Dynamic stiffness compensation with active aerostatic thrust bearings

Transcription

1 Dynamic stiffness compensation with active aerostatic thrust bearings G. Aguirre, F. Al-Bender, H. Van Brussel K.U.Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 B, B-3001, Heverlee, Belgium Abstract Aerostatic thrust bearings are used in linear guideways as an alternative to contact bearings. The absence of friction in the motion makes aerostatic guideways most suitable for high precision and high speed machines. The low stiffness and damping of the aerostatic support, however, limits their application in machining processes. Active compensation, applied to the bearing films, is proposed as a solution to overcome this limitation and to achieve further performance improvements in terms of stiffness and damping, adding positioning capabilities. Previous research shows that gap shape control is the most promising activation method. However the need for a flexible bearing surface increases the complexity of the design. A simulation model that considers the coupling between the structural, fluid dynamics and piezoelectric aspects has been developed in order to aid in the design of active air bearings. The simulation results show the validity of the coupling strategy and the relevant influence of the flexibility of the bearing plate on the bearing performance. However, the simplified models used, in order to keep short simulation times, and the influence of some phenomena, such as surface roughness, which are not modeled, require that the model accuracy be validated by experimental investigation. For this purpose, a test setup capable of identifying the bearing performance with high bandwidth and resolution has been built. Experimental results show that high dynamic stiffness can be obtained by active compensation. The final performance is highly dependent on the bearing design and the working conditions, and thus the need of a coupled simulation model is confirmed. 1 Introduction Aerostatic thrust bearings are widely used in high precision guideways. Their main advantage over conventional bearings (roller, ball bearings) comes from the absence of contact between the moving and the stationary parts. Thus, linear motion control is not affected by the highly nonlinear friction forces, which characterize contacting bearings, so that high precision positioning and high speeds can be achieved. Such guideways are often found in high precision applications, such as coordinate measuring machines, wafersteppers, etc. On the other hand, aerostatic guideways suffer from low stiffness, and sometimes poor damping, compared to contacting guideways, so that their application in other sectors, e.g. machine tools, is less popular. However, continuously increasing performance requirements, in particular, sub-micrometer precision and high positioning speeds, are pushing traditional guideways to their limits, so that new solutions are needed. Active compensation is a viable candidate. It can be applied to aerostatic bearings, which guide the moving part of the slideway system, in the five degrees of freedom to be blocked, leaving free the linear motion, which is controlled by the motor. If the force in each bearing can be controlled independently, deviations in any of these degrees of freedom can be compensated, and high dynamic stiffness can thus be added to the advantages of aerostatic guideways. Different methods are found in the literature to control the pressure distribution in the air gap, and therefore, 105

