracy in the flow force approximation, which is the main closing force in case of the non-spring returned check valves. The estimated parameters are given in Tab Valve type M-SR 15 KE M-SR 15 KE2 M-SR 15 KE5 M-SR 3 KE p cv-cr bar.28 bar.54 bar bar p cv-end bar.65 bar 1.5 bar bar p cv-n 1.16 bar 1.16 bar 1.16 bar 2.4 bar Q cv-n 7 L min 7 L min 7 L min 35 L min D cv 15 mm 15 mm 15 mm 3 mm D ori 1 mm 1 mm 1 mm 2 mm m cv 26.7 g 26.7 g 26.7 g g x cv,max 3 mm 3 mm 3 mm 6 mm β Table 6.1: Estimated check valve parameters based on 3D model of outer contour and 2D illustration in [Bosch Rexroth AG, 26, p. 4].

70 6 Control and Experimental Evaluation of SvSDP Concept Simplified Model Model incl. Check Valve Dynamics 22 2 Position [mm] Velocity [mm/s] Time [s] A side Pressure [bar] (a) Piston position x Time [s] (c) A-side chamber pressure p A Time [s] B side Pressure [bar] (b) Piston velocity ẋ Time [s] (d) B-side chamber pressure p B. Figure 6.15: Simulated response of both the simplified model and the model including check valve dynamics. The input is a step sequence in ω m-ref As seen from Fig the difference between the simplified and model with check valve dynamics is limited. The simplified model has therefore been used in the control design developed prior to the experimental evaluation. Based on observations obtained just before the project submission, the developed dynamic check valve model may, however, not accurately describe the observed check valve behavior. In the test setup a mix of spring loaded and non spring loaded check valves have been used, as specified in Section Undesirable flow in the reverse direction of the non spring loaded check valves have been observed, which may be a result of an overestimation of the flow force. It is therefore suggested that spring loaded check valves should exclusively be used in a possible next iteration of the SvSDP concept.

71 Chapter 7 - Control Specification and Model Linearization 61 7 Control Specification and Model Linearization This and the following chapters presents the development, parameterization and evaluation of a generally applicable control strategy for the SvSDP system, which is developed based on a linearized version of the dynamic model presented in Chapter 6. The purpose, limitations and goals of the control system are outlined in Section 7.1. The model linearization is documented in Section 7.2 and validated in Section 7.3 through comparison with simulation results of the non-linear model. Since the SvSDP system is a Multiple Input Multiple Output (MIMO) system, the choice of control strategy is based on an investigation of the couplings between the various inputs and outputs presented in Section 7.4. This analysis is followed by an interpretation of the results and a discussion of proper choice of control strategy. The results of the coupling analysis indicates that decentralized control may not readily be suitable, why decoupling methods are presented in Chapter 8. Finally linear feedback controllers are designed for the decoupled system in Chapter 9, and the performance of the control system(s) is evaluated throughout Chapter Purpose, Considerations and Limitations of Control System The purpose of the control system for the SvSDP system has been briefly described in Chapter 1, and the primary objective is to achieve good tracking performance with respect to a position reference. Since the match ratio χ > 1 in the majority of the operating range, both chamber pressures will tend to increase. The secondary objective is therefore to maintain the chamber pressures at reasonable values through use of the proportional valves, without significant influence on the motion control. Reasonable pressures are in this case considered as maintaining the lowest of the chamber pressure at 25 bar. This corresponds to the bulk modulus of the oil being approximately constant at the maximum value, thus giving the maximum obtainable system stiffness. The set-point for the minimum pressure also determines the losses associated with throttling of excess flow, i.e. if the same flow must be throttled, the associated losses are proportional to the pressure difference across the valve. The immediate limitations for the control system are mainly the actuator limitations. The servo-drive has a high bandwidth ( 12 Hz) closed loop shaft velocity control. In addition, the simple functional principle of the gear pumps, makes it reasonable that the desired pump flow may be accurately achieved with approximately the same bandwidth as the servo-drive velocity loop. The bandwidth of the proportional valves is significantly lower ( 15 Hz), and without closed loop spool position control. On the contrary, the valves are pressure compensated, such that for a given command value, the steady state flow through the valve is approximately independent of the pressure difference. Since the flow characteristic through the valve (see Fig. 6.11) is non-linear with respect to the command value, the actual flow through the valve can only be approximately estimated by using the inverse of the valve flow characteristic. For control analysis and development, it is assumed that the control signals for the valves are desired flow references.

72 62 Control and Experimental Evaluation of SvSDP Concept It is furthermore assumed, that the model of the flow characteristic may be used well enough to determine the necessary physical valve command signal, such that the relation between valve flow reference and actual valve flow is unity for sufficiently low frequency references. The utilization of the inverse valve flow characteristic is illustrated in Fig Physical Valve IFC Flow Characteristic Valve Flow Reference Valve Command Value Valve Dynamics Actual Valve Flow Pressure Difference Figure 7.1: Change of control signal for the proportional valves, where the new control signal is a flow reference. An inverse flow characteristic (IFC) is used to approximate the command value required to realize the desired flow. A further limitation of the proportional valves is that they can only be used to sink flow from the cylinder. As a consequence, the controller must not actively attempt to utilize the valves for supplying flow to the cylinder. Due to the described limitations of the utilized actuators, it may generally be concluded that the servo-drive should primarily be used for high frequency control signals, whereas only low frequency control signals should be applied to the valves, to ensure that the use of the IFC is reasonable. Additionally, how the valves are utilized, depending on the piston velocity and chamber pressures, influences the overall energy efficiency of the system. The valve utilization, resulting in the highest energy efficiency during steady state operation, has been derived in [Grønkjær and Rahn, 214b, p. 38, 39] and repeated in Section 9.3. It is therefore desirable to be able to independently utilize either or both of the valves without otherwise influencing the control system. These aspects are further investigated in Chapter 8 and Chapter 9, and the results are shown in Chapter 9. The analysis is initiated with the development of a linear system model as presented in the following sections. 7.2 Model Linearization This section presents the development of a linear model of the SvSDP system. Since the purpose of the linear model is to investigate control strategies for the SvSDP concept, the control of the load system is considered ideal. As the dynamics of the actuators (servo-drive and proportional valves) are considered decoupled from the remaining system, a separate small signal model is developed in Section to describe this. Depending on the purpose of a given analysis, the actuator dynamics may thus be either included or neglected for simplicity. The combination of the two models to form a single complete linear system model is presented in Section The model linearizion is carried out using first order Taylor approximation of the nonlinear system equations from Chapter 6. The s are used to denote small signal variables and the denotes that the term is evaluated in the linearizion point. By disregarding the external load force and the non-linear friction terms, the linear mechanical

73 Chapter 7 - Control Specification and Model Linearization 63 system equation is given by Eq ẍ = 1 M (A p p A α A p p B B ẋ) (7.1) The pressure dynamics are linearized by assuming constant chamber volumes and oil bulk modulus. The validity of these assumptions are seen from the comparison of simulation results between the non-linear and the linear model in Section 7.3. The pressure dynamics are then described by Eq. 7.2, where the flow from the switched displacement pump unit (the combination of P1 and P2) are joined into a single flow denoted Q P12. p A = β V A ( Q P12 Q AV A p ẋ) p B = β V B ( Q P3 Q BV + α A p ẋ) (7.2) The pump flows are linearized by neglecting the second order pressure term from the polynomial description of the flow characteristics (see Eq. 6.18) and separating the signs from the coefficients as seen in Eq Q P1 = K P1ω ω m K P1p p A Q P2 = K P2ω ω m K P2p p A Q P3 = K P3ω ω m + K P3p p B (7.3) Due to the "switched displacement pump", the A-side pumps P1 and P2 are joined together into a single A-side pump unit. For ease of reference, the linearized constants for the B-side pump P3 are named using the same notation, as seen in Eq Q P12 = K Aω ω m K Ap p A Q P3 = K Bω ω m + K Bp p B (7.4) Due to the switched displacement pump, the flow gain K Aω and pressure gain K Ap will change size based on whether the linearizion point is taken for ω m or ω m <, as either both pumps or only the P1 pump is exchanging flow with the cylinder chamber. This dependency is shown in Eq K Aω = { KP1ω + K P2ω ω m K P1ω ω m < K Ap = { KP1p + K P2p ω m K P1p ω m < (7.5) Linear Plant Model The linear model of the plant is presented as a state-space model of the form shown in Eq. 7.6, where the s are omitted for simplicity. ẋ = A x + B u (7.6) y = C x Where the state vector x and the input vector u are chosen as shown in Eq ẋ ẍ ẋ = ṗ A ṗb x ẋ x = p A p B ω m u = (7.7) Q AV Q BV

74 64 Control and Experimental Evaluation of SvSDP Concept The system matrix A, input matrix B and output matrix C are given by Eq B A p M M α Ap M A = β A Ap V β A K Ap A V B = β A K Aω A β B α A p V β B K V β A A V A Bp β B K Bω B V V β B B B V B x 1 y = p A C = 1 (7.8) 1 p B The system represented by Eq. 7.6 is referred to as a whole by the name S, which is shown schematically in Fig u B ẋ x C y S A Figure 7.2: System S without actuator dynamics. The transfer function matrix representation of S is denoted by G(s) (s is the Laplace Operator), and is given in terms of the state space matrices as shown in Eq. 7.9 [Glad and Ljung, 2,p. 35]. G(s) = C (s I A) 1 g 11 (s) g 12 (s) g 13 (s) B = g 21 (s) g 22 (s) g 23 (s) Y(s) = G(s) U(s) (7.9) g 31 (s) g 32 (s) g 33 (s) Linear Actuator Model A linear small signal model of the actuator dynamics is given by Eq ẋ u = A u x u + B u u ref (7.1) u = C u x u Where the state vector x u and the input vector u ref are chosen as shown in Eq The subscript ref is used here to denote the desired value of u. Disregarding the actuator dynamics thus corresponds to u = u ref. ω m ω m Q ẋ u = AV Q AV Q BV x u = ω m ω m Q AV Q AV Q BV ω m,ref u ref = (7.11) Q AV,ref Q BV,ref Q BV Q BV The system, input and output matrices of the actuator model are given in Eq and Eq The coefficients of the matrices are directly given by considering only the small signal response,

75 Chapter 7 - Control Specification and Model Linearization 65 and therefore neglecting any slew-rate limitations. 1 ω n,m 2 2 ζ m ω n,m A u = 1 ω 2 2 ζ n,av AV ω n,av 1 (7.12) ω 2 n,bv 2 ζ BV ω n,bv ωn,m 2 B u = ω n,av 2 ωn,bv 2 1 C u = 1 (7.13) 1 The system represented by Eq. 7.1 is denoted S u and is shown schematically in Fig u ref B u ẋ u x e C u u S u A u Figure 7.3: System S u describing actuator dynamics. The corresponding transfer function matrix representation is denoted G u (s), and it is found similarly to Eq The elements of G u (s) are denoted by g u,ij, where i and j are the row and column of the element respectively Extended Linear Model The linear plant model and the model of the actuator dynamics are combined into an extended state-space model given in Eq The combination is achieved by concatenating the various matrices and vectors as given in Eq ẋ e = A e x e + B e u ref (7.14) y = C e x e [ ] x x e = x u [ ] A B Cu A e = A u [ ] B e = B u C e = [ ] C (7.15) The extended linear system represented by Eq is denoted S e and it is shown in Fig u ref u S u S y S e Figure 7.4: Block diagram of the extended linear model denoted S e with included actuator dynamics. The corresponding transfer function matrix representation is denoted G e (s). It is found similarly to Eq. 7.9, and the elements of G u (s) are denoted by g e,ij,

76 66 Control and Experimental Evaluation of SvSDP Concept 7.3 Verification of Linear Model The linear model is verified through comparison of the response of Eq and the non-linear model of Chapter 6 to the same step input sequence. The simulation is started from the equilibrium point given by Eq. 7.16, which is found by solving the system equations for steady state, i.e. x e =. mm 1 mm/s 94 RPM x = u 3 bar = 4.79 L/min (7.16) L/min bar The equilibrium point is chosen to indicate typical conditions where all the inputs may be used. Additionally, one of the valves are initially closed, to investigate the accuracy of the linear model within the most non-linear part of the valve flow characteristic. The input sequence consists of steps in the different inputs at different times. At time t = s the servo-drive shaft speed reference is increased by 1 RPM, at time t = 1 s the A-side valve flow reference is increased by 1 L/min. Finally at time t = 1.5 s the B-side valve flow reference is also increased by 1 L/min. Velocity [mm/s] Non linear Model Linear Model Pistonside Pressure [bar] Rodside Pressure [bar] Time [s] Time [s] Time [s] Figure 7.5: Comparison of response of non-linear and linear model to a step input sequence in the various inputs. As evident from Fig. 7.5, the response of the linear model is very similar to that of the non-linear model. The slight discrepancies, which are most apparent in the last part of the response, are due

77 Chapter 7 - Control Specification and Model Linearization 67 to the pressure dependence of the valve flow which is completely neglected in the linear model. Due to the ability to capture the dynamic behavior, the linear model is considered sufficient for control analysis and design. 7.4 Cross-Coupling Analysis In this section it is investigated through Relative Gain Array (RGA) analysis, whether decentralized control is readily applicable to the original system. The RGA is an analytical method to quantify the amount of input-output cross coupling, why it may be used to determine if decentralized control is applicable. The RGA for the linear model is defined in Eq [Glad and Ljung, 2,p. 22], with ". " denoting element-by-element multiplication. This definition requires that the investigated transfer function matrix is square and invertible. RGA(G(s)) = G(s). (G(s) 1) T (7.17) The closer the RGA, evaluated at a specific frequency, approximates that of the identity matrix, the less cross coupling exist between the different inputs and outputs at that specific frequency. For the considered transfer function matrix G(s), the outputs are the piston position and two chamber pressures. With the available inputs, it is not possible to independently control both chamber pressures as well as the piston position. This results in G(s) being singular, why the RGA of the original system is undefined. A method to circumvent this limitation is to disregard an input-output set, which means that the number of inputs and outputs are, in this case, reduced to 2. To investigate the reasonable input output combinations, the system is divided into smaller subsystems as shown in Eq It is chosen to always use the shaft velocity ω m as an input, as this is the only input which may supply flow to the cylinder. [ ] [ ] [ ] [ ] [ ] [ ] x ωm x ωm x ωm = G1(s) = G2(s) = G3(s) p A [ ] x = G4(s) p B Q AV [ ωm Q BV p A ] [ pa p B ] = G5(s) Q BV [ ωm Q AV p B ] [ pa p B ] = G6(s) Q AV [ ωm Q BV ] (7.18) The resulting sub-transfer function matrices are given by Eq. 7.19, based on the original transfer function matrix G(s) from Eq [ ] [ ] [ ] g11 (s) g G1(s) = 12 (s) g11 (s) g G2(s) = 13 (s) g11 (s) g G3(s) = 12 (s) g 21 (s) g 22 (s) g 21 (s) g 23 (s) g 31 (s) g 32 (s) [ ] [ ] [ ] g11 (s) g G4(s) = 13 (s) g21 (s) g G5(s) = 22 (s) g21 (s) g G6(s) = 23 (s) (7.19) g 31 (s) g 33 (s) g 31 (s) g 32 (s) g 31 (s) g 33 (s) To visualize the RGA elements, the magnitude of the complex RGA elements may be found as a function of frequency. This however, renders it difficult to distinguish between a desirable RGA element of 1 and an undesirable RGA element of -1. Instead, the RGA-number, defined in Eq. 7.2 [Skogestad and Postlethwaite, 21,p ], is utilized to visualize the properties of the RGA. RGA number(ω) = RGA(Gx(j ω)) I 2 sum (7.2) "x" is a placeholder for the number of the transfer function submatrix, and the sum norm is defined to be A sum = [ ] 1 i,j a i,j. As Eq. 7.2 only calculates the diagonal pairing, J 2 = must 1 be used instead of the identity matrix I 2, to determine the off-diagonal pairings for a 2x2 RGA.

