Local and Global Stability and Dynamics of a Class of Nonlinear Time-Delayed One-Degree-of-Freedom Systems

Transcription

1 Local and Global Stability and Dynamics of a Class of Nonlinear Time-Delayed One-Degree-of-Freedom Systems Nader A. Nayfeh Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering William T. Baumann, Chair Daniel J. Stilwell Chris L. Wyatt Peter Athanas Saad A. Ragab July 17, 2006 Blacksburg, Virginia Keywords: Container cranes, delayed feedback control, lathe cutting tools, hopf bifurcation Copyright 2006, Nader A. Nayfeh

2 Local and Global Stability and Dynamics of a Class of Nonlinear Time-Delayed One-Degree-of-Freedom Systems Nader A. Nayfeh ABSTRACT We investigate the dynamics and stability of nonlinear time-delayed one-degree-of-freedom systems possessing quadratic and cubic nonlinearities and subjected to external and parametric disturbances. Due to the time-delay terms, the trivial solution of the unforced system undergoes Hopf bifurcations. We use the method of multiple scales to determine the normal forms of the Hopf bifurcations and hence determine whether they are locally supercritical or subcritical. Then, we use a combination of a path following scheme, the normal forms, and the method of harmonic balance to calculate and trace small- and large-amplitude limit cycles and use Floquet theory to ascertain their stability and hence generate global bifurcation diagrams. We validate these diagrams using numerical simulations. We apply the results to two important physical problems: machine-tool chatter in lathes and control of the sway of container cranes using time-delayed position feedback. We find that the Hopf bifurcations in machine tools are globally subcritical even when they are locally supercritical. We find multiple large-amplitude solutions coexisting with the linearly stable trivial solution. Consequently, there are three operating regions for machine tools: an unconditionally stable region, an unconditionally unstable region, and a conditionally stable region. In the latter region, the multiple responses lead to hysteresis. Then, we investigate the use of bifurcation control to transform the subcritical bifurcations into supercritical ones. We find that cubic-velocity feedback with appropriate gains can shrink or even eliminate the conditionally stable region. Then, we find that time-delayed acceleration feedback with an appropriate gain can completely eliminate the linear instability region. In contrast, we find that the Hopf bifurcations in controlled cranes are locally and globally supercritical. Finally, we investigate the effectiveness of time-delayed position feedback in rejecting external and parametric disturbances in ship-mounted cranes.

3 Dedication To my parents and wife. iii

4 Contents 1 Introduction Motivation Objective Contributions Literature Review Pendulation Control Open-Loop Techniques Closed-Loop Techniques Machine-Tool Chatter in Lathes Modeling Modeling of Container Cranes Fully Nonlinear Model Time-delayed Position Feedback Controller Linearized Model iv

5 3.1.4 Cubic Model Lathe-Tool Model Comparison of three feedback controllers Controller design Time-delayed position feedback Classical controller Linear quadratic regulator (LQR) Numerical simulations Conclusions Implementation on a container crane System overview Sway sensors Trolley position Operator cabin Results Analysis of the General Equation Linear Analysis Nonlinear Analysis Analysis of the cutting tool on a lathe Linear Analysis v

6 7.2 Nonlinear Analysis Global Analysis Control Bifurcation control Time-delayed control Conclusions Nonlinear analysis of the crane controller Local Analysis Numerical results Global Analysis Disturbance rejection Conclusions Conclusions Conclusions and Recommendation for Future Work Recommendation for Future Work Bibliography 148 vi

7 List of Figures 1.1 Diagrams of hopf bifurcations in a two dimensional representation: (a) supercritical and (b) subcritical An overhead container crane Qualitative features of experimental data shown in Hanna and Tobias, A schematic model of a container crane A simplified schematic model of a container crane One-degree-of-freedom model of a cutting tool Solutions of (4.4) and (4.5): (a) variation of σ with k when τ = 0.3 and (b) variation of ω with k when τ = Solutions of (4.4) and (4.5): (a) variation of σ with τ when k = 0.4 and (b) variation of ω with τ when k = Variation of the dominant damping or growth rate σ with k and τ: (a) variation of the dominant damping or growth rate with k for several values of τ and (b) variation of the dominant damping or growth rate with τ for several values of k vii

8 4.4 A contour plot of the damping as a function of the controller gain k and delay τ where T is the natural period of the uncontrolled system (Henry et al., 2001) A block diagram of the time-delayed feedback controller A block diagram of the classical feedback controller A block diagram of the LQR feedback controller Acceleration, velocity, and position profiles for the horizontal and vertical maneuvers of the trolley and payload Payload sway for the no hoist simulation Trolley accelerations for the no hoist simulations Payload sway for the hoist simulations Trolley accelerations for the hoist simulations Jeddah Shanghai Zenhua Port Machinery (ZPMC) crane number 25 preparing to begin transfer operations over a panamax class container ship An overview of the system architecture Complete sway sensor design Isometric view of the bottom part of the sway sensor Top view of the bottom part of the sway sensor Isometric view of the top part of the sway sensor Both sway sensors mounted on the underside of the trolley Complete sway sensor attached to the underside of the trolley Bottom assembly of the sway sensor, demonstrating how the cable is lined up Trolley encoder viii

9 5.11 Trolley proximity calibration switch Trolley proximity calibration trigger Operator cabin with equipment installed Bifurcation diagram obtained for τ = 1/75 and varying values of w Time trace and PSD of the tool response corresponding to point A from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point B from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point C from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point D from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point E from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point F from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point G from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point H from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point I from Fig ix

10 7.11 Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point J from Fig Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point K from Fig Bifurcation diagram obtained for τ = 1/60 and varying values of w Bifurcation diagram obtained for τ = 1/75 and varying values of w, with a bifurcation control gain κ = Bifurcation diagram obtained for τ = 1/75 and varying values of w, with a bifurcation control gain κ = Bifurcation diagram obtained for τ = 1/75 and varying values of w, with a bifurcation control gain κ = Cutting tool model Time-delay control of cutting tools on lathes Stability boundary Variation of χ 1 and χ 3 with τ f, the fraction of time delay Variation of χ 1 /χ 3 with τ f Bifurcation diagram and representative time traces and phase portraits obtained for τ = 0.2 and varying values of k PSD of the response at k = 1.0, which is beyond the secondary Hopf bifurcation Bifurcation diagram and representative time traces and phase portraits obtained for τ = 0.3 and varying values of k PSD of the response obtained at k = 0.9, which is beyond the secondary Hopf bifurcation x

11 8.8 Bifurcation diagram and representative time traces and phase portraits obtained for τ = 0.4 and varying values of k Bifurcation diagram and representative time traces and phase portraits obtained for k = 0.2 and varying values of τ Bifurcation diagram and representative time traces and phase portraits obtained for k = 0.4 and varying values of τ Bifurcation diagram and representative time traces and phase portraits obtained for k = 0.6 and varying values of τ Frequency-response curves obtained in the case of direct excitation when F = 0.1. The red curve is the uncontrolled response and the blue curve is the controlled response when k = 0.4 and τ = Frequency-response curves obtained in the case of parametric excitation when F = 0.2. The red curve is the uncontrolled response and the blue curve is the controlled response when k = 0.4 and τ = Time history response with parametric excitation at ω = 0.62 and controller parameters k = 0.4 and τ = The top curve is the uncontrolled response and the lower curve is the controlled response xi

12 Chapter 1 Introduction 1.1 Motivation Container cranes are used all over the world to facilitate the transfer of cargo containers from either ship-to-shore or shore-to-ship. Due to the massive size, speed, and power of the crane, it is very easy for an operator to induce large swinging motions, or payload oscillations. These payload oscillations cause a significant increase in overall transfer time, increasing the time that a ship is required to be in port and the cost of unloading it. In addition, operating a container crane requires highly experienced operators, which also increases the cost of unloading. Consequently, many researchers have proposed, tested, and implemented many different control systems and algorithms for container cranes in an attempt to reduce the overall transfer time. While most of the control systems show promise in simulation, they tend to not work as well in experiments or in practice. One particular problem is that such massive systems tend to exhibit time delays between operator or automatic controller commands and the execution of the command. Although these time delays are small, they make controller design and implementation very difficult in most circumstances. 1

13 CHAPTER 1. INTRODUCTION 2 Prior work by Henry et al. (2001), Masoud et al. (2000), Nayfeh (2002), Masoud et al. (2003), and Masoud and Nayfeh (2003) showed that for plants exhibiting pendulating motions, like container cranes, time-delay could be used as part of a feedback controller which can effectively reduce payload pendulations. In this way, the actual time delay of the crane system could be absorbed into the delay required by the controller, eliminating its deleterious affect on most control designs. As further proof of the performance of the time-delayed feedback control system, we implemented the system on a real crane used in a port in Jeddah, Saudia Arabia. The control system enabled the operators to double their typical transfer per hour. However, all of the results were based on simulation and experimental work. There was some doubt in the community about the behavior of the system in a global sense, because time delays typically introduce multiple large-amplitude oscillations, including limit cycles and chaos. These large-amplitude oscillations can cause dramatic and sometimes catastrophic system responses. Typically, the instabilities caused by time delays are caused by Hopf bifurcations. A Hopf bifurcation is a transition from an equilibrium solution to a limit-cycle (oscillatory) solution. Depending on the parameters of the system, the generated limit cycles maybe unstable or stable. There are two types of Hopf bifurcations: supercritical and subcritical. A supercritical bifurcation, Fig. 1.1(a), occurs when the system transitions from a stable equilibrium to an unstable equilibrium and two stable limit cycles. In this type of bifurcation, the limit cycles start small and are stable, hence it is easy to adjust the parameter causing the bifurcation, such as controller gain, and return to a stable equilibrium. A subcritical bifurcation, Fig. 1.1(b) occurs when a system transitions from a stable equilibrium with two coexisting unstable limit cycles to an unstable equilibrium with no limit cycles. If a subcritical bifurcation occurs, no amount of parameter adjusting will return the system to a stable equilibrium. We show, analytically and numerically, that the crane control system is globally robust to

14 CHAPTER 1. INTRODUCTION 3 (a) (b) Figure 1.1: Diagrams of hopf bifurcations in a two dimensional representation: (a) supercritical and (b) subcritical. changes in the system parameters, such as cable length or controller gain. Furthermore, we show that the crane control system has excellent disturbance rejection behavior to both direct and parametric excitations. As further proof of the performance of the time-delayed feedback control system, we implemented the system on a real crane used in a port in Jeddah, Saudia Arabia. The control system enabled the operators to double their typical transfer per hour. We then extended the analysis to a cutting tool on a lathe as it has a similar equation of motion to the container crane with time-delayed feedback. However, due to the parameters of the cutting tool on the lathe, it exhibits vastly different dynamics and is an excellent example of the negative effects of time delays. Say something about these dynamics, why they are bad and what you tried to do about them. The instabilities in a cutting tool on a lathe produce rough and inaccurate machined pieces. Typically, these instabilities occur when an operator attempts to cut a wide section of material in one pass. We proposed two different feedback control options to mitigate or eliminate these instabilities: nonlinear velocity feedback and a time-delay feedback system. The nonlinear velocity feedback system attempts to increase the damping in the dynamics

15 CHAPTER 1. INTRODUCTION 4 of the system, where a linear velocity feedback controller would cause the system to become even more unstable. The time-delayed feedback system attempts to counter the time delays inherent in the lathe dynamics, much like feedback linearization techniques for nonlinear systems. Numerically, both approaches appear to be effective. 1.2 Objective The objective of this work is to investigate the linear and nonlinear stability and dynamics of specific members of the following class of retarded one-degree-of-freedom systems with quadratic and cubic nonlinearities: ü + 2µ u + ω 2 u + δ 1 u 2 + δ 2 u 2 + δ 3 uü + α 1 u 3 + α 2 u u 2 + α 3 u 2 ü + κ 1 u(t τ) + κ 2 [u(t τ) u(t)] 2 + κ 3 [u(t τ) u(t)] 3 + γ 1 ü(t τ) + γ 2 u(t τ) u 2 (t τ) + γ 3 u 2 ü(t τ) + γ 4 u 2 (t τ)ü(t τ) = κ u 3 + f 1 (t) + uf 2 (t) (1.1) This system describes many physical systems, depending on the parameters δ i, κ i, α i, γ i, and f i (e.g., Hale, 1977; Stépán, 1989; Kuang, 1993). The term κ u 3 is used for bifurcation control. For example, we show in Section 3.1 that (1.1) models a cubic approximation of the sway of a container crane with time-delayed position feedback when the δ i, κ i, and f i are set equal to zero. We also show in Section 3.2 that it models the vibrations of a lathe cutting tool when the γ i and f i are set equal to zero. It models also the sway of a container on a ship-mounted crane in waves when the ship is rolling, pitching, and heaving and the f i are different from zero. In this work, we focus on chatter in a lathe cutting tool and control of the pendulations of cranes. Time-delay in a lathe cutting tool produces a phenomenon called chatter. For some machining operations, tool vibrations decay with time, the trivial solution is a stable equilibrium solution, and hence the machined surface is smooth. For other operating conditions, the

16 CHAPTER 1. INTRODUCTION 5 trivial solution is unstable and the tool undergoes large oscillations, which might be periodic, quasiperiodic, or chaotic, and hence the surface finish will not be smooth. There are three operating regions for the cutting tool on a lathe due to chatter: an unconditionally stable region, a conditionally stable region, and an unconditionally unstable region. Although there exists many theoretical studies of chatter, there are few systematic approaches for investigating local and global dynamics and stability of cutting tools on lathes and methods of controlling the instabilities. In this Dissertation, we follow a systematic approach using a combination of path following, the method of multiple scales, the method of harmonic balance, Floquet theory, and numerical simulations. We show that the system loses stability via a Hopf bifurcation. Next, we use the method of multiple scales to construct the normal form of the Hopf bifurcation and show that it is supercritical and hence there are only two operating regions: an unconditionally stable region and an unconditionally unstable region. We use a global analysis to show that this local result is erroneous. To this end, we use a combination of a path following scheme, the method of harmonic balance, Floquet theory, and numerical simulations to continue the branch of limit cycles generated by the Hopf bifurcation. We show that there are several other branches of large-amplitude solutions coexisting with the linearly stable trivial solution. Some of these solutions emanate from the larger linearly critical bifurcation values. Even the branch of small-amplitude limit cycles generated by the supercritical Hopf bifurcation may encounter a cyclic fold, which generates a branch of unstable large-amplitude solutions. This branch encounters yet another cyclic fold, generating stable limit cycles coexisting with the linearly stable trivial solution. We investigate whether cubic-velocity feedback can be used to eliminate the branches of large-amplitude solutions, which coexist with the linearly stable trivial solution, and hence can eliminate or shrink the conditionally stable region. Then, we turn our attention to investigating the possibility of widening the region of linearly stable trivial solutions using time-delay feedback. Pendulation control of cranes is an important field of study for two primary reasons: increased productivity and safety. We have shown in previous work that the time-delayed po-

17 CHAPTER 1. INTRODUCTION 6 sition feedback controller works extremely well numerically, experimentally, and physically on several classes of cranes. Unfortunately, a detailed theoretical analysis of the controller and its robustness has been lacking. Furthermore, due to time-delay being used as control, there was a concern that the crane control system can introduce complex dynamic behavior similar to that of the cutting tool on a lathe; that is, large-amplitude solutions coexisting with the linearly stable trivial solution. In this Dissertation, we undertake such an analysis. We start by comparing the time-delayed position controller with two feedback control strategies: a linear proportional-integrator (PI) controller and a linear quadratic regulator (LQR) controller. Using simulations, we show that, while all three controllers are effective in damping transient and residual pendulations, the time-delayed position feedback controller is faster than the other two. Furthermore, the LQR controller is very sensitive to the feedback gains, and small deviations may cause the system performance to degrade significantly. Then, we provide a detailed implementation of the time-delayed position feedback controller on a full-scale (real) 60-ton crane. Because time delays introduce infinite modes into control systems, we first examine how the additional roots (modes) introduced by the time delay affect the linearized system as the gain, time delay, and inherent damping vary. Second, we show that the system loses stability via a Hopf bifurcation. Third, we use the method of multiple scales to construct the normal form of the Hopf bifurcation and show that it is supercritical. Fourth, we use a combination of a path following scheme, the method of harmonic balance, Floquet theory, and numerical simulations to continue the branch of limit cycles generated by the Hopf bifurcation. We show that, in contrast with machine tools on a lathe, there are no large-amplitude solutions coexisting with the linearly stable trivial solution and hence there are only two regions of operation: an unconditionally stable region and an unconditionally unstable region. Furthermore, we investigate the frequency- and force-response curves of the nonlinear system to study robustness of the control system to direct disturbances like wind and operator inputs. Furthermore, we investigate the response of the nonlinear system to parametric disturbances.

18 CHAPTER 1. INTRODUCTION Contributions I. Modeling of Container Cranes A. We derived fully nonlinear equations governing the dynamics of container cranes by representing the system as a double pendulum and including time-delayed position feedback. B. We extracted an approximate polynomial model that includes up to cubic terms. The polynomial model is used to determine the normal form of the bifurcation and its type. II. Control of Machine-Tool Chatter in Lathes A. We investigated bifurcation control of chatter using cubic-velocity feedback. We found that using a proper gain shrinks and may even eliminate the conditionally stable region. B. We investigated chatter control using time-delay acceleration feedback. We found that time-delay feedback can widen the region of linearly stable trivial solution. III. Linear Container-Crane System A. We determined the stable and unstable regions of the crane controller in parameter space, as a function of the gain and delay, and hence the boundary separating them. B. Then, we used perturbation methods to derive closed-form expressions for the growth (damping) rate and frequency near the stability boundary, used these expressions to show that the system loses stability as a result of a pair of complexconjugate eigenvalues moving transversely from the left-half to the right-half of the complex plane, and hence proved that the trivial solution loses stability via a Hopf bifurcation.

19 CHAPTER 1. INTRODUCTION 8 C. As a baseline comparison of the performance of the time-delayed position feedback controller, we showed that it outperforms a classical linear controller and a linear quadratic regulator in suppressing pendulations in container cranes via simulations. D. We implemented the controller on a real operational container crane at the Jeddah Port, Saudi Arabia with very impressive results. IV. Nonlinear Analysis of Time-Delayed Feedback of Container-Crane System A. We used the method of multiple scales to determine the normal form of the Hopf bifurcation and used the normal form to show that the Hopf bifurcation is supercritical. B. Then, we used a combination of a path following scheme, the method of multiple scales, the method of harmonic balance, Floquet theory, and numerical simulations to continue the branch of stable limit cycles generated by the Hopf bifurcation. Using these results, we generated bifurcation diagrams and found that the Hopf bifurcation is locally as well as globally supercritical and hence the time-delayed feedback controller is robust. C. Further, we discovered that the post-bifurcation limit-cycle response encounters a secondary Hopf bifurcation, leading to the birth of modulated responses. D. We generated frequency-response curves to investigate the effectiveness of timedelay position feedback in rejecting external and parametric disturbances.

