Semi active Control of Adaptive Friction Dampers for Structural Vibration Control

Transcription

1 Semi active Control of Adaptive Friction Dampers for Structural Vibration Control Jens Becker and Lothar Gaul Institute of Applied and Experimental Mechanics, University of Stuttgart Pfaffenwaldring 9, 5 Stuttgart, Germany ABSTRACT Reduction of structural vibrations is of major interest in mechanical engineering for lowering sound emission of vibrating structures, improving accuracy of machines and increasing structure durability. Besides design optimization and passive damping treatments, active structural vibration control can be applied to reduce unwanted vibrations. They become more and more important as lightweight constructions evolve. In this contribution, two semi-active control laws for control of friction dampers are derived and investigated. Purely semi-active control has the advantage to yield intrinsically stable closed-loop systems and low energy consumption compared to active vibration control. In the experimental implementation, the control make use of piezoelectric stack actuators to apply adjustable normal forces between structure and attached friction damper elements. Roughly speaking, the normal forces are controlled accordingly to the measured structural vibrations in order to achieve optimal damping effect. The control algorithms make use of reduced finite-element models of the structural dynamics for the estimation of the not observable relative displacement beneath the normal force actuator based on acceleration measurements. Experimental results of the control algorithms for a beam test structure with an attached friction damper with one adjustable normal force show the effectiveness of the algorithms. The control laws are compared with respect to their effectiveness, need of actuator bandwidth, applicable frequency bandwidth and energy consumption. Nomenclature x Coordinate along the beam length F,F C,F N,F N,min,F N,max Fiction force, sticking force, normal force, minimal and maximum normal force W d Dissipated energy in one period k T,kT Tangential stiffness, control parameter derived from tangential stiffness µ Friction coefficient u rel (t),û rel (t) Tangential displacement u rel, (t),û rel, (t) Tangential displacement amplitude z,ẑ States in first order state space A System matrix of state space representation a meas Measured acceleration y meas,ŷ meas Measured and estimated system output, here the velocity c rel,c meas Output matrices of relative displacement output and measurement output l Correction vector for observer M,K,F Mass and stiffness matrix, load vector Φ i, Ψ i Eigenvectors x Displacement vector of second order representation x m Vector of modal amplitudes T Reduction base I Unit matrix Ω Spectral matrix ε Boundary layer control parameter F exc Amplitude of the sweep shaker excitation U P,U min,u max Actuator voltage, minimal force and maximal force actuator voltage

2 a meas shaker F exc force cell U P stack actuator insulators friction damper structure a T fixture washer F N,meas force cell x Figure 1: Sketch of the investigated structure with adaptive friction dampers (amplifiers are not shown for sake of clarity). Beam structure Friction damper Length 775 mm 1 mm Width 4 mm 4 mm Thickness 3 mm 3 mm Material steel steel Stack actuator Type PI 36.2 Maximum displacement 2 µm Stiffness 18 N/µm Operating voltage -7 V.. 2 V Table 1: Properties of investigated structure and applied piezoelectric stack actuator (cf. Fig. 1). 1 Introduction Semi active control strategies offer interesting alternatives to passive means for vibration reduction and active vibration control (AVC). Hereby the term semi active means, that passive system properties are actively controlled. From this, it directly follows the advantage that semi active control concepts in contrast to AVC concepts can not feed energy into the structure under control. This in turn eliminates the problem of system destabilization due to spillover effects well known in AVC of flexible structures [1, 4] and guarantees stable closed loop systems. Moreover, semi active control concepts generally lead to very energy efficient controllers. On the other hand, the achievable performance is limited because semi active control concepts