78 68 Control and Experimental Evaluation of SvSDP Concept Due to the definition of the RGA number, the ideal diagonal value is obtained with an RGA number of, as this corresponds to an RGA equal to the identity matrix as seen in Eq The resulting off-diagonal value for this ideal case will result in an RGA number of 4 as seen in Eq [ ] RGA number id = I I sum = = (7.21) [ ] sum 1 1 RGA number iod = I J sum = = 4 (7.22) 1 1 sum As the RGA s are all 2x2 matrices, it is not necessary to check for other pairings than the diagonal and off-diagonal pairings. The RGA number for the diagonal and off-diagonal elements are shown in Eq. 7.6 for the transfer function sub-matrices defined in Eq RGA Number x(ω m ), p A (Q AV ) x(q AV ), p A (ω m ) RGA Number x(ω m ), p A (Q BV ) x(q BV ), p A (ω m ) Frequency [rad/s] (a) RGA numbers of G Frequency [rad/s] (b) RGA numbers of G2. RGA Number x(ω m ), p B (Q AV ) x(q AV ), p B (ω m ) RGA Number x(ω m ), p B (Q BV ) x(q BV ), p B (ω m ) Frequency [rad/s] (c) RGA numbers of G Frequency [rad/s] (d) RGA numbers of G4. RGA Number p A (ω m ), p B (Q AV ) p A (Q AV ), p B (ω m ) RGA Number p A (ω m ), p B (Q BV ) p A (Q BV ), p B (ω m ) Frequency [rad/s] (e) RGA numbers of G Frequency [rad/s] (f) RGA numbers of G6. Figure 7.6: RGA numbers for diagonal and off-diagonal pairing of various combinations of inputs and outputs.

79 Chapter 7 - Control Specification and Model Linearization 69 Similar behavior is exhibited for all of the pairings shown in Fig. 7.6 around the natural frequency of the system, where large spikes in the RGA numbers are seen. This indicates that if either of these pairings are to be used, the controller bandwidth must be sufficiently low, such that the influence of the cross-couplings is reduced [Glad and Ljung, 2, p. 222]. In general it is indicated in Fig. 7.6a to Fig. 7.6d that the piston position x should be controlled by the servo-drive. This pairing is also desirable to achieve good dynamic performance of the system, as the bandwidth of servo drive is significantly higher than that of the proportional valves. Either of the chamber pressures can be controlled by the proportional valves, as long as the position is controlled by the shaft velocity as seen by Fig. 7.6a to Fig. 7.6d. The remaining two RGA number plots shown in Fig. 7.6e and Fig. 7.6f, which corresponds to controlling both pressures, shows that decentralized control is not applicable for this control strategy. These plots therefore further supports the idea to pair the position and the shaft velocity. 7.5 Discussion of Control Strategies From the analysis in the previous section, decentralized control is not considered sufficient, due to the large degree of cross-couplings in the frequency band around the system bandwidth. In addition, the proper pairing scheme often changes for high frequencies. An alternative strategy is to accept the cross-couplings, and use e.g. pole placement or LQR for determination of a non-diagonal control law. This method has the following disadvantages for the system under consideration. The unidirectional flow constraint on the proportional valves can not be included, i.e. the resulting controller may attempt to utilize the valves for supplying flow to the system. The controlled chamber pressure changes depending on the operating conditions, why the controller parameters may have to change in discrete steps during operation. Full state feedback is required, which is generally not desirable as some difficulties are typically involved in the estimation of the piston velocity. The design process must be repeated depending on the valve utilization scheme, i.e. if it is desired to use only one valve under some operating conditions, then a controller must be designed where the opposite valve is excluded. An alternative strategy is to attempt to decouple the system, both to simplify the design process, and such that the controller may be separated from the valve utilization scheme. This was attempted in a specialized case in [Grønkjær and Rahn, 214a, p. 6,7] and [Grønkjær and Rahn, 214b, p ], and the concept is generalized in the following chapters.

80

81 Chapter 8 - System Decoupling Methods 71 8 System Decoupling Methods In this chapter, the linear system model is transformed into a system more suitable for decentralized control, i.e. where the RGA matrix is as close as possible to the identity matrix in a larger frequency range. The input and output variables are changed to ũ and ỹ respectively using the transformation matrices W 1 and W 2 as defined in Eq. 8.1, which is based on [Glad and Ljung, 2,p. 226]. ỹ = W 2 y ũ = W 1 1 u (8.1) The original transfer function matrix G(s) is thereby transformed into the new transfer function matrix G(s) as shown in Eq. 8.2 [Glad and Ljung, 2,p. 226]. ỹ = G(s) ũ G(s) = W2 G(s) W 1 (8.2) Since the model of the actuator dynamics has unity DC-gain, the transformation matrices W 1 1, W 1 and W 2 found for S may be readily applied to the extended linear system S e. This transformed system is denoted S e and shown in Fig The transformed system without actuator dynamics is denoted S. ũ ref S e W 1 u ref S e y W 2 ỹ Figure 8.1: The complete system with input and output transformation matrices. The transfer function matrix representation G e (s) is given in Eq. 8.5, which is derived in Eq. 8.3 and Eq. 8.4 (see definition of the linear systems in Section 7.2). u = G u u ref u ref = W 1 ũ ref u = G u W 1 ũ ref (8.3) ỹ = W 2 y y = G u ỹ = W 2 G u (8.4) ỹ = G e ũ ref Ge = W 2 G G u W 1 (8.5) For simplicity, the derivation of proper choices of the input and output transformation matrices is based on the system equations describing S, i.e. is assumed that the control signals are u. However, as seen in Eq. 8.5, the transfer function matrix of the actuator dynamics G u appears between G and W 1. This means that a choice of transformation matrices that results in a decoupled G does not necessarily give a decoupled G e. For evaluation of the different transformation types, the extended system S e is thus used to obtain a more accurate measure of the applicability of the developed decoupling method. All RGA-number studies in this chapter are thus made using the extended model. As different input transformation methods are investigated, variables may be appended with an index M1, M2 etc. when it is necessary to distinguish between the different transformation method, e.g. W 1,M1, S e,m1 or G e,m1. For simplicity the indices are omitted in the derivations and in general discussion. 8.1 Output Transformation This section presents the derivation of the developed output transformation, which is based on the expression for the chamber pressure gradients. These can be described by Eq. 8.6 and Eq.

82 72 Control and Experimental Evaluation of SvSDP Concept 8.7 respectively, when the bulk modulus of the two chambers are assumed to be constant and equal β A = β B = β. The assumption of equal bulk modulus is valid if both pressures are above 2 bar, which is the case when both the motion control and the pressure control are active. ṗ A = β A V A ( K Aω ω m A p ẋ Q AV K Ap p A ) ṗ B = β B V B ( K Bω + αa p ẋ Q BV K Bp p B ) (8.6) (8.7) The load pressure p L and the pressure level p H are introduced in Eq The reason for these definitions becomes apparent from the following derivations. p L = p A α p B p H = p A + H p B (8.8) The load pressure p L describes the theoretically available cylinder force (when multiplied with the piston area A p ), whereas the pressure level p H is a form of weighted sum of the chamber pressures. The parameter H is (initially) a non-physical factor, which simply makes p H a linear combination of the two chamber pressures. The physical interpretation of H follows from the presented derivation. From the definitions in Eq. 8.8, the actual chamber pressures may similarly be described in terms of the load pressure and pressure level by Eq p A = H α + H p + α L α + H p p H B = 1 α + H p + 1 L α + H p (8.9) H The pressure level gradient is derived in Eq ṗ H = ṗ A + H ṗ B ( 1 ) = β (K Aω ω m A p ẋ Q AV K Ap p A + H ) ( K ) Bω ω m + α A p ẋ Q BV K Bp p B VA VB (( KAω = β H K ) ( Bω 1 ω m A p H α ) ẋ... V A V B V A V B ( QAV + H Q ) ( KAp BV p V A V B V A + H K )) Bp p A V B (8.1) B If the parameter H is selected to be H = V B, the influence of the velocity ẋ on the pressure level α V A gradient ṗ H is removed. As V A and V B are functions of the piston position x, the parameter H can be estimated directly from the measured piston position. As will be apparent from the remainder of this chapter, the use of an estimated H will in some sense correspond to gain scheduling, where the scheduling follows directly from the choice of H. Allowing H to vary in time introduces an additional term in Eq. 8.1 (Ḣ p ), however as H varies slowly with the piston position, this term B is considered neglectable. The sensitivity of H to the estimation accuracy of hose volume etc. is investigated in Section With the selected H, Eq. 8.1 may be written as Eq ṗ H = β ((K V Aω V ) ( V B A K A α V A V Bω ω m Q AV + V ) ( V B A Q B α V A V BV K Ap p A + V )) V B A K B α V A V Bp p B B = β ((K V Aω K ) ( Bω ω m Q A α AV + Q ) ( BV K α Ap p A + K )) Bp α p (8.11) B ( To obtain an expression for the pressure level gradient in terms of the new variables p L and p H, Eq. 8.9 is inserted into Eq. 8.11, giving the expression for the pressure level gradient in Eq ṗ H = β V A (K Aω K ) ( Bω ω m Q α AV + Q ) BV K HpH α α + H p K ) HpL H α + H p L }{{} K w (8.12)

83 Chapter 8 - System Decoupling Methods 73 The leakage coefficients with respect to p H and p L are given in terms of the original leakage coefficients in Eq K HpH = α K Ap + K Bp α K HpL = H K Ap K Bp α The final expression for the pressure level gradient may thus be written as Eq β ṗ H = (α + H) V A ( ( ( (α + H) K w ω m Q AV + Q BV α )) K HpH p H K HpL p L ) (8.13) (8.14) Similarly, the load pressure gradient is given in Eq by inserting Eq. 8.6 and Eq. 8.7 into the definition of the load pressure. ṗ L = ṗ A α ṗ B (8.15) = β ( ( ) K V Aω ω m A p ẋ Q AV K Ap p A α V ( ) ) A K A V Bω ω m + α A p ẋ Q BV K Bp p B B = β ((K V Aω + K ) ( Bω ω m Q A H AV Q ) ( BV A p 1 + α ) ( ẋ K H H Ap p A K )) Bp H p B To describe the load pressure gradient in terms of p L and p H, Eq. 8.9 is again inserted into Eq. 8.15, giving Eq ) ṗ L = β ( (K V Aω + K Bω A H }{{} ΛK ω ( ω m Q AV Q BV H ) ( A p 1 + α ) ẋ H K LpL α + H p K ) LpH L α + H p H (8.16) The leakage coefficients in Eq with respect to p H and p L are given in terms of the original leakage coefficients in Eq K LpL = H K Ap + K Bp H K HpL = α K Ap K Bp H The final expression for the load pressure gradient is thus given by Eq ṗ L = β (α + H) V A H ( H α + H ( ( ΛK ω ω m Q AV Q BV H )) A p ẋ (8.17) H K LpL (α + H) 2 p L H K LpH (α + H) 2 p H (8.18) From equation Eq. 8.18, it is interesting to note the following when considering the steady state condition, i.e. ṗ L =. Neglecting the leakage terms, and forcing the valve flows to zero, gives the steady state equation in Eq ẋ ss = H ΛK ω A p (α + H) ω m = H K Aω + K Bω A p (α + H) ω m (8.19) When considering the ideal case of the original SvDP concept, i.e. the displacement of the A and B side pumps are matched to the cylinder area ratio, gives K Bω = α K Aω. In this case, Eq reduces to ẋ = K Aω A p ω m. However, for the SvSDP concept, the pumps are purposefully mismatched with respect to the cylinder area ratio, why the steady gain will generally vary with the piston position. )

84 74 Control and Experimental Evaluation of SvSDP Concept Based on the above derivation, a proper choice of output transformation matrix W 2 is shown in Eq. 8.2, which is given directly by the definition of the load pressure p L and the pressure level p H. x 1 x p L = 1 α p A p H 1 H p B }{{}}{{}}{{} ỹ W 2 y H = V B α V A (8.2) Analysis of Output Transformation To analyze the effect of the output transformation, the RGA number is utilized to visualize the degree of cross-couplings, similar to the analysis in Section 7.4. This is shown in Fig. 8.2 where the input transformation is chosen as W 1 = I RGA Number x(ω m ), p H (Q AV ) x(q AV ), p H (ω m ) RGA Number x(ω m ), p H (Q BV ) x(q BV ), p H (ω m ) Frequency [rad/s] (a) Frequency [rad/s] (b) Figure 8.2: RGA number for the output transformed system, where leakage and actuator dynamics is included in the model. As evident from Fig. 8.2, the cross-couplings of the transformed system are reduced significantly (recall that the ideal values are and 4). It is expectable however, that the transformation may be sensitive to the estimation accuracy of H. To investigate this, the variation of H with the piston position is shown in Fig. 8.3a. Furthermore the estimation of H is also shown for the case where the A and B side volumes are estimated to be 1 dl higher than the actual value. This corresponds to approximately 1 m of hose length, which is considered an upper bound for the estimation error. H Estimate [ ] Actual Value Estimation error in V A Estimation error in V B H Estimation Error [%] Actual Value Estimation error in V A Estimation error in V B Position [mm] Position [mm] (a) (b) Figure 8.3: Variation of H and estimation error due to unknown hose volume.