20 Chapter 2 Literature Review 2.1 Pendulation Control A crane consists of a hoisting mechanism that is attached to a support mechanism. Typically, the hoisting mechanism consists of one or more cables and a hooking mechanism. Typical support mechanisms are trolley-girders, trolley-jibs, and booms. The hoisting assembly is suspended from one or more points on the support mechanism. The support mechanism moves the suspension point or points around the crane work space. The hoisting mechanism lifts and lowers the payload to avoid obstacles in its path and deposits the payload at the target point. Cranes can be classified by the degrees of freedom the support mechanism offers the suspension point. A boom crane has the suspension point fixed at the end of a boom. It has two degrees of freedom consisting of rotations around two orthogonal axes located at the base of the boom. A gantry crane is composed of a trolley moving on a girder along a single axis. In some gantry cranes, the girder is mounted on a second set of orthogonal railings, adding another degree of freedom in the horizontal plane. This configuration allows two translational degrees 9

21 CHAPTER 2. LITERATURE REVIEW 10 Figure 2.1: An overhead container crane. of freedom in the horizontal plane. A rotary crane has a suspension point moving along a girder, or jib, which is rotating in the horizontal plane about a fixed vertical axis. Thus, it has two degrees of freedom in the horizontal plane, translation and rotation. Ship-mounted cranes are widely used to transfer cargo from large container ships to other ships, either for a tactical sea-basing operation or when deep-water ports are not available. The ships that are unloaded during these operations include commercial container ships, large, medium-speed, roll-on/roll-off (LMSR) ships, heavy-lift barge carriers, and deep-draft tankers. Typically, ship-mounted cranes are of the boom type. Ship-mounted cranes are sensitive to the six-degree-of-freedom (DOF) motions of the ship. These motions, or base-excitations, can cause very large swinging motions, or pendulations, in the payload. The frequency content of these motions may contain significant energy at/or near the natural frequency and/or twice the natural frequency of the hoisted payload, which could initiate an external and/or a parametric resonance. This sensitivity to the frequency content of the base excitation is present regardless of the crane type. Therefore, when a ship is being excited due to waves and wind, it can become very dangerous to operate the crane. Container cranes are large dockside cranes used in loading and unloading of large container ships, Fig A container ship can hold hundreds of containers. A standard 8 ft by 8 ft by 40 ft container can weigh up to 65 tons. Shipyards and dockside facilities are continuously

22 CHAPTER 2. LITERATURE REVIEW 11 searching for ways to improve the speed of unloading and loading of containers to reduce the operating costs and to improve the efficiency, but not at the expense of safety (Abdel- Rahman et al., 2003). A container crane consists of a trolley moving along a single axis, a spreader bar that has a locking mechanism for capturing the payload (the container), and a four-cable hoisting mechanism. The cables are configured in two parallel sets, two cables on each side of the spreader bar. Motion of the trolley can induce large inertial forces on the payload, which may result in large sway angles. The payload oscillations may result in damage to the crane structure, the payload, and even the surroundings. To mitigate these oscillations, crane operators often choose to slow down operations, resulting in a reduced efficiency and an increased operating cost. Average transfer operations are on the order of 30 containers per hour. In this section, we discuss crane control strategies available in the literature. Pendulation control for cranes can be divided into two primary classifications: open-loop and closed-loop techniques. For a detailed review of the modeling and control of cranes, we refer the reader to Abdel-Rahman et al. (2003). Many models are presented and several approaches to crane control are also examined, including modifications to crane structures and computerized control systems not modifying the structure. Next, we present an updated review of the control literature where no modifications are made to the crane structure Open-Loop Techniques Open-loop control techniques are very common. However, due to the lack of feedback, these strategies are almost all applied to fixed-crane structures. In other words, cranes without base excitations of either primary or parametric forces. Some of these strategies can be applied to bidirectional gantry cranes.

23 CHAPTER 2. LITERATURE REVIEW 12 Input Shaping A common control strategy today is based on an open-loop algorithm designed to shorten or automate cycle times for gantry cranes. Usually, this algorithm requires a predefined path. Input shaping is the most common open-loop control strategy. Input shaping is a technique primarily used to move a crane a set distance along a predefined path. There is a version, produced by Smartcrane, LLC, that is designed to achieve maximum acceleration and deceleration. The acceleration profile of the trolley is designed to induce minimum payload pendulation during both of the acceleration and deceleration stages of the maneuver, and to minimize the residual pendulations at the end point of the maneuver. The first to propose an input-shaping control strategy for payload pendulations were Alsop et al. (1965). Their controller accelerates the trolley in steps of constant acceleration. When the payload reaches zero pendulation angle, the acceleration stages are terminated, and the trolley travels at constant velocity. The same process is repeated for the deceleration stage. They use an iterative procedure to calculate the acceleration profile of the trolley, assuming two constant acceleration steps, and use a linear approximation of the cable-payload period. Their results show that, at the end of maneuvering, there are no residual pendulations, however during the motion the pendulation angle is on the order of 10. Carbon (1976) used one-step and two-step versions of the strategy of Alsop et al. in the design of a controller for commercial gantry cranes used in ship unloading. Alzinger and Brozovic (1983) used an input-shaping strategy similar to that of Carbon (1976) to demonstrate, via a numerical example, that a two-step acceleration profile produces faster travel time than a one-step acceleration profile. They used a two-step acceleration profile to design a control system for commercial gantry cranes used in ship unloading. Tests on an actual crane show that the two-step acceleration yields fast travel and small residual pendulations at the target point. However, the tests also show that there were significant pendulations, on the order of 5, due to deviations from the planned acceleration profile.

24 CHAPTER 2. LITERATURE REVIEW 13 Hazlerigg (1972) proposed a different input-shaping strategy using symmetric two-step acceleration/deceleration profiles to move the trolley to the target in a time period equal to the period of the payload. The sizes of the steps are determined based on the travel distance, the maximum available acceleration, and the period of the payload. Experiments show that this strategy can dampen payload pendulations. However, the performance of the strategy is highly sensitive to changes in the cable length. Kuntze and Strobel (1975) modified this strategy by introducing steps of zero-acceleration into the profile (i.e., constant travel speed intervals), thereby relaxing the constraint of optimal travel time to one period of the payload. Furthermore, this modification allows a constraint on the maximum travel velocity to accommodate the capabilities of the trolley motors. Simulations show that this strategy is very sensitive to disturbances and variations in the crane parameters. Yamada et al. (1983) proposed an input-shaping strategy to achieve the minimum transfer time with no residual pendulations by generating the acceleration profile using Pontryagin s maximum principle. The optimal profile was approximated with a suboptimal profile, using one or two steps of constant acceleration, to simplify the control effort. The suboptimal profile was then used to generate a look-up table containing the acceleration profiles for different payload initial positions and speeds. The strategy was verified experimentally on a scaled model of a gantry crane. The results show that residual payload pendulations wire less than 1.5. Takeuchi et al. (1988) proposed an input-shaping strategy for either a rotary or a boom crane. The strategy is designed to achieve time-optimal slew motion only while minimizing residual pendulations. This strategy uses an acceleration profile similar to that of Yamada et al. (1983), except that it is applied to the slew motion of a rotary or a boom crane instead of the trolley motion of a gantry crane. Simulations show that this strategy is effective in reducing out-of-plane pendulations but cannot suppress in-plane pendulations, which continue well after the slew maneuver is complete. Jones and Petterson (1988) extended the work of Alsop et al. (1965) by using a nonlinear

25 CHAPTER 2. LITERATURE REVIEW 14 approximation of the payload period. They used this approximation to generate an analytical expression for the duration of the constant velocity stage as a function of the amplitude and duration of the constant acceleration steps. A two-step acceleration profile was then generated from the analytical expression. Simulations using different profiles show that this strategy can reduce residual pendulations to between 0.1 to 0.3. However, the strategy is not able to reduce initial disturbances of the payload and, in some cases, may amplify them. Noakes et al. (1990) and Noakes and Jansen (1990, 1992) applied a one-step acceleration variation of this strategy to an actual crane. Their test results matched the numerical results of Jones and Petterson (1988). Karnopp et al. (1992) proposed another input-shaping strategy that is computed from the desired trolley travel and the pendulation natural frequency. The strategy gives a prescribed input position of the trolley to deliver the payload with no pendulations at the target. However, the minimum travel time has to be at least 1 n of the payload pendulation period, 2 where n = 3, 5, 7,. Using the minimum travel time results in very large intermediate pendulations, where the amplitude is the ratio of the travel distance divided to the cable length. Increasing the travel time reduces the pendulation, however that can introduce higher harmonics in the pendulations during the maneuver. Kress et al. (1994) showed analytically that input shaping is equivalent to a notch filter applied to a general input signal and centered around the natural frequency of the payload. Based on this analysis, they proposed applying a second-order notch filter, or robust notch filter, to the acceleration input command. Simulations and experimental results of an actual crane, moving at arbitrary step accelerations and slow cable-length changes, show that this strategy is effective in suppressing residual payload pendulations. Parker et al. (1995a,b) applied various optimization techniques to input shaping of the acceleration profile of the jib in order to eliminate residual pendulations for a jib maneuver along a predefined path. Their experiments showed that significant pendulations develop during the maneuver, reaching as much as 10 for the given maneuver.

26 CHAPTER 2. LITERATURE REVIEW 15 Parker et al. (1996) proposed another strategy using a quasi-static notch filter to avoid exciting the payload at its natural frequency due to operator commands. The frequency of the notch filter is a function of the natural frequency of the payload; that is, inversely proportional to the cable length. The notch filter roll-off coefficient is held constant for any cable length, and hence it is only optimal at a single cable length. Furthermore, the filter characteristics change as the cable length changes. Experiments show significant reductions in pendulations throughout the maneuver. However, the notch filter introduces a varying time delay, depending on the cable length, of up to 2.5 seconds between the operator input and the motor drive. Lewis et al. (1998) extended the control strategy proposed by Parker et al. (1996) to boom cranes. Simulations show reductions in in-plane and out-of-plane pendulations. Parker et al. (1999a,b) modified this strategy by changing the roll-off coefficient of the notch filter as a function of the natural frequency employed in the notch-filter design. Their experiments showed an 18 db reduction in the payload pendulations at the end of the prescribed maneuver. Lewis et al. (1999) performed simulations with this strategy. Their results show that, while this notch filter is more consistent and demonstrates a slight improvement in the pendulation reduction over the strategy of Parker et al. (1996), its design imposes more time delays and larger-amplitude changes in the operator input. As a correction, Lewis et al. (1999) modified the strategy by using a roll-off coefficient that is a function of the accelerations in the notch filter. Their simulation results did not show significant performance improvement compared to that of Parker et al. (1999a,b). Optimal Control Field (1961) was the first to propose an automatic crane-operation control strategy. He used an analog computer to simulate the dynamics of a crane. He produced optimum velocity profiles for the trolley and cable motions by trial-and-error. The motions minimized the travel time while avoiding obstacles. The control strategy did not attempt to reduce payload

27 CHAPTER 2. LITERATURE REVIEW 16 pendulations. Beeston (1969) used Pontryagin s maximum principle to generate time-optimal trolley acceleration profiles designed to minimize the hoisting and travel time for various target points. Bang-bang control of the trolley was used, with three switching points for each acceleration profile. Then, regression analysis was used to express each of the switching points in terms of the initial payload and trolley positions and velocities. There was no attempt at controlling payload pendulations in this strategy. Manson (1982) extended the work of Hazlerigg (1972) by relaxing the restrictions on the travel time. He used Pontryagin s maximum principle to generate a time-optimal acceleration profile. Three switching points on the acceleration profile and total travel time were evaluated as a function of the travel distance. The cable length was assumed to be constant. These optimal solutions are not practical in application and were intended to be used as a benchmark for the performance of other control strategies. Sakawa et al. (1981) proposed an optimal control strategy to generate the torque profile of a crane along a predefined path. The strategy minimizes the payload pendulations during the maneuver and at the target point. The technique was simulated on a model of a slewing boom crane, with constant luff angle, while hoisting the cable. Their results show that there are no residual pendulations, but there are pendulations along the path proportional to the slewing angle. Sakawa and Shindo (1982) applied the optimal control strategy of Sakawa et al. (1981) to a gantry crane model linearized around the payload equilibrium position. The predefined payload path was divided into three stages: hoisting up, trolley travel, and hoisting down. The control strategy was applied to each stage separately. Their simulations show significant pendulations, as much as 7 during the hoisting up and hoisting down stages. However, there were no residual pendulations at the target point. Kimiaghalam et al. (1998, 1999) used genetic algorithms to solve the optimal control formulation of Sakawa and Shindo (1982). Their simulations show similar results with a shorter

28 CHAPTER 2. LITERATURE REVIEW 17 travel time. However, their controller drives the crane at speeds higher than the given constraints. Auernig and Troger (1987) proposed a strategy using Pontryagin s maximum principle to minimize the transfer time for a gantry crane traveling and hoisting at constant speeds by producing acceleration profiles for the trolley and hoist. The system is subject to constraints on the maximum speeds of the trolley and hoist. Simulations of the optimal control profiles show that the paths generated by this technique are not always superior to the performance of manually operated cranes in use in European ports. Furthermore, while there are no residual pendulations, there is significant pendulation during the trolley travel. Hämäläinen et al. (1995) proposed a strategy where a predefined crane path was divided into five stages: reeling in-place, reeling and trolley acceleration, coasting at a constant travel velocity, unreeling and trolley deceleration, and unreeling in-place. The velocity profiles of the acceleration (second and fourth) stages were generated by minimizing the energy demand on the motors. The time required for these stages was determined by trial-anderror. Simulations and experiments on a scaled model showed no residual pendulations at the target, but pendulations of as much as 6 developed during the travel (second, third, and fourth) stages. They found that their strategy provides faster and smoother performance than that of a skilled operator on the same crane. Summary The open-loop techniques mentioned above are extremely sensitive to variations in parameter values of the plant, such as cable length and payload mass, and to changes in initial conditions (Zinober and Fuller, 1973; Virkkunen and Marttinen, 1988; Yoon et al., 1995). In fact, Singhose et al. (1997) showed that input-shaping techniques are particularly sensitive to the natural frequency of the payload oscillations. As a result, they suffer significant degradation in crane maneuvers that involve hoisting.

29 CHAPTER 2. LITERATURE REVIEW 18 Many of the above studies assume an undamped crane system. Due to the unaccounted Coulomb damping, in practice, the payload will not come instantaneously to a stop at the target position. This behavior actually produces residual pendulations. Furthermore, all of the above techniques, except for that of Hämäläinen et al. (1995), use bang-bang control profiles, which may apply excessive stresses to the crane motors. Moreover, the motors cannot accurately produce the bang-bang accelerations. Closed-loop control may be used to overcome many of the problems in the open-loop techniques at the expense of travel time. van de Ven (1983) showed that combining a closed-loop control technique with time-optimal control techniques can lead to the development of limit cycles. Furthermore, Zinober and Yang (1988) showed that the use of closed-loop control with either input shaping or optimal control requires a very detailed and accurate plant model and does not offer significant improvements over the open-loop techniques Closed-Loop Techniques While open-loop techniques are, by definition, designed to suppress pendulations due to inertia excitations, all available closed-loop techniques are by design restricted to counter inertia excitations only. In these control strategies, the control input is the force or torque applied to the trolley and girder motor (where available) in order to suppress pendulations due to the acceleration and deceleration of the trolley. Linear Control Hazlerigg (1972) was the first to propose a feedback control strategy. The control strategy employs a second-order lead compensator to dampen payload pendulations. Their experiments show that, while payload pendulations are reduced at the natural frequency, pendulations are introduced at higher frequencies.

30 CHAPTER 2. LITERATURE REVIEW 19 Ridout (1987, 1989) proposed a feedback controller that uses a combination of negative feedback of the trolley position and velocity and positive feedback of the pendulation angle at constant cable length. Experiments on a scaled model showed that residual pendulations were under 0.3. However, pendulations of as much as 10 developed during travel. Furthermore, the controller was found to be insensitive to external disturbances, changes in payload mass, and small changes in cable length. Moustafa and Ebeid (1988) proposed a pendulation suppression strategy for bidirectional cranes by controlling both directions. The strategy uses three feedback controllers for the different stages of motion: acceleration, coasting, and deceleration. The controllers are based on the linearized equilibrium position of the crane model at each stage. Their simulations show that this technique mitigates disturbances due to inertial motion. However, transient pendulations, as large as 20, and whirling motions, as large as 110, are found during the acceleration and deceleration stages. Hara et al. (1989) proposed an LQR state feedback controller for a telescoping boom crane. The strategy uses the telescoping motion to control planar payload pendulations induced by the telescoping motion. The controller is constrained by a saturation condition to keep it within available control authority. Simulations and testing on an actual crane showed that the controller was successful in suppressing pendulations. Nguyen et al. (1992) proposed a state feedback control strategy for a boom crane to hoist and stabilize the payload and boom position. Two independent controllers were used: one to control the boom luff angle and the pendulations and the other to control payload hoisting. Experiments on a scaled model show boom oscillations around the reference and steady-state errors in the luff angle and cable length. However, transient pendulations are kept below 4. Souissi and Koivo (1992) proposed a two-tier control strategy for a boom crane against inertia-induced pendulations. A proportional-integrator-derivative (PID) controller is used to track a reference trajectory, while a proportional derivative (PD) controller is used to mitigate payload pendulations. Simulations of the boom performing a predefined maneuver

31 CHAPTER 2. LITERATURE REVIEW 20 show significant pendulations, as much as 15. This indicates that the PD controller is ineffective. Moustafa and Emara-Shabaik (1992) proposed a PD controller using an augmented linearized model of the crane with a linear model of the trolley and girder motors, accounting for the motor dynamics. Simulations representing trolley travel only showed that the controller is effective in suppressing payload pendulations. Moustafa (1994) proposed a linear feedback controller using the trolley motor force. The controller was designed to suppress pendulations due to the trolley motion and cable hoisting. Simulations on a time-varying linear model of the crane show that the strategy is effective in suppressing payload pendulations. However, static errors in the trolley position are present. Nguyen (1994) proposed a state feedback control strategy with two independent controllers. The first controls the trolley position and payload pendulations. The second controls the payload hoisting position. Experiments on a scaled model showed good tracking of the trolley and hoist positions, no residual pendulations, and good rejection of external disturbances. However, transient pendulations of as much as 12 occurred. Yu et al. (1995) proposed a control strategy consisting of two independent PD controllers. They used the method of averaging to separate the slow and fast dynamics of a gantry crane model. One PD controller was applied as a slow-input control system to maintain tracking of a predefined motion profile (slow dynamics). The other PD controller was applied as a fast-input controller to mitigate payload pendulations (fast dynamics). Their simulations show that this strategy can smoothly move the trolley along a predefined trajectory with a maximum transient pendulation angle of 5. Lee et al. (1997) proposed a control strategy consisting of a proportional-integrator (PI) controller to track the trolley position and a PD controller to eliminate payload pendulations. The control strategy is similar to a notch filter centered around the payload natural frequency. Their experiments show transient pendulations of 3 and no residual pendulations. However, the experiments show that the added damping from the controller during travel is low,

32 CHAPTER 2. LITERATURE REVIEW 21 leading to poor disturbance rejection. Lee (1997) modified this strategy by compensating for the payload mass, cascading a PI trolley velocity controller with the PI trolley position controller, and cascading a lag compensator with the PD controller in order to increase the added damping. Experiments on a scaled model show that transient pendulations are less than 1 and there are no residual pendulations. This strategy is more effective in disturbance rejection and significantly increases damping. However, the results show that the PD controller is sensitive to cable-length changes and needs to be gain-scheduled to optimize its performance. Joshi and Rahn (1995), Martindale et al. (1995), and Rahn et al. (1999) proposed a linear parallel distributed compensation (PDC) feedback controller assuming that the cable is flexible and the payload mass is the same order of magnitude as that of the cable. They validated their controller experimentally. They reported robust response to wind loading and time-varying cable length. Omar and Nayfeh (2001) proposed a control strategy for a rotary crane using two independent full-state feedback controllers for the trolley travel and crane slew. Experiments and simulations showed that this strategy is effective in damping payload pendulations within one cycle of oscillation. However, the feedback gains have to be tuned to each specific payload mass and cable length. Changes or deviations in the parameters lead to marked degradation in the controller effectiveness. Yong-Seok et al. (2003, 2004) developed a state feedback control system for quay-side container cranes. To measure the cable angle, they mounted an inclinometer on the top surface of the spreader bar and used inverse kinematics to calculate the cable angle. The inclinometer results are in good agreement with vision-based systems. Experiments on a 1/4th scaled model crane show that the system is effective in eliminating residual pendulations, with transient pendulations being below 2. However, there is steady-state error in the trolley position.