rely generally on passive damping mechanisms. Though, they outperform passive vibration reduction means due to their ability to adapt to the instantaneous vibration state of the structure. This property links semi active control concepts like the presented one to the context of adaptive structures and adaptronics [7]. In this contribution, strategies for the semi active vibration control by controlling the normal force between frictional interfaces between a beam structure and a damper element are investigated. Previous work has shown the efficiency of such damper elements that are small compared to the structure dimensions [2]. The idea of using friction in joints to damp structural vibrations by semi active normal force control is reported first in [8], which inspired several researchers, e.g. [1]. In these references, only discrete frictional joints and strongly idealized structures are considered whereas the considered structure in this work is more general. Semi active control concepts are probably most often encountered in the context of magnetorheological, electrorheological dampers and variable stiffness dampers, see e.g. [12, 11]. The test structure investigated in the sequel consists of a beam as main structure and an attached adaptive friction damper element as depicted in Fig. 1 and shown partly in Fig. 3. The attached friction damper is fixed to the main structure by two screws. One screw is strongly tightened whereas the normal force of the other screw can be controlled by a hollow piezoelectric stack actuator. This screw with adaptive clamping force is shortly referred to as the adaptive screw. The remainder of the paper is organized as follows: First, two different control strategies based on simplified structural models and appropriate friction models are derived. After that, the experimental setup is presented and the piezoelectric normal force actuation is investigated. Next, experimental results of the control algorithms are shown. Finally a short conclusion closes the paper. 2 Semi active Vibration Control Design In the following, control design procedures based on discrete friction models and finite-element (FE) models of the mechanical structure are proposed. Two control laws are deduced in the following. Furthermore, the design of appropriate

3 F F C k T F u rel, k T u rel u rel, u rel Figure 2: Left: Masing friction model with tangential stiffness k T and F C = µf N. Right: Hysteresis loops of Masing (thick) and Coulomb (thin) friction model. F C observers to estimate the non measureable quantities needed for the derived control laws is shown. 2.1 Friction Model Experimental investigations of structures with friction joints have shown that relatively simple dynamical friction models are capable to model the dominant effects, see e.g. [6, 5]. The Masing model has proven its usability to capture the dominant friction behavior for bolted joints in various experiments, e.g. see [5]. The model consists of an elastic spring in series with a Coulomb element The model and the obtained hysteresis for sinusoidal tangential displacements are depicted in Fig. 2. For the tangential stiffness goint to infinity, the hysteresis of the Masing element approximates the Coloumb model hysteresis. 2.2 Hysteresis optimal Control For derivation of the first control law, it is assumed that the structure is predominantly excited by mono frequent excitations, which holds for many practical applications. Furthermore, it is assumed that the dominant damping effects originate from the contact area below the normal force actuator and hence can be captured by one discrete friction model. The goal of this control law is to maximize the frictional work W d during one vibration cycle. This corresponds to maximizing the enclosed hysteresis area hence the resulting control law is shortly denoted as hysteresis optimal control. The normal force F N that maximizes the dissipated work W d can be determined off line with respect to the friction model, its parameters and the vibration amplitude. For the chosen Masing model (see Fig. 2), the dissipated energy W d per cycle evaluates to W d = 4 ( u rel, F C k T ) F C with F C = µ F N. (1) Maximizing this friction work yields the optimal normal force F N as function of the tangential contact stiffness k T, the friction coefficient µ and the amplitude of the relative sliding oscillation amplitude u according to F N = f(u rel, ) = k T u rel, 2µ = k T u rel,. (2) Note that similar algebraic expressions could be derived also for hysteresis loops of more sophisticated dynamic friction models. Eq. (2) is the control law, which allows the controller to adjust the normal force accordingly to the structural vibration state. 