85 Chapter 8 - System Decoupling Methods 75 As seen from Fig. 8.3a the error does not influence the shape of the estimate significantly. The corresponding estimation errors in percent are shown in Fig. 8.3b, and the maximum error is seen to be 2%. To investigate the influence of an estimation error of 2%, Fig. 8.2 is repeated in Fig. 8.4, but where the estimated H is only 8% of the actual value. RGA Number x(ω m ), p H (Q AV ) x(q AV ), p H (ω m ) RGA Number x(ω m ), p H (Q BV ) x(q BV ), p H (ω m ) Frequency [rad/s] (a) Frequency [rad/s] (b) Figure 8.4: RGA number for the output transformed system with 2% estimation error in H, where leakage and actuator dynamics is included in the model. Fig. 8.4 shows that an estimation error does influence the RGA number, however the result, particularly that of Fig. 8.4b, is still a significant improvement compared to the analysis of the original system. In order to obtain further decoupling, possible input transformations are derived and analyzed in the remainder of this chapter Input Transformation Based on Eq and Eq. 8.18, new inputs, respectively the load flow Q L and the level flow Q H, are defined in Eq and Eq The load flow relates to the motion of the piston, while the level flow relates to the pressure level. The choice of Q L and Q H are such that the influence of the shaft velocity and proportional valve flows are "hidden" from the pressure level and load pressure gradients. Control structures using these new inputs can thereby be designed independent of the value and sign of the shaft velocity, and of the specific valve utilization method. Q L = H ( ( ΛK ω ω m Q α + H AV Q )) BV (8.21) H ( ( Q H = (α + H) K ω ω m Q AV + Q )) BV (8.22) α The flow gain K Aω of the combined switched displacement pump (pump P1 and pump P2) changes value based on the sign of the shaft velocity, as the pump P2 is either delivering flow to the actuator (ω m > ) or idling its flow through the check valve (ω m < ). This means that the sign of K ω is changed based on the sign of ω m. ΛK ω is always positive regardless of the sign of ω m, although its magnitude is changed. These partial results are shown in Eq. 8.23, Eq and Eq for ease of reference. H(x) = V B (x) α V A (x) = V B α A p x α (V A + A p x) > for x min x x max (8.23)

86 76 Control and Experimental Evaluation of SvSDP Concept K P1ω + K P2ω K P3ω K ω (sign (ω m)) = α > ω m K P1ω K P3ω α < ω m < K P1ω + K P2ω + K P3ω ΛK ω (H(x), sign (ω m)) = H(x) > ω m K P1ω + K P3ω H(x) > ω m < (8.24) (8.25) With the load flow and the level flow inserted, Eq and Eq are simplified as shown in Eq and Eq. 8.27, which describes the gradients of the transformed pressure states. ( β (α + H) ṗ L = Q V A H L A p ẋ H K LpL (α + H) 2 p H K ) LpH L (α + H) 2 p (8.26) H β ( ) ṗ H = Q (α + H) V H K HpH p H K HpL p L (8.27) A The relation between the original inputs u and the new inputs ũ is described by the inverse input transfer function matrix W 1 1 given by Eq. 8.28, which can be constructed from Eq and Eq Q L Q H Q }{{} ũ H ΛK ω H 1 ω α + H α + H α + H m = (α+h) K ω (α+h) α+h α Q AV v 31 v 32 v 33 Q BV }{{}}{{} W 1 u 1 (8.28) The third flow Q is denoted as the flow constraint and is an equation, which may be used to achieve different valve utilization methods. The simplest flow constraint equation is formed by defining Q. Due to the flow constraint, various input transformation matrices may be utilized, as discussed in the following sections. For ease of reference, the general (non-inverted) input transformation matrix W 1 is shown symbolically in Eq ω m w 11 w 12 Q L Q AV = w 21 w 22 Q H (8.29) Q BV w 31 w 32 Q }{{}}{{}}{{} u W 1 ũ For simplicity, the elements of the last column in W 1 are marked by due to the definition that Q, which makes them irrelevant. Prior to the development of the specific input transformation methods, the influence of the unidirectional valve flow is investigated. 8.2 Influence of Unidirectional Valve Flow This section presents an analysis of the influence of the sign constraint on the valve flows Q AV and Q BV, i.e. that the valves can only sink flow from the cylinder chambers. This constraint is not directly a part of the system transformation, but must be considered for proper performance. Eq assumes that the transformed flows are realizable in order to properly distribute the physical control signals. As a consequence, the transformed flows must be limited to realizable values prior to the evaluation of Eq to ensure proper values of u. Since the performance of

87 Chapter 8 - System Decoupling Methods 77 the motion control is considered of higher priority than that of the pressure level control, the load flow Q L should ideally not be influenced, while the achievable level flow Q H may be constrained. The equivalent constraint on Q H is given in Eq. 8.3, which may be obtained by inserting the sign constraints on the valve flows into the definition of Q H given in Eq Q AV, Q BV Q H (α + H) K ω ω m (8.3) Since sign ( K ω ) = sign (ω), the limitation on Q H will always be positive, and the magnitude of the constraint will depend on the shaft speed. The two distinct values of K ω, which depends on the sign of ω m, are denoted K ω and K + ω as seen in Eq K ω = { K + ω ω m K ω ω m < (8.31) The expressions for these gains are shown in Eq K ω = K P1ω K P3ω α < K+ ω = K P1ω + K P2ω K P3ω α > (8.32) For analysis purposes the problem is turned around, i.e. the upper limit on Q H in Eq. 8.3 is translated into limits on the shaft velocity, by isolating for ω m. Since sign ( Kω ) = 1 the inequality is flipped when dividing by a negative number as seen in Eq. 8.34, whereas sign ( K ω + ) = 1 does not influence the inequality as seen in Eq ω m ω m 1 (α + H) K ω + 1 (α + H) Kω Q H = f b+ (Q H ) ω m (8.33) Q H = f b- (Q H ) ω m < (8.34) The feasibility boundaries, f b+ (Q H ) and f b- (Q H ), are illustrated graphically in Fig. 8.5 for an arbitrary value of H, where the enclosed area is denoted as the "Infeasible Region". For a given desired Q L and Q H, this constraint must be imposed prior to the evaluation of the corresponding ω m, Q AV and Q BV to obtain correct results. However, as the constraint changes with the sign of ω m, and ω m may in turn be a function of both Q L and Q H, this results in an implicit problem. However, conclusions may be drawn, depending on the magnitude of ωm Q H relative to the constraint. The two possible cases are shown in Fig. 8.5a and Fig. 8.5b respectively. ω m Feasible Region Q L = const. ω m Feasible Region Q L = const. ω m,i Infeasible Region Q H,c ω m,i Infeasible Region Q H,i Q H Q H,i Q H ω m,c (a) Improper ω m mapping. (b) Suitable ω m mapping. Figure 8.5: Illustration of ω m-mappings, ω m = w 11 Q L +w 12 Q H, where the matrix elements w 11 and w 12 may be functions of x and sign(ω m).

88 78 Control and Experimental Evaluation of SvSDP Concept As stated in the beginning of this section, the desired load flow should always be realized. Both cases thus represents an arbitrary piecewise linear function ω m = w 11 Q L + w 12 Q H with Q L = const.. This function may in general be discontinuous around ω m =, as the elements w 11 and w 12 may be functions of sign (ω m ). The value of Q H corresponding to crossing the feasibility bound is denoted as Q H,i. In Fig. 8.5a, the gradient ωm Q H = w 12 is such that the function may "escape" the constraints. This in turn implies that a critical value Q H,c exists, such that if Q H,i Q H < Q H,c, the constraint will reduce Q H such that the point ( Q H,i, ω m,i ) is used. Contrary, if Q H = Q H,c, the point becomes feasible, and the slight increase in Q H may thus oppose the sign of ω m, rendering the solution improper for control purposes. In Fig. 8.5b, the gradient ωm Q = w 12 is such, that once the function enters the infeasible region, H it remains there for any higher value of Q H, i.e. any value Q H Q H,i will all lead to the solution point ( Q H,i, ω m,i ). This mapping is thus suitable, since no sudden jumps can occur in ω m, as the value will be stuck at the boundary. Based on the presented analysis, for an input transformation to be feasible, the inequality given in Eq must be fulfilled. This ensures that the ω m -mapping may never escape the boundary to the infeasible range for increasing Q H, and thus no sudden jumps will occur in the ω m value. f b- Q H ω m Q H f b+ Q H 1 K ω (α + H) w 12 }{{} K gc 1 K + ω (8.35) As seen from Eq. 8.32, the lower bound on K gc is always strictly negative, while the upper bound is always strictly positive Determination of implicit input transformation As mentioned above, the input transformation coefficients w 11 and w 12 may generally depend on sign (ω m ). To determine ω m from a given Q L and Q H thus requires the solution of the implicit expression in Eq ω m = w 11 (x,sign(ω m)) Q L + w 12 (x,sign(ω m)) Q H (8.36) As a result, it is necessary to evaluate sign (ω m ) prior to the evaluation of Eq This is possible if Eq may be written on the form of Eq. 8.37, with the common coefficient c. ( ) ω m = c(x,sign(ω m)) w11 (x) Q + L w 12 (x) Q H (8.37) If c can be selected such that only the magnitude but not sign (c) changes based on sign (ω m ), then sign (ω m ) may be determined from Eq The validity of this assumption should be checked for each input transformation method. sign (ω m ) = sign (c) sign (w 11 Q L + w 12 Q H ) (8.38) Evaluating Eq prior to the evaluation of Eq. 8.36, thus solves the implicit problem, but requires that the desired Q H is realizable. This issue is investigated further in the remainder of this section. Analysis of sign (ω m ) for Q H < As the "Infeasible Region" in Fig. 8.5 exists only for Q H, any desired negative Q H may be realized directly. Eq may thus be used directly for evaluation of sign (ω m ).

89 Chapter 8 - System Decoupling Methods 79 Analysis of sign (ω m ) for Q H The analysis of sign (ω m ) for Q H is slightly more involved, as an arbitrary Q H > cannot necessarily be realized due to the "Infeasible Region" shown in Fig This means that the desired Q H cannot immediately be used in Eq. 8.38, without first checking whether it is feasible or not. To initiate the analysis, the problem is considered for the specialized case where Q H =. If it is valid that sign (c) is independent of sign (ω m ) as described above, Eq may be used to determine sign (ω m ) in this specialized case. sign (ω m ) = sign (c) sign (w 11) sign (Q L ) (8.39) This corresponds to the point (, ω m, ) shown in Fig. 8.6, for the case where sign (ω m, ) = +1. ω m =Possible Gradient =Feasible Solution Curve =Infeasible Solution Curve =Q H < ω m, ω m,b+ =Infeasible Region =Feasible Region Q H,b+ Q H Figure 8.6: Example of the possible solutions of ω m corresponding to a desired Q L = const. and Q H. The non-trivial case where Q H is increased along the solution curve shown in Fig. 8.6 is now considered, where Q L = const. and ω m, >. Due to the gradient limitations from Eq (illustrated by the red area in Fig. 8.6), it is known that for any Q H >, the solution point will be either above the positive feasibility boundary or in the point ( Q H,b+, ω m,b+ ). Similarly, for the case where ω m, <, the solution point will be either below or on the negative feasibility boundary. As a result, for any Q H, sign (ω m ) may be determined from Eq if the gradient condition is fulfilled. Result of analysis of sign (ω m ) The two cases described above can be summarized by Eq sign (ω m ) = { sign (c) sign (w 11 Q L + w 12 Q H ) Q H < sign (c) sign (w 11 ) sign (Q L ) Q H (8.4) From Eq. 8.4 the implicit solution problem may be circumvented as sign (ω m ) may be determined directly from the desired value of the two control signals Q L and Q H. The solution may be used to evaluate the numerical values of K ω and ΛK ω, which are in turn used to obtain the actual control signals (ω m, Q AV and Q BV ) from the transformed control signals Q L and Q H. For Eq. 8.4 to be valid, it must be verified for each proposed transformation method, that the gradient condition Eq is fulfilled and that the expression for ω m in terms of Q L and Q H may be written in the form of Eq. 8.37, where sign (c) is independent of sign (ω m ).

90 8 Control and Experimental Evaluation of SvSDP Concept Enforcing feasibility boundaries In the case of Q H, the feasibility boundaries must be enforced. To implement the limitations on Q H, the intersection between the solution curve and the feasibility boundary is expressed as a function of Q L. The inequalities describing the feasible region are given by Eq Q AV = w 21 Q L + w 22 Q H Q BV = w 31 Q L + w 32 Q H (8.41) The inequalities in Eq can be expressed as the limitations on Q H assuming that w 22, w 32 <. given in Eq by Q H w 21 w 22 Q L Q H w 31 w 32 Q L (8.42) If the gradient condition Eq is fulfilled, enforcing the feasibility boundaries thus corresponds to limiting the value of Q H to Q H,max which is given by Eq ( Q H,max = min w 21 Q w L, w ) 31 Q 22 w L 32 Q H (8.43) Eq is evaluated and simplified for each of the presented transformation method. Refer to Eq for the symbolic definition of w 21, w 22, w 31, w Input Transformation Method 1 The purpose of the first transformation method is to control the servo drive ω m using only the load flow Q L. In this way, the dynamic behavior of the load is only coupled to the dynamics of the servo drive and not to the slower proportional valves. The disadvantage of this method is that it requires utilization of both valves simultaneously, which is not the ( most energy efficient valve utilization method. This method is achieved by canceling the term Q AV Q ) BV H in Eq. 8.21, which gives the constraint equation in Eq Q = Q AV Q ] BV H = Q = [ 1 1 H u = (8.44) The transformation matrices W 1 1 and W 1 are therefore given by Eq W 1 1 = H ΛK ω H 1 α+h α + H α + H α + H (α+h) K ω (α+h) α+h α α α W 1 = K ω H ΛK ω 1 1 }{{} H K H c ω 1 H ΛKω (α + H) 2 H2 ΛK ω (α + H) 2 As seen from Eq. 8.45, the servo drive velocity ω m is only a function of Q L with the gain Furthermore, it can be seen that Q AV and Q BV depends on both Q L and Q H. Gradient Condition For this transformation method w 12 =, why the gradient condition is readily fulfilled. Shaft Sign Determination (8.45) α+h H ΛK ω. The coefficients c 1 and w 11 are always positive as H, ΛK ω >. With these conditions, sign (ω m ) may be determined from Eq. 8.46, which is the reduced form of Eq sign (ω m ) = sign (Q L ) (8.46)

91 Chapter 8 - System Decoupling Methods 81 Feasibility Boundary It is noted from Eq. 8.45, that the coefficients w 22 and w 32 are strictly negative. The limitation on Q H may thus be expressed by Eq. 8.47, which is determined from Eq Q H,max = (α + H)2 H K ω ΛK ω Q L Q H (8.47) Influence of actuator dynamics and sensitivity to H estimate The RGA numbers for the transformed system using both input and output transformation is shown in Fig. 8.7a, and again in Fig. 8.7b, where H is estimated with an estimation error of 2% RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) Frequency [rad/s] (a) Perfect H estimate Frequency [rad/s] (b) 2% error in H estimate Figure 8.7: RGA number for the transformed system with input transformation 1. Actuator dynamics are included in the model. Fig. 8.7a shows that with a perfect H estimate, transformation method 1 combined with the output transformation gives perfect decoupling. Additionally, transformation method 1 shows low sensitivity to estimation errors with respect to H. 8.4 Input Transformation Method 2 In this transformation, it is attempted to control both proportional valves using only Q H and thereby decouple the load flow from the proportional valves. This constraint can be formulated as shown in Eq and Eq. 8.49, where the matrix coefficients w 21 and w 31 are set to zero. The coefficients are calculated as the elements of the inverse of W 1 1. α + H (α + H) K ω v 33 v 31 w 21 = = ( ) α (8.48) det W 1 1 w 31 = = (α + H) K ω v 32 + (α + H) v ( ) 31 (8.49) det Eq and Eq constitutes two equations with three unknowns. As W 1 1 may not be singular, the trivial solution v 31 = v 32 = v 33 = is not allowed. Considering v 31 as the free variable, then v 32 and v 33 are given by Eq W 1 1 v 32 = v 31 K ω v 33 = v 31 α K ω (8.5)

92 82 Control and Experimental Evaluation of SvSDP Concept In Eq it is however shown, that the determinant of W 1 1 is always zero, regardless of the selected v 31, why a transformation matrix W 1 fulfilling the criteria for this method cannot be found. ( det W 1 1 ) ( ) H = v 31 α + 1 v 32 ( H ) α ΛK ω K ω + v 33 ( H ΛK ω + H K ω ) ( H = v 31 α + 1 H ΛK ω 1 + H ΛK ω H ) = (8.51) α K ω α K ω α In other words, it is not possible to make the proportional valve flow independent of Q L. 8.5 Input Transformation Method 3 Due to the nonlinear flow characteristics of the proportional valves, a small model error can lead to incorrect throttling flow, if the proportional valve signals are not the same. In this transformation, the same amount of flow is therefore throttled from both chambers at the same time, which means that Q AV = Q BV. In this way, a model error in the flow characteristic of the valves will only lead to an increased control signal for both valves, why an imbalance in flows is avoided. The flow constraint for this method is given by Eq [ ] Q AV = Q BV Q = 1 1 u = (8.52) The resulting transformation matrices are shown in Eq H ΛK ω H 1 1+α α W 1 α + H α + H α + H 1 = (α+h) K ω (α+h) α+h α W α 1 = K ω K Aω + K Bω 1 1 }{{} K c ω 3 1 H (α + H) 2 H ΛKω (α + H) 2 H ΛKω (α + H) 2 (8.53) Gradient Condition For this transformation method w 12, why the fulfillment of the gradient condition must be verified. K gc is in this case given by Eq α K gc = (α + H) w 12 = K Aω + K Bω 1 H (α + H) (8.54) Assuming H to vary in the interval [, [, corresponding to the extreme case where V A and V B consists solely of the volume of the cylinder chambers with no auxiliary hose volume etc.. The minimum of Eq corresponds to the limit for H going towards, while the maximum corresponds to the limit for H going toward. These extreme cases are given by Eq and Eq K gc,min = lim H (α + H) w 12 = K gc,max = lim (α + H) w H + 12 = α K Aω + K Bω (8.55) 1 K Aω + K Bω (8.56) The inequality Eq. 8.35, may be split into two inequalities and reduced as given by Eq and Eq K gc,min 1 Kω K gc,max 1 K ω + (1 + α) (8.57) (1 + α) (8.58) Since both these inequalities are always fulfilled, the input transformation is feasible.