33 CHAPTER 2. LITERATURE REVIEW 22 Adaptive Control Kuntze and Strobel (1975) proposed an input-shaping strategy using a linear crane model to predict the payload and trolley motions. Using the predicted motions, they modify the acceleration profile accordingly and absorbed the disturbances. Furthermore, the acceleration profile is updated during operation to account for parameter changes, such as cable length. Their simulations and experiments on a scaled model show that this strategy is effective in reducing travel time and eliminating residual pendulations. Ackermann (1980) proposed a linear state feedback controller with a robust gain-scheduling scheme as a backup controller to be activated in case of a sensor failure or a large change in a state. The scheme sets the feedback gains so that the system poles are restricted to a region of stability, instead of specific points. Performance is sacrificed during a system emergency to insure stability. Hurteau and Desantis (1983) proposed an adaptive linear state feedback controller to eliminate residual pendulations. The adaptive strategy uses a gain-scheduling module to tune pole placement due to changes in the cable length. Marttinen (1989), Salminen et al. (1990), and Virkkunen et al. (1990) proposed similar strategies with time-varying parameter strategy to aid in the adjustment for cable-length changes over time. Experiments of both strategies were performed on a scaled model. The results show that residual pendulations persist, pendulations of as much as 10 develop during travel, and the time-varying strategy has a steady-state error in the trolley position. Also, the results show that the strategy of Hurteau and Desantis (1983) is insensitive to measurement errors. Corriga et al. (1998) proposed an LQR controller for a linear time-varying model of a crane with constant velocity hoist. This implicit gain-scheduling procedure produced a gain vector that was a function of the time-varying length of the cable. Their simulations show that this strategy is effective in rejecting initial disturbances. However, it is excessively slow in approaching the target point and develops a steady-state error in the trolley position.

34 CHAPTER 2. LITERATURE REVIEW 23 d Andrea Novel and Boustany (1991) and Boustany and d Andrea Novel (1992) extended the nonlinear controller proposed by d Andrea Novel and Lévine (1989) by using an adaptive controller for a wider range of payload masses. However, this strategy was shown to be only locally stable. Butler et al. (1991) proposed a combined control strategy consisting of a classical feedback controller (primary) and an adaptive controller (secondary) to account for unmodeled dynamics in the linear model used for designing the classical feedback controller. The adaptive controller is based on an unmodeled dynamics transfer function to minimize plant-model error. Their experiments on a scaled model show significant reduction in residual pendulations after the trolley traveled along a predefined path. Nguyen and Laman (1995) proposed a control strategy with three independent H controllers: trolley position, hoist position, and payload pendulation. Experiments using a scaled model showed that the combined strategy produces a small steady-state error in the trolley position and good rejection of external disturbances. However, the performance degrades as the acceleration of the trolley increases. Lee (1998) extended his control strategy (Lee, 1997) by introducing a PI controller to track the cable length and gain scheduling to adapt the gains to slow changes in the cable length. The gains for optimum damping are found at several cable lengths. Then, curve fitting is used to express the gains as functions of the cable length. His experiments on a scaled bidirectional crane model running at low speeds show transient pendulations of less than 2, no residual pendulations, and some external disturbance rejection. Méndez et al. (1998, 1999) proposed a state feedback controller with neural networks to tune the feedback gains online as the cable length changes. Two neural networks are used to model the dynamics and generate the gains applied independently to the state of the trolley and payload. The neural networks use an LQR architecture to find optimal gains at each time step. Simulations and experiments on a scaled model show that this strategy is effective in suppressing residual pendulations and provides a smooth position of the trolley

35 CHAPTER 2. LITERATURE REVIEW 24 at low travel speeds. Fuzzy Logic Control Yasunobu and Hasegawa (1986, 1987) and Yasunobu et al. (1987) proposed a predictive fuzzy control strategy to minimize payload pendulations and travel time, while moving along a predefined path with obstacles. The strategy breaks the operation into seven stages and decides which fuzzy control rule to use based on simplified models of the plant. The control rules are linear feedback controllers that command the trolley motion and cable hoist. Experiments on a scaled model and an actual crane showed that this strategy is more effective than most skilled operators in minimizing travel time, payload pendulations, and accuracy at the target point. Kim and Kang (1993) proposed a control strategy using two fuzzy models of the crane dynamics to generate reference velocities for the trolley travel and cable hoist. Then, two more fuzzy controllers are used to track these velocities. The system is designed to minimize travel time and pendulations while moving along a predefined path. Their simulations show that the control strategy is comparable to that of a skilled operator. Itoh et al. (1993, 1995) proposed a control strategy using fuzzy logic to imitate an inputshaping acceleration profile with one-step acceleration and two-step deceleration. This strategy is designed to minimize residual pendulations and improve trolley positioning at the target. This strategy uses a fixed cable length. Experiments on an actual crane showed that this strategy is more effective in minimizing payload pendulations than an input-shaping strategy or a skilled operator. Nally and Trabia (1994) proposed a distributed fuzzy logic control strategy for a bidirectional gantry crane. They used two independent fuzzy inference engines (FIE). One FIE is used to track the desired position, while the other corrects for payload pendulations. Each FIE is designed to handle two perpendicular planes of crane motion. The outputs of the FIE

36 CHAPTER 2. LITERATURE REVIEW 25 are superimposed to obtain the total control input. The controller is used to adjust a path generated by an input-shaping strategy. Their simulations show good damping of pendulations. Liang and Koh (1997) proposed a fuzzy logic controller to eliminate residual pendulations using a heuristic approach. The trolley decelerates as it reaches the target point, producing inertia-induced pendulations. Then, the trolley accelerates to bring the trolley directly above the payload when it reaches a maximum point on the upward swing, causing the payload to be temporarily at rest. This procedure is repeated until the payload is at rest. Simulations showed that significant pendulations develop in the process, but after a few cycles the payload is brought to rest. Méndez et al. (1999) proposed a similar fuzzy controller employing the position of the trolley and pendulation angle. Experiments showed that the controller eliminates residual pendulations and has a smooth travel, however the trolley was very slow. Almousa et al. (2001) proposed a fuzzy logic control system with four fuzzy inference engines (FIE). A pair of FIEs were used for the radial direction, one for tracking and one for pendulation control. The other pair of FIEs were used for the tangential direction. Each FIE operates independently. Simulations show that the control strategy can mitigate and limit the in-plane and out-of-plane pendulations due to external or inertia-induced disturbances. However, the control strategy slows down operations of the crane. Nonlinear Control Zinober (1979) proposed a sliding-mode control strategy to minimize the travel time, eliminate residual pendulations, and avoid obstacles along the travel path. The strategy employs a linear switching function of the system states to switch a bang-bang controller of the torque applied to the trolley motor. A low-pass filter is applied to the control input to eliminate high-frequency components. Their simulations showed that the travel time is 10% longer than the optimal travel time, but the strategy can reject external disturbances. Furthermore, since the controller is independent of the crane parameters, it is not sensitive to cable

37 CHAPTER 2. LITERATURE REVIEW 26 length and payload mass changes. d Andrea Novel and Lévine (1989) showed that static state feedback linearization works only at slow velocities and can ensure local stability only when starting from a stable configuration. On the other hand, dynamic state feedback linearization can handle any initial configuration and higher speeds. They demonstrated this result on an actual crane traveling and hoisting along a predefined path. Fliess et al. (1991, 1993) proposed a nonlinear dynamic state feedback technique, dubbed flatness-based control, to linearize the dynamics of a crane. This technique is applicable only to systems where the input and state variables can be expressed in terms of the output variable and their time derivatives can be expressed in closed form. Based on inverse dynamics analysis of the nonlinear plant model, they expressed the hoisting and trolley accelerations in terms of the payload position. The required input accelerations are produced by substituting the mathematical representation of the desired path into the nonlinear expressions. A PI tracking controller is then used to drive the trolley and hoist motors along the predefined input accelerations. Their simulations show enhanced performance in trolley and payload positioning with less operation time. Payload pendulations never exceeded 1.7 during the maneuver. Bourdache-Siguerdidjane (1993, 1995) extended the Fliess linearization technique to an extended model that includes the dynamics of the trolley. The nonlinear model is linearized by matching it to a linearized version of the model for a single equilibrium. Then, an LQR controller is applied to the new linear system, generating gains for a state feedback controller, which drives the motors and tracks the predefined path. Simulations show that this strategy eliminates residual pendulations. DeSantis and Krau (1994) proposed a sliding-mode control strategy for a bidirectional crane. Two independent, planar state feedback controllers designed to suppress inertia-induced payload pendulations estimate the control inputs to each motor. Then, a sliding-mode controller is used to produce the actual inputs to the motors. Simulations comparing the

38 CHAPTER 2. LITERATURE REVIEW 27 sliding-mode control to linear state feedback control showed that both strategies are able to stabilize the payload at low speeds. However, sliding-mode control is better at coping with changes in the crane parameters and external disturbances. On the other hand, sliding-mode control is less effective in dealing with the feedback delays inherent in the motor drives than linear state feedback. Martindale et al. (1995) proposed two feedback control strategies to track a predefined path: one uses backstepping control, and the other adds an adaptive gain matrix to account for uncertainty in the model. Experiments on a scaled model show that both control strategies are effective in mitigating pendulations at low speeds. It should be noted that backstepping control uses fourth-order derivatives of the output, thereby forcing a smooth trajectory. Burg et al. (1996) used the Teel saturation control approach to design a third feedback controller. Experiments on a scaled model show that the controller can suppress pendulations at low speeds, but significant pendulations develop at higher speeds. Cheng and Chen (1996) proposed a control strategy using feedback linearization and time delay control to move along a predefined path. Their simulations show that this strategy is able to deliver the payload with no residual pendulations and transient pendulations under 3. Furthermore, the controller is robust enough to handle changes in the payload mass and unmodeled forces. However, this approach is sensitive to external disturbances, which significantly increase transient pendulations. The controller performs better than the adaptive feedback linearization of d Andrea Novel and Boustany (1991) and Boustany and d Andrea Novel (1992). Chin et al. (1998) proposed a nonlinear feedback control scheme for a ship-mounted boom crane to suppress parametric instabilities in payload motions due to wave-induced base excitations. Analytically, they showed that introducing a control effort in the form of a harmonic change in the cable length at the same frequency as the base excitations can suppress the parametric instability and result in a smooth response.

39 CHAPTER 2. LITERATURE REVIEW 28 Time-Delayed Feedback Henry et al. (2001) developed a time-delayed position feedback controller using a planar model of a ship-mounted crane. The time-delayed position feedback controller measures the pendulation angle, then actuates the suspension point by a fraction of the payload displacement delayed in time. Simulations and experiments show an effective suppression of payload pendulations due to in-plane, roll, and parametric, heave, excitations. Masoud et al. (2000) extended this strategy to the three-dimensional case. Simulations and experiments showed that the controller suppresses payload pendulations due to both in-plane and outof-plane base excitations. Nayfeh (2002), Masoud et al. (2003), and Masoud and Nayfeh (2003) extended the time-delayed position feedback controller to container and rotary cranes. Simulations and experiments of the rotary crane showed effective suppression of payload pendulations. Simulations, experiments, and actual tests of the container crane system further confirm the effectiveness of this strategy for cranes. Hybrid Control Ohnishi et al. (1981) proposed a hybrid two-phase control strategy to dampen payload pendulations. The first phase is a linear feedback controller designed to reduce payload motion around its equilibrium position. The second phase is an input-shaped deceleration stage. This strategy was implemented on an overhead crane in a mill. It was able to minimize the pendulation angles, but they reported that the system was 30% slower than the manually operated system. Sakawa and Nakazumi (1985) proposed a hybrid control strategy for a boom crane with a predefined path. An open-loop controller tracks the trajectory, while an LQR controller regulates the slew, luff, and hoist to eliminate inertia-induced residual pendulations. Their simulations show significant pendulation angles, up to 21.6 during the maneuver.

40 CHAPTER 2. LITERATURE REVIEW 29 Virkkunen and Marttinen (1988) and Vähä and Marttinen (1989) proposed a combined control strategy to mitigate residual pendulations at the target point encountered by timeoptimal controllers due to unmodeled forces and disturbances. The strategy uses the acceleration profile of Yamada et al. (1983) to drive the trolley until it is close to the target point, then switches to an LQR controller to eliminate residual pendulations. The strategy was shown to be successful on a scaled model of a gantry crane. Caron et al. (1989) proposed a strategy using a one-step acceleration profile to generate trajectories designed to minimize payload pendulations. Then, a PI controller is used to track the path. Their simulations show good tracking with minimal transient pendulations of 1.7. Grassin et al. (1991) extended this work by replacing the PI controller with a LQR controller to track the one-step acceleration profiles. Their simulations and scaled experiments show smooth operations with transient pendulations less than 3.5. However, both strategies of Caron et al. (1989) and Grassin et al. (1991) are not able to reject disturbances in the payload angle. Yoshida and Kawabe (1992) proposed a saturation linear state feedback controller to perform predefined maneuvers. The controller decreases the transfer time at the cost of incurring larger pendulations compared to a traditional linear feedback controller. Golafshani and Aplevich (1995) proposed a time-optimal control scheme to generate trajectories for trolley travel and hoist. Then, a sliding-mode controller is used to track these trajectories. Simulations show that the time-optimal trajectories induce pendulations. They relaxed the constraint on time to 110% of the optimal value. Simulations using these relaxed trajectories show a reduction in pendulations, however significant pendulations still exist and persist at the end point. Yoon et al. (1995) proposed a modification of an input-shaping two-step acceleration profile. The second acceleration step and coasting stage are replaced with feedback of the payload angular velocity. The strategy aims at reducing the sensitivity of input shaping to external disturbances and changes in cable length. Their simulations and experiments of a scaled

41 CHAPTER 2. LITERATURE REVIEW 30 model traveling at slow speeds show that this strategy rejects disturbances better than pure input shaping. However, the controller cannot reject disturbances during the deceleration stage. Furthermore, the control strategy cannot eliminate residual pendulations. Alli and Singh (1999) proposed an optimal feedback controller to two different models: rigid cable and flexible cable. The controller was optimized to minimize the integral over time of the product of time and the magnitude of the error. Their simulations show good mitigation of payload pendulations, however the inertia forces in the simulation are small. Daqaq and Masoud (2005) proposed a hybrid control system with a nonlinear input-shaping controller and a time-delayed position feedback controller. The nonlinear input-shaping controller generates bang-bang acceleration profiles from an approximation of the pendulation frequency. Then, the time-delayed position feedback controller is applied at the end of the maneuver to alleviate any unmodeled dynamics or disturbances. Their simulations show that the performance of the combination of the two control strategies is superior to the performance of the two controllers operating separately. Summary The time-delay feedback control system has been shown to be very effective in simulations, experiments, and in practice. While there are many near perfect feedback and feedforward control systems in the literature, in practice they do not perform as well. Due to the massive size and power of a typical crane and its actuators, there are inherent time delays between the operator command and the actuators responses, which degrade controller that do not account for these delays. With the time-delay controller, these delays are absorbed. Furthermore, in the case of many feedforward controllers, while the time delays may be overcome, operators and safety inspectors dislike the predefined trajectories because there is no way to suddenly and safely stop once a maneuver has begun. Due to the fact that time delays introduce infinite modes or roots in most systems, there

42 CHAPTER 2. LITERATURE REVIEW 31 was a concern that the controller may cause the crane to have other stable solutions in the region where only the trivial solution is stable. Systems such as high speed cutting tools and aircraft wings exhibit this behavior. In this Dissertation, we undertake the task of showing that the time-delayed position feedback controller does not induce these extra solutions and that it is globally robust in the region of interest. 2.2 Machine-Tool Chatter in Lathes For some machining operations, tool vibrations decay with time, the trivial solution is a stable equilibrium solution, and hence the machined surface is smooth. For other operating conditions, the trivial solution is unstable and the tool undergoes large oscillations, which might be periodic, quasi-periodic, or chaotic, and hence the surface finish will not be smooth. These oscillations are called machine-tool chatter and must be avoided or mitigated to maintain machining tolerances, preserve surface finish, and prevent tool breakage. Therefore, extensive research has been devoted to the understanding of the mechanisms of, modeling, and methods of controlling machine-tool chatter. In this review, we concentrate on the studies that addressed nonlinearities. Tobias (1965) documents much of the pioneering work in the field of machine-tool chatter. Along with Tobias and Fishwick (1958) and Koeingsberger and Tlusty (1970), they identified two mechanisms for machine-tool chatter known as regeneration and mode coupling. Regenerative chatter occurs when a cut overlaps a previous cut that left some waves in the material. At each subsequent pass of the tool, the waves are regenerated. This mechanism is considered to be the predominant cause of machine-tool chatter. Mode coupling occurs when the relative vibration of the cutting tool and material exists in at least two directions in the plane of the cut. As the width of cut or the feed rate increases, the frequencies of two modes in two different directions approach each other and coalesce, resulting in an instability and causing the tool tip to trace an elliptical path. This mechanism most likely occurs in nearly