2 106 PROCEEDINGS OF ISMA2008 F+ F bearing platen air flow air gap piezo (3x) flexible plate air supply Figure 1: Sketch of an active aerostatic bearing the bearing force, ranging from controlling the feeding pressure [1] to modifying the shape of the bearing gap, either by acting locally around the restrictor [2], or by acting on the periphery of the bearing plate and modifying the conicity [3]. Previous research shows that conicity control is a very effective activation method for both position control and stiffness compensation. Fig. 1 shows a typical implementation of an active aerostatic bearing with conicity control. The bearing face is made as a (thin) plate with a central air feed hole, which is clamped centrally onto a cylindrical column that connects to the back of the bearing. The circular plate is supported on three piezoelectric stack actuators at its periphery so that the bearing face will deform to a concave shape (acquire conicity) when the actuators expand. This actuation produces two benefits: a small conicity in the bearing gap increases the passive stiffness of the bearing [4], and a change in conicity produces a change in the pressure distribution, and thus, in the bearing force. A high precision displacement sensor (e.g capacitive) can be embedded in the bearing or be placed next to it to measure the variation of the air-gap height. The deviations from the desired distance to the platen can be compensated for by controlling the bearing force through acting on the piezo actuators. Previous research on active aerostatic bearings based on conicity control has proved its potential, offering quasi-infinite static stiffness, increased dynamic stiffness up to a few hundred Hz and positioning resolutions of around ten nanometer [3]. These results have been obtained on laboratory test beds, and no design procedure has been defined to optimize their performance, and so to promote their industrial application. Active air bearings are complex mechatronics systems wherein the final performance is governed by different physical phenomena, namely, fluid dynamics for the air flow and pressure distribution in the air gap, structural flexibility for the deformation of the bearing plate due to the piezo actuators and the air pressure, piezoelectricity for the actuators, and control of the system as whole. All these fields need, thus, to be considered in the design of active air bearings. Various parameters need to be tuned for an optimal design, namely, bearing area, feeding pressure, actuator dimensions, plate thickness and geometry, restrictor diameter, etc. With previously existing simulation models, active bearings can be designed sequentially, that is, first considering the fluid dynamics in the same way as for passive bearings, where the bearing plate is rigid, thereafter choosing the thickness of the bearing plate and selecting the actuators, and finally tuning a controller for the system. This strategy can be misleading, as it neglects the influence of the flexibility of the bearing plate on the performance of the bearing. On the one hand, the plate should be as flexible as possible, in order to achieve the maximum stroke from the piezoelectric actuators. On the other hand, a stiff plate is desirable, as the pressure in the air gap will otherwise deform the plate, reducing the conicity (eventually making it even negative) and producing a wavy profile in between the actuators. This paper aims to present the tools needed in order to optimize the design of active air bearings. Simulation tools are needed in order to estimate the final performance and the influence of the different design parameters, and experiments with real prototypes are required in order to validate the simulation results or detect any unmodeled phenomena.

3 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES 107 The simulation model must be able to represent the coupling between the different disciplines relevant for the final performance. A multiphysics model with strongly coupled formulation, considering the structural flexibility, the fluid dynamics, the piezoelectricity and the control is presented in this paper. The focus of the model is not on the detailed modeling of all the phenomena present in the bearing, but to provide meaningful results that can be used in the design stage in order to optimize the performance of the bearing prototypes. The modeling of such bearings encounters some practical limitations, arising from the difficulty to consider the influence of manufacturing errors, such as roughness of the bearing surface or the geometry of the feeding hole, and also from the need of reduced models for fast simulation. Therefore, the simulation work needs to be validated with experimental tests, in order to ascertain the validity of the model. A test setup for single air bearings and a number of active air bearing prototypes have been built in order to perform these tests. In the following, Section 2 describes the FE coupled multiphysics model that has been developed, and the validity of the modeling approach is. Section 3 describes the test setup and the active bearing prototype that have been built, and some experimental results are presented. Finally, conclusions are drawn in Section 4. 2 Simulation work A multiphyisics coupled model for active air bearings is needed in order to solve the design trade-offs and apply an optimal design strategy. Such a model was presented in detail in [5], and is briefly introduced here. 2.1 A multiphysics coupled model for active air bearings The model for active air bearings considers the different physical disciplines relevant to the problem, namely, structural flexibility, fluid dynamics, piezoelectricity and control, representing the interaction between these fields with a strongly coupled formulation. The main goal of the model is to help in optimizing the design of active air bearings. Therefore, the focus is not on representing in detail all the phenomena present in the bearing, but to analyze the effects that are relevant for the global performance of the bearing, with the lowest possible calculation effort. The model is implemented in finite element (FE) formulation, and can be used to solve static equilibrium configurations (by following an iterative procedure due to the nonlinear characteristic of the model), harmonic excitations around an equilibrium configuration for obtaining the linearized dynamic response to small disturbances, and time analysis for considering the nonlinear effects. It can also be incorporated into numerical optimization algorithms. Static and harmonic analysis are discussed in this paper General formulation of the model The finite element formulation of the model, considering the change in pressure distribution δp, the deformation of the bearing plate δh s, the displacement of the counter surface δh cs, and the change in voltage applied to the piezo actuators δv as the unknowns of the problem, has the following general form. [M] δp δh s δh cs δv + [C] δp δh s δh cs δv + [K] δp δh s δh cs δv = {f} (1) The contributions to M, C and K (inertia, stiffness and damping matrices, respectively) and to f (vector of external forces on each degree of freedom) are now presented.

4 108 PROCEEDINGS OF ISMA Structural flexibility The deformation of the bearing plate due to the elongation of the piezoelectric actuators and to the pressure in the air gap is a key element in the performance of the bearing. In order to minimize the simulation time, the plate is represented as a 2D mesh based on triangular shell elements, standard in many finite element software. The formulation of only the vertical displacement of the nodes is presented here for a simplified representation of the full bearing model. [M s ] { δhs } + [C s ] { δh s } + [K s ] {δh s } = 0 (2) where the contributions to the structural degree-of-freedom-equations in (1) are M s, C s and K s, the inertia, stiffness and damping matrices, respectively, of the structural flexibility problem. No external force contribution is considered, as forces coming from the air pressure and from the piezos will appear as internal coupling forces Fluid dynamics The pressure distribution in the bearing can be solved most accurately with computational fluid dynamics (CFD) analysis, solving the Navier-Stokes equations. This is computationally too demanding for the purpose of this model, and such a detailed solution is not necessary. Simplified models have proved successful in representing the global response of the bearing [3]. The flow in the bearing film is assumed to be predominantly viscous, so that the Reynolds equation for fluid film lubrication can be used. This equation assumes that the flow is constrained to the two dimensions of the bearing plane. This is valid for representing the behavior of the air film layer in between the non-permeable bearing surfaces, but not in the area under the restrictor, where there is an air flow coming in perpendicular to the film. The air entrance region is relatively small, so that the error committed by assuming the flow to be viscous everywhere may be negligible, and the Reynolds equation can consequently be applied in all the bearing surface, adding the perpendicular mass flow rate per unit area f r in the area under the restrictor: ( ρh 3 12µ p ρh 2 V ) + f r = ρh (3) t where the nabla operator acts only on the bearing plane co-ordinates, ρ is the density of the air, p is the pressure, h is the air gap height, µ is the dynamic viscosity of the air, V is the relative velocity between the bearing surfaces, t is time and f r takes value zero for all the bearing surface except for the area under the air entrance, where, for an inherent restrictor, it can be calculated as [6]: f r = 2πr 0h 0 πr 2 0 2κ p C d s Φ e (4) κ 1 RTa where r 0 is the restrictor radius, h 0 is the air gap height at the restrictor, C d is the discharge coefficient, κ is the ratio of specific heats (=1.4 for air), p s is the feeding pressure, and φ e is the nozzle function. The fluid dynamics model is represented by a second order non-linear partial differential equation (3). The model can be linearized on the two unknowns of the problem, the pressure and the air gap height distribution, and solved applying a standard finite element procedure. The following system is obtained. { } { } [C p ] δp + [K p ] {δp} = {f p } + c ps δh + k ps {δh} (5)