93 Chapter 8 - System Decoupling Methods 83 Shaft Sign Determination The coefficients c 3 and w 11 are always positive as H, K Aω, K Bω >. With these conditions, sign (ω m ) may be determined from Eq. 8.59, which is the reduced form of Eq sign (ω m ) = { sign (w 11 Q L + w 12 Q H ) Q H < sign (Q L ) Q H (8.59) Feasibility Boundary It is noted from Eq. 8.53, that the coefficients w 22 and w 32 are strictly negative. The limitation on Q H may thus be expressed by Eq. 8.6, which is determined from Eq Q H,max = (α + H)2 H K ω ΛK ω Q L Q H (8.6) Influence of actuator dynamics and sensitivity to H estimate The RGA numbers for the transformed system using both input and output transformation is shown in Fig. 8.8a, and again in Fig. 8.8b, where H is estimated with an estimation error of 2%. RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) Frequency [rad/s] (a) Perfect H estimate Frequency [rad/s] (b) 2% error in H estimate Figure 8.8: RGA number for the transformed system with input transformation 3. Actuator dynamics are included in the model. Fig. 8.8a shows that with a perfect H estimate, transformation method 3 combined with the output transformation results in very good decoupling at low frequencies. However, with inclusion of the actuator dynamics, cross-couplings becomes significant at frequencies above the actuator bandwidth. Compared with transformation method 1, method 3 is also significantly more sensitivity to estimation errors with respect to H in the high frequency range. 8.6 Input Transformation Method 4 The purpose with this transformation is to realize the level flow using only the servo-drive and the A-side valve, while keeping the opposite proportional valve closed. The flow constraint for this method is given by Eq [ ] Q BV = Q = 1 u = (8.61)

94 84 Control and Experimental Evaluation of SvSDP Concept The resulting transformation matrices are shown in Eq W 1 1 = H ΛK ω H 1 α + H α + H α + H (α+h) K ω (α+h) α+h α W 1 = α 1 K Bω }{{} c 4 1 H (α + H) 2 K ω H ΛKω (α + H) 2 (8.62) Gradient Condition For this transformation method w 12, why the fulfillment of the gradient condition must be verified. K gc is in this case given by Eq K gc = (α + H) w 12 = α K Bω H (α + H) (8.63) Again assuming H to vary in the interval [, [, the minimum of Eq corresponds to the limit for H going towards, while the maximum corresponds to the limit for H going toward. These extreme cases are given by Eq and Eq K gc,min = lim 12 = α H K Bω (8.64) K gc,max = lim H + 12 = (8.65) As the upper bound of Eq is strictly positive and as K gc,max =, this condition is automatically fulfilled. The lower condition may be reduced as given by Eq K gc,min 1 K ω α (8.66) Since both these inequalities are always fulfilled, the input transformation is feasible. Shaft Sign Determination The coefficients c 4 and w 11 are always positive as K Bω >. With these conditions, sign (ω m ) may be determined from Eq. 8.67, which is the reduced form of Eq sign (ω m ) = { sign (w 11 Q L + w 12 Q H ) Q H < sign (Q L ) Q H (8.67) Feasibility Boundary As the B-side proportional valve is not utilized for this transformation method, it is unnecessary to check that the B-side flow is positive. Instead, only the A-side condition of Eq is used. Since the coefficient w 22 is strictly negative, the limitation on Q H may be expressed by Eq Q H,max = (α + H)2 H K ω ΛK ω Q L Q H (8.68) Influence of actuator dynamics and sensitivity to H estimate The RGA numbers for the transformed system using both input and output transformation is shown in Fig. 8.9a, and again in Fig. 8.9b, where H is estimated with an estimation error of 2%.

95 Chapter 8 - System Decoupling Methods 85 RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) Frequency [rad/s] (a) Perfect H estimate Frequency [rad/s] (b) 2% error in H estimate Figure 8.9: RGA number for the transformed system with input transformation 4. Actuator dynamics are included in the model. Similar to the results of transformation method 3, Fig. 8.9a shows that with a perfect H estimate, transformation method 4, combined with the output transformation results, in very good decoupling at low frequencies. However with inclusion of the actuator dynamics, cross-couplings becomes significant at frequencies above the actuator bandwidth. Additionally, transformation method 4 is significantly more sensitivity to estimation errors with respect to H in the high frequency range than both transformation method 1 and Input Transformation Method 5 The purpose with this transformation is to realize the level flow using only the servo-drive and the B-side valve, while keeping the opposite proportional valve closed. The flow constraint for this method is given by Eq [ ] Q AV = Q = 1 u = (8.69) The resulting transformation matrices are shown in Eq W 1 1 = H ΛK ω H 1 α + H α + H α + H (α+h) K ω (α+h) α+h α W 1 = α 1 K Aω }{{} c α (α + H) 2 K ω H ΛKω (α + H) 2 (8.7) Gradient Condition For this transformation method w 12, why the fulfillment of the gradient condition must be verified. K gc is in this case given by Eq K gc = (α + H) w 12 = α 1 K Aω (α + H) (8.71) Again assuming H to vary in the interval [, [, the minimum of Eq corresponds to the limit for H going towards, while the maximum corresponds to the limit for H going toward. These extreme cases are given by Eq and Eq K gc,min = lim H (α + H) w 12 = (8.72) K gc,max = lim H + (α + H) w 12 = 1 K Aω (8.73)

96 86 Control and Experimental Evaluation of SvSDP Concept As the lower bound of Eq is strictly negative and as K gc,min =, this condition is automatically fulfilled. The upper condition may be reduced as given in Eq K gc,max 1 K + ω 1 α (8.74) Since both these inequalities are always fulfilled, the input transformation is feasible. Shaft Sign Determination The coefficients c 5 and w 11 are always positive as K Aω >. With these conditions, sign (ω m ) may be determined from Eq. 8.75, which is the reduced form of Eq sign (ω m ) = { sign (w 11 Q L + w 12 Q H ) Q H < sign (Q L ) Q H (8.75) Feasibility Boundary As the A-side proportional valve is not utilized for this transformation method, it is unnecessary to check that the A-side flow is positive. Instead, only the B-side condition of Eq is used. Since the coefficient w 32 is strictly negative, the limitation on Q H may be expressed by Eq Q H,max = (α + H)2 H K ω ΛK ω Q L Q H (8.76) Influence of actuator dynamics and sensitivity to H estimate The RGA for the transformed system using both input and output transformation is shown in Fig. 8.1a, and again in Fig. 8.1b, where H is estimated with an estimation error of 2%. RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) RGA Number x(q L ), p H (Q H ) x(q H ), p H (Q L ) Frequency [rad/s] (a) Perfect H estimate Frequency [rad/s] (b) 2% error in H estimate Figure 8.1: RGA number for the transformed system with input transformation 5. Leakage and actuator dynamics are included in the model. Similar to the results of transformation method 3 and 4, Fig. 8.9a shows that with a perfect H estimate, transformation method 5 combined with the output transformation results in very good decoupling at low frequencies. However with inclusion of the actuator dynamics, cross-couplings becomes significant at frequencies above the actuator bandwidth. Additionally, transformation method 5 is significantly more sensitivity to estimation errors with respect to H in the high frequency range than both transformation method 1 and 3, but the decoupling is slightly better than for transformation method 4.

97 Chapter 8 - System Decoupling Methods Discussion of the Results of the Decoupling Analysis Based on the preceding analysis, the following aspect are concluded and utilized in the further development of the control system: Compared to the original system, the use of the output transformation of Eq. 8.2 and either of the presented feasible input transformations will result in a transformed open loop system that is significantly more decoupled in the frequency range up to the actuator bandwidth frequency. Using Input Transformation Method 1 (Eq. 8.45) minimizes the influence of the actuator dynamics, such that the transformed system may be considered decoupled even beyond this frequency. This is not the case for the remaining transformation methods, where the inclusion of actuator dynamics generally result in increased cross-couplings at high frequencies. The introduction of cross couplings may be contributed to the fact that the valve flows cannot be realized at the same rate as the pump flows. For instance, a constant Q H may be desired while Q L is changed. The change in Q L triggers a change in the pump flow depending on the dynamics of the servo-drive, and the valves must be able to accurately compensate for this to avoid a change in Q H. Since the valves have lower bandwidth than the servo-drive this is not possible, and hence the introduction of cross-couplings at high frequencies. Based on these results, it is expectable that Transformation Method 1 will generally yield the best results with respect to dynamic performance. However, this is not the most energy efficient method, which would correspond to method 4 and 5, depending on which chamber pressure is the highest. In the following chapter, the actual control system for the transformed system will be designed with the purpose of minimizing the high frequency cross-couplings, allowing equivalent use of either input transformation, as well as the possibility for switching between these. The conditions for a smooth translation between method 4 and method 5 are further analyzed in Section 9.3.

98

99 Chapter 9 - Design and Evaluation of SvSDP Control Strategies 89 9 Design and Evaluation of SvSDP Control Strategies Based on the results of Chapter 8, decentralized control of the transformed plant is adequate in the frequency range below the actuator bandwidth. As described in Section 7.1, there are no strict requirements to the bandwidth of the control system that attempts to maintain the minimum chamber pressure at the specified set point. As a consequence, the control system for the pressure level may be designed such that the bandwidth of the closed loop pressure level system is significantly below both the natural frequency of the system and the bandwidth of the actuators. Since the control signal of the pressure level control is Q H only (i.e. decentralized), the low closed loop bandwidth in effect limits the high frequency content of Q H, and thus the high frequency cross-couplings will not be excited. Similar to the approach in Chapter 8, the principles and considerations are investigated based on the decoupled plant model, i.e. actuator dynamics are disregarded. However, unless otherwise specified, actuator dynamics are included when evaluating frequency responses to ensure stability of the closed loop system. The investigated transformation methods are 1, 4 and 5. Method 1 represents the "ideal" transformation with respect to the dynamic performance, whereas method 4 and 5 represent the "ideal" energy efficient valve utilization methods. 9.1 Design of the Pressure Level Control System Prior to the design of the pressure level control system, the actual purpose of this loop is reinvestigated. As stated in Section 7.1, the purpose of the pressure level control system is to maintain the steady state value of the minimum actual chamber pressure (i.e. the minimum of p A and p B ) at a specified set point. The term steady state is used, as the control system should not otherwise interfere with the load pressure, but simply shift the actual chamber pressures to a higher value, while maintaining the same load pressure. Since the signal p H does not relate to a single physical property, but is simply the result of a decoupling attempt, the reference p H,ref, that the pressure level p H should ideally follow is defined in the following section Pressure Level Reference Generator This subsection describes how the measured chamber pressures p A and p B are mapped into a pressure level reference p H,ref, such that a minimum chamber pressure min (p A, p B ) = p set = 25bar can be achieved. The minimum chamber pressure may be identified from the load pressure as described in Section 1, and the derived conditions (Eq. 1.3, Eq. 1.4 and Eq. 1.5) are rewritten and inserted in Tab Additionally, the desired references for the chamber pressures are translated into a reference for the pressure level from the definition in Eq. 8.8, which is shown again in Eq. 9.1 for ease of reference. p H = p A + H p B H = V B α V A (9.1)

100 9 Control and Experimental Evaluation of SvSDP Concept Pressure condition p L p set (1 α) p L < p set (1 α) Chamber A reference p A,ref = p L + α p set p A,ref = p set Chamber B reference p B,ref = p set p B,ref = pset p L α Pressure level reference p H,ref = p L + p set (α + H) p H,ref = H α p L + p set ( ) 1 + H α Table 9.1: Pressure references to obtain a minimum chamber pressure of p set at all times. Since it may not be transparent from Tab. 9.1, why the prescribed pressure level reference corresponds to controlling only the minimum chamber pressure, this is proved in the following. Defining the pressure level error as e H = p H,ref p H, then the pressure level error for both cases can be rewritten in terms of the actual chamber pressures as shown in Eq. 9.2 and Eq. 9.3 respectively. The chamber pressures are inserted so it can easily be seen that only a single chamber pressure at a time is responsible for the pressure level error and hence the control of the pressure level. e H = p L + p set (α + H) p H p L p set (1 α) = p A α p B + p set (α + H) p A H p B (9.2) p L < p set (1 α) = (α + H) (p set p B ) e H = H α p L + p set ( 1 + H ) p α H = H ( α p + A H p B + p set 1 + H α = α + H α (p set p A ) ) p A H p B (9.3) Both Eq. 9.2 and Eq. 9.3 are similar expression of only a single chamber pressures, and in effect simply scales the gain of the controller depending on which pressure is controlled. Furthermore, from insertion of the switching condition, i.e. p L = p set (1 α) into the two cases of p H,ref given in Tab. 9.1, it can be shown in that there are no jumps in the pressure reference as seen in Fig Pressure level reference [bar] x = 35 mm (H = 8.2 [ ]) x = 175 mm (H = 2.7 [ ]) x = mm (H = 1.3 [ ]) x = 175 mm (H =.7 [ ]) x = 35 mm (H =.3 [ ]) Load pressure [bar] Figure 9.1: Pressure level reference p H,ref as function of load pressure p L for different piston positions. The set pressure p set = 25 bar gives the switching load pressure p L,sw = 7.7 bar. The pressure level reference is highly dependent upon the piston position, as a result of the nonlinear expression for the parameter H. Due to the definition of p H as a weighted sum of the chamber pressures, the reference values does not directly relate to any physical pressures, why the possibly high numerical values of p H,ref is not of concern.