2.3 Simplified Model based Observer For the proposed control law, the tangential movement u rel (t) in the friction interface has to be known. For the investigated structure, there is no possibility to directly measure this quantity, hence a model based approach is used for its estimation. Model based observers are generally capable to estimate non measureable system outputs from measureable ones. In the sequel, such an observer is designed based on a FE model in order to estimate the tangential movement beneath the normal force actuator from measurements of other signals, e.g. displacements, strains or accelerations at other locations of the structure. For accurate estimation, the underlying model has to be verified and its parameters adjusted. This task is performed by use of the modal assurance criteria (MAC) that correlates numerical modes shapes with experimental ones. This criteria reads MAC ij = φt i ψ j φ i ψ j [,1]

4 Figure 3: Part of experimental setup of beam with attached damper, stack actuator, force cell, shaker and accelerometers. Figure 4: Modal assurance criterium (MAC) plot correlating numerical and experimental modes (obtained by 1 N clamping force for adaptive screw) for the beam structure (cf. Fig. 1 and Tab. 1). and yields a measure of the match quality of two mode shape sets φ i and ψ i. The modes φ i are obtained by FE analysis whereas the modes ψ i are found by experimental modal analysis (EMA). For good agreement, values of.85 and higher are generally demanded. For the considered structure, the obtained MAC values are shown as a 3-D plot in Fig. 4 indicating good agreement by the high diagonal terms and the low off-diagonal terms for the first 6 structural modes. Hereby the experimental modes shapes are obtained by a roving impact EMA with 52 points. Please note that although strictly orthogonality with respect to the mass matrix is demanded from theory, for the considered beam structures with relatively uniform mass distribution, the additional incorporation of a (reduced) mass matrix in Eq. (2.3) yields no noticeable difference. The observer is obtained from the simplified linear FE model by modal reduction (see [2] for an alternative modal filtering approach). Please note that due to the fact that the excitation force is unknown to the controller, the simulation model of the observer possesses only the correction part as input. The observer has two outputs: the measurement signal y meas (t) used for the correction of the estimated state vector ẑ and the output u rel (t), whose estimate is fed to the controller, i.e. ẑ = Aẑ + l(ŷ meas y meas ), (3) ŷ meas = c T measẑ, û rel = c T rel ẑ, where the hat denotes estimated quantities. Hereby, the velocity y meas (t) is obtained by integration of the accelerometer signal according to y meas (t) = t a meas(τ)dτ. All matrices in Eq. (3) are derived from a finite element (FE) model for the linear case of no friction beneath the actuator, which gives a good approximation. This approximate FE model is given by y meas = c T meas x, M ẍ + K x = F, (4) ŷ rel = c T rel x, with mass and stiffness matrices M,K and load vector F. The corresponding eigenvalue problem ( K ω 2 M ) φ = (5)

5 excitation force F exc F exc mechanical structure fixed F N F exc F N a meas F N normal force nonlinear control law t û t T rel(t) dt u π 2T F N = k T u 2 µ û rel observer ẑ = Aẑ + l(ŷ meas t ŷ meas = c T measẑ, measured acceleration a meas a meas(τ)dτ) û rel = c T rel ẑ. Figure 5: Closed control loop schematic of the proposed hysteresis optimal control law. yields the eigenfrequencies ω k and eigenvectors φ k (k N + ) which allows a modal transformation (mass normalized eigenvectors) by x = T x m with T = [φ 1,φ 2,...,φ N ] if the first N bending modes are retained. It follows Ω = T T K T = diag{ω 1,ω 2,...,ω N }, (6) c T rel = c rel