5 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES 109 where C p and K p are the stiffness and damping matrices of the pressure model, f p is the vector containing the components not dependent on δh or δp, and c ps and k ps are the air gap height to pressure coupling coefficients. Deformation to pressure coupling In order to obtain a strongly coupled formulation of the interaction between the plate deformation and the pressure distribution, δh needs to be redefined in terms of the degrees of freedom of the problem, with the following definition. The following coupled model is obtained. δh = δh cs δh s (6) { { } [C p ] δp δh [C ps ] s δh cs } + [K p ] {δp} [K ps ] { δhs δh cs } = {f p } (7) where the contributions to the pressure degree of freedom equations in (1) are C p and C ps to C, K p and K ps to K, and f p to f. Pressure to deformation coupling The pressure in the bearing gap applies a force on both bearing surfaces that leads to deformations and displacements. Therefore, a pressure to deformation coupling contribution needs to be defined. Pressure can be converted to force on each node using an equivalent area A eq as conversion factor. A coupled stiffness contribution K sp can be defined, which multiplies the pressure degrees of freedom with the area and adding it to the equations of the structural degrees of freedom. [K sp ] = {A eq } [I] (8) where I is the identity matrix and the sum of all the equivalent areas A eq is equal to the bearing area. In case of static analysis, an iteration is performed to find the equilibrium between deformation and pressure distribution. In this case, the force that is applied to the bearing plate is the sum of the pressure distribution in the previous step p 0 and the change in pressure obtained in the current one δp. Therefore, an external force vector must be added to the general formulation to consider the force due to p 0. where p a is the ambient pressure. {f s } = {A eq } [I] {p 0 p a } (9) Counter surface dynamics The counter surface to the bearing is assumed to be rigid compared to the bearing plate, thus it can be represented by a single node. The resultant force due to the pressure in the bearing gap is applied to this node, and thus inertia, stiffness, damping and external force contributions can be added for this node to the global matrix formulation of the model. Therefore, the influence of any dynamic system standing on the bearing can be represented by this node.

6 110 PROCEEDINGS OF ISMA Piezoelectricity Piezoelectric materials present a coupling between the electric field applied on them, the stress and the strain they produce. Piezoelectric stack actuators are designed to produce strain and force in one direction, and they are quite brittle in the others. If mounted properly, taking only axial loads, a one dimensional model is sufficient to represent the influence of the piezos in the bearing performance. The constitutive equations of piezoelectric material [7] can be rewritten in terms of voltage, elongation and force, which are the degrees of freedom in this model. f = A s E L (δh s nd 33 δv) (10) where f is the force taken by the piezo, A is the area of the piezo in contact with the structure, s E is the compliance for constant electric field, L is the length of the actuator, d 33 is the piezoelectric constant, n is the number of disks in the stack, and δv is the voltage degree of freedom. The stack can then be represented in FE as an added stiffness to the equations of the structural degrees of freedom of the model, with a coupling to the voltage. [K z ] {δh s } + [K zv ] {δv} (11) where K z and K zv are the contributions to the stiffness matrix of the problem Controller The active compensation principle is based on changing the voltage applied to the piezo actuators in function of the change of some measured value, normally the relative displacement between the bearing surfaces. The bearing is assumed to be clamped in this model. Therefore, the element representing the controller couples the displacement of the counter surface node to the change in the electric potential applied to the piezos. The equations of the voltage degree of freedom on the piezos will have the following structure. δv + m c δh cs + c c δh cs + k c δh cs = 0 (12) For example, a PID controller acting on position feedback in the harmonic analysis has the following FE formulation. m c = 0; c c = k D k I ω 2 ; k c = k P (13) with k P, k I and k D the proportional, integral and derivative gains of the controller. 2.2 Simulation Results In order to show the validity of the modeling approach and justify the need for such a coupled multiphysics model for the optimal design of active aerostatic bearings, some simulation results are reported here. All the simulations are performed on a bearing made of steel, with a plate diameter of 80 mm, feeding pressure of 5 bar with a feeding hole diameter of 1 mm, and three 5 mm x 5 mm x 20 mm piezoelectric stack actuators with 50 V applied on them, supporting a preload of 600 N. Bearing plate thicknesses between 3 and 10 mm are compared.

7 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES Harmonic Analysis The harmonic analysis gives the response of the system to the variation of a parameter considering the dependence on the frequency. In this analysis the model is linearized around the static configuration, thus the results are only valid for small disturbances, where the influence of the nonlinearities is not relevant. The dynamic response of the system to a change in the air gap height, to a force applied on a mass standing on the bearing and to the voltage applied to the piezos is presented next Variation of air gap height The dynamic stiffness is defined as the ratio between the applied force and the variation of the gap height it produces. In a first step, a displacement is imposed on the counter surface node, and the change in the bearing force is calculated. This analysis shows the response of the aerostatic support to a variation of the air gap height, considering the influence of the deformation of the bearing plate. The dynamic properties of the system standing on the bearing are not considered. Fig. 2 shows the dynamic stiffness of the aerostatic support coupled with the flexibility of the bearing for different bearing plate thicknesses. The shape of the function qualitatively agrees with experimental and simulation results from previous research works. The flexibility of the bearing plate shows a clear influence on the results. At low frequencies, the dynamic stiffness increases with the plate thickness, from 34 to 52 N/µm, but for high frequencies, even if the bearing with the thickest plate offers the highest stiffness, the thinnest bearing can be stiffer than thicker ones. The damping is also influenced by the plate thickness, and it is maximized for the thinnest and thickest plates. Stiffness (N/µm) mm 5 mm 7 mm 10 mm Phase ( ) Frequency (Hz) Figure 2: Dynamic stiffness of the aerostatic support for bearings with plates of different thicknesses under a static load of 600N External force In a real application, the system standing on the bearing has a dynamic behavior and the disturbance will be a force applied to it. The response to this force will couple the dynamics of the aerostatic support and the flexibility of the bearing plate with the dynamics of the counter surface. The proposed model can represent this phenomenon by defining some dynamic properties and an external force for the counter surface node. Fig. 3 shows the dynamic stiffness of the bearing for an axial load applied to a mass of 20 kg standing on the bearing for different plate thicknesses. At low frequencies, the result is the same as in the previous case. The

8 112 PROCEEDINGS OF ISMA2008 effect of the inertia is noticed at higher frequencies, with a resonance between 300 Hz and 400 Hz, higher for the stiffest bearings. 150 Stiffness (N/µm) Phase ( ) mm 5 mm 7 mm 10 mm 0 Frequency (Hz) Figure 3: Dynamic stiffness of the aerostatic support for plates of different thicknesses under a static load of 600N, supporting a 20 kg inertia Variation of voltage on actuators The voltage that can be applied to the actuators is limited, either by the piezo stacks or by the power supply, so the stiffness of the bearing plate limits the deformation, and thus the force, that can be obtained for compensation. As this force also depends on the dynamic properties of the counter surface, Fig. 4 shows the change in the bearing force that is generated per Volt applied to the piezo actuators, when the counter surface is fixed. 6 Force (N/V) Phase ( ) mm 5 mm 7 mm 10 mm 8 Frequency (Hz) Figure 4: Change in bearing force for a variation in the voltage applied to the piezo actuators For low frequencies, the force generation varies between the 5.8 N/V of the thinnest plate and the 3.1 N/V of the thickest. For high frequencies, the force decreases slightly. A phase lead in the low and medium frequencies, and phase lag for high frequencies is observed. The thinnest and thickest plates have lower

9 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES 113 phase leads, but in any case these values are low and should not influence much the performance of the bearing. The thickness of the bearing plate clearly limits the compensation capabilities of the bearing, and when relatively high dynamic forces need to be compensated, this will be one of the key criteria to optimize the design. 3 Experimental work Active air bearings are complex high precision mechatronic systems, with different physical fields relevant for the final performance, and with some phenomena that are quite difficult to model (e.g. bearing surface shape and roughness, geometry of the air gap,...). The coupled multiphysics simulation model presented above uses simplified models of the physical fields present in the system in order to obtain short calculation times so that it can be used for design optimization. The use of simplified models and the unmodeled phenomena will lead to deviations from the real performance in the simulation results. Therefore, experimental tests need to be carried out in order to analyze the validity of the model. The design of a test setup for single air bearings and of an active bearing prototype and the results obtained with them are presented and discussed next. 3.1 Description of test setup The test setup (see Fig. 5) has been designed to obtain the most information possible (displacement, acceleration, force) of the response of the bearing to different inputs (external force, voltage on the piezos) with the highest possible resolution (displacements in the order of the nanometer) and bandwidth (up to 1 khz), avoiding the influence of the dynamics of systems external to the bearing itself (e.g. the structures that hold the shaker and the displacement sensors). The air bearing stands above the air gap, with its motion constrained vertically by the aerostatic support itself, and by a ball joint above it, which connects the bearing to an electrodynamic shaker that is used to apply dynamic disturbances to the bearing in vertical direction, and that constrains the horizontal motion (not of interest in this analysis). The shaker also has a pneumatic actuator that can provide high static preloads, avoiding the use of high masses that affect the dynamic behavior. Three capacitive sensors hang above the bearing and measure the vertical displacement and tilt of the bearing. A piezoelectric force cell is placed in the force line between the bearing and the shaker, in order to measure the forces that are applied to the bearing. An accelerometer can also be placed on top of the bearing. 3.2 Description of the bearing prototypes An active bearing prototype has been designed and built (see Fig. 6) in order to validate the model results. The bearing is made of steel, with a plate diameter of 80 mm and a uniform thickness of 7 mm. The diameter of the restrictor is 1 mm. Three piezoelectric stack actuators of 5mmx5mm cross section and 20 mm length axisymmetrically disposed are used. Each piezo stack is preloaded with a screw, threaded into the base plate of the bearing, thicker and thus stiffer than the bearing surface, so that most of the elongation of the piezos is converted into deformation of the plate. 3.3 Experimental results Using the test setup and the active air bearing prototype presented above, different tests can be performed to analyze the behavior of the bearing. The passive dynamic stiffness, the displacement generation and the active dynamic stiffness with position feedback are presented here.

10 114 PROCEEDINGS OF ISMA2008 Figure 5: Test setup for single air bearings Figure 6: Active air bearing prototype Passive dynamic stiffness The first step to analyze the performance of the bearing prototype is to analyze the dynamic stiffness when no active compensation is applied. A disturbance force, between 1 Hz and 1 khz, is applied with the shaker, and the vibration of the bearing is measured, keeping the voltage on the piezos constant. Fig. 7 shows the dynamic stiffness of the bearing with a feeding pressure of 6bar and 50 V applied to the piezoelectric actuators, for static loads between 100N and 400N. At low frequencies, the stiffness increases with the static preload, from 5 N/µm with 100 N preload to 22 N/µm with 400 N. As a consequence, the axial resonance mode of the bearing vibrating as a rigid body above the aerostatic support happens at different frequencies, between 250 Hz for 100 N preload and 800 Hz for 400 N. These results lead to the conclusion that the highest preload is desirable, but it must be considered that high preloads lead to small air gaps, and thus a higher risk of losing the aerostatic support Displacement generation An active air bearing needs to be able to generate forces in order to compensate external disturbances. These forces are produced by the elongation of the piezo actuators, which deform the bearing plate, modifying the air gap shape and the pressure distribution. The actual force and displacement obtained by acting on the piezos depend on the dynamics of the moving system, i.e. a clamped system will show no displacement but

11 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES Stiffness (N/µm) Phase ( ) N 300 N 200 N 100 N 0 Frequency (Hz) Figure 7: Passive dynamic stiffness under static loads between 100N and 400N the highest force, and a free system would get high displacements and lower forces. In order to characterize the force and displacement generation capabilities of the bearing prototype, a voltage variation is applied to the piezos, and the displacement of the bearing, only constrained by the static force of the pneumatic preload and the inertia of the bearing, is measured. Results are shown in Fig. 8 The displacement produced at low frequencies increases with the static preload, from 25 nm/v for 100 N to 58 nm/v for 400 N. The deformation of the plate due to the elongation of the piezos is similar in all the cases, but a higher change in the pressure distribution, and thus force and displacement, is obtained when the air gap is small (high static load) due to the higher relative importance of the deformation compared to the air gap thickness. At higher frequencies the dynamics of the moving system becomes dominant, and resonances appear following the same trend as for the passive stiffness (see Fig. 7) Closed-loop performance The most straightforward approach to active compensation is based on position feedback, and the goal is to increase the dynamic stiffness against small disturbances around a working point, where the system can be considered linear. For these experiments, a proportional integral (PI) controller is used, tuned for achieving the highest dynamic stiffness. Fig. 9 shows the active dynamic stiffness, compared to the passive, for static loads of 100 N and 400 N. Due to the integral action in the controller, the static stiffness of the system is quasi-infinite (in practice only limited by sensor resolution) for both cases. In all the compensation bandwidth (265 Hz for the 100 N preload and 300 Hz for 400 N), the bearing with 400 N load shows higher stiffness, 15 times more on average. Stiffnesses of 15 N/nm at 1Hz or 130 N/µm at 100 Hz are achieved with 400 N load. Above the crossover frequency, the dynamic stiffness of the active bearing is lower that for the passive system, and this effect is more remarkable with 400 N preload. Better controller design should minimize this effect.

12 116 PROCEEDINGS OF ISMA2008 Displacement (µm/v) Phase ( ) N 300 N 200 N 100 N 150 Frequency (Hz) Figure 8: Displacement generation under static loads between 100N and 400N 4 Conclusions and future work Active aerostatic thrust bearings are complex mechatronics systems in which several physical disciplines are relevant to the final performance. In order to obtain the full potential of this technology, new simulation tools are needed that consider the coupled interaction between the structural flexibility, the fluid dynamics and the piezoelectricity. Such a model has been presented, and the simulation results show that this coupling is relevant for the performance of the bearing and needs to be considered in the design. The model represents properly the influence of the different design parameters, and can thus be used to aid in the design of new prototypes. However, the accuracy is limited by the difficulty to consider in the model the influence of some effects related to manufacturing limitations, such as the surface roughness. Therefore, validation with experimental results is needed in order to analyze the accuracy of the model. A test setup and an active bearing prototype have been built. The results prove that the setup is valid for testing the performance of the active bearings with high resolution and bandwidth, and that the compensation principle works and high dynamic stiffness can be obtained. Further work on control techniques is needed in order to optimize the performance. These knowledge and tools will be used in the design of active air bearings for a new linear guideway that is expected to offer sub-micrometer accuracy for high precision machine tools. Acknowledgements The authors would like to acknowledge the support of the EU-project NEXT- no. IP This research is also partially sponsored by the Fund for Scientific Research - Flanders (F.W.O.) under Grant FWO4283. The scientific responsibility is assumed by its authors.

13 ACTIVE VIBRATION CONTROL AND SMART STRUCTURES N 100 N 10 4 Control ON Control OFF 10 4 Control ON Control OFF Active Stiffness (N/µm) Active Stiffness (N/µm) Frequency (Hz) 10 0 Frequency (Hz) Figure 9: Active dynamic stiffness under static loads of 100N and 400N References [1] F. Al-Bender and H. Van Brussel, Active dynamic compensation of aerostatic bearings, in Proceedings of ISMA19, Leuven, Belgium, 1994, pp [2] H. Mizumoto, S. Arii, Y. Kami, K. Goto, T. Yamamoto, and M. Kawamoto, Active inherent restrictor for air bearing spindles, Precision Engineering, vol 19, no 2, pp , [3] F. Al-Bender, On the Modelling of the Dynamic Characteristics of Aerostatic Bearing Films: from Stability Analysis to Active Compensation, Precision Engineering (2008), doi: /j.precisioneng [4] R. Snoeys and F. Al-Bender, Development of improved aerostatic bearings, KSME Journal, vol 1, pp. 8188, [5] G. Aguirre, F. Al-Bender and H. Van Brussel, A Multiphysics Coupled Model for Active Aerostatic Thrust Bearings, Proceedings of the IEEE-ASME International Conference on Advanced Intelligent Mechatronics, Xi an, China, 2008 [6] F. Al-Bender, Contributions to the Design Theory of Circular Centrally Fed Aerostatic Bearings, PhD Thesis, Katholieke Universiteit Leuven, Leuven, Belgium, [7] A. Preumont, Mechatronics. Dynamics of Electromechanical and Piezoelectric Systems, Springer, 2006.

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