101 Chapter 9 - Design and Evaluation of SvSDP Control Strategies Design of Pressure Level Control Based on the pressure level reference described in Section 9.1.1, a feasible control law for the pressure level control is given by Eq Q H = G c,h (p H,ref p H ) (9.4) where G c,h is the compensator of the pressure level control system. Since p H,ref is a function of the load pressure p L, this basically corresponds to a feedback of the load pressure, however due to the insignificant cross-couplings at low frequencies (K HpL p L ), this is considered acceptable without further investigation. The pressure level control structure is seen Fig p set H Pressure level reference generator p H,ref e H G c,h Q L,ref Q H,ref Q,ref H ω m,ref Q AV,ref W 1 S e W 2 Q BV,ref Decoupler x p A p B H x p L p H Figure 9.2: Schematic of the closed loop pressure level control structure. The closed loop system including actuator dynamics is denoted S ehcl. The closed loop system including actuator dynamics is denoted S ehcl, where the inputs of this system is Q L,ref and the pressure set-point p sec. Fig. 9.2 is for generality shown with the extended system S e inserted. As mentioned in the beginning of this chapter, the actuator dynamics are disregarded in the controller design, but reintroduced in the evaluation of the controller performance. To avoid the cross couplings present at high frequencies, the pressure level controller should have a steep roll off, such that any high frequency in the error signal is dampened as much as possible. The compensator G c,h may thus be designed based on the transfer function from Q H to p H only. This transfer function is given in Eq p H (s) Q H (s) = 1 K HpH 1 V A (α + H) β K HpH s + 1 (9.5) The control system is designed based on the piston position resulting in the highest possible time constant of the first order system in Eq. 9.5, i.e. the position resulting in the most significant amount of uncompensated phase lag. The time constant τ H of the pressure level transfer function may be described in terms of the chamber volumes V A and V B in Eq τ H = V A (α + H) β K HpH = 1 β K HpH ( α + V ) ( B 1 V A = α V A β K HpH α V A + V B α ) (9.6) Considering the bulk modulus and the leakage coefficients as constant, and inserting the chamber volumes given in Eq. 9.7 in to Eq. 9.6, the time constant τ H may be described in terms of the piston position as given in Eq V A = V A + A p x V B = V B α A p x (9.7) τ H = 1 β K H,pH ( αv A + V ) B α (1 α) A p x (9.8) The maximum value of τ H thus corresponds to the minimum possible value of x. In Fig. 9.3a, the frequency response of Eq. 9.5 is shown along with the equivalent element of G e,m1, Ge,M4

102 92 Control and Experimental Evaluation of SvSDP Concept and G e,m5, i.e. the transfer function from Q H,ref to p H including actuator dynamics for input transformation methods 1, 4 and 5 respectively. Simplified model. No actuator dynamics or cross couplings Extended model. Input transformation method 1 Extended model. Input transformation method 4 Extended model. Input transformation method 5 Magnitude [db] Phase [ ] Frequency [rad/s] Frequency [rad/s] (a) Uncompensated open loop transfer function Magnitude [db] Phase [ ] Frequency [rad/s] Frequency [rad/s] (b) Compensated open loop transfer function Figure 9.3: Pressure level flow reference to pressure level, Q H,ref p H, at linearizion point x = x min = 35 mm (H = 8.23), ẋ = 1 mm/s, p A = p B = p set = 25 bar. As seen from Fig. 9.3a, the transfer function of Eq. 9.5 accurately describes the dynamics of p H in the frequency range below the bandwidth of the actuators. An interesting behavior is seen in the frequency analysis of the extended model with input transformation 4 above approximately 1 rad/s. This behavior is the result of a zero in the right half plane, introduced by this particular transformation method. When selecting the controller design for the pressure level control, three main considerations should be accounted for. It is desirable to achieve no steady state error in the pressure level to step inputs, why a free integrator is needed in the controller [Philipps and Parr, 211, p. 173]. To counter the introduced phase lag, a zero is included, such that acceptable stability margins can be achieved (yielding effectively a conventional PI-controller). The pressure level error is given directly based on the difference between a single chamber pressure and the constant set pressure, see Eq. 9.2 and Eq. 9.3, why it is expected that the error will tend to oscillate with the natural frequency of the plant. To avoid the propagation of the oscillating pressures into the pressure level control signal, a 2nd order low pass filter is introduced in the controller design shown in Eq G c,h = K p,h s ( 1 ω 2 filt s + ω z,h s ζ filt ω filt s + 1 ) ω z,h = K i,h (9.9) K p,h The choice of the natural frequency for the low pass filter is a compromise between attenuating the high frequency content of the level flow reference signal as much as possible, while achieving acceptable stability margins of the closed loop system. To ease the parameterization of the compensator, the procedure is specified as follows: Choose the cross-over frequency ω c,h of the compensated system as less than a tenth of the valve bandwidth. Place the compensator zero, such that the phase lag at the desired cross-over frequency is 18 + P M where P M is the desired phase margin in this case chosen as 6, to ensure

103 Chapter 9 - Design and Evaluation of SvSDP Control Strategies 93 sufficient stability margins even when the phase lag from the actuator dynamics is included. Place the natural frequency of the filter one decade above the compensator zero, such that the eigen frequency of the plant is attenuated, with neglectable influence on the stability margins. Choose the compensator proportional gain such that the compensated open loop gain is db at ω c,h. The selected ratio between the cross-over frequency of the compensated system and the valve bandwidth is a compromise between minimizing the difference between the different input transformation methods and maximizing the pressure level control bandwidth. The resulting compensated open loop transfer function is shown in Fig. 9.3b for the considered plant. With the above procedure, the parameterization of the pressure level control system may be carried out with minimal knowledge of the plant parameters, as the simplified model in Eq. 9.5 can be used for the control design. ω c,h ω z,h ω filt ζ filt K p,h 7.23 rad/s 2. rad/s 2 rad/s L/min bar Table 9.2: Selected G c,h controller parameters. To verify the performance of the pressure level control system in the nonlinear model, a step sequence is given to Q L,ref as shown in Fig. 9.4, for the different input transformation methods. The similarity between velocity and chamber pressure responses are clearly seen for all three transformation methods. Fig. 9.4c and Fig. 9.4d shows that the minimum chamber pressure is kept at the selected set pressure of p set = 25 bar without influencing the piston motion regardless of input transformation method. The piston velocity and chamber pressure responses are virtually indistinguishable for the different input transformation methods. Furthermore it is seen that the pressure level controller does not influence the dynamics of the piston motion, i.e. the chamber pressures are free to oscillate, which is the desired behavior of the pressure level control. The difference between the transformation methods are seen in the control signals for the servo drive and proportional valves. Due to the inverse flow characteristics of the proportional valves, i.e. the required offset in voltage to open the valve, the shown bias of.22 in the normalized u AV and u BV signals corresponds to zero valve flow. This means that the desired utilization of only the A-side valve in method 4 and only the B-side valve vale in method 5 are obtained. The pressure level controller performance is deemed acceptable, to proceed to design of the closed loop motion control.

104 94 Control and Experimental Evaluation of SvSDP Concept Input transformation method 1 Input transformation method 4 Input transformation method 5 Q L,ref [L/min] Time [s] 5 4 (a) Load flow reference Q L,ref. ẋ [mm/s] Time [s] (b) Piston velocity ẋ. 5 4 p A [bar] 3 2 p B [bar] Time [s] Time [s] (c) A-side chamber pressure p A. (d) B-side chamber pressure p B. 3 Q H,ref [L/min] Time [s] (e) Pressure level flow reference Q H,ref. Normalized ω m,ref [ ] Time [s] (f) Shaft velocity reference ω m,ref..4.4 Normalized u AV [ ] Normalized u BV [ ] Time [s] Time [s] (g) A-side proportional valve signal u AV,ref. (h) B-side proportional valve signal u BV,ref. Figure 9.4: Simulated response of the nonlinear model using the pressure level controller G c,h. The input is a step in Q L,ref using both input transformation method 1, 4 and Design of Motion Controller As described in the beginning of this chapter, the purpose of choosing a low bandwidth for the pressure level control system is mainly to minimize the influence of the high frequency crosscouplings introduced by the various input transformations. If this is achievable, the closed loop

105 Chapter 9 - Design and Evaluation of SvSDP Control Strategies 95 motion control system may be designed from a simplified model of the hydro-mechanical system, independent of both the pressure level control system and the chosen input transformation. To investigate whether this has been achieved, the simplified linear dynamic model of the hydromechanical system is derived in Section The validity of this simplified model is shown in Section 9.2.2, where the frequency response of the simplified model is compared to that of the extended linear model before and after closing the pressure level loop Simplified Motion Transfer Function As verified in Section 9.2.2, the choice of control law for the pressure level control system has neglectable influence on the validity of the transfer function from Q L to x. To ease the design procedure for the motion control system, a simplified model usable for control design is therefore derived. The model is derived in the general case, and specialized for the case of lowest eigenfrequency. The linear dynamic equations describing the motion are given again in Eq. 9.1 and Eq ẍ = 1 M (A p p L B ẋ) ( ) (9.1) β (α + H) ṗ L = Q V A H L A p ẋ H K LpL (α + H) 2 (9.11) Through Laplace transformation, the equations may be written as Eq and Eq H α + H (M s + B) ẋ = A p p L (9.12) ( VA β s + K ) LpL p α + H L = Q L A p ẋ (9.13) Insertion of Eq into Eq gives Eq H α + H By solving Eq for ẋ Q L = = = ( VA β s + K ) LpL (M s + B) ẋ = A p Q α + H L A 2 p ẋ (9.14) ẋ Q L, the transfer function is given by Eq H α+h ( ( VA M β s 2 VA B + β ( V A M β s 2 VA B + β ( s 2 B + M + β K LpL V A (α+h) A p + M K LpL α+h α+h A p H) + M K LpL α+h α+h H V ) A β A p M s + α+h H V A ) s + B K LpL α+h β M s + B K LpL α+h ( + α+h H ) + A 2 p A2 p A 2 p + B K LpL α+h B K LpL ) (9.15) Since hydraulic systems typically have poor inherent damping, i.e. α+h A2 p, finding the piston position that yields the lowest natural frequency approximately corresponds to finding the value of H that minimizes α + H (assuming constant mass M and oil bulk modulus β). The H V A A-side volume V A may be described in terms of H by addition of Eq and Eq and

106 96 Control and Experimental Evaluation of SvSDP Concept substitution of the definition of H in the form V B α = V H. The result is given in Eq A V A = V A + A p x (9.16) V B α = V B α A p x (9.17) V A = V 1 + H V = V A + V B α (9.18) Finding the H value corresponding to the lowest eigen frequency thus correspond to finding the minimum of Eq ω 2 n (α + H) (1 + H) H V (9.19) The minimum of this function may be found by finding the derivative and setting it equal to zero. The result is given by Eq H ef,min = α (9.2) The piston position x ef,min corresponding to Eq. 9.2 may be found from insertion of Eq into Eq The result is given in Eq ( 1 x ef,min = (1 + VB α) A p α ) α V A (9.21) Using Eq. 9.2 in Eq. 9.15, reduces the transfer function to Eq ẋ Q L = ( s 2 B + M + β K LpL α V ) (1+ α) 2 V β A p M s + (1+ α) 2 V β M ( ) (9.22) A 2 p + B K LpL α (1+ α) It is interesting to note that Eq is general in the sense that it is the transfer function from Q L to x. Since a typical valve operated system may also be described in such a way through e.g. active gain compensation, it is reasonable to assume that the same control parameters may be utilized independent of whether a SvSDP system or a valve is used to realize Q L. This means that the dynamic performance of a SvSDP system might theoretically be completely equivalent to that of a valve operated system Influence of pressure level control system To investigate if the desired simplification has been achieved, the frequency response from Q L,ref to x is evaluated for input transformation methods 1, 4 and 5. Fig. 9.5a and Fig. 9.5b shows the frequency response prior to and after closing the pressure level loop respectively.

107 Chapter 9 - Design and Evaluation of SvSDP Control Strategies 97 Simplified model. No actuator dynamics or cross couplings Extended model. Input transformation method 1 Extended model. Input transformation method 4 Extended model. Input transformation method 5 Magnitude [db] Phase [ ] Frequency [rad/s] Frequency [rad/s] (a) Uncompensated open loop transfer function Magnitude [db] Phase [ ] Frequency [rad/s] Frequency [rad/s] (b) Compensated open loop transfer function Figure 9.5: Frequency response of the open loop transfer function from load flow reference to piston position for input transformation method 1, 4 and 5. As seen from the phase plots of Fig. 9.5, closing the pressure level loop has only slightly increased the damping of the system, and the simplified model is thus still quite applicable for control design. This is investigated further in the next section Parameterization of Motion Controller Due to the generality of Eq. 9.22, various compensators may be designed, without consideration of how the transformed control signal Q L is distributed between the physical control signals, i.e. which input transformation method is utilized. In this project, only a simple PI compensator is considered to investigate whether the theoretical performance may be achieved. More advanced control laws are considered beyond the scope of this project. The design procedure for the PI compensator is outlined in the following. The frequency response of the transfer function Eq is shown in Fig. 9.6a.

108 98 Control and Experimental Evaluation of SvSDP Concept p A, p B = 2 bar with β A, β B = 1186 bar p A, p B = 25 bar with β A, β B = 5944 bar Magnitude [db] Frequency [rad/s] Magnitude [db] Frequency [rad/s] 9 9 Phase [ ] Phase [ ] Frequency [rad/s] Frequency [rad/s] (a) Transfer function from Q L to x. (b) Transfer function from Q to x, where static load L pressure feedback has been applied. Figure 9.6: Uncompensated open loop frequency response Due to the poor damping of the open loop system, static feedback of the load pressure is utilized to realize the open loop frequency response from Q to x shown in Fig. 9.6b. The control law L giving this relation is given by Eq Q L,ref = Q L,ref K ad p L (9.23) Q is thus the new control signal for which the motion compensator is designed. The value of L K ad is chosen to obtain a damping coefficient of.5. This choice results in a open loop gain at the natural frequency which is 6 db below the open loop gain at the frequency corresponding to a open loop phase of 12. This choice makes it possible to obtain a gain margin of 6 db and a phase margin of 6 using a simple proportional compensator. To visualize the influence of the pressure feedback, simulation results of the response to steps in Q is shown in Fig L The step magnitude is equivalent to those applied for the simulation in Fig. 9.4.

109 Chapter 9 - Design and Evaluation of SvSDP Control Strategies 99 Input transformation method 1 Input transformation method 4 Input transformation method 5 Q L,ref [L/min] Time [s] (a) Load flow reference Q L,ref. 5 4 ẋ [mm/s] Time [s] 5 4 (b) Piston velocity ẋ. p A [bar] 3 2 p B [bar] Time [s] Time [s] (c) A-side chamber pressure p A. (d) B-side chamber pressure p B. Q H,ref [L/min] Normalized ω m,ref [ ] Time [s] (e) Pressure level flow reference Q H,ref Time [s] (f) Shaft velocity reference ω m,ref..4.4 Normalized u AV [ ] Normalized u BV [ ] Time [s] Time [s] (g) A-side proportional valve signal u AV,ref. (h) B-side proportional valve signal u BV,ref. Figure 9.7: Simulated response of the nonlinear model using the pressure level controller G c,h, based on step in Q L,ref for input transformation method 1, 4 and 5. From Fig. 9.7 it is seen that the feedback of the load pressure has significantly reduced the oscillations in both pressure and piston velocity, and slightly reduced the steady state gain from Q to ẋ. The response with respect to the pressure level and the level flow reference Q L,ref H,ref is not altered significantly with the exception of a slightly increased high frequency content at the beginning and the end of the input sequence. This is caused by the feasibility boundary discussed in Section 8.2, which is enforced when the load flow reference Q L,ref is close to zero.

110 1 Control and Experimental Evaluation of SvSDP Concept A closed loop PI position controller is designed based on the transfer functions shown in Fig. 9.6b. Feedforward of the reference velocity is furthermore used to improve the servo properties of the closed loop system. The final control law for the motion control system including load pressure feedback is thus given by Eq t Q L,ref = A p ẋ ref + K p e x + K i e x ( t) d t K ad p L e x = x ref x (9.24) The feedforward term is scaled with the piston area to approximately correspond to the static load flow required to realize the given velocity (see Eq. 9.11). The closed loop motion control system is shown in Fig ẋ ref x ref e x A p G c,l Q L,ref Q L,ref p set S ehcl x p L p H K ad Figure 9.8: Schematic of motion control structure. See Fig. 9.2 for definition of S ehcl. The parameters for the PI motion controller are chosen to yield sufficient stability margins, both for low and high pressure operation. Due to the higher gain and lower bandwidth of the transfer function at low pressures, the low pressure operation poses the most restrictive limitations to the choice of compensator parameters. However, as the pressures will increase during control activity, it is considered reasonable to accept slightly lower stability margins during low pressure operation, which on the other hand results in a slightly higher closed loop bandwidth of the motion control system during both low and high pressure operation. As a result, the compensator parameters are chosen as given in Tab. 9.4, such that the stability margins given in Tab. 9.3 are realized. Operating Pressure Gain Margin [db] Phase Margin [ ] Low pressure (p A = p B = 2 bar) 6 4 High pressure (p A = p B = 25 bar) 18 5 Table 9.3: Stability margins of the motion control system K ad ω z K p K i 1.5 L/min bar 8 rad/s 4 L/min mm Table 9.4: Selected parameters for the motion controller. 32 L/min mm s The frequency responses of the resulting compensated open loop transfer functions are shown in Fig. 9.9a, and the resulting closed loop transfer function is shown in Fig. 9.9b for both high and low pressure operation.