T, c T meas = c meas T. Then, the observer system matrix in Eq. (3) corresponding to the state vector z = [ x m,x m ] T R 2N is given by [ ] Ω A =. (7) I For implementation, Eq. (3) is transformed into a numerically better form. The correction vector l is determined by providing appropriate state and measurement noise variance matrices by the Kalman filter design procedure, i.e. solution of the corresponding Riccati equation [9]. From the estimated signal û rel (t), the actual vibration displacement amplitude can now calculated as û rel, (t) π t û rel (t) dτ. (8) 2T t T This formula implies that the signal u rel (t) has zero mean, which is automatically fulfilled if u meas (t) is measureed by an accelerometer. Because the evaluation of the integral would need large memory storage for the signal uˆ rel (τ),τ [t T,t] for a reasonable integration time T much larger than the vibration period, Eq. (8) is approximated for efficient implementation by a PT 1 element according to T ûrel, (t) + û rel, (t) = π 2 û rel(t). (9) Hereby, the time constant T prescribes how fast the signal û rel, (t) (and by this, also the controlled normal force) reacts to a change in the vibration amplitude. The overall control loop for the hysteresis optimal control algorithm is depicted in Fig Lyapunov type Control In [1], Lyapunov s direct method is applied to derive an optimal control law based on a discrete friction model. Following this procedure based on the Masing friction model, a control law is found depending on the friction force, which is not measureable or observeable for the investigated test structure in practical implementation. Hence the Coulomb model is used as an approximation of the Masing model as basis for the control design since the results from the hysteresis optimal

6 2 3 force FN [N] 1 1 force FN [N] voltage U P [V] (a) N clamping force - voltage U P [V] 1 (b) 1 N clamping force 4 force FN [N] force FN [N] voltage U P [V] (c) 2 N clamping force 1 - voltage UP [V] (d) 2 N clamping force 1 Figure 6: Measured voltage-force relationsships of the stack actuator for different clamping forces. 2 stroke [N] clamping force [N] 2 Figure 7: Stack actuator stroke as function of clamping force (determined from measurements as shown in Fig. 6). control indicate high values for the tangential stiffness values k T. From the Coulomb model, a velocity dependent bang bang controller can be deduced as optimal control law following [1]. Though, the bang bang behavior leads to actuator chattering, which can be avoided by regularization by a velocity boundary layer ε yielding the modified sub optimal law { ( ) F F N = N,min 1 û rel (t) u ε + F rel N,max ε for û rel (t) < ε F N,max for û (1) rel (t) ε With the applied piezoelectric stack voltage U P as manipulated variable, Eq. (1) transforms into { ( ) U U P = min 1 û rel (t) u ε + U rel max ε for û rel (t) < ε U max for û rel (t) ε Again, the tangential relative movement in the interface beneath the normal force actuator is needed. It is estimated in an analogous way to Sec. 2.3, the only difference being the estimation of the relative tangential velocity u(t) instead of the relative displacement u(t). A schematic of the overall control loop is omitted due to space limitations it is similar to Fig. 5 with the control law is replaced with Eq. (1). Note that the minimal and maximal normal forces F N,min and F N,max appearing in the control law as control parameters are actually determined by the mechanical properties of the normal force actuation principle. (11) 3 Experimental Results The proposed control laws are implemented and experimentally investigated for the beam like test structure depicted in Figs. 1 and 3 with the properties from Tab. 1. The control algorithms are implemented on a dspace system running at a sampling speed of 5 khz in real time. The implemented modal observers are based on the first low frequency five bending modes (N = 5), hence 5 modes can be controlled at maximum. For the measurement input of the control algorithm (y meas ), the lateral acceleration is measured close to the tip (at x =.765 m). The control output is filtered by a low pass with Hz cut off frequency to eliminate digitization noise before it is amplified by a high voltage amplifier for piezoelectrics.