43 CHAPTER 2. LITERATURE REVIEW 32 Figure 2.2: Qualitative features of experimental data shown in Hanna and Tobias, symmetric slender tools, such as boring bars. This mechanism is similar to the phenomenon of aeroelastic flutter. Hanna and Tobias (1974) presented the first mathematical theory of nonlinear chatter. They contend that linear theory does not adequately capture the behavior of chattering cutting tools for a cutting tool on a lathe with constant speed and varying cutting width. Figure 2.2 shows the qualitative features of the experimental data presented by Hanna and Tobias (1974). It is clear from Fig. 2.2 that there are three operating regions: a region of unconditional stability, a region of unconditional instability, and a region of conditional stability where multiple responses and hysteresis exist. Linear theory can predict the regions of unconditional stability and instability, but cannot predict the region of conditional stability. Moreover, linear theory predicts that the amplitude of vibration increases indefinitely in the region of unconditional instability at variance with the experimental results, which show a finite-amplitude response. Hanna and Tobias (1974) contend that the linear stability charts are not accurate because nonlinear effects may, in fact, stabilize or dampen a linearly

44 CHAPTER 2. LITERATURE REVIEW 33 unstable system. Their nonlinear theory predicts the three observed regions of operation. Shi and Tobias (1984) extended the work of Hanna and Tobias (1974) to examine multiple regenerative chatter, a process whereby the cutting tool leaves the surface of the material. Both papers include structural nonlinearities. Nayfeh et al. (1997) presented a method to analyze the nonlinear dynamics of a lathe cutting tool using the model of Hanna and Tobias (1974). They used the method of multiple scales to examine the behavior of the Hopf bifurcations that occur when the system loses linear stability. They found that the bifurcation is locally supercritical, in agreement with the experimental results of Hanna and Tobias, qualtitatively shown in Fig To ascertain the global behavior of the system, they presented a bifurcation diagram, generated by a sixth-term harmonic-balance solution, showing variation of the amplitude of the fundamental harmonic with the width of cut. Using Floquet theory and Hill s determinant, they found that the system undergoes two cyclic-fold bifurcations, resulting in a branch of large-amplitude periodic solutions, hysteresis, jumps, and subcritical instability, in qualitative agreement with the experimental results. They also found that, as the width of cut increases, quasi-periodic solutions begin to appear, eventually leading the system into chaos. Pratt and Nayfeh (1996) confirmed the mathematical results using an analog computer. Kalmár-Nagy and Pratt (1999) examined the behavior of a single-degree-of-freedom dynamic cutting fixture near the linear stability boundary. They showed experimentally the hysteretic nature of the instability. Using parameters from the experiment, they derived a singledegree-of-freedom model consisting of a nonlinear second-order ordinary-differential equation with time delay. Then, using center manifold theory, they showed the existence of a local subcritical Hopf bifurcation. Finally, using numerical integration techniques, they found that their model agrees with the experimental results. Moon and Kalmár-Nagy (2001) presented a review of the literature predicting complex, unsteady, and chaotic dynamics associated with cutting processes. They examined several nonlinear models in the literature and drew the conclusion that single-degree-of-freedom

45 CHAPTER 2. LITERATURE REVIEW 34 models are not good predictors of low-level chaos and that more complex multi-degree-offreedom models derived from experiments are needed. Kalmár-Nagy et al. (2001) presented an analysis of a nonlinear second-order ordinarydifferential equation with time delay that models a machine tool with regenerative effects. They analytically showed the existence of a local subcritical bifurcation by reducing the infinite-dimensional model to a two-dimensional dynamical system on the center manifold. Then they transformed this system into its normal form. Their simulations show good agreement with the analytical results. Gilsinn (2002) presented another analysis of a nonlinear second-order delay differential equation with application to cutting tool chatter due to regenerative effects. He demonstrated the existence of a Hopf bifurcation using center manifold theory, characterized the critical eigenvalues on the stability boundary, and further examined the period of the bifurcating solution and associated Floquet exponents. He also presented simulations that show the Hopf bifurcation occurring at critical points on the stability boundary. He also reduced the infinite-dimensional model to a two-dimensional dynamical system on the center manifold and then transformed it into its normal from. The analyses of Kalmár-Nagy et al. (2001) and Gilsinn (2002) are so involved that neither of them compared his result with the other. Insperger and Stépán (2004) investigated a single-degree-of-freedom model of a turning cutting tool with periodically modulated spindle speed. The model includes time-varying delay. They analyzed the model using a semi-discretization method and generated stability charts and chatter frequencies. They showed that the cutting efficiency can be increased when either the spindle is modulated at high frequency or the spindle is turned at low speed. Further, they detected that Hopf bifurcations were present from eigenvalue analysis. Stépán et al. (2005) presented a review of the development of stability analysis for an oscillating delayed dynamical system, focusing on cutting and milling tools. They presented a linear analysis of parametrically excited systems by comparing the dynamics of the system to a Mathieu equation and using a semi-discretization method to determine the location of

46 CHAPTER 2. LITERATURE REVIEW 35 the Hopf bifurcations. They also discussed the local and global behavior of a high-speed milling process with parametric excitation. Summary Three mechanisms have been proposed for chatter: regenerative effect, mode coupling (fluttertype instability), and velocity-dependent effect. In the regenerative mechanism, the cuts overlap and the initial tool vibrations produce small waves in the material, which are regenerated with each subsequent pass of the tool (Tlusty and Spacek, 1954; Tobias, 1965). This mechanism can be activated even if the tool vibrates in one direction. In the mode coupling mechanism, tool vibrations in two directions are necessary. These vibrations are coupled asymetricaly by even the linear part of the cutting force. Therefore, if the frequencies of two modes in the two directions coalesce or near coalesce, the trivial solution loses stability and the tool tip undergoes large oscillations, as in the flutter of aircraft wings. In the velocitydependent mechanism, variation of the cutting force with the speed has a negative slope (Arnold, 1946), and hence it is a source of negative damping similar to those of the van der Pol and Rayleigh oscillators (Nayfeh and Mook, 1979). Because the regenerative mechanism does not require vibrations in two directions and leads to a time-delayed system, we only consider the regenerative mechanism consistent with the motivation of the present work. In most of the research that examined the type of Hopf bifurcation, with the exception of Nayfeh et al. (1997), the analysis is local and restricted to points at the bottom of the stability notch. These points do not capture the dynamics of the entire system and the local analysis may be misleading. The local analysis may predict a supercritical bifurcation, whereas the global analysis predicts a subcritical behavior because the generated limit cycles by the Hopf bifurcation undergo cyclic folds. Moreover, the normal form analyses of Kalmár- Nagy et al. (2001) and Gilsinn (2002) are involved and based on an operational differential form of the governing equation. They are so involved that neither of them compared his result with the other although they acknowledge each other. Furthermore, the multiple scales

47 CHAPTER 2. LITERATURE REVIEW 36 analysis of Nayfeh et al. (1997) has an error. In this Dissertation, we use a systematic approach combining a path following scheme, the method of multiple scales, the method of harmonic balance, Floquet theory, and numerical simulations to investigate the local and global dynamics and stability of cutting tools on lathes due to the regenerative mechanism. First, we use the method of multiple scales to determine the normal form of the Hopf bifurcation at all of the stability boundary and calculate the limit cycles generated by the bifurcation. Then, we use a combination of a path following scheme and the method of harmonic balance to continue the branch of generated limit cycles. Thus, we calculate small- and large-amplitude limit cycles and ascertain their stability using Floquet theory. We validate these results using numerical simulations. Then, we search for isolated branches of large-amplitude solutions coexisting with the linearly stable trivial solution. We use all of the results to generate bifurcation diagrams consisting of multiple large-amplitude stable and unstable branches of limit cycles coexisting with the trivial response, indicating three regions of operation, as in the experimental observations. Then, we investigate bifurcation control using cubic-velocity feedback and show that the region of unconditionally stable region can be expanded at the expense of the conditionally stable region. Finally, we show that the region of linearly stable trivial solution can be widened using time-delay acceleration feedback.

48 Chapter 3 Modeling 3.1 Modeling of Container Cranes The crane can be modeled either using rectangular coordinates (a four-bar mechanism model, Fig. 3.1) or using polar coordinates (a double-pendulum model, Fig. 3.2). The four-bar mechanism model, Fig. 3.1, has been used and validated by Nayfeh (2002) and Masoud et al. (2003). In this work, we use the double-pendulum formulation to model the trolley, cable, and hoist assemblies. The double-pendulum model captures the dynamics of the trolley-cablehoist assembly without the complexity of a four-bar mechanism (Nayfeh, 2002). We first derive the full nonlinear equations, including the time-delayed position feedback controller, then we linearize the equations in Section 3.1.2, and finally extract an approximate equation, including up to cubic nonlinear terms in Section The linearized equation is used to perform the controller design and the nonlinear cubic model is used to perform local and global bifurcation analysis. We ignore torsional effects, because the cable rigging system minimizes this effect. 37

49 CHAPTER 3. MODELING 38 Figure 3.1: A schematic model of a container crane Fully Nonlinear Model We represent the crane by a double-pendulum system with two fixed-length links and a kinematic constraint relating the angles φ and θ, as shown in Fig To derive the governing equations, we let OP = l. Then, the closing constraints of the loop AOPB in Figure 3.1 can be written as l sin φ 1 2 w cos θ d = L sin φ 1 (3.1) l cosφ 1 2 w sin θ = L cosφ 1 (3.2)

50 CHAPTER 3. MODELING 39 Figure 3.2: A simplified schematic model of a container crane. Similarly, the closing constraints of the loop ODCP can be written as l sin φ w cosθ 1 2 d = L sin φ 2 (3.3) l cosφ w sin θ = L cos φ 2 (3.4) Squaring and adding (3.1) and (3.2) yields ( l cosφ 1 ) 2 ( 2 w sin θ + l sin φ 1 2 w cosθ + 1 ) 2 2 d = L 2 (3.5) and squaring and adding (3.3) and (3.4) yields ( l cosφ + 1 ) 2 ( 2 w sin θ + l sin φ w cosθ 1 ) 2 2 d = L 2 (3.6)

51 CHAPTER 3. MODELING 40 Eliminating L 2 from (3.5) and (3.6) and manipulating the result, we obtain the following relation between φ and θ: which for small oscillations reduces to d sin φ = w sin(θ + φ) (3.7) θ = d w φ = aφ (3.8) w where a = d w. It follows from (3.7) that the present model is valid only when φ w sin 1 (w/d). Using (3.7) to simplify (3.6), we obtain l 2 = L ( d 2 + w 2 2dw cosθ ) (3.9) Next, we write the potential and kinetic energies of the system. To this end, we note that the coordinates of Q are x = f(t) + l sin φ R sin θ (3.10) y = l cos φ R cosθ (3.11) where f(t) represents the position of the trolley. Then, differentiating (3.10) and (3.11) once with respect to time yields the velocity components ẋ = f + φl cosφ + l sin φ R θ cosθ (3.12) ẏ = l φsin φ + R θ sin θ (3.13) The kinetic and potential energies, K and V, are given by K = 1 2 mẋ mẏ2 (3.14) and V = mg(y R b) (3.15) where b 2 = L (d w)2 = L a2 w 2 (3.16)

52 CHAPTER 3. MODELING 41 and the definition d = wa + w has been used. Using (3.11)-(3.13), we express the kinetic and potential energies as ( K = 1 2 m f R cos(θ) θ (1 + a)w2 sin(θ) sin(φ) θ φ) 2 + l cos(φ) (3.17) 4l ( m R sin(θ) θ + (1 + a)w2 sin(θ) cos(φ) θ ) 2 + l sin(φ) 4l φ V = mg [ b R + R cos(θ) + l cos(φ)] (3.18) Substituting (3.17) and (3.18) into the augmented Lagrangian L = K V + λ [d sin(φ) w sin(θ + φ))] (3.19) where λ is the Lagrange multiplier, we obtain L = g ( b R + R cos(θ) + l cos(φ)) + wλ ((1 + a) sin(φ) sin(θ + φ)) ( + 1 f R cos(θ) 2 θ (1 + a)w2 sin(θ) sin(φ) θ φ) 2 + l cos(φ) (3.20) 4l ( + 1 R sin(θ) 2 θ + (1 + a)w2 sin(θ) cos(φ) θ ) 2 + l sin(φ) 4l φ Writing the Euler-Lagrange equations corresponding to (3.20) for both θ and φ yields the following nonlinear equations of motion: θ 16l 2 [ 32l 2 R 2 + w 4 (1 + a) 2 (1 cos 2θ) 8lRw 2 (1 + a) cos(2θ + φ) ] 2 φlr cos(θ + φ) + θ(1 + a) 2 w 4 [ 3(1 + a)w 2 sin θ + 8l 2 sin 2θ (1 + a)w 2 sin 3θ + 4lR sin(θ φ) 128l 4 ] +8lR sin(θ + φ) + 64l3 R sin(2θ + φ) 4lR sin(3θ + φ) (1 + a)w 2 + φ 2 2 [(1 + θ φ [ a)w2 sin θ + 4lR sin(θ + φ)] + (1 + a)rw 2 cos(θ + φ) sin θ ] + 2wλ cos(θ + φ) l + f [ 4lR cosθ + (1 + a)w 2 sin θ sin φ ] + g(4lr + (1 + a)w2 cosφ) sin θ = 0 2l 2l (3.21)

53 CHAPTER 3. MODELING 42 φl 2 + φ θ 2 (1 + a)w2 sin θ + θlr cos(θ + φ) + θ 2 lr sin(θ + φ) +gl sin θ + λw [cos(θ + φ) (1 + a) cosφ] + fl cosφ = 0 (3.22) These two equations are supplemented by the constraint (1 + a) sin(φ) sin(φ + θ) = 0 (3.23) Therefore, we have three equations, (3.21)-(3.23), in the three unknowns θ, φ, and λ. These three equations are used in deriving a linear model, a cubic model, and for simulations. The linear model will be used for linear stability analysis. The cubic model will be used for nonlinear analysis Time-delayed Position Feedback Controller The time-delayed position feedback controller has the form of f = k sin(φ(t τ)) (3.24) where k is a gain and τ is a time-delay. The feedback variable is φ(t), or the cable angle measured from the vertical with respect to the trolley. Time delays in systems can either increase damping or cause instabilities. Careful combination of k and τ yields excellent results. The parameters of k and τ are developed in Section Linearized Model We linearize (3.21) and (3.22), use (3.16), and obtain R 2 θ ( + g R a ) 4b w2 θ br φ R f + λw = 0 (3.25) b 2 φ br θ + bgφ + b f awλ = 0 (3.26)

54 CHAPTER 3. MODELING 43 Then, solving (3.25) for λ yields λ = g [4bR + (1 + a)w2 ] θ + 4b(R 2 θ Rb φ + R f) 4bw (3.27) Substituting (3.27) into (3.26) and using (3.8) and (3.16) yields (b ar) 2 4b φ + (b ar) f 2 + a 2 [4bR + (1 + a)w 2 ] + g φ = 0 (3.28) 4b Finally, manipulating (3.28) produces φ + f b ar + g4b2 + a 2 [4bR + (1 + a)w 2 ] 4b(b ar) 2 φ = 0 (3.29) Adding a linear damping term for φ, we rewrite (3.29) as φ + 2µ φ + f b ar + g4b2 + a 2 [4bR + (1 + a)w 2 ] 4b(b ar) 2 φ = 0 (3.30) Using the linear form of the delayed-position feedback controller f = ˆkφ(t τ) (3.31) we rewrite (3.30) as φ + 2µ φ ˆk φ(t τ) b ar + g4b2 + a 2 [4bR + (1 + a)w 2 ] 4b(b ar) 2 φ = 0 (3.32) Cubic Model For the bifurcation analysis, we need a nonlinear approximation to the governing equation (cubic model) that includes up to cubic terms. An easy approach to developing the cubic model is to expand the kinetic and potential energies (3.17) and (3.18), keep up to quartic terms in γ(t) and φ(t), write the Lagrangian, and then write the Euler-Lagrange equation. To this end, we expand (3.7) and (3.9) in Taylor series for small, but finite θ and φ and obtain θ aφ a(1 + a)(2 + a)φ3 (3.33)

55 CHAPTER 3. MODELING 44 ( l b a2 dw 8b φ2 a2 dw 1 4b a a2 + dw ) φ 4 (3.34) 32b 2a2 Further, substituting the nonlinear form of the controller into (3.10) and expanding the result up to cubic terms yields where f(t) = ˆk sin γ (3.35) x(t) kγ kγ3 + (b ar)φ 1 3 ˆα 5φ 3 (3.36) ˆα 5 = 4b2 + 4a(2 + 3a)bR + 3a 2 (1 + a)w 2 8b (3.37) Expanding (3.11) and keeping up to quartic terms yields y(t) = (b + R) + 1 ) (4b + 4a 2 R + a2 (1 + a)w 2 φ b 4 ˆα 3φ 4 (3.38) where ˆα 3 = 16b4 + 16a 2 (8 + 12a + 3a 2 )b 3 R + 4a 2 (2 + 14a + 15a 2 + 3a 3 )b 2 w 2 + 3a 4 (1 + a) 2 w 4 96b 3 (3.39) We note that y(t) has been expanded up to quartic terms rather than up to cubic terms because this is needed in the potential energy. Then, the kinetic energy up to quartic terms can be rewritten as K t = 1 2ˆk γ 2 ˆkˆα 2 γ φ ˆα2 2 φ 2 1 2ˆkγ 2 γ ˆkˆα 2 γ 2 γ φ where 3ˆkˆα 5 φ 2 γ φ + ( ) ˆα b + 3ˆα 2ˆα 2 5 φ 2 φ2 (3.40) ˆα 1 = 4b 2 + 4a 2 br + a 2 (1 + a)w 2 (3.41) ˆα 2 = b ar (3.42) Similarly, up to quartic terms, the potential energy takes the form [ V t = g 2(b + R) + ˆα ] 1 8b φ2 + ˆα 3 φ 4 (3.43)

56 CHAPTER 3. MODELING 45 Using (3.40) and (3.43), we write the Lagrangian of the motion as L = K t V t = 1 γ 2ˆk 2 ˆkˆα 2 γ φ ˆα2 φ 2 2 2ˆkγ 1 2 γ 2 + 2ˆkˆα 1 2 γ 2 γ φ ( ) [ 3ˆkˆα 5 φ 2 γ φ ˆα b + 3ˆα 2ˆα 2 5 φ 2 φ2 g 2(b + R) + ˆα ] 1 8b φ2 + ˆα 3 φ 4 (3.44) Substituting the Lagrangian (3.44) into the the Euler-Lagrange equation [ ] d L dt φ L φ = 0 (3.45) and setting γ(t) = φ(t τ) yields the following equation of motion for small but finite angles: φ(t) + α 1 φ(t) + ˆkα 2 φ(t τ) = ǫα3 φ 3 (t) ǫα 4 φ(t) φ 2 (t) ǫα 4 φ 2 (t) φ(t) ǫˆkα 2 φ(t τ) φ 2 (t τ) ǫˆkα 5 φ 2 (t) φ(t τ) 1 2 ǫˆkα 2 φ 2 (t τ) φ(t τ) (3.46) where the small nondimensional parameter ǫ has been introduced to scale the nonlinear terms and α 1 = gˆα 1 4bˆα 2 2 (3.47) α 2 = 1ˆα 2 (3.48) α 3 = 4gˆα 3 ˆα 2 2 α 4 = ˆα ˆα 2ˆα 5 16b 2ˆα 2 2 α 5 = 3ˆα 5 ˆα 2 2 (3.49) (3.50) (3.51) Next, we let k = ˆkα 2 and α 6 = α 5 /α 2 and rewrite (3.46) as φ(t) + α 1 φ(t) + k φ(t τ) = ǫα 3 φ 3 (t) ǫα 4 φ(t) φ 2 (t) ǫα 4 φ 2 (t) φ(t) ǫkφ(t τ) φ 2 (t τ) ǫkα 6 φ 2 (t) φ(t τ) ǫ 1 2 kφ2 (t τ) φ(t τ) (3.52) Equation (3.52) is a special case of (1.1). Hence, (1.1) is a cubic approximation of the crane dynamics.

57 CHAPTER 3. MODELING 46 Figure 3.3: One-degree-of-freedom model of a cutting tool. 3.2 Lathe-Tool Model The machine tool of a lathe can be modeled as a cantilever beam subjected to nonlinear boundary conditions at the cutting end. For an inextensible tool, its motions are governed by two nonlinear coupled partial-differential equations subject to nonlinear boundary conditions. The nonlinearities in the governing equations are of two types: elastic and inertia nonlinearities. The nonlinearity in the boundary conditions is due to the nonlinear dependence of the cutting force on the tool-tip displacement. Usually, the machine tool is modeled as a cantilever beam vibrating only in the direction normal to the machined surface. Moreover, the mode of vibration that might become unstable is represented by a single-degree-of-freedom system with an equivalent mass m, an equivalent dashpot, an equivalent restoring force R, and a cutting force F, as shown in Fig Hence, the governing equation of motion is written as m v + c v + R = F (3.53) where v is the displacement of the equivalent mass and the overdot denotes the derivative with respect to time t. The restoring force consists of two parts: an elastic part and an inertia part. To third order, the elastic part has the form R 1 = ˆβ 1 v + ˆβ 3 v 3 (3.54)

58 CHAPTER 3. MODELING 47 whereas the inertia part has the form R 2 = ˆα 2 v v 2 + ˆα 3 v 2 v (3.55) Dependence of the cutting force on the displacement is determined empirically by measuring variation of the cutting force with the chip thickness h. There are two approaches for determining an analytical form for the measured cutting force. In one approach, Hanna and Tobias (1974) and Shi and Tobias (1984) fit the empirical data with a third-order polynomial, F = ˆκ 1 h + ˆκ 2 h 2 + ˆκ 3 h 3 (3.56) Substituting (3.54)-(3.56) into (3.53) yields the equation of motion m v + c v + ˆβ 1 v + ˆβ 3 v 3 + ˆα 2 v v 2 + ˆα 3 v 2 v = ˆκ 1 h ˆκ 2 h 2 ˆκ 3 h 3 (3.57) In the steady state, (3.53) becomes ˆβ 1 v 0 + ˆβ 3 v 3 0 = ˆκ 1h 0 ˆκ 2 h 2 0 ˆκ 3h 3 0 (3.58) where h 0 is the nominal chip thickness in steady state and v 0 is the static displacement of the equivalent mass. In this case, the mean or steady-state cutting force is given by F 0 = ˆκ 1 h 0 + ˆκ 2 h ˆκ 3h 3 0 In the unsteady case, the displacement would be the sum of the steady-state component v 0 and an unsteady component u(t) and the chip thickness would be also the sum of the mean thickness h 0 and an unsteady component h; that is, v(t) = v 0 + u(t) and h = h 0 + h (3.59) Moreover, the chip thickness variation h can be expressed as the difference between the current tool edge position v(t) and its delayed position v(t τ); that is, h = v(t) v(t τ) = u(t) u(t τ) (3.60)

59 CHAPTER 3. MODELING 48 Substituting (3.59) and (3.60) into (3.57), expanding the result, using (3.58), and rearranging, we obtain ü + 2µ u + ω0u 2 + δ 1 u 2 + δ 2 u 2 + δ 3 uü + α 1 u 3 + α 2 u u 2 + α 3 u 2 ü + κ 1 [u(t) u(t τ)] + κ 2 [u(t) u(t τ)] 2 + κ 3 [u(t) u(t τ)] 3 = 0 (3.61) where {2µ, ω0 2, δ 1, δ 2, δ 3, α 1 } = {c, ( ˆβ 1 + 3v0 2 ˆβ 3 ), 3v 0 ˆβ3, v 0 ˆα 2, 2v 0 ˆα 3, ˆβ3 }/(m + v0 2 ˆα 3) {α 2, α 3, κ 1, κ 2, κ 3 } = { ˆα 2, ˆα 3, (ˆκ 1 + 2ˆκ 2 h ˆκ 3 h 2 0), ( ˆκ ˆκ 3 h 0 ), ˆκ 3 }/(m + v0 2 ˆα 3 ) (3.62) Hanna and Tobias (1974) and Shi and Tobias (1984) modeled the tool dynamics with an equation having the form (3.61) but without the inertia terms. In the second approach, Taylor (1906) fitted his empirical data with a power-law expression of the form F = Kwh 3/4 (3.63) where w is the width of cut and K depends on technological parameters independent of the chip thickness. Kalmár-Nagy and Pratt (1999) also fitted their empirical data with a power-law form, but with the exponent 0.41 instead of the 0.75 found by Taylor. Therefore, they replaced (3.63) with F = Kwh s (3.64) where s varies from case to case. Using (3.64), we replace the equation of motion (3.57) with m v + c v + ˆβ 1 v + ˆβ 3 v 3 + ˆα 2 v v 2 + ˆα 3 v 2 v = Kwh s (3.65) In this case, the steady-state displacement is given by ˆβ 1 v 0 + ˆβ 3 v0 3 = Kwhs 0 (3.66) and the mean cutting force is given by F 0 = Kwh s 0

60 CHAPTER 3. MODELING 49 Substituting (3.59) and (3.60) into (3.65), expanding the result, using (3.66), and rearranging, we obtain (3.61), where all of the parameters are given in (3.62) except that { {κ 1, κ 2, κ 3 } = skwhs 1 0 1, s 1 } (s 1)(s 2), m + v0 2 ˆα 3 2h 0 6h 2 0 (3.67) Kalmár-Nagy and Pratt (1999) and Kalmár-Nagy et al. (2001) neglected the nonlinear elastic and inertia terms and hence considered the following governing equation: ü + 2µ u + ω0u 2 + skwhs 1 0 m { [u(t) u(t τ)] + s 1 } [u(t) u(t τ)] 2 2h 0 + skwhs 1 0 m (s 1)(s 2) 6h 2 0 Next, we introduce nondimensional quantities defined by in (3.68) and rearranging, we obtain ˆt = ω 0 t, ˆτ = ω 0 τ, u = 3h 0 2 sû [u(t) u(t τ)] 3 = 0 (3.68) ü + 2ξ u + u + p [u(t) u(t τ)] 3p(1 s) 2(2 s) where the hat has been removed for ease of notation and { [u(t) u(t τ)] 2 [u(t) u(t τ)] 3} = 0 (3.69) p = skwhs 1 0 mω0 2 Kalmár-Nagy and Pratt (1999), Kalmár-Nagy et al. (2001), and Gilsinn (2002) used (3.69) in their analyses.

61 Chapter 4 Comparison of three feedback controllers In this chapter, we compare the performance of the time-delayed position feedback controller with two feedback schemes for controlling container cranes: a classical feedback controller and a linear quadratic regulator (LQR) controller. The controllers are developed for a linearized version of the crane model. The performance, however, is evaluated using the full nonlinear model. The controllers are simulated for two maneuvers: no-hoist and variablehoist maneuvers. The trolley motion is assumed to be independent of the dynamics of the payload and friction effects are neglected. 4.1 Controller design Time-delayed position feedback We start with (3.32) and rewrite it as φ(t) + 2µ φ(t) + k φ(t τ) + ω0 2 φ(t) = 0. (4.1) 50

62 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 51 where ω 2 0 = g 4b2 + a 2 [4bR + (1 + a)w 2 ] 4b(b ar) 2 and k = ˆk b ar (4.2) Whereas (3.30) represents a two-dimensional dynamical system (a one-degree-of-freedom system), (4.1) represents an infinite-dimensional system. We seek the general solution of (4.1) in the form φ = Ae σt cos(ωt + η) (4.3) where A, σ, ω, and η are real constants. Substituting (4.3) into (4.1) and setting each of the coefficients of sin(ωt + η) and cos(ωt + η) equal to zero yields k(σ 2 ω 2 ) cos(ωτ) + 2kσω sin(ωτ) + e στ (ω µσ + σ2 ω 2 ) = 0 (4.4) k(σ 2 ω 2 ) sin(ωτ) 2kσω cos(ωτ) 2e στ (µ + σ)ω = 0 (4.5) For a given gain k and time delay τ, (4.4) and (4.5) are solved numerically for a pair ω and σ. Then A and η can be determined from initial conditions. Due to the delay term, there are an infinite number of solutions. When σ > 0 the system grows exponentially with time and the equilibrium is unstable; when σ < 0 the system decays exponentially with time and the equilibrium is stable; and σ = 0 defines the stability boundary. To locate this boundary, we let σ = 0 in (4.4) and (4.5) and obtain ω 2 + kω 2 cos(ωτ) ω 2 0 = 0 (4.6) kω sin(ωτ) + 2µ = 0 (4.7) The solution of (4.6) and (4.7) when µ 0, a good approximation for cranes, yields the following critical values ω c and k c separating stable from unstable trivial solutions: k c = ω c = nπ τ ω2 0 ω 2 c ω 2 c cos(ω cτ) (4.8) (4.9)

63 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 52 for n = 1, 2, 3,. As discussed below, (4.8) and (4.9) define the locus of the Hopf bifurcation in the ω k plane. Near any boundary given by k = k c, we determine an analytical expression for σ and ω for a given τ. To this end, we introduce a detuning parameter δ to express the nearness of k to k c ; that is, k = k c + ǫδ, where ǫ is a small nondimensional parameter. Moreover, since σ is small near the boundary, we let σ ǫσ 1, ω ω c + ǫω 1, and µ = ǫµ in (4.4) and (4.5), expand the results in Taylor series, equate the coefficients of ǫ to zero, and obtain 2ω 1 ω c + 2ω 1 k c ω c + δωc 2 σ 1τk c ωc 2 = 0 (4.10) 2σ 1 ω c 2σ 1 k c ω c ω 1 τk c ωc µω c = 0 (4.11) Solving (4.10) and (4.11) for σ 1 and ω 1 yields σ 1 = δτk cω 2 c 4µ(1 k c) 4 8k c + 4k 2 c + τ2 k 2 c ω2 c ω 1 = 2δ (1 k c)ω c + 2µτk c ω c 4 8k c + 4k 2 c + τ2 k 2 c ω2 c (4.12) (4.13) Equations (4.12) and (4.13) define the growth rate and the change in the frequency near the boundary, respectively. It follows from (4.3), (4.12), and (4.13) that d(σ + jω) d(σ + jω) dk = k=kc ǫdδ = τk cωc 2 + 2j (1 k c)ω c k=kc 4 8k c + 4kc 2 + τ 2 kcω 2 c 2 when µ = 0, which is different from zero. Therefore, the trivial solution loses stability with two complex conjugate eigenvalues transversely crossing the imaginary axis, and hence the trivial solution undergoes a Hopf bifurcation at k = k c, and the stability boundary is the locus of Hopf bifurcations. Figure 4.1 shows variation of the solutions of (4.4) and (4.5) with k for a constant τ = 0.3. We note that there are an infinite number of solutions (modes), yielding an infinite number of poles σ + jω. When k < 0.642, all of the poles are in the left-half of the complex plane and hence the system is stable. On the other hand, when k > 0.642, the second pole-pair

64 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS σ vs k (τ = 0.3) mode 1 mode 2 mode 3 mode 4 mode ω vs k (τ = 0.3) σ 1 ω k k (a) (b) Figure 4.1: Solutions of (4.4) and (4.5): (a) variation of σ with k when τ = 0.3 and (b) variation of ω with k when τ = 0.3. is in the right-half of the complex plane and hence the system is unstable. Therefore, for stability, we need to choose k < when τ = 0.3. In fact, Figure 4.1(a) shows that, for 0 < k 0.642, all modes are highly damped except the first and second modes, and hence the dynamics are dominated by these two modes. In the range 0 < k < 0.37, the first mode has less damping than the second mode and hence the dynamics are dominated by the first mode. On the other hand, in the range 0.37 < k < 0.642, the second mode has less damping than the first mode and hence it dominates the dynamics of the crane. We note from Fig. 4.1(b) that the second mode has a higher frequency than the first mode. Figure 4.2 shows variation of the solutions of (4.4) and (4.5) with τ for a constant k = 0.4. Once again, we note that there are an infinite number of solutions (modes), yielding an infinite number of poles σ + jω. Figure 4.2(a) shows that, when τ < 0.38, all of the poles are in the left-half of the complex plane and hence the system is stable. When τ < 0.28, the dynamics are dominated by the first mode, and when τ > 0.28, the dynamics are dominated by the second mode. Again, it follows from Figure 4.2(b) that the frequency associated with

65 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS σ vs τ (k = 0.4) mode 1 mode 2 mode 4 mode 5 mode ω vs τ (k = 0.4) σ ω τ τ (a) (b) Figure 4.2: Solutions of (4.4) and (4.5): (a) variation of σ with τ when k = 0.4 and (b) variation of ω with τ when k = 0.4. the second mode is higher than that associated with the first mode, which might be favorable for disturbance rejection. Figure 4.3 shows the dominant damping or growth rate σ as a function of k and τ. Figure 4.3(a) shows the least damping as a function of k for several values of τ. We note that as k increases, the system loses stability (σ > 0). As τ increases, the range of values of k for which the system is stable decreases. Figure 4.3(b) shows the least damping as a function of τ for different values of k. We note that as τ increases, the system loses stability (σ > 0). Again, as k increases, the range of values of τ for which the system is stable decreases. We note that linear stability of an equilibrium position, when σ is away from zero, guarantees its local stability when the nonlinear terms are included according to the Hartman-Grobman theorem (Hartman (1960); Grobman (1959)). However, linear analysis is inadequate, even locally, when σ is near zero, the Hopf bifurcation. In this case, a nonlinear analysis, as carried out in the next section, is needed. Combining Fig. 4.3(a) and Fig. 4.3(b), we obtain the contour plot in Fig The darker the

66 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS τ = 0.20 τ = 0.25 τ = 0.30 τ = 0.35 σ vs k for different values of τ k = 0.2 k = 0.3 k = 0.4 k = 0.5 k = 0.6 σ vs τ for different values of k σ σ k τ (a) (b) Figure 4.3: Variation of the dominant damping or growth rate σ with k and τ: (a) variation of the dominant damping or growth rate with k for several values of τ and (b) variation of the dominant damping or growth rate with τ for several values of k. shading in Fig. 4.4 is, the more damping is introduced by the controller. We can use Fig. 4.4 to easily choose a gain-time-delay combination to produce the desired controller damping. A block diagram of the time-delayed feedback controller implementation is shown in Figure Classical controller To develop a classical controller, we use the linearized model, (3.30) of a simple pendulum. The equation of motion is given in the form φ + 2µ φ + ω 2 0 φ = β f (4.14)

67 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 56 Figure 4.4: A contour plot of the damping as a function of the controller gain k and delay τ where T is the natural period of the uncontrolled system (Henry et al., 2001) x o - 1 s TD control plant l delay calc φ f Figure 4.5: A block diagram of the time-delayed feedback controller.

68 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 57 x o + - k i 1 s s plant k c φ f Figure 4.6: A block diagram of the classical feedback controller where ω 0 is the natural frequency of the pendulum. The transfer function of the system in (4.14) can be expressed as G(s) = Φ F = s 2 s 2 + 2µs + ω 2 0 (4.15) Figure 4.6 shows a proposed block diagram of the control system. The outer loop has an integrator to track the trolley position based off the commanded input. The transfer function of the controlled system now becomes G c (s) = Ks s 2 + 2µs + ω 2 0 (4.16) where K is a linear combination of k i and k c, Fig The controller gains, k c and k i, affect the damping and settling time of the controlled system Linear quadratic regulator (LQR) In order to use LQR, we rewrite (3.30) as a standard linear state-space system ψ = Aψ + B f (4.17) y = Cψ

69 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 58 where ψ = [φ, φ], A is the state matrix, B is the input vector, and C is the output vector; they are given by A = B = C = 0 1 g(l+a2 R) 2µ (l ar) [ l ar 1 0 ] (4.18) The control input f is given by f = u c + x 0 (4.19) where u c is the control input determined by the LQR implementation. The LQR formulation provides a controller that minimizes the sway, but it does not provide for tracking. Hence, we add an integrator tracking controller to the LQR formulation by adding another state. Since this controller is supposed to be between the operator and the trolley and the operator commands velocity, the tracking should be on the commanded velocity. Therefore, we choose the integrator control of the error of the trolley velocity in the form ξ = ( x 0 f)dt (4.20) However, f is not a state or an input to the system in (4.17). Therefore, we differentiate (4.20) twice with respect to time and obtain ψ 3 = d2 ξ dt 2 = ẍ 0 f (4.21) Combining (4.21) and (4.17) yields ψ = g(l+a2 R) 2µ 0 (l ar) 2 ψ + [ 0 ] 0 1 y = ψ 0 1 l ar 1 f (4.22)

70 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 59 k ẋ s - plant k 1 φ f k 2 φ Figure 4.7: A block diagram of the LQR feedback controller Now, we apply the LQR cost function J = 1 2 (x T Q c x + u T R c u)dt (4.23) to (4.22), where Q c is some positive definite matrix and R c is some positive semi-definite matrix. Substituting the result into (4.19) produces a feedback control law in the form f = k 1 ψ 1 k 2 ψ 2 k 3 ψ 3 (4.24) where k 1, k 2, and k 3 are gains derived by solving the Riccati equation. A block diagram of the LQR feedback controller implementation is shown in Figure Numerical simulations Numerous test cases were simulated to compare the three control strategies. We present two of them here. In the first test case, the payload is held at a constant height of 35 m below the trolley. The trolley is moved 50 m in 21.5 s, Fig. 4.8(a). In the second test case, the trolley is moved as in the first case, Fig. 4.8(a). However, the payload is now raised 15 m in the first 10 s. Then it is held at 20 m below the trolley for 2.5 s. Finally, the payload

71 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 60 is lowered back to 35 m below the trolley in 16 s, for a total maneuvering time of 28.5 s, Fig. 4.8(b). The controllers are evaluated based on the resulting payload sway, the settling time of the sway, and the trolley acceleration. The control requirement is that within 5 seconds of the end of the trolley maneuver, the payload sway should drop below 5 cm. Furthermore, the error in the trolley end position should be the same order as the payload sway. The trolley acceleration determines how much torque is required from the trolley motors. Higher accelerations mean increased power requirements and a need for larger motors. The controllers are implemented as follows. The classical controller with a desired damping coefficient of results in k i = 0.24 and k c = The time-delayed position feedback controller is gain scheduled based on the period, by changing the delay used according to the length of the hoisting cables. The LQR gains are given by k 1 = 0.073, k 2 = 0.924, and k 3 = For all three controllers, µ = The classical controller and LQR controllers are not gain scheduled, because gain scheduling actually reduced the performance of the controllers. The LQR cost function requires the choice of Q c and R c. The values chosen for the simulations are Q c = , R c = 2 (4.25) Changes in Q c and R c affect the system performance. Small changes in Q c and R c have large effects on the system response. Figures 4.9 and 4.10 summarize the results for the first case. Figure 4.9(a) shows that all three controllers provide the same amount of sway control during the maneuver. Figure 4.9(b) depicts the settling time of the three controllers. It shows that the oscillations settle the fastest in the case of the time-delayed position feedback control, with a settling time of 28 s. The performances of the classical and LQR controllers are almost identical, with a

72 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS 61 acceleration (m/s 2 ) Horizontal Acceleration, Velocity, and Position Profile acceleration (m/s 2 ) Vertical Acceleration, Velocity, and Position Profile velocity (m/s) 2 0 velocity (m/s) position (m) position (m) time (seconds) time(seconds) (a) Commanded horizontal motion profile (b) Commanded vertical motion profile Figure 4.8: Acceleration, velocity, and position profiles for the horizontal and vertical maneuvers of the trolley and payload. 4 3 Overall Payload Sway uncontrolled linear control time delay LQR settling time spec. Overshoot of the Payload Sway settling boundary sway (meters) 1 0 sway (meters) time (seconds) 0.08 linear control time delay LQR time (seconds) (a) Payload sway. (b) Settling time of the payload sway. Figure 4.9: Payload sway for the no hoist simulation.