111 Chapter 9 - Design and Evaluation of SvSDP Control Strategies 11 p A, p B = 2 bar with β A, β B = 1186 bar p A, p B = 25 bar with β A, β B = 5944 bar Phase [ ] Magnitude [db] Frequency [rad/s] Frequency [rad/s] Magnitude [db] Phase [ ] Frequency [rad/s] Frequency [rad/s] (a) Compensated open loop motion transfer function. (b) Closed loop motion transfer. Figure 9.9: Compensated motion frequency response The performance of the final closed loop system is verified by simulation to a position ramp input including feedforward of the velocity reference. The results are shown in Fig. 9.1.

112 12 Control and Experimental Evaluation of SvSDP Concept Input transformation method 1 Input transformation method 4 Input transformation method 5 x [mm] Q L,ref [L/min] Time [s] (a) Piston position x Time [s] 5 4 (c) Load flow reference Q L,ref. e x [mm] ẋ [mm/s] Time [s] (b) Piston position error e x Time [s] (d) Piston velocity ẋ. p A [bar] 3 2 p B [bar] Time [s] Time [s] (e) A-side chamber pressure p A. (f) B-side chamber pressure p B. 4.4 Q H,ref [L/min] Time [s] Normalized ω m,ref [ ] Time [s] (g) Pressure level flow reference Q H,ref. (h) Shaft velocity reference ω m,ref. Normalized u AV [ ] Normalized u BV [ ] Time [s] Time [s] (i) A-side proportional valve signal u AV,ref. (j) B-side proportional valve signal u BV,ref. Figure 9.1: Simulated response of the nonlinear model using the pressure level controller G c,h, based on step in Q L,ref for input transformation method 1, 4 and 5.

113 Chapter 9 - Design and Evaluation of SvSDP Control Strategies 13 From Fig. 9.1, the closed loop motion control system is seen to yield very good tracking performance, with a peak tracking error of max (e x ) < 3 mm and a settling time of 25 ms. Rather large pressure fluctuation are seen at the acceleration points which is considered acceptable, as the applied position reference sequence is non-realizable (infinite acceleration), and will thus not be present for smoother position trajectories. The designed control structure is thus considered suitable for practical implementation and evaluation, which is treated in the following chapter. 9.3 Energy Efficient Valve Utilization In order to fully utilize the developed control strategy, it should be possible to smoothly transition between the different input transformation methods during operation. In this section a set of criteria for the smooth transition between input transformation method 4 and 5, such that most energy efficient valve utilization is achieved. Recall that in input transformation method 4 only the A-side proportional valve is used, whereas only the B-side proportional valve is used for method 5. The optimal energy efficient valve utilization method was original derived in [Grønkjær and Rahn, 214b, p. 39] and the key points are summarized here for easy reference. The steady state continuity equations for the two cylinder chambers assuming zero leakage are formulated in Eq = K Aω ω m Q AV A p ẋ = K Bω ω m Q BV + α A p ẋ (9.25) Setting ω m equal each other in Eq and solving for either Q AV or Q BV yields Eq (Q BV = ) Q AV = A p ẋ α K Aω K Bω K Bω Q BV = A p ẋ α K Aω K Bω K Aω (Q AV = ) (9.26) The throttling loss through a restriction is P throttle = pq and when assuming zero tank pressure, the inequality Eq can be set up describing when to use the B-side proportional valve. p A Q AV >p B Q BV p A A p ẋ α K Aω K Bω K Bω >p B A p ẋ α K Aω K Bω K Aω (9.27) As the value of K Aω changes with sign of ω m, the factor shown in Eq is always positive regardless of the shaft rotational direction. (ẋ >) (α K Aω K Bω ) (A p ẋ) }{{}}{{} > > > (ẋ <) (α K Aω K Bω ) (A p ẋ) > (9.28) }{{}}{{} < < This means that Eq transforms into the conditions shown in Tab Pressure condition p A < K Bω K Aω p B p A > K Bω K Aω p B Throttling side Use A-side valve (method 4) Use B-side valve (method 5) Table 9.5: Pressure conditions for switching between input transformation method 4 and 5 to achieve energy efficient proportional valve utilization. When switching between the two different input transformation methods, the gain from the Q L and Q H inputs to the three signals ω m, Q AV and Q BV changes. It is undesirable to have sudden jumps in the reference signals, as that would affect the dynamic performance of the system. The only possible solution for a smooth switch between the transformation methods is when both

114 14 Control and Experimental Evaluation of SvSDP Concept valve flows are equal to zero (if necessary see definition of W 1 shaft velocity mappings are given in Eq in Eq and Eq. 8.7). The Input transformation method 4: Input transformation method 5: ω m = α α H Q L K Bω (α + H) 2 Q ω H m = 1 α Q K L + Aω K Aω (α + H) 2 Q (9.29) H K Bω For Q AV = Q BV = then Eq becomes Eq. 9.3 and Eq ω m = ω m = α H K Bω (α + H) H K Aω (α + H) ( K Aω + K ) Bω ω m H ( K Aω + K ) Bω ω m + H α H (α + H) K Bω (α + H) 2 α (α + H) K Aω (α + H) 2 ( K Aω K ) Bω α ( K Aω K ) Bω α ω m ω m (9.3) ω m ω m (9.31) As both transformation methods yields the natural solution that ω m = ω m for Q AV = Q BV =, smooth switching between the input transformation methods can only occur when both valve flows are zero. This results corresponds exactly to the ω m -mapping being on the feasibility boundary, which is a condition, that has already been calculated for both of these transformation methods. To sum up these findings: The pressure condition in Tab. 9.5 should at all times be checked and when the ω m -mapping result in a solution on the feasibility boundary, then the transformation method should be switched to the one indicated in Tab Due to spikes in the chamber pressure this described switching method might result in selecting the wrong transformation method. A better solution could be to introduce a band outside the feasibility bound, in which a weighted average of method 4 and method 5 could be used.

115 Chapter 1 - Conclusion 15 1 Conclusion The main objective of this project has been to experimentally evaluate the Speed-variable Switched Differential Pump (SvSDP) concept. A second objective of the project has been to develop an application-independent control strategy for the SvSDP concept, which allows for easy parameterization, good dynamic tracking performance and high energy efficiency. Test Bench Design The basis for the dimensioning of the SvSDP system, is the possibility for reuse of the main components from the original SvDP setup, especially the servo-drive and the motor. The continuous torque of the motor is mainly limited by the possible cooling of the motor, why this limitation is used to roughly estimate the possible continuous load force. The found load force is approximately 25 kn, whereas the remaining main specifications for the system are based on values that are considered reasonable. These remaining specification are: realization of a nominal piston velocity of 25 mm/s at a shaft speed 15 2% below the maximum pump speed of 3RPM. The desired stroke length is chosen to be 7mm, such that steady state operation at the nominal piston velocity may be tested continuously for a few seconds. Different test bench topologies have been considered, where the evaluation criteria are: As linear a load characteristic as possible, such that the intrinsic properties of the SvSDP concept may be isolated from the properties of the particular load. Flexibility in loading conditions, such that the concept may be evaluated in the complete operating range. Small physical size to fit into the available space, and with minimum safety issues. Ease of design and assembly as well as the possibility for reuse of the test bench in future, and possibly non-related projects. Through generation and evaluation of the test bench topologies, the most suitable is found to be a general purpose test bench utilizing a sliding inertia load and two cylinders; one for the system under evaluation, and another for emulation of various load conditions. Back-of-an-envelope calculations are made for each topology to roughly estimate the required dimensions, supporting the selection. Detailed mechanical design, as well as dimensioning of hydraulic and mechanical components, has been carried out, giving a rated load force for the test bench of 5 kn, with a target safety factor of 3. This load force corresponds to the buckling resistance of the utilized cylinders. The Data Acquisition and Control (DAC) system for the test bench has been implemented on a LabVIEW Real-Time machine, which is responsible for evaluation of the control signals for both the SvSDP system and the load system, as well as measurement and logging of all sensor data. The DAC currently transmits the desired SvSDP control signals to the servo-drive through analog channels, however a topology, which should allow for an EtherCAT fieldbus based system, has been proposed. The designed test bench has been constructed and the SvSDP system has been connected to it. The hydraulic components for the load system have not yet been implemented on the test setup. Control Strategy Development A non-linear dynamic model of the SvSDP and load system has been developed, and the obtained experimental results have verified the general model structure, and only slightly changed the parameterization compared to the initially estimated values. Test sequences have been designed to allow for isolated determination of the unknown parameters. From the results of these test sequences, the parameters relating to friction, oil bulk modulus, pump displacement and leakage have been determined from experiments. Experimental results that may be used to verify the dynamic response and flow characteristic of the proportional valves have not yet been acquired.

116 16 Control and Experimental Evaluation of SvSDP Concept From the modeling of the utilized check valves, is has been investigated whether the dynamics of these may be neglected in the simulations. From comparison of simulation results with and without included check valve dynamics, no significant influence has initially been found. Based on the observations from experiments, slight delays have however been observed in the system response, which is believed to be due to the closing time of the non-spring returned check valves. A higher-than-estimated reverse flow is apparently required before the check valves close, why an improvement proposal is to implement small springs in all the by-pass check valves such that the valve closing time is reduced. A linearized version of the initially developed dynamic model has been developed, and the amount of cross-couplings have been determined through use of Relative Gain Array (RGA) analysis. The result of this analysis is that decentralized control is not immediately the most suitable control strategy. Instead, decoupling is attempted, where the motion of the piston is decoupled from the pressure level. An RGA analysis of the decoupled system reveals significantly better possibilities for decentralized control of this transformed system. Through input transformation, two new control signals are defined, which allows for design of closed loop controllers independent of the valve utilization scheme. Different valve utilization schemes results in changes in the energy efficiency of the system, why this a desired property. Is has furthermore been found that the active valve utilization scheme may be changed on-line without inducing discontinuities in the control signals, as long as the change occurs while both valves are closed. A design procedure for parameterization of the closed loop feedback controllers has been developed, allowing the design to be based on simple linear models requiring a minimum of parameter and model knowledge. No experimental results have been obtained for the designed closed loop controllers, however, simulation results have shown consistent tracking performance, independent of the valve utilization scheme. The maximum position tracking error during the simulation has been found to be < 3 mm even for a non-realizable ramp trajectory. This, combined with the currently verified model parameters, indicates that the proposed SvSDP concept may remedy the unacceptable closed loop performance of the original SvDP system.

117 Appendices 17

118

119 Appendix A - Dimensioning and Mounting Design of Cylinders 19 A Dimensioning and Mounting Design of Cylinders This appendix presents the calculations of the load force which is realizable with the utilized hydraulic cylinder actuators, as well as the considerations behind the choice of cylinder mounting design. The permissible cylinder force is mainly limited by the buckling strength of the cylinder, why buckling calculations based on [DNV, 29] are presented in Section A.1. The buckling strength is, among other things, influenced by the choice of mounting design, which in turn influences the possibility for compensation of misalignments due to component tolerances. The considered mounting design topologies are presented in Section A.2 along with a short description and a list of the pros and cons of the specific topology. A.1 Cylinder Buckling The buckling of the cylinder must be considered, when a pushing force is present in the cylinder. The cylinder is designed in accordance with [DNV, 29], why formulas presented in [DNV, 29] are used for the strength calculation. The buckling calculation presented in [DNV, 29,p. 6] accounts for the combined buckling of both the cylinder tube and the piston rod. The buckling load F E and the corresponding maximum permissible load force F max are given in Eq. A.1. F E = E π2 L Z F b,max = F E SF b (A.1) SF b is the safety factor w.r.t. buckling (chosen as the target safety factor of 3), E is the modulus of elasticity of the cylinder material (Steel S355) and L is the total length of the part of the cylinder subjected to buckling. The resistance towards buckling depends on the equivalent area moment of inertia, Z, of both the cylinder tube and the piston rod. This may be described by Eq. A.2 [DNV, 29], where the utilized lengths are shown in Fig. A.1. Z = L 1 + L ( ) I 1 I 2 I 2 I 1 ( L 2 π sin 2 π L ) 1 L (A.2) The area moments of inertia I 1 and I 2 of respectively the cylinder tube and the piston rod are described by Eq. A.3 [Gere and Goodno, 29]. I 1 = π ( Do 4 Di 4 ) I 2 = π ( d 4 o d 4 ) i (A.3) The utilized diameters are shown in Fig. A.1. D o D i d o d i L 1 L 2 Figure A.1: Dimensions used in the evaluation of buckling load of a cylinder. The length L 1 denotes the length of the cylinder tube starting from the mounting point, and the length L 2 is the extended part of the piston rod. Eq. A.1 through Eq. A.3 are applicable for various mounting topologies, and a discussion of the considered topologies are presented in Section A.2 for completeness. The chosen mounting

120 11 Control and Experimental Evaluation of SvSDP Concept topology is the trunnion mount sketched in Fig. A.1, where the resulting dimensions and material properties used for the buckling evaluation are listed in Tab. A.1. Description Value Unit Source SF b Safety factor wrt. to buckling 3 - [Norton, 214, p ] E Modulus of Elasticity of Steel S GPa [DS, 27, p. 28] L 1 Length of the cylinder part from the center of the mounting point 1 mm L 2 Visible length of the piston rod in fully extracted position from the 885 mm center of the mounting point L Total length of cylinder from mounting trunnion to rod eye 985 mm D o Outer diameter of cylinder tube 73 mm D i Inner diameter of cylinder tube 63 mm d o Outer diameter of piston rod 35 mm d i Inner diameter of piston rod 14 mm I 1 Area moment of inertia of cylinder tube mm 4 I 2 Area moment of inertia of piston rod mm 4 Z Combined length-to-inertia ratio mm 3 F E Buckling load 154 kn F b,max Maximum permissible load force 51 kn Table A.1: Cylinder dimensions of the chosen cylinder w. trunnion mount. A.2 Cylinder Mounting In this section the investigated cylinder mounting possibilities are presented and discussed. Due to tolerance issues, it is expected that the cylinders are not aligned perfectly with the sliding element on the rails. The mounting systems must therefore be able to compensate for the misalignments, such that any shear forces in the cylinders are reduced. Additionally, the location of the cylinder mountings influences the buckling strength of the cylinder. Mountings placed in both ends of a cylinder has reduced buckling strength compared to a cylinder of equal stroke length, but with the mounting points placed closer together. This dependency is shown through the example in Fig. A.2, where the location of the cylinder mount L 1 is changed, and the buckling strength is evaluated based on Eq. A.1. Common for all the considered mounting types is that the rod end eye has a spherical plain bearing, which allows for free rotation of the rod end in all directions. Buckling strength F E [kn Tube mounting location L [mm] Figure A.2: Theoretical buckling strength for cylinder with fixed stroke length and therefore fixed L 2. Influence on increasing the L 1 distance. See Fig. A.1 for variable definitions.

121 Appendix A - Dimensioning and Mounting Design of Cylinders 111 A.2.1 Mounting Topology 1: Spherical Tube Eye Cylinders with spherical plain bearings in both ends forming two ball joints, allows for free rotation of both ends in all directions, as shown in Fig. A.3. Bolted enclosure for built-in position transducer Figure A.3: Spherical eye on both cylinder tube and piston rod. To incorporate a position transducer, the length of the bottom end is increased. With respect to reducing stresses and shear forces in the cylinder due to mounting misalignment, this is the ideal mounting system. Since the chosen position transducers are integrated within the cylinder, this mounting topology however requires special attention to accommodate this. The housing for the electronics of a piston position transducer is typically too large to be installed completely within the cylinder [Balluff GmbH, 215a, p. 7], [MTS Sensors, 215, p. 7], why an end enclosure, surrounding the sensor, is required as illustrated in Fig. A.3. Pros: Spherical plain bearings in both ends ideally removes any radial forces acting between the cylinder tubes and the pistons. Cons: Long frame is required, since the cylinders must be completely within the frame. The possible stroke length must thus be reduced, to fit the construction within the space available for the test setup. The cylinder complexity and length increases further due to the necessity of an end enclosure for the built-in position transducer. The buckling strength of the cylinders is reduced due to the long distance between the bearings, which reduces the operating range of the cylinders. However, having the cylinders completely within the frame removes the necessity of any rod overhang, which slightly mitigates this problem. A.2.2 Mounting Topology 2: Spherical Tube Eye - Cylinders Beside Each Other To decrease the total length of the setup compared to Mounting Topology 1, the cylinders may be mounted side-by-side or on top of each other as shown in Fig. A.4. Figure A.4: Cylinders with spherical eyes in both ends mounted on top of each other to reduce the length of the setup. With this approach, the frame length can be reduced to 2/3 the length of Mounting Topology 1. The vertical frame profile must only account for the turning moment induced by the cylinders

122 112 Control and Experimental Evaluation of SvSDP Concept forces, where a diagonal strut can be installed as shown in Fig. A.4 to increase the stiffness of the frame. As the cylinders are not connected in the same point, the sliding bearing will be subjected to vertical forces, which is transfered onto the rails. This makes the friction between the sliding element and the rail depend of the load condition. Furthermore the rails are exposed to bending stresses, why it must be ensured that the rails do not deform during operation. The piston rod ends may also be mounted in the same point on the slide, but the issue would persist, and the cylinder force components parallel to the rails would additionally become position dependent. Pros: Shorter and simpler frame compared to Mounting Topology 1. Spherical plain bearings ideally removes any radial forces in the cylinders. Cons: The slide is turned during operation, which increases the forces on the sliding bearings and rails, why the rail friction is also a function of the cylinder forces. A.2.3 Mounting Topology 3: Frontal Flange on Tube The simplest mounting type for the cylinder tube is the frontal flange, where a flange is welded directly on to the cylinder tube. The flange is then bolted to the frame as shown in Fig. A.5. Figure A.5: Frontal flange on cylinder tube which is bolted to the frame. Spherical eye on piston rod. Parallel misalignments may be compensated for during the assembly, by moving the cylinder vertically up/down and horizontally left/right within the clearance for the bolts. However, the cylinder cannot, be tilted to compensate for angular misalignments. The installation of the position transducer is simpler compared to Mounting Topology 1 and 2, as no enclosure must be made for the housing of the sensor electronics. Pros: Reduced length of the frame compared to Mounting Topology 1 and 2. The reduced distance between the mounting points of each cylinder increases the buckling strength, which allows for longer stroke lengths or increased push force. Some rod overhang may, however, be necessary, which in turn slightly reduces the buckling strength. Simple sensor mounting at the bottom end of the cylinders. Simple and inexpensive mounting system with no moving parts. Cons: Reduced possibility to properly align the cylinders to the rails. No possibility for the cylinder to compensate for angular misalignments. A.2.4 Mounting Topology 4: Trunnion on Tube with Slide Bearing A less constrained variant of Mounting Topology 3 is the trunnion mount design shown in Fig. A.6. With this topology, trunnion pins are welded onto the cylinder and mounted in eyes attached

123 Appendix A - Dimensioning and Mounting Design of Cylinders 113 to the frame. Figure A.6: Cylinder tube with trunnion mount and sliding bearings, spherical eye on piston rod. The trunnion pins are fitted with simple sliding bearing, which allows the cylinder tube to rotate around the trunnion pins to compensate for angular misalignments in that direction. Rotation in the horizontal plane is still not possible with this mounting type. Parallel misalignment can be adjusted by inserting shims between the eye plates connecting the trunnion to the frame and moving the eyes up/down within the clearance for the bolts. Pros: Reduced length of the frame compared with Mounting Topology 1 and 2. 1 additional degree of freedom compared with Mounting Topology 3 to compensate for angular misalignment. Simple and inexpensive sliding bearings for the trunnion pins. Reduced distance between cylinder mounting points increases the buckling resistance of the cylinder. Slight overhang may however be required. Easy/simple mounting of the position transducer. Cons: Not possible to account for angular misalignment in the horizontal plane. A.2.5 Mounting Topology 5: Trunnion with Axial Plain Bearing To both reduce the required frame size, improve the buckling resistance by reducing the distance between the cylinder mounting points, and maintain the flexibility of a spherical joint (i.e. Mounting Topology 1), two axial spherical plain bearings can be installed on the trunnion as shown in Fig. A.7. Due to the width of the trunnion mount, the corresponding ball diameter must be relative large to ensure that the center point is located on the center axis of the cylinder. A A-A A Figure A.7: Cylinder with trunnion and two axial plain bearing forming a ball joint on the cylinder tube. Free to rotate in all directions. The grey elements in Fig. A.7 are the ball elements of the axial spherical plain bearings. The spherical joint allows for rotation about all three axes on the cylinder tube, which minimizes the bending stress in the piston rod. Pros:

124 114 Control and Experimental Evaluation of SvSDP Concept Spherical joints in both mounting points of the cylinder allows for free rotation to compensate for misalignments. Trunnion mount on the cylinder tube allows for short distance between the cylinder mounting, which increases the buckling resitance of the cylinder. Easy/simple position transducer mounting. Cons: Necessity of large and expensive axial spherical bearings (Unit price for a single Schaeffler ELGES GE 1-AW is approximately 125e [kugellager-online GmbH & Co. KG, 215]). A.2.6 Mounting Topology 6: Trunnion with Gimbal Mounting An alternative to the spherical joint in Fig. A.7 is the gimbal joint shown in Fig. A.8. A A-A A Figure A.8: Cylinder with trunnion and gimbal mount with plain bearings. Gimbal design allows for up-down and left-right rotation. The two axial spherical plain bearings in Mounting Topology 5 may be replaced with four simple plain bearings as shown in Fig. A.8, at the expense of an additional intermediate component between the trunnion and the frame. The plain bearings on the trunnion allows for rotation around the horizontal axis, whereas the plain bearings on the intermediate element allows for rotation around the vertical axis. In order to mount the intermediate element on the cylinder, the element must be split into at least two different parts. Depending on where the element is split, alignment errors of the pins can be a problem depending on the assembly method. A design using guide pins with fine tolerances must thus be used to ensure that the pins are concentric within the required specifications. The stiffness of the intermediate element could also be a concern, which may increase its size. The gimbal mechanism should be able to handle the bending stresses caused by the cylinder forces. Pros: Rotational freedom similar to that of Mounting Topology 1, 2 and 5 is achieved. Frame size is similar to Mounting Topology 3, 4 and 5, without the use of expensive axial spherical plain bearings. Cons: The intermediate element must be split into at least two components, which introduces the possibility of alignment errors of the axes in the gimbal bracket. Increased mechanical complexity of the mounting system. Sufficient stiffness and strength of the gimbal elements must be ensured. A.2.7 Mounting Topology Selection Mounting Topology 4 is selected due to the following considerations: It is desirable to achieve as a long stroke length as possible, such that the system can achieve steady state operation for a period of time. This is limited however, by the space available in the laboratory where the test

125 Appendix A - Dimensioning and Mounting Design of Cylinders 115 bench is to be set up. Mounting Topology 1 is therefore disregarded as the overall length of the frame would be too long for the desired stroke length. The reduced buckling strength of Mounting Topology 1 is also a concern, which would limit the operational range of the cylinders as seen in Fig. A.2. Mounting Topology 3, although mechanically the simplest, is considered to be too sensitive to angular misalignments in both the horizontal and vertical plane. As a compromise, the cylinder with the trunnion pins on the tube is selected, as angular misalignments in the vertical plane is compensated for. This reduces the possible misalignment that is not compensated for to the horizontal plane. Due to the clearance in bolt holes, it is considered reasonable that any misalignment in this plane may be minimized through proper tolerance specification and during the assembly process. An investigation of the effect of a horizontal misalignment is still carried out in Section A.3, to prove that the mounting topology is suitable. A.3 Stresses due to Misalignment Since the cylinder tube is mounted on the frame using a trunnion mount, a misalignment between the piston rod eye and the slide may result in stresses in both the piston rod and the bearing strips within the cylinder. This section investigates both the effect of this phenomenon, and the corresponding limitations on the allowable misalignment. The situation is illustrated in Fig. A.9, where it is assumed that only the piston rod bends due to an misalignment. In reality both the cylinder tube and the frame will bend, but in this section only the worst case scenario is accounted for, which will result in the maximum stress on the bearing strip and the piston rod. It is assumed that the reaction forces acting on the bearing strips can be modeled as point loads, which means that the piston rod may bend within the cylinder gland. As the bearing strips are made from a PTFE fabric they can deform to allow for a slightly bended rod [Hallite, 215, p. 111], which justifies the assumption. Bearing strip Bearing strip Piston rod Eye Piston ɛ Cylinder tube v Cylinder gland v(ɛ) v(l + a) = δ c V A F L L V B a Figure A.9: Rod eye deflected by δ c piston rod. due to misalignment, which causes bending of the For a given piston position x, v(ɛ) describes the deflection of the piston rod at the distance ɛ from the center of the piston bearing strip. The moment and force equilibrium of Fig. A.9 yields the reaction forces V A and V B, and the moment M B in point B given in Eq. A.4. V A (x) = a(x) ( L(x) F V L B (x) = 1 + a(x) ) L(x) F L M B = 1 a(x) F L (A.4) Where the distances between the center of the interactions points as functions of the piston position x are given in Eq. A.5. L(x) = L x x min a(x) = a + x x min (A.5)

126 116 Control and Experimental Evaluation of SvSDP Concept The area moment of inertia and cross sectional area of the piston rod are given in Eq. A.6, [Gere and Goodno, 29, p. 397]. I = π ( d 4 o d 4 ) i 64 A rod = π ( d 2 o d 2 ) i 4 (A.6) Lastly the deflection at the tip of the piston rod δ c is translated into an equivalent shear force F L, as given by Eq. A.7, [Krex, 24, p. 337]. F L = 3 E I δ c (a(x)) 2 (L(x) + a(x)) (A.7) The allowed misalignment is determined from three constraints: The possible deflection angle of the rod within the piston and cylinder gland. The allowed compression stresses within the bearing strips of the piston and cylinder gland. The allowed stresses within the rod. The dimensions for this analysis are given in Tab. A.2. Description Value Unit Source a Minimum distance center gland bearing strip to center eye 216 mm x = x min = 35 mm d i Inner diameter piston rod 14 mm d o Outer diameter piston rod 35 mm d Abs Inner diameter piston bearings strip 58 mm d Bbs Outer diameter gland bearings strip 39 mm L Distance center to center of bearing strips at x = x min = 35 mm 853 mm L θ Distance center gland bearing strip to edge in gland 24.5 mm r θ Radial distance piston rod to gland.2 mm w Abs Width piston bearing strip 9.7 mm w Bbs Width gland bearing strip 15 mm σ bs Yield strength bearing strip at 8 C 58 MPa [Hallite, 215, p. 111] σ pr Yield strength piston rod 2MnV6 45 MPa [Heléns Rör AB, 215,p75] A.3.1 Table A.2: Parameters associated with the misalignment analysis. Rod deflection angle Only the cylinder gland (point B in Fig. A.9) is investigated with respect to deflection angle, as the deflection rate is twice as high at the cylinder gland compared to within the piston [Krex, 24, p. 337]. The deflection angle θ B at point B is given by Eq. A.8 [Krex, 24, p. 337]. ( ) θ B (x) = tan 1 FL a(x) L(x) 3 E I ( ) θ B-max = tan 1 rθ =.46 (A.8) L θ Assuming the deflection angle to be constant within the cylinder gland, the maximum allowed deflection angle θ B-max is determined from the geometry of the cylinder gland as sketched in Fig. A.1.

127 Appendix A - Dimensioning and Mounting Design of Cylinders 117 w Bbs d Bbs θ B-max L θ (a) Deflection θ B = r θ (b) Deflection θ B = θ B-max Contact point Figure A.1: Piston rod inside cylinder gland. Steel against steel contact at θ max. The corresponding deflection angle as a function of both the piston position x and the deflection δ c of the rod tip is evaluated based on Eq. A.8, and the result is shown in Fig. A.11. The deflection angle resulting in steel against steel contact in the cylinder gland is marked with the θ B-max line in Fig. A.11. θ B = θ B max Piston Position [mm] Rod Deflection [mm] Deflection angle [ ] Figure A.11: Deflection angle at point B From Fig. A.11 it can be concluded that to avoid steel against steel contact in the cylinder gland, the misalignment of the eye when the cylinder is fully retracted can at most be approximately 2 mm. When the piston is extracted the allowable misalignment increases. A 2 mm misalignment is viewed as a very large misalignment when working with machined steel parts. Compared with the distance from the tube mounting mount to eye mounting point of 285 mm, a misalignment of 2 mm corresponds to a very coarse tolerance according to ISO A.3.2 Compression stress in bearing strips As a worst case scenario, the compressive yield strength of the bearing strips is evaluated at the elevated temperature of 8 C, which is well above the normal operating temperature of 4 C. It is assumed that half of each bearing strip is carrying the reaction forces in the points A and B, as the bearing strip material is flexible and thus allows for uniform distribution of the load across the full surface. The compressions stresses σ Abs and σ Bbs are given in Eq. A.9. σ Abs = 2 V A (x) w Abs d Abs π σ Bbs = 2 V B (x) w Bbs d Bbs π (A.9) The compressive stresses in the bearing strips due to the shear reaction forces acting on the piston and cylinder gland are shown in Fig. A.12 as a function of piston position x and rod tip deflection

128 118 Control and Experimental Evaluation of SvSDP Concept δ c. It is here assumed that the stresses are uniformly distributed across half the cylindrical surface area of the bearing strips. Piston Position [mm] Stress [MPa] Piston Position [mm] Stress [MPa] Rod Deflection [mm] Rod Deflection [mm] (a) Piston bearing strip (point A) (b) Gland bearing strip (point B) Figure A.12: Compressive stresses in bearing strips of piston and gland due to shear reaction forces. The compressive stresses acting of the bearing strips are well within the acceptable level, with corresponding safety factors above 9 for the full mapping shown in Fig. A.12. The bearing strip dimensions are therefore adequate and are not the limiting factor with respect to possible misalignment of the piston rod. A.3.3 Combined stresses in piston rod The final check concerns the bending stresses introduced in the piston rod due to the possible misalignment. The largest bending moment will occur at point B, [Krex, 24, p. 337], why the stress analysis is carried out at this point. The shear stress τ VB and the bending stress at the outer surface σ MB of the piston rod at point B are given by Eq. A.1, [Gere and Goodno, 29, p. 35, 364]. τ VB (x) = V B (x) A rod σ MB (x) = M do (x) B 2 I (A.1) The Von Mises stress in the piston rod is given in Eq. A.11 based on the shear and bending stresses from Eq. A.1. σ (x) = (σ MB (x)) (τ VB (x)) 2 (A.11) The Von Mises stress in the rod from Eq. A.11 is shown in Fig. A.13. Please note that the rod is considered unloaded in the axial direction, why the σ max limit does not corresponds to the limit when the cylinder in use.

129 Appendix A - Dimensioning and Mounting Design of Cylinders 119 σ = σ max Piston Position [mm] Stress [MPa] Rod Deflection [mm] Figure A.13: Von Mises stress in rod. Indication of safety factor 3 against the yield point σ max = σpr = 15 MPa. Cylinder unloaded in the axial direction. SF The yield limit (incl. desired safety factor) is reached with a misalignment of approximately 3 mm, when the cylinder is fully retracted. By comparison of the possible rod deflection angle in Fig. A.11 and the yeild limit in Fig. A.13 it can be concluded that steel against steel contact in the cylinder gland will occur before the yield limit of the piston rod is reached. The overall conclusion is therefore, that misalignment of up to 2 mm, when the cylinder is fully retracted, will not have critical effect on the static operation of the test bench. Fine tolerances based on ISO are used in the tolerance chain for the dimensions on the parts which accounts for the horizontal alignment of the cylinders to the rails. The screw clearances in the rails allow for some additional adjustment, to minimize any misalignment misalignment. The overall test bench design is therefore deemed feasible.

130

131 Appendix B - Strength Analysis of Mechanical Components 121 B Strength Analysis of Mechanical Components This appendix contains detailed strength analysis and calculations for the mechanical components of the test bench. Each section is referenced to individually in the main report, particularly in Chapter 4. Furthermore, each section is, to a large extent, self-contained and with a minimal number of references to other sections. B.1 Strength of Bolted Friction Joints The different frame parts are joined through bolted joints to allow for adjustment of the alignment of the cylinders, as well as to allow for assembly without the necessity of welding equipment and trained personnel. To avoid the use of dowel pins to support the shear loads, the joints are dimensioned such that the friction force between the joint surfaces is sufficient by itself. The evaluation of the friction joints is based on [Krex, 24, p ] and [DS, 21]. In order to determine the forces acting on the friction joint, the frame type has been compared to case studies found in literature [Krex, 24] as shown in Fig. B.1. Upper square tube B U-beam Cylinder A Foot Plate Lower square tube (a) Side view of left half frame P b a B A I 2 h I 1 I 1 l D C (b) Frame approximation P B A (c) Single beam approximation Figure B.1: Comparison of frame types found in literature [Krex, 24,p. 346, 371] The actual frame structure is very similar to the approximation shown in Fig. B.1b, where the assumptions are respectively neglectable influence of the feet, and that the middle of the upper and lower square tubes is fixed. Due to the similarity between the frame construction in Fig. B.1a and Fig. B.1b, it is initially investigated whether the governing equations for Fig. B.1b gives an accurate description of particularly the bending moments acting in the two joints "A" and "B". Based on [Krex, 24,p. 346, 371], the joint moments are described by Eq. B.1, where P is considered as the maximum permissible cylinder force of 51 kn. M A = P a b l ( 1 k β α ) 12 k + 2 M B = P a b l b a ( 1 k + 2 β α ) 12 k + 2 l (B.1) The relation coefficient k is defined in Eq. B.2 to be a function of the stiffness of the two different members and the geometry. k = I 2 h I 1 l α = a l β = b l (B.2) Mounted on the lower square tube in the neighborhood of point "A" are one of the feet and the plate on which the rails are mounted. Both of these two elements contributes to the stiffness of

132 122 Control and Experimental Evaluation of SvSDP Concept the lower square tube, why it cannot be described by the stiffness of the tube only. For increasing stiffness of the lower square tube, the relation coefficient k as I 1, which in turn makes the moments M A and M B go toward the maximum values. This means that the moments acting on the joints tends to increase for increased stiffness of the joint. For the extreme case where the relation coefficient becomes k =, the governing equations for Fig. B.1b reduces to that of Fig. B.1c. The equations, which are given in Eq. B.3 and Eq. B.4, describes the reaction forces R A and R B and the moments M A and M B acting in the joints of a single beam, which is fixed at both ends and subjected to a external load P. R A = P b2 l 2 ( a l ) = 41.4kN R B = P b2 l 2 ( b l ) = 8.6kN (B.3) M A = P a b2 l 2 = 4.93 knm M B = P a2 b l 2 = 1.78 knm (B.4) As seen from Eq. B.3 and Eq. B.4, the load in joint "A" is significantly larger than that in joint "B", why joint "A" is chosen as the basis for dimensioning. For simplicity, the dimensions for joint "B" are however chosen to be the same as those of joint "A". The forces acting between the friction surfaces due to the joint forces and moments of Eq. B.3 and Eq. B.4 depend on the location of the bolts. The hole layout of the joint shown in Fig. B.2b must comply with the requirement for hole layouts for friction joints as defined in [DS, 21,p. 23]. These requirements depend on the thickness of the joint elements as given by Eq. B.5 and Eq. B.6, which are fulfilled with the utilized dimensions (see Tab. B.1 for numerical values). 1.2 d e 1 4 t + 4 mm 1.2 d e 2 4 t + 4 mm (B.5) 2.2 d p 1 min(14 t, 2 mm) 1.2 d p 2 min(14 t, 2 mm) 2.4 d L (B.6) A conservative assumption is that only the friction surface between the two elements directly underneath the bolt washers carries the entire load. The friction surface A fric is shown in Fig. B.2a. In reality some of the pretension will propagate to a larger friction area giving rise to reduced stresses. The reaction force, R A, which is assumed to be distributed evenly between the five bolts, gives the per-bolt-forces F V. The per-bolt-forces F M due to bending moment in the joint are considered to be distributed between the friction areas of each of the four corner bolts based on the distance L from the center bolt as shown in Fig. B.2b [Norton, 214,p. 942]. The center bolt is a tight-fitting bolt, which is not pretensioned to the same extent as the corner bolts, why it is not assumed to carry any of the bending moment load. As seen from Fig. B.2b, the resulting per-bolt-force F R is thus different for each bolt, where the largest is used for dimensioning. d wo Friction area d d wi d e 2 p 2 p 2 e 2 e 1 L p 1 F V F M F R A fric (a) Illustration of friction area A fric. (b) Friction joint exposed to both reaction forces F V and moment forces F M giving the resultant forces F R. Figure B.2: Friction joint of the frame.

133 Appendix B - Strength Analysis of Mechanical Components 123 Description Value Unit Source A b Tensile stress area (M16) 156 mm 2 ISO 898 part 1 A net Load carrying cross-sectional area of the square tube (16 mm) 2125 mm 2 d Nominal bolt diameter (M16) 16 mm d b,2 Basic pitch diameter (M16) 14.7 mm d Hole diameter 17.5 mm [DS, 211,p.42] d km Friction diameter mm d wi Washer inner diameter (M16) 17 mm ISO 789 / DIN 125 A d wo Washer outer diameter (M16) 3 mm ISO 789 / DIN 125 A e 1 Distance hole to end 5 mm e 2 Distance hole to side 4 mm k s Hole factor - normal hole 1. - [DS, 21,p. 3] L Distance diagonal hole-hole 64 mm n s Number of friction surfaces per joint 1 - n b Number of bolts per joint 1 - p b Thread pitch (M16) 2 mm p 1 Distance hole-hole along profile 1 mm p 2 Distance hole-hole across profile 4 mm t u Thickness U-beam (UPN 2) 8.5 mm DIN 126-1: 2 t s Thickness square tube (16 mm) 1 mm γ M Partial factor for resistance of [Erhvervs- og cross-sections, arbitrary class Byggestyrelsen, 21,p. 4] γ M2 Partial factor for resistance of bolts [Erhvervs- og Byggestyrelsen, 21,p. 4] γ M3 Partial factor for slip resistance at ultimate limit state [DS, 21,p. 18] µ g Friction coefficient bolt thread.5 - [Dow Corning Corporation, 22, p. 1] µ h Friction coefficient bolt head.1 - [Dow Corning Corporation, 22, p. 1] Friction coeff. surface "Steel cleaned µ s with steel brush, loose rust removed".3 - [DS, 211,Class C, p. 64] σ ub Ultimate strength bolt (mat. 1.9) 1 MPa [DS, 21,p. 2] σ uf Ultimate strength frame (S355J2) 47 MPa [DS, 24,p. 24] σ yb Yield strength bolt (mat. 1.9) 9 MPa [DS, 21,p. 2] σ yf Yield strength frame (S355J2) 355 MPa [DS, 24,p. 24] Table B.1: Dimensions and parameters for the friction joints. To ensure that no sliding occurs between the friction areas, the design force is chosen to be the maximum resultant per-bolt-force F R shown in Fig. B.2b. Based on the geometry, this maximum resultant force corresponds to Eq. B.8. F V = R A n b = 4.14 kn F M = F R,max = M A (n b 2) L = 9.64 kn θ = arctan ( p1 2 p 2 ) =.675 rad (B.7) (F V + cos(θ) F M ) 2 + (sin(θ) F M ) 2 = 13.1 kn (B.8) The bolt dimension, material property and pretension must be selected, such that acceptable safety margins can be obtained for the friction joint. The minimum conditions a friction joint

134 124 Control and Experimental Evaluation of SvSDP Concept must obey are summarized in Eq. B.9 [DS, 21,p. 22], with F s,rd the design slip resistance at the ultimate limit, F b,rd the design bearing resistance, and N net,rd the design plastic resistance to normal forces of the net cross-sectional area. F R,max F s,rd F R,max F b,rd F R,max N net,rd (B.9) In other words the three conditions relates to the shear strength of the friction joint, the strength of the "eye" material at the bolt contact point, and the strength of the cross-section of the profile carrying the load. Design Slip Resistance The design slip resistance F s,rd is given in Eq. B.1 [DS, 21,p. 3], with F p,c being the bolt pretension force. It is common practice to pretension the bolt up to 7 % of the ultimate strength of the bolt. As stated in [DS, 21,p. 2] only bolts of material type 8.8 or 1.9 should be used in applications where pretension is required. The actually utilized bolts are of material type 12.9, however they are pretensioned as if 1.9 bolts were used instead, to ensure that no yield limitations are exceeded. As an additional check, it is verified that the frame material is within the yield limit, when the pretension F p,c is applied to the bolts. No reference has been found in the literature stating whether such a check is required, why the check is carried out to ensure that the frame material does not undergo plastic deformation in the contact point. F p,c =.7 σ ub A b = 19.7 kn F s,rd = k s n µ s γ M3 F p,c = 26.3 kn (B.1) SF additional = F s,rd F R,max = 2. (B.11) As seen in Eq. B.1 the first condition in Eq. B.9 is satisfied with an additional safety margin given by Eq. B.11. The compression stress under the washer and corresponding safety factor is given in Eq. B.12. A fric = ( d 2 wo d 2 ) π 4 = 466 mm 2 σ comp = F p,c A fric = 235 MPa SF yield = σ yf σ comp = 1.5 (B.12) Based on Eq. B.12 it can be seen that there is a safety margin against plastic deformation directly underneath the washer. As no guidelines has been found in the standard, it is chosen to compare SF comp with γ M2 = 1.35, which describes "The resistance of plates in bearing". As SF comp > γ M2 it is concluded that acceptable safety against plastic deformation is found in the friction joint design. It is normal practice to tighten the bolt and nut using a torque wrench, why the pretension force is translated to an assembly torque using Eq. B.13 derived in [Mortensen, 213,p. 18]. [Mortensen, 213,p. 18] describes a special case for metric screws with standard rough thread, where the formula originates from the procedure described in [VDI, 23]. It is assumed that the thread and the bolt head are lubricated by the product specified in Tab. B.1 prior to the assembly. Lubrication is carried out both to reduce the assembly torque and to minimize the spread of the actual pretension force in the bolts due to uncertainty in the friction coefficient of steel against steel. ( M assembly = F p,c.16 p b +.58 d b,2 µ g + d ) km 2 µ h = 195 Nm (B.13)

135 Appendix B - Strength Analysis of Mechanical Components 125 Design Bearing Resistance The design bearing resistance per bolt is investigated in Eq. B.14 based on the procedure in [DS, 21,p 27]. F b,rd = k 1 α b σ uf d t γ M2 = 67.6 kn SF additional = F b,rd F R,max = 5.15 (B.14) With the two coefficients k 1 and α b respectively described by the relations between the geometry and the ultimate strength given in Eq. B.15 and Eq. B.16. ([ k 1 = min α b = min 2.8 e 2 ([ e1 ] 1.7 d ] 3 d,, [ 1.4 p 2 [ p1 3 d 1 4 ] 1.7 d ] ], [ σub σ uf ), [2.5] ), [1.] = 1.5 (B.15) =.95 (B.16) As seen in Eq. B.14 the second condition in Eq. B.9 is fulfilled with an acceptable additional safety factor. Design Plastic Resistance The plastic resistance to normal forces is shown in Eq. B.17 [DS, 27,p. 49]. N net,rd = A net σ yf γ M = 8.7 kn SF additional = N net,rd F R,max = 6.15 (B.17) With A net being the load carrying cross-sectional area taking in an arbitrary zig-zag curve across the profile. As a conservative assumption, it is assumed that only two of the four sides of the square tube profile are load carrying, which gives the area A net seen in Tab. B.1. As seen in Eq. B.17 the third condition in Eq. B.9 is also fulfilled with acceptable additional safety factor. This concludes the strength analysis of the friction joints. B.2 Stresses in the Frame Profile The static strength and the fatigue strength of the U-beam is investigated in this section. Due to the short distance between the eye mounting and the lower square tube it is deemed necessary to conduct a finite element (FEM) analysis of the stresses in the beam, as opposed to analytical evaluation. The frame is two times reflectional symmetric, why only a quarter of the frame is meshed in SolidWorks Simulation as seen in Fig. B.3. Fine grid is used near the bolt holes to more accurately simulate the stress distribution. The frame is fixed at the midpoint of the square tubes (yellow arrows) and mirrored at the green arrows. The cylinder force is applied in the eye for the trunnion shown with purple arrows. When the cylinder is pulling (the frame sections are "pulled together"), the hole in the eye for the trunnion is not allowed to rotate due to the stiffness of the trunnion bracket. The bolts are pretensioned directly with the pretension force given by Eq. B.13. The friction coefficient between the steel profiles is set to the value µ s as defined in Tab. B.1. The mounting holes for the plate for rails in the lower square tube are omitted in the simulation to reduce the simulation time. The lower square tube is only very lightly stressed, 1 MPa, why this simplification should not give rise to a faulty conclusion regarding the strength of the frame.

136 126 Control and Experimental Evaluation of SvSDP Concept Figure B.3: Quarter of the frame. Mesh with nodes and elements. Frame fixed at the midpoints of the two square tubes (yellow arrows). Symmetry constraints (green arrows) along the splitted upper and lower square beam. Force applied distributed in the trunnion eye hole (purple arrows). The effect of both a pushing and a pulling load is simulated for the frame construction as shown in Fig. B.4a and Fig. B.4b respectively. Due to the relative low stresses in the U-beam compared with the stress caused by the pretensioned bolts, the scale is limited to 4 MPa, to improve the visual resolution. It should be noted that the pretensioned area directly underneath the washers do not experience changes in the compression stress, why fatigue failure is not an issue at these points. von Mises [MPa] (a) The cylinder is pushing with F max = 5 kn - the frame segments are "pushed apart". (b) The cylinder is pulling with F max = 5 kn - the frame segments are "pulled together". Figure B.4: Von Mises stress in MPa in the U-beam. Stress range is limited to 4 MPa to show the stress concentration in the U-beam. The pretension stress under the washers are above the shown scale.