7 (a) mode 2, excitation F exc, = 1 N (b) mode 2, excitation F exc, = 2 N (c) mode 2, excitation F exc, = 3 N (d) mode 3, excitation F exc, = 1 N (e) mode 3, excitation F exc, = 2 N (f) mode 3, excitation F exc, = 3 N (g) mode 4, excitation F exc, = 1 N (h) mode 4, excitation F exc, = 2 N (i) mode 4, excitation F exc, = 3 N Figure 8: Measured accelerance FRFs of the test output for controlled sine sweep excitation of different amplitudes with control off (dashed line) and hysteresis optimal control (solid line) for k T = 5 17 N/m and a clamping force of F N = N. 3.1 Normal Force Actuation For application of the normal force, a piezoelectric ring stack actuator clamped by a bolt is used as shown in Fig. 1 and Fig. 3 (technical data are given Tab. 1). The applied normal force is measured by a strain gage based force cell sensor, that allows measuring static force signal components, which is essential for this application. The maximum achieveable actuator force stroke depends on the stiffness of the clamping and the actuator stiffness itself, for details on modeling of piezoelectric materials see [4]. It is important to note that the actuator block force of 3 N is only theoretically obtained for infinite clamping stiffness, i.e. zero displacement. The force voltage dependency for different clamping forces applied by tightening of the screw are determined experimentally, examples of such a measurements are presented in Fig. 6. These measurements are evaluated to obtain the actuator stroke depending on the clamping force shown in Fig. 7. The measurements clearly reveal a strong nonlinear stiffness in normal direction depending on the clamping conditions which can not be explained by linear theory, where the stiffness should be independ of the actual normal force. This effect is mainly contributed to the multiple contact pairs introduced in normal direction, namely the force measurement cell, washers, structure and damper element. This nonlinear effects are also visible in the measured indidual hysteresis shapes (Fig. 6). For small normal forces the stiffness nonlinearities change the shape of the hysteresis whereas at higher force levels, the hysteresis shape known from ferroelectricity is obtained. Especially for the hysteresis optimal control, which has proven to be sensitive to errors in the absolute normal force values, these nonlinear relations make an underlying control of the applied normal force necessary. This is realized by a proportional-integral (PI) controller that makes the force F N follow the prescribed one from the hysteresis optimal control. However, for the Lyapunov type control, the high actuation dynamics makes this underlying control based on the strain gage force cell impossible (due to their inherent noise limiting the maximum amplification). Such a controller could be implemented in future work based on an additional piezoelectric force cells with much higher sensitivity with the drawback

8 (a) mode 2, excitation F exc, = 1 N (b) mode 2, excitation F exc, = 2 N (c) mode 2, excitation F exc, = 3 N (d) mode 3, excitation F exc, = 1 N (e) mode 3, excitation F exc, = 2 N (f) mode 3, excitation F exc, = 3 N (g) mode 4, excitation F exc, = 1 N (h) mode 4, excitation F exc, = 2 N (i) mode 4, excitation F exc, = 3 N Figure 9: Measured accelerance FRFs of the test output for controlled sine sweep excitation of different amplitudes with control off (dashed line) and Lyapunov type control (solid line) for 1/ε = s/m and a clamping force of F N = N. of adding additional contact pairs and thereby decreasing the stiffness. 3.2 Results For investigation of the presented control algorithms for experimental test structure, the parameters for the control laws are determined experimentally and are kept constant for all presented measurements. Both control algorithms are compared by measuring accelerance frequency response functions (FRFs) between a test accelerometer a T located at position x =.45 m (cf. Fig. 1) and the measured excitation force from an attached shaker at x =.325. For nonlinear mechanical structures such as the beam with friction damper, measuring and comparing FRFs needs special attention. The main obstacle is the dependence of the obtained (equivalent) eigenfrequencies and the peak amplitudes on the excitation amplitude. Hence, sine sweep measurements of low speed (.1 Hz/s) with continously controlled shaker excitation force amplitude F exc, = 1,2,3 N are conducted. The shaker controller eliminates the back EMF effect [3] by controlling the sinusoidal excitation amplitude. For the test structure, the chosen low sweep rate has shown to approximates extremely time consuming stepped sine measurements. Figs. 8 and 9 show the accelerance FRFs for the structure under semi active control and without control (i.e. passive) for both control laws around the resonances of mode 2, 3 and 4. For the passive case, the minimal possible force F N = F N,min is applied to the adaptive screw which still introduces significant structural damping. The measurements clearly show that both controllers strongly reduce the peak resonance amplitudes. The hysteresis optimal control significantly shifts the resonance amplitudes whereas this effect is less pronounced for the Lyapunov type control. In most examined cases escecially at mode 2 and mode 3 the Lyapunov type control is more effective in reduction of the resonance vibration than the hysteresis optimal control. The Lyapunov type control with F exc, = 1 N is

9 the only case where the control decreases the vibration reduction. This effect is most probably contributed to estimation errors. Therefore in ongoing research the Lyapunov type control is modified for the small vibration regime by introduction of a lower velocity boundary, below which the minimal normal force is set by the controller, i.e. F N = F N,min. This is motivated by the fact, that the estimated tangential velocity is a blurred version of the exact tangential velocity, especially the velocity during sticking in the frictional joint is expected to be overestimated. Compared to the hysteresis optimal control, the high output dynamics of the Lyapunov type controller require high bandwith high voltage amplifiers with high power supply. Note that the power consumption of the stack is roughly a linear function of the squared vibration frequency and the vibration amplitude. Furthermore, this controller is well suitable to actively suppress mono frequent as well as broadband vibrations. The hysteresis optimal control on the other hand is restricted in theory to mono frequent vibrations, but has the advantage that it can also be implemented with actuators of low dynamics, i.e. actuators of different working principles than piezoelectricity. In summary, the choice which controller type suits best for a certain application depends mainly on the frequency range of interest, the used actuator and the primary excitation type. 4 Conclusion and Outlook Control algorithms for semi active vibration control of structures with attached friction dampers whose normal force is adjusted by a piezoelectric stack actuator are derived and experimentally investigated for a a generic test structure. It is experimentally verified that semi active control of friction dampers is an effective means for active reduction of structural vibrations at multiple resonance frequencies and varying amplitude. Thereby, in contrast to fully active control methods, semi active control is inherently fail safe and guarantees stability, i.e. spillover effects can not destabilize the system. Moreover significant passive damping is always introduced into the mechanical system by the friction damper element. Ongoing research evaluates the control algorithms with respect to different excitation types and frequency ranges. Furthermore the damper geometry and location could be optimized and multiple adaptive screws could be used to improve the control performance. Finally, research towards application of the presented concepts to real world structures is conducted. Acknowledgment The support of the DFG (German Research Foundation) with project SPP 1156 is grateful acknowledged. References [1] Balas, M. Feedback control of flexible systems. IEEE Transactions on Automatic Control 23, 4 (1978), [2] Becker, J., and Gaul, L. Semi-active damping of vibrations of mechanical structures with friction dampers. In Proceedings of the 13th International Conference of Sound and Vibration (26). [3] Ewins, D. J. Modal Testing: Theory and Practice. John Wiley & Sons Inc., New York, [4] Fuller, C., Elliott, S., and Nelson, P. Active Control of Vibration. Academic Press, [5] Gaul, L., and Lenz, J. Nonlinear dynamics of structures assembled by bolted joints. Acta Mechanica 125 (1997), [6] Gaul, L., and Nitsche, R. Role of friction in mechanical joints. Applied Mechanics Reviews 54 (21), [7] Hurlebaus, S., and Gaul, L. Adaptive structures - an overview. In Proceedings of 24th IMAC (26). [8] Lane, J. S., Ferri, A. A., and Heck, B. S. Vibration control using semi-active friction damping. In Proceedings of the ASME (1992), vol. 49, pp [9] Lunze, J. Regelungstechnik 2, 3 ed. Springer, 25. [1] Nitsche, R., and Gaul, L. Lyapunov design of damping controllers. Archive of Applied Mechanics 72 (23), [11] Nitzsche, F., Zimcik, D. G., Wickramasinghe, V. K., and Yong, C. Control laws for an an active tunable vibration absorber designed for rotor blade damping augmentation. Aeronautical Journal 18, 179 (23), [12] Patten, W. N., Mo, C., Kuehn, J., and Lee, J. A primer on design of semiactive vibration absorbers (sava). ASCE Journal of Engineering Mechanics 124, 1 (1998),