73 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS Trolley Accelerations uncontrolled linear control time delay LQR 0.2 acceleration (m/s 2 ) time (seconds) Figure 4.10: Trolley accelerations for the no hoist simulations. settling time of 29 s. The settling time for all three controllers are beyond the settling time specification, but the time-delayed position feedback is the best. Finally, Fig shows the trolley accelerations for the three controllers. All three controllers have smooth acceleration profiles. The results of the classical and LQR controllers are almost identical. If we let the accelerations of the classical and LQR controllers approach the maximum allowed value, the settling time of the system increases drastically. Figures 4.11 and 4.12 compare the simulation results for the second case. Similar to the nohoist case, all three controllers have a similar performance with respect to the payload sway, as shown in Fig. 4.11(a). It shows that the pendulations for all three controllers settle below 5 cm before the settling time requirement of 33.5 s. In this case, the time-delayed position feedback controller has a significant edge (4.5 s) over the classical and LQR controllers. Figure 4.12 shows the trolley accelerations for the three controllers. Similar to the no-hoist case, the classical and LQR controllers have very smooth profiles. The time-delay controller does not have as smooth a profile, but it is below the maximum acceleration. Similarly to the no-hoist case, if the accelerations of the classical and LQR controllers approach the maximum acceleration, the settling time increases.

74 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS Overall Payload Sway uncontrolled linear control time delay LQR Overshoot of the Payload Sway sway (meters) sway (meters) settling time spec time (seconds) settling boundary 0.08 linear control time delay LQR time (seconds) (a) Payload sway. (b) Settling time of payload sway. Figure 4.11: Payload sway for the hoist simulations Trolley Accelerations uncontrolled linear control time delay LQR 0.2 acceleration (m/s 2 ) time (seconds) Figure 4.12: Trolley accelerations for the hoist simulations.

75 CHAPTER 4. COMPARISON OF THREE FEEDBACK CONTROLLERS Conclusions We compare the performance of the time-delayed position feedback controller with two feedback schemes for controlling container cranes: a classical feedback controller and a linear quadratic regulator controller. The performance metrics used to evaluate the controllers are the payload sway, the settling time of the sway, and the trolley acceleration. It follows from two test cases in this dissertation that the performances of the classical and LQR controllers with respect to the payload sway and settling time are nearly identical. However, the performance of the time-delayed position feedback controller using the same metrics is better. On the other hand, the classical and LQR controllers have better acceleration profiles than the time-delayed position feedback controller. The difficulty in implementing the LQR controller, with the trial-and-error approach to determining the best gains, is not worth the effort to produce a performance that can be easily obtained with the classical controller for this system.

76 Chapter 5 Implementation on a container crane Installing a control system on a real crane is a daunting task. Simulations and experiments give a good foundation for theory and what to expect, however many things that work in the laboratory do not work in an industrial setting. The following work described in this chapter was performed in collaboration with Dr. Ziyad Masoud of Amman, Jordan and Mr. Said Daour of Jeddah, Saudi Arabia. The first thing that strikes a person is the scale and size of the actual crane. When running a simulation, a 100 m track and a 50 m hoist does not seem that massive. For reference, Fig. 5.1 shows the Shanghai Zenhua Port Machinery (ZPMC) crane on which we installed the control system. 5.1 System overview Preparing for the system installation, we determined that there were five major components: sway sensors, trolley position, hoist length, operator interface, and standalone control hardware. The sway sensors, trolley position, and hoist length are used to measure the feedback parameters for the control system. The operator interface is the unit that allows the op- 65

77 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 66 Figure 5.1: Jeddah Shanghai Zenhua Port Machinery (ZPMC) crane number 25 preparing to begin transfer operations over a panamax class container ship. erator to engage the control system and provides some virtual memory positions. Finally, the standalone control hardware is used for putting the system together so that it is small, dummy-proof, and maintenance free. Figure 5.2 shows an overview of the system architecture. A signal from the crane hoist cam, essentially a large potentiometer, Fig. 5.2(1), is used to measure the hoist length. The trolley position is measured using an optical encoder, Fig. 5.2(3), and a proximity switch, Fig. 5.2(2), is used for calibration. Two sway sensors, Fig. 5.2(4,5), are used to measure the angles of the front and rear hoisting cables. The signals from these sensors are routed to a junction box, Fig. 5.2(6), which is mounted on the outside of the trolley. The cargo status indicators (locked, unlocked, and landed) are taken directly from the crane programmable logic controller (PLC), Fig. 5.2(8). The operator commands are taken from the optical encoder in the trolley joystick, Fig. 5.2(10). Finally, all of these signals, along with the operator interface panel, Fig. 5.2(9), are input to the control processing unit (CPU), Fig. 5.2(7).

78 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 67 Figure 5.2: An overview of the system architecture Sway sensors The sway sensors we decided to use are high-resolution hoist-cable riders installed underneath the trolley as a pair to continuously measure the sway angles of the hoisting cables. The sway sensors measure the cable angle through mechanical linkages attached to the crane hoist cables. In a live, full-scale test of a quay-side container crane, a contact sway sensor must be rugged and able to withstand very large forces from multiple sources. Therefore, we custom designed these sensors and made them from stainless steel, high strength aluminum alloy, and a few nylon components. The forces on the sensors arise from the hoist cables. The cables are 30mm thick greased, braided, and galvanized steel. The braided cables generate, whenever the crane moves, a motion similar to that of a very high-speed screw on anything touching the cables. Also, whenever the crane base is moved perpendicular to the motion of the trolley (gantry), oscillations can be induced in that direction, which are not controllable. Furthermore, due to day-to-day operations, other uncommon forces can be exerted on the system. For example, if the operator lets the hoist cables slack and then lifts cargo, the cables snap much like a whip.

79 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 68 Figure 5.3: Complete sway sensor design. Design Figure 5.3 shows the complete design of the sway sensor assembly. All of the materials used were either high-strength stainless steel (316) or aircraft-grade aluminum (7075-T6). The bottom assembly can be seen in Figures 5.4 and 5.5, whereas the top assembly can be seen in Figure 5.6. The sensor has two degrees of freedom: in-plane with the trolley (and sway motion of the cargo) and out-of-plane with the trolley to compensate for the gantry motion. The hoist cable passes through two spring-loaded pulleys in the bottom assembly. The entire assembly weighs approximately 85 lbs.

80 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 69 Figure 5.4: Isometric view of the bottom part of the sway sensor. Figure 5.4 shows the configuration for the bottom assembly. There are two spring-loaded pulleys sandwiched between two aluminum plates, held together by stainless steel rods. The spring loaded pulleys keep the sensor firmly attached to the hoist cable and, at the same time, absorb most of the forces from the braided cable. The bronze colored rollers, made out of nylon, are used to prevent any snapping force from the cables striking the rest of the assembly and to make sure that the sensor alignment is correct in the horizontal axis. Figure 5.5 shows a top view of the bottom assembly. The plate behind the assembly is attached to a four-bar mechanism, which is connected to the top assembly. The four-bar mechanism is used to strengthen the sensor assembly and to keep it properly aligned. Figure 5.6 shows the top assembly. An absolute encoder is mounted on the shaft go between the Y shaped plate and arms on the bottom part. This is the direction of motion that we want to measure. This assembly is mounted directly to the crane structure using a dovetail mount. The materials in the dovetail are stainless steel. Aluminum is used on the lower Y shaped plate and arms to minimize weight.

81 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 70 Figure 5.5: Top view of the bottom part of the sway sensor. Figure 5.6: Isometric view of the top part of the sway sensor.

82 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 71 Figure 5.7: Both sway sensors mounted on the underside of the trolley. Implementation The two sway sensors are located on the bottom side of the trolley next to the left hoisting cable opening. Figure 5.7 shows this configuration on the trolley. Figure 5.8 shows a close up of the sensor attached underneath the trolley on the rear hoist cable. Figure 5.9 shows one of the sway sensors attached to the hoist cable. The pulleys clamping onto the hoist cable are visible in Fig. 5.9(1,2). The stainless steel reinforcements for the pulleys are visible in Fig. 5.9(3,4).

83 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 72 Figure 5.8: Complete sway sensor attached to the underside of the trolley. Figure 5.9: Bottom assembly of the sway sensor, demonstrating how the cable is lined up.

84 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 73 Figure 5.10: Trolley encoder Trolley position The easiest method to measure the trolley position is to use an optical encoder mounted to the drive shaft of one of the two trolley motors. This can be seen in Fig Because optical encoders reinitialize every time they are powered on, a calibration method is needed. To this end, we installed a capacitive proximity switch, Fig onto the trolley. The switch is triggered by a large bolt on a custom mount along the trolley track, Fig Operator cabin Figure 5.13 shows the operator cabin with the operator interface panel and the CPU installed. The operator interface includes two virtual memory locations, which place stops on the operator input. For example, the operator moves the trolley over a truck lane and stores that location in memory. The operator now cannot move backwards past the stop, enabling the operator to essentially hold the joystick in full reverse to return to the truck. Similarly, a memory position can be stored over the ship for a specific stack. This cuts down the time wasted in positioning a container over its drop position and reduces operator stress. The operator remains in the loop the entire time. That is, for the trolley to move, the

85 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 74 Figure 5.11: Trolley proximity calibration switch. Figure 5.12: Trolley proximity calibration trigger.

86 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 75 Figure 5.13: Operator cabin with equipment installed.

87 CHAPTER 5. IMPLEMENTATION ON A CONTAINER CRANE 76 operator has to command it via joystick. This ensures the ability of the operator to suddenly stop or make very small incremental changes in position. The operator controls the hoisting maneuvers independently of the control system. The operator can perform simultaneously hoisting and trolley maneuvers without compromising the controller performance. 5.2 Results The time-delayed position feedback control system dramatically reduces payload oscillations during and at the end of transfer maneuvers. Using the control system, the trolley can be operated safely at full speed without the need for any special maneuvering techniques to reduce the sway. A side benefit of the control system is that it actually smooths the trolley motion and hence reduces the force and wear and tear on the trolley motors and hoist cables. Furthermore, it reduces the stress on crane operators and gives them the confidence to utilize the full speed of the crane. At the end of any transfer maneuver, the control system brings the payload to a complete stop within 50 mm. Inching maneuvers for payload position adjustments are smooth, precise, and effortless. Trolley travel and payload hoisting maneuvers can be performed simultaneously without compromising the controller performance. Even when suddenly releasing the trolley joystick while the trolley is moving at full speed, the controller guarantees smooth and sway-free payload stop. Immediately following installation of the system, operators found no less than 50% increase in crane productivity. That is, before installation, the average operator could perform an average of 25 moves per hour. After installation, the average operator can achieve moves per hour. Effectively, 50% more crane is obtained without occupying additional berth space and without requiring additional power or support personnel.

88 Chapter 6 Analysis of the General Equation In this chapter, we consider the linear and locally nonlinear behavior of the homogeneous equation (1.1). First, we investigate the stability of the trivial solution and determine the set of parameters κ 1, γ 1, and τ defining the boundary separating the stability and instability regions. Then, we determine the damping (growth) near this boundary and show that the trivial solution loses stability via a Hopf bifurcation. Finally, we use the method of multiple scales (Nayfeh, 1973, 1993a,b) to determine the normal form of the bifurcation. 6.1 Linear Analysis In this section, we examine the stability of the trivial solution as a function of the parameters κ 1, γ 1, and τ. The linearized system (1.1) can be written as ü(t) + 2µ u(t) + ω0u(t) 2 + κ 1 u(t τ) + γ 1 ü(t τ) = 0 (6.1) Theorem Let σ and ω be the linear damping and frequency of (6.1), then A. the linear stability boundary is given by ω 2 0 ω2 + ( κ 1 γ 1 ω 2) cosωτ = 0 77

89 CHAPTER 6. ANALYSIS OF THE GENERAL EQUATION 78 2µω ( κ 1 γ 1 ω 2) sin ωτ = 0 B. the analytical expressions for σ and ω near the stability boundary σ = 0, ω = ω c, κ 1 = κ 1c + ǫκ 11, and γ 1 = γ 1c + ǫγ 11 are given by σ = ǫσ 1 and ω = ω c + ǫω 1 where σ 1 = [2µ cos(ω cτ) 2ω c sin(ω c τ) τκ 1c + τω 2 c γ 1c](κ 11 ω 2 c γ 11) and ω 1 = 2 [µ sin(ω cτ) + ω c cos(ω c τ) + ω c γ 1c ] (κ 11 ω 2 c γ 11) = 4µ 2 + 4ω 2 c + τ2 κ 2 1c + 4ω2 c γ2 1c 4µτ cos(ω cτ)κ 1c + 8 sin(ω c τ)µω c γ 1c + 4τω c sin(ω c τ)κ 1c +8ω 2 c cos(ω c τ)γ 1c + 4µτω 2 c cos(ω c τ)γ 1c 4τω 2 c sin(ω c τ)γ 1c 2τ 2 ω 2 cκ 1c γ 1c + τ 2 ω 4 cγ 2 1c Proof of Theorem A. We seek the solution of (6.1) in the form u = Ae (σ+jω)t (6.2) Substituting (6.2) into (6.1) and separating real and imaginary parts yields σ 2 + ω 2 0 ω2 + 2µσ + [ κ 1 + γ 1 (σ 2 ω 2 ) ] e στ cosωτ + 2σωγ 1 e στ sin ωτ = 0 (6.3) 2σω + 2µω [ κ 1 + γ 1 (σ 2 ω 2 ) ] e στ sin ωτ + 2σωγ 1 e στ cosωτ = 0 (6.4) For a given τ, γ 1, and κ 1, (6.3) and (6.4) yield σ and ω. When σ > 0, the origin is unstable; and when σ < 0, the origin is asymptotically stable. The special value σ = 0 corresponds to neutral stability. Letting σ = 0 in (6.3) and (6.4) yields the following equations defining the stability boundary: ω0 2 ω2 + ( κ 1 γ 1 ω 2) cosωτ = 0 (6.5) 2µω ( κ 1 γ 1 ω 2) sin ωτ = 0 (6.6)

90 CHAPTER 6. ANALYSIS OF THE GENERAL EQUATION 79 Proof of Theorem B. Next, we introduce a small nondimensional parameter ǫ as a bookkeeping device and perturb κ 1 and γ 1 by small amounts and obtain κ 1 = κ 1c + ǫκ 11 and γ 1 = γ 1c + ǫγ 11 (6.7) where κ 1c and γ 1c correspond to points on the stability boundary. Moreover, we let σ = ǫσ 1 and ω = ω c + ǫω 1 (6.8) Substituting (6.7) and (6.8) into (6.3) and (6.4) and equating the coefficient of ǫ in each equation equal to zero, we obtain ( 2µ Cτκ1c + 2Sω c γ 1c + Cτω 2 cγ 1c ) σ1 ( 2ω c + Sτκ 1c + 2Cω c γ 1c Sτω 2 cγ 1c ) ω1 = C ( κ 11 ω 2 c γ 11) (6.9) ( 2ωc + Sτκ 1c + 2Cω c γ 1c Sτωc 2 γ ) 1c σ1 + ( 2µ Cτκ 1c + 2Sω c γ 1c + Cτωc 2 γ ) 1c ω1 = S ( κ 11 ω 2 cγ 11 ) (6.10) where C and S stand for cos(ω c τ) and sin(ω c τ). Solving (6.9) and (6.10) yields where σ 1 = (2µC 2ω cs τκ 1c + τωcγ 2 1c )(κ 11 ωcγ 2 11 ) ω 1 = 2 (µs + ω cc + ω c γ 1c )(κ 11 ωcγ 2 11 ) (6.11) (6.12) = 4µ 2 +4ω 2 c + τ2 κ 2 1c + 4ω2 c γ2 1c 4µτCκ 1c + 8Sµω c γ 1c + 4τω c Sκ 1c + 8ω 2 c Cγ 1c +4µτω 2 c Cγ 1c 4τω 2 c Sγ 1c 2τ 2 ω 2 c κ 1cγ 1c + τ 2 ω 4 c γ2 1c (6.13) It follows from (6.7), (6.8), (6.11), and (6.12) that d dκ 1 (σ + jω) = d dγ 1 (σ + jω) = d dκ 11 (σ 1 + jω 1 ) (6.14) d dγ 11 (σ 1 + jω 1 ) (6.15) are different from zero. Hence, the trivial solution loses stability with two complex conjugate eigenvalues transversely crossing from the left-half to the right-half of the complex plane. Therefore, the stability boundary is the locus of Hopf bifurcations.

91 CHAPTER 6. ANALYSIS OF THE GENERAL EQUATION Nonlinear Analysis Theorem The normal form for the Hopf bifurcation of the trivial solution of (6.1) is a = σ 1 a χ 3a 3 β = ω χ 2a 2 where a and β are measures of the amplitude and phase of the generated limit cycle, σ 1 and ω 1 are defined in (6.11) and (6.12), and χ 3 = ( 2µ Cτκ 1c + 2Sω c γ 1c + Cτω 2 cγ 1c ) Λr / + ( 2ω c + Sτκ 1c + 2Cω c γ 1c Sτω 2 cγ 1c ) Λi / χ 2 = ( 2µ Cτκ 1c + 2Sω c γ 1c + Cτω 2 cγ 1c ) Λi / ( 2ω c + Sτκ 1c + 2Cω c γ 1c Sτω 2 cγ 1c ) Λr / Proof of Theorem We use the method of multiple scales and seek an approximate solution of (1.1) in the form u(t; ǫ) = ǫu 1 (T 0, T 2 ) + ǫ 2 u 1 (T 0, T 2 ) + ǫ 3 u 2 (T 0, T 2 ) + (6.16) u(t τ; ǫ) = ǫu 1 ( T0 τ, T 2 ǫ 2 τ ) + ǫ 2 u 2 ( T0 τ, T 2 ǫ 2 τ ) + ǫ 3 u 3 ( T0 τ, T 2 ǫ 2 τ ) + (6.17) where T 0 = t and T 2 = ǫ 2 t. We note that there is no dependence on T 1 because as shown below secular terms appear at O(ǫ 3 ) and not at O(ǫ 2 ). The time derivative is transformed as follows: d dt = + ǫ 2 + = D 0 + ǫ 2 D 2 + (6.18) T 0 T 2 Substituting (6.16)-(6.18) into (1.1), expanding the result for small ǫ, and equating coefficients of like powers of ǫ, we obtain Order ǫ D 2 0 u 1 + ω 2 0 u 1 + κ 1c u 1τ + γ 1c D 2 0 u 1τ = 0 (6.19)

92 CHAPTER 6. ANALYSIS OF THE GENERAL EQUATION 81 Order ǫ 2 D 2 0 u 2 + ω 2 0 u 2 + κ 1c u 2τ + γ 1c D 2 0 u 2τ = δ 1 u 2 1 δ 2 (D 0 u 1 ) 2 δ 3 u 1 D 2 0 u 1 κ 2 (u 1τ u 1 ) 2 (6.20) Order ǫ 3 D0u ω0u κ 1c u 3τ + γ 1c D0u 2 3τ = 2D 0 D 2 u 1 2γ 1c D 0 D 2 u 1τ κ 11 u 1τ γ 11 D0u 2 1τ +κ 1c τd 2 u 1τ + γ 1c τd0 2 D 2u 1τ 2δ 1 u 1 u 2 2δ 2 (D 0 u 1 ) (D 0 u 2 ) δ 3 u 1 D0 2 u 2 δ 3 u 2 D0 2 u 1 2κ 2 (u 1τ u 1 )(u 2τ u 2 ) α 1 u 3 1 α 2 u 1 (D 0 u 1 ) 2 α 3 u 2 1D0u 2 1 κ 3 (u 1τ u 1 ) 3 γ 2 u 1τ (D 0 u 1τ ) 2 γ 3 u 2 1 D2 0 u 1τ γ 4 u 2 1τ D2 0 u 1τ + κ (D 0 u 1 ) 3 (6.21) where u mτ stands for u m (T 0 τ, T 2 ). The general solution of (6.19) can be expressed as u 1 = A(T 2 )e jωct 0 + Ā(T 2)e jωct 0 + Σ m=1 [ Am (T 2 )e (σm+jωm)t 0 + Ām(T 2 )e (σm jωm)t 0 ] (6.22) where ω c is the critical frequency corresponding to σ = 0 on the stability boundary and it is given by (6.5) and (6.6); the σ ± jω m are the remaining roots of (6.3) and (6.4); and the function A(T 2 ) is determined by eliminating the secular terms at O(ǫ 3 ). Near the stability boundary, all of the eigenvalues have negative real parts except the eigenvalue corresponding to ω c, which changes sign as the stability boundary is crossed. Hence, as time increases all of the terms in (6.22) decay with time, leaving only the first two terms. Therefore, the long-time behavior of the system is given by u 1 = A(T 2 )e jωct 0 + Ā(T 2)e jωct 0 (6.23) Substituting (6.23) into (6.20) yields D 2 0 u 2 + ω 2 0 u 2 + κ 1c u 2τ + γ 1c D 2 0 u 2τ = ( δ 1 + 2κ 2 2κ 2 C + δ 2 ω 2 c) A Ā ( δ 1 + κ 2 2κ 2 e jωcτ + κ 2 e 2jωcτ δ 2 ω 2 c jδ 3 ω 3 c) A 2 e 2jωcT 0 + cc (6.24)

93 CHAPTER 6. ANALYSIS OF THE GENERAL EQUATION 82 where cc stands for the complex conjugate of the preceding terms. Ignoring the homogeneous solution, we write the solution of (6.24) as u 2 = Γ 1 A 2 e 2jωcT 0 + 2Γ 2 AĀ + Γ 1 Ā 2 e 2jωcT 0 (6.25) where Γ 1 = δ 1 + κ 2 2κ 2 e jωcτ + κ 2 e 2jωcτ δ 2 ω 2 c jδ 3 ω 3 c ω 2 0 4ω 2 c + κ 1c e 2jωcτ 4ω 2 cγ 1c e 2jωcτ (6.26) Γ 2 = δ 1 + 2κ 2 2κ 2 C + δ 2 ω 2 c ω κ 1c (6.27) Substituting (6.23) and (6.25) into (6.21) and setting the terms that produce secular terms equal to zero, we obtain the complex-valued normal form ( modulation equation) [ 2jωc + ( 2jω c γ 1c C + τω 2 c γ 1c τκ 1c ) e jω cτ ] A = where [ 2jµω c + ( κ 11 γ 11 ω 2 c) e jω cτ ] A + ΛA 2 Ā (6.28) Λ = 3α 1 + 9κ 3 4δ 1 Γ 2 2δ 1 Γ 1 2κ 2 Γ 1 α 2 ωc 2 + 3α 3 ωc 2 4δ 2 ωcγ 2 1 2jδ 3 ωc 3 Γ 1 3jωc 3 κ ( 2κ 2 Γ 1 + 9κ 3 + γ 2 ωc 2 2γ 3ωc 2 3γ ) 4ωc 2 e jω cτ + ( 2κ 2 Γ 1 3κ 3 + γ 3 ωc) 2 e jω cτ + (2κ 2 Γ 1 + 3κ 3 )e 2jωcτ (6.29) Next, we introduce the polar representation A(T 2 ) = 1 2 a(t 2)e jβ(t 2) (6.30) into (6.28), separate real and imaginary parts, and obtain the real-valued normal form a = σ 1 a χ 3a 3 (6.31) β = ω χ 2a 2 (6.32) where σ 1 and ω 1 are defined in (6.11) and (6.12) and χ 3 = ( ) 2µ Cτκ 1c + 2Sω c γ 1c + Cτωcγ 2 1c Λr / + ( ) 2ω c + Sτκ 1c + 2Cω c γ 1c Sτωcγ 2 1c Λi / (6.33)

94 CHAPTER 6. ANALYSIS OF THE GENERAL EQUATION 83 χ 2 = ( 2µ Cτκ 1c + 2Sω c γ 1c + Cτω 2 c γ 1c) Λi / ( 2ω c + Sτκ 1c + 2Cω c γ 1c Sτω 2 c γ 1c) Λr / (6.34) Corollary The bifurcation is supercritical with respect to the parameter κ 1 when χ 3 is negative and it is subcritical when χ 3 is positive, and the bifurcation is supercritical with respect to the parameter γ 1 when χ 3 is positive and it is subcritical when χ 3 is negative. Proof of Corollary Equations (6.31) and (6.32) constitute the full normal form of the Hopf bifurcation. When κ 1 is the bifurcation parameter, it follows from (6.11) that σ 1 is always positive when κ 11 > 0. Consequently, the trivial solution is stable when κ 11 < 0 and unstable when κ 11 > 0. If χ 3 > 0, it follows from (6.31) that nontrivial solutions exist only when κ 11 < 0. The stability of these solutions can be ascertained by the eigenvalue of the Jacobian in (6.31) evaluated at these equilibrium solutions. The eigenvalue is given by σ 1 + 3χ 3 a 2 = 2σ 1 which is positive when κ 11 < 0. Hence, the nontrivial equilibrium solutions are unstable and the Hopf bifurcation is subcritical. In this case, the unstable limit cycles coexist with the stable trivial solution. On the other hand, if χ 3 < 0, nontrivial solutions of (6.31) exist only when κ 11 > 0 and the eigenvalue is again given by 2σ 1, which is negative when κ 11 > 0. Hence, the nontrivial equilibrium solutions are stable and the Hopf bifurcation is supercritical. In this case, the trivial solution loses stability as κ 1 increases beyond κ 1c, giving way to two small stable (180 0 out-of-phase) limit cycles whose amplitudes increase as κ 1 increases. When γ 1 is the bifurcation parameter, it follows from (6.11) that σ 1 is always negative when γ 11 > 0. Consequently, the trivial solution is stable when γ 11 > 0 and unstable when γ 11 < 0. If χ 3 > 0, it follows from (6.31) that nontrivial solutions exist only when γ 11 > 0. The stability of these solutions can be ascertained by the eigenvalue of the Jacobian in (6.31)

95 CHAPTER 6. ANALYSIS OF THE GENERAL EQUATION 84 evaluated at these equilibrium solutions. The eigenvalue is given by σ 1 + 3χ 3 a 2 = 2σ 1 which is positive when γ 11 > 0. Hence, the nontrivial equilibrium solutions are unstable and the Hopf bifurcation is subcritical. In this case, the unstable limit cycles coexist with the stable trivial solution. On the other hand, if χ 3 < 0, nontrivial solutions of (6.31) exist only when γ 11 < 0 and the eigenvalue is again given by 2σ 1, which is negative when γ 11 > 0. Hence, the nontrivial equilibrium solutions are stable and the Hopf bifurcation is supercritical. In this case, the trivial solution loses stability as γ 1 decreases below γ 1c, giving way to two small stable limit cycles whose amplitudes increase as γ 1 decreases. It follows from (6.11) and that, when both of κ 1 and γ 1 are used as control parameters, the bifurcation is supercritical when χ 3 < 0 and κ 1 ωcγ 2 1 > 0 and subcritical when χ 3 > 0 and κ 1 ωc 2γ 1 > 0. Because there are too many parameters in the general equation whose values are not known to do numerical analysis, we present numerical solutions for the two special cases of machinetool lathes and time-delayed feedback controlled cranes in Chapters 7and 8.

96 Chapter 7 Analysis of the cutting tool on a lathe For some machining operations, tool vibrations decay with time, the trivial solution is a stable equilibrium solution, and hence the machined surface is smooth. For other operating conditions, the trivial solution is unstable and the tool undergoes large oscillations called chatter, which might be periodic, quasiperiodic, or chaotic, and hence the surface finish will not be smooth. Therefore, chatter must be avoided or mitigated to maintain machining tolerances, preserve surface finish, and prevent tool breakage. As discussed in Section 2.2, the predominant mechanism for machine-tool chatter is regeneration, which occurs when a cut overlaps a previous cut that left some waves in the material, which are regenerated at each subsequent pass of the tool. Due to chatter, there are three operating regions for the cutting tool on a lathe due to chatter: an unconditionally stable region, a conditionally stable region, and an unconditionally unstable region. To avoid chatter, machinists usually restrict their operation to the unconditionally stable region, which might be small. In this chapter, we use a systematic approach to analyze the local and global stability and bifurcation of a cutting tool on a lathe as a function of the width of cut w and the spindle speed 1/τ or time delay τ and investigate methods of expanding the unconditionally stable region and shrinking the conditionally stable region. 85

97 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 86 First, we use perturbation methods (Nayfeh, 1973, 1993a) to derive closed-form expressions for the growth (damping) rate and frequency near the stability boundary, use these expressions to show that the machine tool loses stability as a result of a pair of complex-conjugate eigenvalues moving transversely from the left-half to the right-half of the complex plane, and hence prove that the trivial solution loses stability via a Hopf bifurcation. Second, we use the method of multiple scales to determine the normal form of the Hopf bifurcation and use it to ascertain whether the Hopf bifurcation is supercritical or subcritical. Third, we use the method of harmonic balance to calculate small- and large-amplitude limit cycles and use Floquet theory to ascertain their stability. We validate the analytical results using numerical simulations. We use the analytical and numerical results to generate bifurcation diagrams, which show that the machine-tool response is globally subcritical even in the cases in which the Hopf bifurcation is locally supercritical. We present results showing multiple coexisting subcritical branches of large-amplitude responses, which undergo cyclic folds, secondary Hopf bifurcations, torus-doubling bifurcations, culminating in chaos. Then, we investigate whether cubic-velocity feedback can be used to expand the unconditionally stable region and shrink the conditionally stable region. Finally, we investigate whether time-delayed acceleration feedback can be used to widen the linearly stable region. 7.1 Linear Analysis The linear equation of motion for a cutting tool on a lathe is given by ẍ(t) + 2ξẋ(t) + p 2 (1 + w)x(t) = p 2 wx(t τ) (7.1) where x(t) is the displacement of the tool-tip, ξ is the damping, p is the natural frequency of the tool, w is the width of cut, and τ is the time delay. We seek the general solution of (7.1) in the form x(t) = Ce σt cos(ωt + η) (7.2)

98 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 87 where C, σ, ω, and η are real constants. Substituting (7.2) into (7.1) and setting the coefficients of sin(ωt + η) and cos(ωt + η) equal to zero yields p 2 (1 + w) + 2ξσ + σ 2 ω 2 e στ cos(ωτ) = 0 (7.3) 2(ξ + σ)ω e στ p 2 w sin(ωτ) = 0 (7.4) For a given w and time delay τ, (7.3) and (7.4) can be solved numerically for σ and ω. Then, C and η can be determined from initial conditions. When σ > 0 the system grows exponentially with time and it is unstable; when σ < 0 the system decays exponentially with time and it is stable; and σ = 0 defines the stability boundary. To locate this boundary, we let σ = 0 in (7.3) and (7.4) and obtain p 2 (1 + w w cos(ωτ)) ω 2 = 0 (7.5) 2ξω + p 2 w sin(ωτ) = 0 (7.6) The numerical solution of (7.5) and (7.6) yields the critical values ω c, critical frequency, and w c, critical width of cut, for a given τ separating stable from unstable trivial solutions. Near the boundary, w = w c, we determine analytical expressions for the damping rate σ and frequency ω for a given time delay τ. To this end, we introduce a detuning parameter w 2 to express the nearness of the width of cut w to its critical value w c ; that is w = w c + ǫw 2, where ǫ is a small nondimensional parameter. Furthermore, we let σ ǫσ and ω ω c + ǫγ in (7.3) and (7.4), expand the results in Taylor series, equate the coefficients of ǫ to zero, and obtain 2ξσ 2γω c + p 2 [w 2 (στw c w 2 )cos(ω c τ) + γτw c sin(ω c τ)] = 0 (7.7) 2(γξ + σω c ) + p 2 [(στw c w 2 ) sin(ω c τ) γτw c cos(ω c τ)] = 0 (7.8) Solving (7.7) and (7.8) for σ and γ yields σ = p2 w 2 [ 2ξ + 2ξ cos(ω c τ) + p 2 τw 2 p 2 τw c cos(ω c τ) 2w c sin(ω c τ)] p 4 τ 2 w 2 c + 4p 2 τw c [ξ cos(ω c τ) ω c sin(ω c τ)] + 4(ξ 2 + ω 2 c) (7.9) γ = p2 w 2 [p 2 τw c sin(ω c τ) + 2ξ sin(ω c τ) + 2ω c (cos(ω c τ) 1)] p 4 τ 2 w 2 c + 4p2 τw c [ξ cos(ω c τ) ω c sin(ω c τ)] + 4(ξ 2 + ω 2 c ) (7.10)

99 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 88 Equations (7.9) and (7.10) define, respectively, the growth rate and change in the frequency of the system near the boundary. It follows from (7.2), (7.9), and (7.10) that d(σ + jω) d(σ + jω) dw = k=kc ǫdw 2 = k=kc { [ p 2 2ξ + 2ξ cos(ω c τ) + p 2 τw 2 p 2 τw c cos(ω c τ) 2w c sin(ω c τ) ] +jp [ 2 p 2 τw c sin(ω c τ) + 2ξ sin(ω c τ) + 2ω c (cos(ω c τ) 1) ]} / {p 4 τ 2 wc 2 + 4p 2 τw c [ξ cos(ω c τ) ω c sin(ω c τ)] + 4(ξ 2 + ωc) } 2 which is different than zero. Therefore, the trivial solution loses stability with two complex conjugate eigenvalues transversely crossing the imaginary axis, and hence the trivial solution undergoes a Hopf bifurcation at w = w c, and the stability boundary is the locus of Hopf bifurcations. 7.2 Nonlinear Analysis The work in this section closely follows that of Nayfeh et al. (1997), correcting some mistakes in the forumalation of the normal form. The nonlinear equation of motion for a cutting tool on a lathe is given by ẍ(t) + 2ξẋ(t) + p 2 [x(t) + β 2 x 2 (t) + β 3 x 3 (t)] = p 2 w [x(t) x(t τ) +α 2 [x(t) x(t τ)] 2 + α 3 [x(t) x(t τ)] 3] (7.11) Next, we use the method of multiple scales to determine an approximate solution of (7.11) and hence the normal form of the Hopf bifurcation and its type (subcritical or supercritical). To this end, we introduce three time scales T 0 = t, T 1 = ǫt, T 2 = ǫ 2 t (7.12)

100 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 89 where ǫ is a small nondimensional bookkeeping parameter that will be set equal to unity in the final analysis. Then, it follows from the chain rule that the time derivative is transformed into d dt = T 0 + ǫ T 1 + ǫ 2 T 2 + = D 0 + ǫd 1 + ǫ 2 D 2 + (7.13) Moreover, we transform x(t; ǫ) and x(t τ; ǫ) as x(t; ǫ) x(t 0, T 1, T 2 ; ǫ) and x(t τ; ǫ) x(t 0 τ, T 1 ǫτ, T 2 ǫ 2 τ; ǫ) (7.14) and seek a uniform first-order expansion in the form x(t; ǫ) = x 0 (T 0, T 1, T 2 ) + ǫx 1 (T 0, T 1, T 2 ) + ǫ 2 x 2 (T 0, T 1, T 2 ) + (7.15) and x(t τ; ǫ) = x 0 (T 0 τ, T 1 ǫτ, T 2 ǫ 2 τ) + ǫx 1 (T 0 τ, T 1 ǫτ, T 2 ǫ 2 τ) + ǫ 2 x 2 (T 0 τ, T 1 ǫτ, T 2 ǫ 2 τ) + (7.16) Furthermore, because secular terms appear at O(ǫ 3, we introduce a parameter w 2 to express the nearness of the width of cut w to the Hopf bifurcation value w c, critical cut width, as w = w c + ǫ 2 w 2 (7.17) We note that ǫ 2 is used to capture the dynamics of the quadratic nonlinearities. Substituting ( ) into (7.11), expanding the result for small ǫ, and equating coefficients of like powers of ǫ, we obtain O(ǫ 0 ): D 2 0x 1 (T 0, T 1, T 2 ) + 2ξD 0 x 1 (T 0, T 1, T 2 ) + p 2 (1 + w c )x 1 (T 0, T 1, T 2 ) p 2 w c x 1 (T 0 τ, T 1, T 2 ) = 0 (7.18)

101 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 90 O(ǫ 1 ): D0 2 x 2(T 0, T 1, T 2 ) + 2ξD 0 x 2 (T 0, T 1, T 2 ) + (1 + w c )p 2 x 2 (T 0, T 1, T 2 ) p 2 w c x 2 (T 0 τ, T 1, T 2 ) = 2D 0 D 1 x 1 (T 0, T 1, T 2 ) p 2 τw c D 1 x 1 (T 0 τ, T 1, T 2 ) 2ξD 1 x 1 (T 0, T 1, T 2 ) p 2 (w c α 2 β 2 )x 2 1 (T 0, T 1, T 2 )+2p 2 w c α 2 x 1 (T 0, T 1, T 2 )x 1 (T 0 τ, T 1, T 2 ) p 2 w c α 2 x 2 1 (T 0 τ, T 1, T 2 ) (7.19) O(ǫ 2 ): D0x 2 3 (T 0, T 1, T 2 ) + 2ξD 0 x 3 (T 0, T 1, T 2 ) + p 2 (1 + w c )x 3 (T 0, T 1, T 2 ) p 2 w c x 3 (T 0 τ, T 1, T 2 ) = 2D 0 D 2 x 1 (T 0, T 1, T 2 ) 2ξD 2 x 1 (T 0, T 1, T 2 ) D1 2 x 1(T 0, T 1, T 2 ) 2D 0 D 1 x 2 (T 0, T 1, T 2 ) 2ξD 1 x 2 (T 0, T 1, T 2 ) p 2 w 2 x 1 (T 0, T 1, T 2 )+p 2 w 2 x 1 (T 0 τ, T 1, T 2 ) p 2 τw c D 2 x 1 (T 0 τ, T 1, T 2 ) p 2 τw c D 1 x 2 (T 0 τ, T 1, T 2 )+ 1 2 p2 τ 2 w c D1x 2 1 (T 0 τ, T 1, T 2 ) 2p 2 β 2 x 1 (T 0, T 1, T 2 )x 2 (T 0, T 1, T 2 ) 2p 2 α 2 w c [x 1 (T 0, T 1, T 2 ) x 1 (T 0 τ, T 1, T 2 )][x 2 (T 0, T 1, T 2 ) x 2 (T 0 τ, T 1, T 2 )] p 2 β 3 x 3 1 (T 0, T 1, T 2 ) p 2 α 3 w c [x 1 (T 0, T 1, T 2 ) x 1 (T 0 τ, T 1, T 2 )] 3 2p 2 τw c α 2 [x 1 (T 0, T 1, T 2 ) x 1 (T 0 τ, T 1, T 2 )]D 1 x 1 (T 0 τ, T 1, T 2 ) (7.20) The solution of (7.18) can be expressed as x 1 = A(T 1, T 2 )e jωct 0 + Ā(T 1, T 2 )e jωct 0 (7.21) where ω c is the chatter frequency corresponding to the Hopf bifurcation at cut width w c. Substituting (7.21) into (7.19) yields D0 2 x 2(T 0, T 1, T 2 ) + 2ξD 0 x 2 (T 0, T 1, T 2 ) + p 2 (1 + w c )x 2 (T 0, T 1, T 2 ) p 2 w c x 2 (T 0 τ, T 1, T 2 ) = ( ) 2ξ + p 2 τw c e jωcτ + 2jω c D1 Ae jωct 0 ( p 2 w c α 2 1 2e jω cτ + e 2jωcτ) A 2 e 2jωcT 0 p 2 β 2 A 2 e 2jωcT 0 p 2 [β 2 + 2w c α 2 2w c α 2 cos(ω c τ)] AĀ + cc (7.22) where cc stands for the complex conjugate of the preceding terms. The solvability condition (annihilation of the terms that produce secular terms) for (7.22) demands that D 1 A(T 1, T 2 ) =

102 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 91 0 or A is a function of T 2. Then, the particular solution of (7.22) can be expressed as where x 2 = Γ 1 A 2 (T 2 )e 2jωcT 0 + Γ 2 A(T 2 )Ā(T 2) + cc (7.23) Γ 1 = p2 [w c α 2 (1 2e jωcτ + e 2jωcτ ) + β 2 ] p 2 + p 2 w c (1 e 2jωcτ ) + 4jξω c 4ω 2 c (7.24) Γ 2 = β 2 2w c α 2 + 2w c α 2 cos(ω c τ) (7.25) Substituting (7.21) and (7.23) into (7.20) yields D0 2 x 3(T 0, T 1, T 2 ) + 2ξD 0 x 3 (T 0, T 1, T 2 ) + p 2 (1 + w c )x 3 (T 0, T 1, T 2 ) + p 2 w c x 3 (T 0 τ, T 1, T 2 ) = { } 2ξ + p 2 τw c [cos(ω c τ) j sin(ω c τ)] + 2jω c A e jωct 0 p 2 w 2 [1 cos(ω c τ) + j sin(ω c τ)] Ae jωct 0 + ΛA 2 Āe jωct 0 + NST + cc (7.26) where NST stands for terms that do not produce secular terms and Λ = p 2 w c α 3 ( 9 9e jω cτ 3e jωcτ + 3e 2jωcτ) 3p 2 β 3 2p 2 β 2 (Γ 1 + 2Γ 2 ) 2p 2 w c α 2 Γ 1 ( 1 + e jω cτ e jωcτ e 2jωcτ) (7.27) Eliminating the terms that lead to secular terms in (7.26) yields the solvability condition { 2ξ + p 2 τw c [cos(ω c τ) j sin(ω c τ)] + 2jω c } A where + p 2 w 2 [1 cos(ω c τ) + j sin(ω c τ)] A + (Λ r + jλ i )A 2 Ā = 0 (7.28) Λ r = Re{Λ} = 3p 2 w c α 3 {3 4 cos(ω c τ) + cos(2ω c τ)} 3p 2 β 3 2p 2 β 2 (Γ r + 2Γ 2 ) 2p 2 w c α 2 Γ r [1 cos(2ω c τ)] (7.29) Λ i = Im{Λ} = 3p 2 w c α 3 [2 sin(ω c τ) sin(2ω c τ)] + 2p 2 β 2 Γ i + 2p 2 w c α 2 Γ i [2 sin(ω c τ) sin(2ω c τ)] (7.30)

103 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 92 and Γ r = Re{Γ 1 } and Γ i = Im{Γ 1 }. Expressing A(T 2 ) in the polar form A(T 2 ) = 1 2 a(t 2)e jβ(t 2) (7.31) where a(t 2 ) and β(t 2 ) are real, and separating (7.28) into real and imaginary parts yields [ 2ξ + p 2 τw c cos(ω c τ) ] a [ p 2 τw c sin(ω c τ) 2ω c ] aβ = p 2 w 2 [1 cos(ω c τ)] a 1 4 a3 Λ r (7.32) [ p 2 τw c sin(ω c τ) 2ω c ] a + [ 2ξ + p 2 τw c cos(ω c τ) ] aβ = p 2 w 2 a sin(ω c τ) 1 4 a3 Λ i (7.33) Solving (7.32) and (7.33) algebraically for a and β yields the normal form for the Hopf bifurcation where a = χ 1 a + χ 3 a 3 (7.34) β = χ 2 + χ 4 a 2 (7.35) χ 1 = p2 w 2 [ 2ξ + 2ξ cos(ω c τ) + p 2 τw 2 p 2 τw c cos(ω c τ) 2w c sin(ω c τ)] D χ 2 = p2 w 2 [p 2 τw c sin(ω c τ) + 2ξ sin(ω c τ) + 2ω c (cos(ω c τ) 1)] D χ 3 = Λ r [2ξ + p 2 τw c cos(ω c τ)] Λ i [p 2 τw c sin(ω c τ) 2ω c ] 4D χ 4 = Λ i [2ξ + p 2 τw c cos(ω c τ)] + Λ r [p 2 τw c sin(ω c τ) 2ω c ] 4D (7.36) (7.37) (7.38) (7.39) D = p 4 τ 2 w 2 c + 4p 2 τw c [ξ cos(ω c τ) ω c sin(ω c τ)] + 4(ξ 2 + ω 2 c) (7.40) 7.3 Global Analysis We used a combination of a path following scheme, the normal form, the method of harmonic balance, Floquet theory, and numerical simulations to characterize the local and global

104 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 93 stability and dynamics of cutting tools on lathes using the model and parameters of Hanna and Tobias. We first used the normal form of the bifurcation to calculate the amplitude and frequency of the limit cycle generated by the Hopf bifurcation at values of w near the linear critical value w c. We used these values as initial guess in the implementation of the method of harmonic balance and used Floquet theory to ascertain the stability of the calculated limit cycle. Then, we generated the phase portrait, Poincare section, and power spectrum of the converged solution. Next, we increased or decreased w slightly and used the converged results for the previous value of w as initial guess and repeated the process. At selected values of w, we validated the results of the method of harmonic balance by integrating the full equation using a Runge-Kutta scheme. Beyond the secondary Hopf bifurcations, we used numerical simulations to calculate the solutions and characterize them using their long-time histories, phase portraits, Poincare sections, and power spectra. Also, we randomly selected other initial conditions away from the branch emanating from the first critical width of cut to find a point on another isolated branch. Then, we used a similar approach to trace the solutions on this branch. This way, we were able to find an isolated branch. We discovered that this branch emanated from a larger critical width of cut. Then, we traced other branches emanating from other larger critical widths of cut, thereby generated multiple large-amplitude solutions coexisting with the linearly stable trivial solution. While the first branch was examined in Nayfeh et al. (1997), they did not extend the analysis to the higher branches. Figure 7.1 shows the bifurcation diagram obtained for τ = 1/75 and varying w. As found experimentally, there are three distinct regions of operation. In the interval 0 < w < 0.043, the machine tool is asymptotically and unconditionally (globally) stable. In the interval < w < 0.117, the machine tool is conditionally stable; that is, the tool response depends on the initial conditions. It is stable for small initial conditions, but unstable for large initial conditions and disturbances. In the interval < w <, the tool is unconditionally unstable; that is, the tool is unstable irrespective of the initial conditions. Let us imagine an experiment in which w is slowly increased from zero while τ is fixed at 1/75.

105 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 94 Figure 7.1: Bifurcation diagram obtained for τ = 1/75 and varying values of w.

106 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 95 4 x 10 5 w 0 = 0.075; τ = 1/75; response at A 3 2 displacement time (seconds) Figure 7.2: Time trace and PSD of the tool response corresponding to point A from Fig Initially, tool vibrations decay to zero even in the presence of small or large disturbances; the trivial motion of the tool is globally stable. As w is increased beyond 0.043, the motion-tool vibrations decay as long as the tool does not encounter large disturbances. An example of the time-history of the tool motion is shown in Fig. 7.2, corresponding to point A. Clearly, the tool vibrations decay to zero. As w is increased beyond 0.117, the trivial motion undergoes a supercritical Hopf bifurcation due to a pair of complex-conjugate Floquet multipliers exiting transversely the unit circle away from the real axis, resulting in the generation of two small stable limit cycles; that is two limit cycles of the same amplitude but with different phases. Figure 7.3 shows a representative tool response at point B. The long-time history indicates an isolated stable periodic motion and its power spectrum consists of a fundamental peak at Hz approximately and an approximate amplitude of The phase portrait consists of a closed curve, confirming the periodic nature of the motion. Also, the Poincare section consists of one point, within the accuracy of the simulations. These local results could indicate that there are only two regions of operation: an unconditionally stable region when w < and an unconditionally unstable region when w > This conclusion is

107 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 96 Figure 7.3: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point B from Fig at variance with the experimental observations and shows the necessity of global analysis. Using global analysis, we find that the limit cycles generated by the Hopf bifurcation undergo a cyclic fold due to the collision of the branch of stable limit cycles with a branch of unstable limit cycles, resulting in their mutual destruction. The unstable branch exists in the region < w < and encounters a cyclic fold at w = as a result of its collision with a branch of stable limit cycles. The long-time history, power spectrum, phase portrait, and Poincare section of the motion at point C and shown in Figure 7.4 confirm the periodic nature of the motion. The power spectrum consists of a fundamental frequency of Hz and an approximate amplitude of The phase portrait is a closed curve and the Poincare section consists of a single point within numerical accuracy. Because this branch coexists with the linearly stable trivial solution, the region < w < is a conditionally stable region of operation, in agreement with the experimental observations. As w is increased along this branch, a secondary Hopf bifurcation occurs at w = 0.157, generating a new frequency. If the new frequency is commensurate with the fundamental frequency, the

108 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 97 Figure 7.4: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point C from Fig response would be periodic with large period. If the new frequency is incommensurate with the fundamental frequency, the response would be quasi-periodic: a two-torus. An example of the response is shown in Figure 7.5. The long-time history indicates a modulated motion. The fundamental frequency of approximately Hz now has asymmetric side bands of an approximate frequency Hz, indicating an amplitude and phase modulated motion. The phase portrait indicates a complex aperiodic motion, but the Poincare section indicates that the motion is quasi-periodic because it consists of a large number of points falling on a closed curve. At w the tool motion exhibits chaotic behavior. There are several other branches of larger-amplitude solutions, which coexist with the linearly stable trivial solution. Two of such branches are shown in Fig These branches emanate from the larger linear critical values of w. Each of these branches terminates on the left due to a cyclic fold. The second branch is terminated on the left at w = Stable limit cycles exist in the interval < w < A representative motion in this interval is shown in Fig. 7.6 corresponding to point E. The phase portrait, Fig. 7.6b, and the Poincare map,

109 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 98 Figure 7.5: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point D from Fig Fig. 7.6c, confirm that the tool motion is a stable limit cycle. It follows from Figs 7.6a,d that the amplitude of the limit cycle is approximately and its fundamental frequency is approximately Hz. We note that the amplitudes and frequencies of the limit cycles on this branch are larger than those of the limit cycles on the first branch. These limit cycles encounter a secondary Hopf bifurcation at w 0.138, generating a new frequency approximately equal to 0.02 Hz. The secondary Hopf bifurcation results in the generation of either a periodic motion with larger period or a quasi-periodic motion. In Fig. 7.7, we show the long-time history, phase portrait, power spectrum, and Poincare section of the motion corresponding to Point F. The time history indicates a modulated motion and the phase portrait indicates a complex aperiodic motion. The power spectrum consists of the fundamental frequency at approximately Hz with two asymmetric smaller peaks on either side, equidistant by approximately 0.02 Hz, Fig. 7.7d. The Poincare section confirms that the motion is quasi-periodic because it consists of infinitely many points falling on a closed curve. The quasi-periodic motion exists in the interval < w < and

110 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 99 Figure 7.6: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point E from Fig bifurcates into chaos at w The third branch in Fig. 7.1 terminates on the left at w = due to a cyclic fold. Stable limit cycles exist on this branch in the interval 0.05 < w < In Fig. 7.8, we present characteristics of such limit cycles at point G. The phase portrait, Fig. 7.8b, and the Poincare map, Fig. 7.8c, confirm that the tool motion is a stable limit cycle. Moreover, it follows from Figs. 7.8a,d that the amplitude of the limit cycle is approximately and its fundamental frequency is approximately 0.18 Hz. Again these values are larger than those of the limit cycles on the second branch and in turn larger than those of the limit cycles on the first branch. These limit cycles suffer a secondary Hopf bifurcation at w 0.109, resulting in either a periodic motion with larger period or a quasi-periodic motion. Characteristics of such a quasi-periodic motion are shown in Fig. 7.9 at point H. Figure 7.9a indicates a modulated motion and Fig. 7.9b indicates a complex aperiodic motion. However, the Poincare map, Fig. 7.9c, indicates a quasi-periodic motion because it consists of infinitely many points falling on a closed curve, which seems to make two loops, indicating a torus doubling bifurcation. The power spectrum, Fig. 7.9d, consists

111 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 100 Figure 7.7: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point F from Fig of a fundamental peak at approximately 0.18 Hz and two smaller asymmetric side bands having a frequency approximately Hz. The quasi-periodic motions bifurcate into chaos at w It seems that the tool response is chaotic for values of w > In Figs , we show the long-time histories, phase portraits, Poincare maps, and power spectra of the tool responses at the three points I, J, and K. The time histories and phase portraits indicate complex aperiodic motions. The dense bounded natures of the Poincare sections indicate chaotic motions. The power spectra confirm that the motions are chaotically modulated in nature. Next, we show in Fig the bifurcation diagram obtained for τ = 1/60 and varying w. It is clear that any conclusions based on only the lower branch would be erroneous. The lower branch indicates that there are only two regions of operation: an unconditionally stable region when w < and an unconditionally unstable region when w > 0.075,

112 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 101 Figure 7.8: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point G from Fig Figure 7.9: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point H from Fig. 7.1.

113 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 102 Figure 7.10: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point I from Fig Figure 7.11: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point J from Fig. 7.1.

114 CHAPTER 7. ANALYSIS OF THE CUTTING TOOL ON A LATHE 103 Figure 7.12: Long-time history, phase portrait, Poincare map, and PSD of the motion corresponding to point K from Fig at variance with the conclusions obtained based on the upper branches. Again, all of the branches emanate from the linear critical values of w due to Hopf bifurcations: w = 0.075, w = 0.248, and w = The upper branches are generated by subcritical bifurcations, whereas the lower branch is generated by a supercritical bifurcation. Moreover, except for the lower branch, all of branches terminate on the left due to cyclic folds at w = and w = On all of the branches, the generated stable limit cycles generated either by the supercritical Hopf bifurcation or the cyclic folds undergo secondary Hopf bifurcations at w = 0.141, w = 0.104, and w = All of the quasi-periodic motions bifurcate into chaos at w = 0.235, w = 0.169, and w = on the first, second, and third branches, respectively.