Application of a passive/active autoparametric cantilever beam absorber with PZT actuator for Duffing systems

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1 Application of a passive/active autoparametric cantilever beam absorber with PZT actuator for Duffing systems G. Silva-Navarro* a, H.F. Abundis-Fong a and B. Vazquez-Gonzalez b a Centro de Investigación y de Estudios Avanzados del I.P.N. Departamento de Ingeniería Eléctrica - Sección de Mecatrónica, A.P , C.P. 736, México, D.F., MEXICO. b Universidad Autónoma Metropolitana, Plantel Azcapotzalco, Departamento de Energía, Av. San Pablo No. 18, Col. Reynosa Tamaulipas, C.P. 22, México, D.F., MEXICO. ABSTRACT An experimental investigation is carried out on a cantilever-type passive/active autoparametric vibration absorber, with a PZT patch actuator, to be used in a primary damped Duffing system. The primary system consists of a mass, viscous damping and a cubic stiffness provided by a soft helical spring, over which is mounted a cantilever beam with a PZT patch actuator actively controlled to attenuate harmonic and resonant excitation forces. With the PZT actuator on the cantilever beam absorber, cemented to the base of the beam, the autoparametric vibration absorber is made active, thus enabling the possibility to control the effective stiffness and damping associated to the passive absorber and, as a consequence, the implementation of an active vibration control scheme able to preserve, as possible, the autoparametric interaction as well as to compensate varying excitation frequencies and parametric uncertainty. This active vibration absorber employs feedback information from a high resolution optical encoder on the primary Duffing system and an accelerometer on the tip beam absorber, a strain gage on the base of the beam, feedforward information from the excitation force and on-line computations from the nonlinear approximate frequency response, parameterized in terms of a proportional gain provided by a voltage input to the PZT actuator, thus modifying the closed-loop dynamic stiffness and providing a mechanism to asymptotically track an optimal, robust and stable attenuation solution on the primary Duffing system. Experimental results are included to describe the dynamic and robust performance of the overall closed-loop system. Keywords: Active vibration control, Autoparametric systems, Cantilever-beam absorber, PZT patch actuator. 1. INTRODUCTION The autoparametric systems are nonlinear systems having at least two nonlinearly coupled subsystems, interacting each other to transfer exogenous energy to a passive vibration absorber. The primary system can be externally excited by some harmonic force and, when this is connected to the absorber then, it can be tuned to get the parametric excitation, which is an internal mechanism that transfers the exogenous energy to the passive absorber. In case the primary system is excited exactly or near its linear parametric frequency then, it is possible to get the principal parametric resonance for the absorber (autoparametric interaction) and, therefore, the response on the primary system can be the attenuated. This inherent nonlinear phenomena, well-known as autoparametric interaction, has been analyzed in the literature since the seminal work by Haxton and Barr 4 as well as by Cartmell, 3 Korenev and Reznikov, 5 Nayfeh and Mook, 6 Tondl 1 and more recently by Vazquez and Silva. 11 Most of the work, however, has been concentrated on passive pendulum-type autoparametric systems (see, e.g., Cartmell 2 and Tondl 1 ) and few of them on active/semiactive pendulum-type autoparametric absorbers (see Cartmell, 21 and Silva-Navarro et al. 8 ). In this paper we extend the Haxton and Barr 4 absorber to consider the design and synthesis of a cantilever type passive/active autoparametric vibration absorber, with a small PZT patch actuator, to be used on primary Duffing systems. The primary system consists of a mass, viscous damping and a cubic stiffness provided by a * Corresponding author. s: gsilva@cinvestav.mx, habundis@cinvestav.mx, bvg@correo.azc.uam.mx Active and Passive Smart Structures and Integrated Systems 213, edited by Henry A. Sodano, Proc. of SPIE Vol. 8688, 86882O 213 SPIE CCC code: X/13/$18 doi: / Proc. of SPIE Vol O-1 Downloaded From: on 2/19/216 Terms of Use:

2 soft helical spring, over which is mounted a cantilever beam with a PZT patch actuator actively controlled. The nonlinear approximate frequency analysis of the overall passive system is performed via multiple scales methods and this is further validated using experimental modal analysis techniques. With the addition of a rectangular PZT actuator to the cantilever beam absorber, cemented to the base of the beam, the autoparametric vibration absorber is made active, thus enabling the possibility to control the effective stiffness and possibly the damping associated to the beam and, as a consequence, the implementation of an active vibration control scheme to preserve, as possible, the autoparametric interaction as well as to compensate varying excitation frequencies and parametric uncertainty. The active vibration absorber employs feedback information from the primary Duffing system and the beam absorber, feedforward information from the excitation force and on-line computations from the nonlinear approximate frequency response, parameterized in terms of the equivalent stiffness of the PZT actuator, thus providing a mechanism to asymptotically tune an optimal and stable attenuation solution. 2. SYSTEM DESCRIPTION In Fig. 1 is described a schematic diagram of the mechanical system. Here, a primary damped Duffing system is attached to a cantilever-beam vibration absorber, with lateral motion restricted to a horizontal plane (i.e., no gravity effects). Nonlinear spring 3 k 1x+k 2x c 1 F(t) Shaker M x Strain gauge u PZT patch actuator L Beam m y Figure 1. Schematic diagram of the primary Duffing system coupled to an autoparametric cantilever beam absorber. The experimental setup is mounted on an arrangement using a rectilinear plant (model 21a) provided by Educational Control Products c. The configuration of the primary Duffing system consists of one mass carriage (M), connected to the base by an helical spring with variable pitch (see Fig. 2). A sort of rubber bands are Figure 2. Details of the experimental platform: accelerometer, strain gage and PZT patch. also connected in parallel to the helical spring to get a realistic nonlinear spring with cubic stiffness function (k 1 x+k 2 x 3 ). The mass carriage suspension has an anti-friction ball bearing system and, therefore, the linear dashpot (c 1 ) is included only to describe the presence of a small (linear) viscous damping. This primary system is affected by an external harmonic force F(t) = F sinωt, with amplitude F and excitation frequency Ω tuned Proc. of SPIE Vol O-2 Downloaded From: on 2/19/216 Terms of Use:

3 close to the principal parametric resonance associated to the primary system. This external force is obtained from a brushless-type servo motor connected to a pinion-rack mechanism. In the mass carriage there exists high resolution optical encoders to measure their actual positions via cable-pulley systems. In order to attenuate the harmonic vibrations F(t) is used a cantilever-beam vibration absorber (secondary system), composed by a thin beam attached over the primary system and with an equivalent mass at the end m. Both primary and secondary subsystems are coupled by means of the inertia resulted from the beam attachment. The length L denotes the beam total length and c 2 is a small viscous damping on the beam. The main actuator on the system is a piezoelectric patch made by Physik Instrumente c. It is a piezoelectric patch model P-876.A15. This is connected to a voltage amplifier (model E-413) type DuraAct c, which can drive the patch. This patch can be seen properly cemented on the base of the cantilever-beam in Fig. 2. The reason for such a place is that on the base of the cantilever-beam the bending stresses achieve the highest values. For control purposes a displacement signal has to be gather and this is accomplished with the use of a strain gage at the bottom of the beam, whose instrumentation is made for a data acquisition system National Instruments c model NI cdaq , then this signal is sent to the high-speed DSP board using the NI 9263 module. This system is controlled through the Matlab/Simulink c platform to compute the control actions that actuate on the piezoelectric patch. The integration of the overall experimental platform is described in Fig NI-CDAQ-9172 t'a > l,1 MI- ". NI PERSONAL COMPUTER DSP BOARD!1 VOLTAGE AMPLIFIER NI 9233 NI 9236 OPTICAL ENCODER AUTOPARAMETRIC VIBRATION ABSORBER Figure 3. Integration of the overall experimental setup. 3. SYSTEM WITH A PASSIVE/ACTIVE CANTILEVER BEAM ABSORBER The equations of motion for the two degrees-of-freedom system consisting of the primary Duffing system and the passive autoparametric cantilever beam absorber are obtained via the Euler-Lagrange formulation as follows (see also Haxton and Barr, 4 Roberts 9 and Cartmell 1 ) Proc. of SPIE Vol O-3 Downloaded From: on 2/19/216 Terms of Use:

4 (M +m)ẍ+c 1 ẋ+k 1 x+k 2 x 3 6m 5L (yÿ +ẏ2 ) = F(t) (1) ( 3EI mÿ +c 2 ẏ + L 3 6m ) y + 36m 5Lẍ 25L 2y( yÿ +ẏ 2) = u(t) (2) where x and y denote the longitudinal motion of the primary system and lateral displacement of the passive cantilever beam absorber, respectively. The active control input u is the equivalent force provided by the PZT patch actuator. The harmonic excitation force is given by F(t) = F cos(ωt), where the excitation frequency Ω is close to the (linear) principal parametric resonance of the primary Duffing system, that is, Ω k 1 /(M +m). Furthermore, the parameters associated to the passive beam absorber are the modulus of Young E (aluminum), the area moment of inertia I and the total length L. The rectangular PZT patch actuator is equivalent to a pair of bending moments M p applied at both ends of the patch. The equivalent control force acting on the Euler-Bernoulli cantilever beam (2) is therefore obtained as u = BM p = Bg a V (3) where V is the voltage applied between the electrodes of the PZT layer, B is the so-called influence vector and g a = e 31 bz m is the actuator gain, which can be calculated from the PZT parameters as material properties and patch size. Here e 31 is a PZTconstant(e 31 = 7.5Coulomb/m 2 ), b is the constantelectrode width (Preumont 7 ). It is important to note the highly nonlinear and coupled system dynamics in (1)-(2). In essence, the beam absorber is inertially coupled to the primary Duffing system in such a way that a proper tuning can lead to the autoparametric condition (two-mode nonlinear operation), where resonant harmonic forces can be attenuated. Moreover, when u results a purely passive cantilever beam absorber (see Haxton and Barr 4 ). Otherwise we have a closed-loop control system (1)-(2), which is strongly nonlinear, underactuated and the output to be controlled x is not controllable from the input u, exactly at the equilibrium points of interest. However, the cantilever beam absorber is controllable from u and, therefore, we propose the application of a suitable feedback and feedforward control scheme to indirectly control the primary Duffing system response. 4. SYSTEM WITH PASSIVE CANTILEVER BEAM ABSORBER The equations of motion for the two degrees-of-freedom system (1)-(2) can be normalized by defining representative parameters and assuming small oscillations to get an approximate analytical solution for the nonlinear frequency response. This procedure results in the following two coupled and nonlinear differential equations for the passive cantilever beam absorber (i.e., when u ): ẍ+2εζ 1 ω 1 ẋ+ω 2 1 x+εαx3 εh(yÿ+ẏ 2 ) = εf cos(ωt) (4) ÿ +2εζ 2 ω 2 ẏ + ( ω 2 2 εgẍ ) y +ε 2 βy(yÿ +ẏ 2 ) = (5) where the normalized system parameters are defined by ω1 2 = k1 M+m, 2εζ 1ω 1 = c1 k2 M+m, εα = M+m, εh = 6 5L ω 2 2 = 3EI ml 3, 2εζ 2 ω 2 = c2 m, εg = 6 5L, ε2 β = 36 25L 2, ( m M+m ), εf = F ε = 6δ 5L, M+m, δ = F k 1 (6) The small perturbation parameter ε considers the internal couplings between the cantilever beam absorber and the Duffing primary system, viscous dampings, nonlinearities and external force into the system. These perturbed equations include the cubic nonlinearity in the restoring force for the primary system, which is multiplied by the small perturbation parameter ε, that is, εα with a constant α > or α < corresponding to a hardening or softening spring, respectively. Proc. of SPIE Vol O-4 Downloaded From: on 2/19/216 Terms of Use:

5 To guarantee the autoparametric interaction between the primary Duffing system and the cantilever beam absorber, by which the attenuation is obtained, the following two-mode autoparametric tuning conditions must be satisfied Ω = ω 1 (7) ω 1 = 2ω 2 (8) where Ω is the excitation frequency, ω 1 corresponds to the principal parametric frequency of the primary system and ω 2 is the natural frequency of the cantilever beam absorber. These two expressions are well-known as the external and internal resonance conditions, respectively (see Cartmell 3 and Nayfeh and Mook 6 ). It is important to note that ω 2 directly depends on the equivalent beam stiffness, which can be actively modified by the PZT actuator. 4.1 Approximate frequency analysis The method of multiple scales is used to compute an approximate solution (frequency response function) for the perturbed system (4)-(5) (Cartmell 3 and Nayfeh and Mook 6 ). Theperturbed solutionsareexpressedbyx = x (T,T 1 )+εx 1 (T,T 1 )+... andy = y (T,T 1 )+εy 1 (T,T 1 )+..., where T = t is the fast time scale, T 1 = εt is the slow time scale and the remaining time scales are related by the perturbation as T n = ε n t, with n =,1,2,... Time derivatives along different time scales lead to differential operators d/dt = D +εd and d 2 /dt 2 = D 2 +2εD D The external and internal resonance conditions, characterizing the autoparametric interaction between the two degrees-of-freedom, are perturbed as Ω = ω 1 +ερ 1 (9) ω 1 = ω 2 +2ερ 2 (1) where ερ 1 and ερ 2 define the external and internal detuning parameters, respectively. Substitution of the proposed first order solutions x(t,t 1 ) and y(t,t 1 ) into (4)-(5) and grouping the zero and first order terms in ε, yields the set of partial differential equations ε : D 2 x +ω 2 1 x = (11) ε 1 : D 2 x 1 +ω 2 1 x 1 = 2ζ 1 ω 1 D x 2D D 1 x αx 3 +hy ( D 2 y ) +h(d y ) 2 +f cos(ωt ) (12) ε : D 2 y +ω 2 2 = (13) ε 1 : D 2 y 1 +ω 2 2y 1 = g ( D 2 x ) y 2D D 1 y 2ζ 2 ω 2 D y (14) The proposed solutions in their polar forms are expressed as x = A(T 1 )e iω1t +Ā(T 1)e iω1t (15) y = B(T 1 )e iω2t + B(T 1 )e iω2t (16) where the amplitudes depend on the slow time scale T 1 and the oscillations on the fast time scale T. Here Ā(T 1 ) and B(T 1 ) denote complex conjugates of the amplitudes A(T 1 ) and B(T 1 ), respectively. Substituting the proposed solutions in equations (12) and (14), removing secular terms and using the polar forms leads to A(T 1 ) = 1 2 a(t 1)e iδ(t1) (17) B(T 1 ) = 1 2 b(t 1)e iγ(t1) (18) iζ 1 ω 2 1a iω 1 a +ω 1 aδ 3 8 αa3 1 2 hω2 2b 2 e iφ feiφ1 = (19) 1 4 gω2 1abe iφ2 iω 2 b +ω 2 bγ iζ 2 ω 2 2b = (2) Proc. of SPIE Vol O-5 Downloaded From: on 2/19/216 Terms of Use:

6 where φ 1 = ρ 1 T 1 δ and φ 2 = 2γ δ 2ρ 2 T 1. Here a,b, δ and γ denote differentiation with respect to the slow time scale T 1. The steady state responses of the overall system are computed for a =, b =, δ = ρ 1 and γ = ρ 1 /2+ρ 2. The steady state responses are obtained by taking real and imaginary parts in (19)-(2) for the steady state conditions. Hence, by solving these equations the approximate amplitude responses for the primary and secondary subsystems are given by where (ερ1 4ω2 2 ) 2 +ω 1 a = (εg)ω (εζ 2 ) 2 (21) 2ω 2 = b 4 +Qb 2 +R (22) Q = 12ω 2(εα)(Ω 2ω 2 ) 3 (εg) 3 (εh)ω ω3 2(εα)(εζ 2 ) 2 (Ω 2ω 2 ) (εh)(εg) 3 ω1 6 [ ( ) ] 2 3 R = 234ω8 2 (εα)2 Ω (εh) 2 (εg) 6 1 +(εζ ω ) 2 768ω4 2 (εα)(ω ω 1) 2ω 2 (εh) 2 (εg) 4 ω [(Ω ω 1 ) 2 +ω 2 1 (εζ 1) 2] (εh) 2 (εg) 2 ω 2 1 8(Ω 2ω 2)(Ω ω 1 ) + 16(εζ 1)(εζ 2 ) (εh)(εg)ω 1 ω 2 (εh)(εg) ) ] 2 2 +(εζ 2 ) 2 [ ( Ω 2ω 2 1 [ ( ) 2 Ω 1 +(εζ 2 ) ] 2 (εf)2 2ω 2 (εh) 2 ω2 4 Note that the primary Duffing system response (21) does not depend on the cubic stiffness and the external force and this expression coincides with that reported by Vazquez and Silva 11 and Cartmell et al., 2,3 where there are only linear elements into the primary system. The secondary system response, however, is influenced by the cubic nonlinearity through the parameters Q and R. In fact, the amplitude (21) is also limited by the beam damping ζ 2. For more details we refer to Vazquez and Silva Experimental results The system parameters for the overall system are given in Table 1. The equivalent stiffness associated to the (first mode) cantilever beam absorber is such that the autoparametric vibration absorber is properly tuned with the excitation force. Table 1. System parameters. M = 4.46kg k 1 = 2466N/m k 2 = 26N/m 3 c 1 = 1.329N(m/s) m =.1kg L =.539m k viga = N/m c 2 =.33N/(m/s) I viga = m 4 A = m 2 E = 69GPa k c ǫ[ 3,3]N/m F = 6N ω 1 = rad/s = 3.723Hz ω 2 = rad/s = Hz The approximate frequency response for the passive Duffing system and its dynamic behavior, without autoparametric interaction (i.e., y ), are described in Fig. 4. Note how the steady-state amplitude a =.226 m is similar in both graphics. The dynamic response for the primary and secondary systems, under autoparametric interaction, are illustrated in Fig. 5, considering the system parameters shown in Table 1. The transient responses are obtained from the nonlinear system (1)-(2), with steady state amplitudes a =.54 m and b =.2415 m. It is important to note that, the percentage of absorption is about 76% (open loop system response) Proc. of SPIE Vol O-6 Downloaded From: on 2/19/216 Terms of Use:

7 a [m] ερ 1 [rad/s] x(t) [m] Figure 4. Experimental (point-to-point) frequency response function and transient experimental responses for the primary Duffing system, without autoparametric interaction, when F = 6N and Ω = Hz. 5. SYSTEM WITH A PASSIVE/ACTIVE CANTILEVER BEAM ABSORBER In case the excitation frequency Ω in the perturbation force F(t) is unknown or time varying, the cantilever beam absorber may not be useful for vibration attenuation in the primary Duffing system. However, when the excitation frequencies change such that Ω = ω 1, one is still able to satisfy the internal tuning condition ω 1 = 2ω 2 in order to get some attenuation of the primary Duffing system response. In our scheme this can be achieved by using an active control law on the PZT patch actuator, thus modifying the equivalent beam stiffness and damping. In Fig. 6 is described the schematic diagram of the proposed passive/active vibration control scheme. Proc. of SPIE Vol O-7 Downloaded From: on 2/19/216 Terms of Use:

8 x(t) [m] y(t) [m] Figure 5. Transient experimental responses for the primary Duffing system with autoparametric interaction and the cantilever beam absorber, when F = 6N and Ω = ω 1 = Hz. It is important to note that, when the frequency response function (21) is parameterized in terms of the equivalent stiffness k c, provided by the PZT actuator, results the approximate frequency response described in Fig. 7. Here, the nonlinear steady state amplitude (21) is shown in terms of ερ 1 and a reasonable range of the PZT actuator stiffness k c, obtained by a simple proportional control law, in such a way that, the internal resonance condition (1) can be satisfied to get the minimal attenuation gain. This information will be used to achieve an optimal attenuation operation for the autoparametric cantilever beam absorber. In fact, there exists Proc. of SPIE Vol O-8 Downloaded From: on 2/19/216 Terms of Use:

9 Shaker Ω F(t)=F cosωt Computation of optimal parameters [k 2(W),c 2(W)] Primary Damped Duffing System x(t) Strain gauge y(t) Active Vibration Control on the PZT patch actuator u(t) PZT Patch Actuator F(t) Autoparametric Cantilever Beam Absorber Figure 6. Block diagram of the passive/active autoparametric cantilever beam absorber with PZT actuator some region with minimal amplitudes, which can be computed to guarantee the optimal attenuation tuning for the passive/active vibration absorber. Figure 7. Parameterized FRF of the primary Duffing system in terms of the PZT actuator stiffness k c [ 5,5] N/m. Proc. of SPIE Vol O-9 Downloaded From: on 2/19/216 Terms of Use:

10 The passive/active control objective for the autoparametric cantilever beam absorber with PZT actuator is stated as follows (see Fig. 6): 1. Given an excitation frequency Ω, compute the optimal attenuation stiffness constant kc (Ω) for the PZT patch actuator, which minimizes the steady state amplitude of the primary Duffing system a for the passive vibration absorber, that is, min a(ω,k c ) (23) k cmin k c k cmax where a(ω,k c ) denotes the steady state amplitude in (21) parameterized in terms of Ω and k c, for the closed interval [k cmin,k cmax ] associated with the physical limitations of the PZT actuator. This solution is computed numerically. For practical purposes the optimal stiffness kc (Ω) can be computed and parameterized in terms of the excitation frequency Ω using curve fitting techniques on the data shown in Fig With the knowledge of the optimal attenuation stiffness k c(ω) is synthesized a proportional state feedback and feedforward control law to get the automatic tuning of the autoparametric cantilever beam absorber: u(t) = k c(ω)y(t) (24) Once the proportional controller (24) is activated, the steady-state response of the passive/active control system converges to the passive performance and, therefore, the control efforts are small compared to a fully active vibration control approach. Note that, the above control law is easy to be implemented and combined with an optimal attenuation criterion. In fact, the main idea is that the equivalent stiffness on the cantilever beam absorber can be controlled in order to get the best tuning condition for resonant vibrations. Remark: In case of large variations on the excitation frequency Ω, the external resonance condition (9) is not longer valid and, hence, the primary system response is affected according to the first-mode operation described by the primary Duffing frequency response (see Fig. 4), thus making useless the cantilever beam absorber. 5.1 Experimental results In order to illustrate the dynamic performance of the passive/active cantilever beam vibration absorber, when the excitation frequency is changing between two different constant values, we use the system parameters in Table 1. The initial conditions are set to x() = m, y() =.1 m, ẋ() = m/s and ẏ() = m/s. The harmonic force F(t) = F cos(ωt) is started with F = 6 N and excitation frequency Ω = ω 1 = Hz (i.e., ερ 1 = rad/s) In Fig. 8 is described the dynamic behavior of the overall closed-loop system (1)-(2) with the Proportional control law (24), which includes the primary Duffing system, the passive/active cantilever beam absorber with PZT actuator. Before t = 45 s the overall system is working in its passive form (i.e., u ), with excitation frequency Ω 1 = ω 1 = Hz and ω 2 = Hz. At t = 45 s the excitation frequency is increased to Ω = Hz (ερ 1 = +.15 rad/s). Here, one can observe that after a transient period about 15 s the primary system achieves the steady-state condition with small amplitudes. In Fig. 9 another experimental result is presented. Again, before t = 45 s the overall system is working in its passive form, with values of excitation frequency of Ω 1 = ω 1 = Hz and ω 2 = Hz, but now the excitation frequency is decreased at t = 45 s to Ω = Hz (ερ 1 =.15 rad/s). The overall system is still able to preserve the autoparametric interaction, that is, the primary system with small and stable amplitude. In Fig. 1 the excitation frequency is changed at t = 45 s, from Ω 1 = Hz to Ω 1 = Hz (ερ 1 = +.2 rad/s). In this case the transient response is bigger than the other ones (about 25 s) and the amplitude of the primary system is increased from a =.54 m to a =.87 m, meaning that the overall system is losing the autoparametric interaction. This phenomenon could be caused by a lack of adequate control effort provided by the small PZT patch actuator. The last experimental result is shown in Fig. 11, here at t = 45 s the excitation frequency changes from Ω 1 = Hz to Ω 1 = Hz (ερ 1 =.2 rad/s). Note how the primary system has a robust steady-state amplitude about a =.42 m. Proc. of SPIE Vol O-1 Downloaded From: on 2/19/216 Terms of Use:

11 x [m] y [m] u [N] Figure 8. Dynamic response of the Duffing system, with autoparametric interaction, using the passive/active beam absorber and PZT actuator switching the stiffness feedback exactly at the frequency change (ερ 1 = +.15 rad/s). 6. CONCLUSIONS The real-time application of a passive/active autoparametric cantilever beam absorber with PZT actuator to dampedduffing systemsisaddressed. Thedesignofthe activevibrationcontrolsystemisbasedonthedesignofa passive vibration absorber and the addition of a small PZT patch actuator to modify the equivalent beam stiffness in order to get an optimal attenuation steady state operation in case of varying excitation frequencies close to the principal parametric resonance. The active vibration scheme employs a simple proportional controller, which uses the measurements of the excitation frequency and the beam deflection. The overall dynamic performance proves the good robustness properties of the proposed control scheme for the attenuation in a highly nonlinear system with cubic stiffness and variable excitation frequencies close to the principal parametric resonant frequency. Further work is being performed to use a PZT actuator with a bigger capacity, using different control schemes in order to improve the transient and steady-state performance of the overall mechanical system. REFERENCES [1] Cartmell, M.P., The equations of motion for a parametrically excited cantilever beam, Journal of Sound and Vibration, Vol. 143, No. 3, (199). [2] Cartmell, M.P., Lawson, J., Performance enhancement of an autoparametric vibration absorber, Journal of Sound and Vibration, Vol. 177, No. 2, (1994). [3] Cartmell, M.P., [Introduction to Linear, Parametric and Nonlinear Vibrations], Chapman and Hall, London, 199. [4] Haxton, R.S., Barr, A.D.S., The autoparametric vibration absorber. Journal of Engineering for Industry, Vol. 94, No. 1, (1972). [5] Korenev, B.G., Reznikov, L.M., [Dynamic Vibration Absorber: Theory and Technical Applications], John Wiley & Sons, London, [6] Nayfeh, A.H., Mook, D.T., [Nonlinear Oscillations], John Wiley & Sons, NY, Proc. of SPIE Vol O-11 Downloaded From: on 2/19/216 Terms of Use:

12 x [m] y [m] u [N] Figure 9. Dynamic response of the Duffing system, with autoparametric interaction, using the passive/active beam absorber and PZT actuator switching the stiffness feedback exactly at the frequency change (ερ 1 =.15 rad/s). [7] Preumont, A., [Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems], Springer, Dordrecht, 26. [8] Silva-Navarro, G., Macias-Cundapi, L., Vazquez-Gonzalez, B., Design of a Passive/Active Autoparametric Pendulum Absorber for Damped Duffing Systems. In: New Trends in Electrical Engineering, Automatic Control, Computing and Communication Science, Edited by C.A. Coello, A. Pozniak, J.A. Moreno and V. Azhmyakov, Logos Verlag Berlin GmbH, Germany, (21). [9] Roberts, J.W., Random excitation of a vibratory system with autoparametric interaction, Journal of Sound and Vibration, Vol. 69, No. 1, (198). [1] Tondl, A., Ruijgrok, T., Verhulst, F., Nabergoj, R., [Autoparametric Resonance in Mechanical Systems], Cambridge University Press, Cambridge, 2. [11] Vazquez-Gonzalez, B., Silva-Navarro, G., Evaluation of the autoparametric pendulum vibration absorber for a Duffing system. Shock and Vibration, Vol. 15, No. 3-4, (28). Proc. of SPIE Vol O-12 Downloaded From: on 2/19/216 Terms of Use:

13 x [m] y [m] u [N] Figure 1. Dynamic response of the Duffing system, with autoparametric interaction, using the passive/active beam absorber and PZT actuator switching the stiffness feedback exactly at the frequency change (ερ 1 = +.2 rad/s) x [m] y [m] u [N] Figure 11. Dynamic response of the Duffing system, with autoparametric interaction, using the passive/active beam absorber and PZT actuator switching the stiffness feedback exactly at the frequency change (ερ 1 =.2 rad/s). Proc. of SPIE Vol O-13 Downloaded From: on 2/19/216 Terms of Use: