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Active vibration control of beam structures using acceleration feedback control with piezoceramic actuators
Journal of Sound and Vibration 331 (2012) 1257–1269
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Active vibration control of beam structures using acceleration feedback control with piezoceramic actuators Changjoo Shin a, Chinsuk Hong b,n, Weui Bong Jeong a a b
School of Mechanical Engineering, Pusan National University, Busan, 609-735, South Korea School of Mechanical Engineering, Ulsan College, Ulsan, 680-749, South Korea
a r t i c l e i n f o
abstract
Article history: Received 25 January 2011 Received in revised form 25 October 2011 Accepted 4 November 2011 Handling Editor: D.J. Wagg Available online 23 November 2011
In this study, the active vibration control of clamped–clamped beams using the acceleration feedback (AF) controller with a sensor/moment pair actuator configuration is investigated. The sensor/moment pair actuator is a non-collocated configuration, and it is the main source of instability in the direct velocity feedback control system. First, the AF controller with non-collocated sensor/moment pair actuator is numerically implemented for a clamped–clamped beam. Then, to characterize and solve the instability problem of the AF controller, a parametric study is conducted. The design parameters (gain and damping ratio) are found to have significant effects on the stability and performance of the AF controller. Next, based on the characteristics of AF controllers, a multimode controllable single-input single-output (SISO) AF controller is considered. Three AF controllers are connected in parallel with the SISO architecture. Each controller is tuned to a different mode (in this case, the second, third and fourth modes). The design parameters are determined on the basis of the parametric study. The multimode AF controller with the selected design parameters has good stability and a high gain margin. Moreover, it reduces the vibration significantly. The vibration levels at the tuned modes are reduced by about 12 dB. Finally, the performance of the AF controller is verified by conducting an experiment. The vibration level of each controlled mode can be reduced by about 12 dB and this value is almost same as the theoretical result. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Active vibration control methods are used for reduction of low frequency vibrations that are the main cause of structural weakness and damage. When any reference signal is not available, the vibration can be controlled by feedback control strategies including direct velocity feedback (DVFB), positive position feedback (PPF), and acceleration feedback (AF). DVFB uses structural velocities for feedback signals to actuators. It is a simple proportional controller. The controller needs only a signal amplifier, so it can be implemented at a low cost. DVFB generates a skyhook damping [1], hence, this controller can reduce vibrations at all frequencies. The collocated sensor/actuator pair in which both the sensor and force actuator are placed at the same point makes the control system unconditionally stable. This means that a control system with an infinite gain margin can be obtained. Elliott [2] and Balas [3] proposed a robust AVC system based on the use of DVFB with a collocated sensor/force actuator pair. However, when a moment pair actuator such as the piezoceramic patch is applied and the feedback velocity is acquired at the center of the moment pair, DVFB shows severe instability problem [4]. This is because of the non-collocated sensor/
n
Corresponding author. Tel.: þ82 52 279 3134; fax: þ82 52 279 3137. E-mail addresses: [email protected], [email protected] (C. Hong).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.11.004
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moment pair actuator configuration. The non-collocated sensor/moment pair actuator induces phase shifts, and this phenomenon leads to instability of the control system, particularly at high frequencies [5,6]. Since the non-collocated configuration of the sensor/actuator holds stable behavior at low frequencies, this configuration is referred to as nearcollocated or quasi-collocated configuration. The configuration with the near-collocated configuration leads, however, to instability. To reduce this instability of the control system with the sensor/moment pairs, Gardonio proposed phase lag compensators to his multichannel smart panel [7]. Hong [8] implemented a multichannel active control system for honeycomb structures. He found that the honeycomb structure works as a low pass filter, so the control system can overcome the instability problem in the high frequencies. PPF controller is an effective active vibration control method that helps to overcome the instability problem of the control system. The PPF controller was developed to suppress the vibration of large flexible structures such as space structures [9,10]. The PPF controller is an electronically implemented dynamic absorber. Hence, the controller is tuned to the structural resonance to be controlled. The PPF control can make the system more stable and so obtained a high performance at the target frequency, because the controller generates the control signal at its resonance. In addition, the instability problem of the sensor/moment pair configuration can be overcome by the roll-off characteristic of the PPF controller: this characteristic results in rapid reduction of the magnitude of the open loop transfer function. Its behavior is similar to that of the low pass filter [11]. However, the PPF controller requires the modal displacement of the structure for the feedback signal and this displacement cannot be measured. Kwak [12] suggested that the modal displacement can be obtained by the block-inverse technique using the measured displacement. The displacement measurement is not easy because the sensors are large and expensive. On the other hand, accelerometers are general equipments that are small, light, and cheap. Juang [13] proposed AF for the second-order compensator. Preumont applied AF to beam structures and investigated its characteristics [14]. Sim and Lee [15] proved the stability of the AF controller in their theoretical study. The sensor and actuator in their study is the collocated configuration. For non-collocated sensor and actuator configuration as in this paper, the proved condition in stability does not hold. The stability of AF controller using the non-collocated configuration needs to be clarified. It is interesting that PPF and AF have the same transfer function, but have different inputs. Input to the PPF is displacement while the input to the AF is acceleration. Thus, AF controller could be defined as the PPF controller with the input of the accelerometer signal. It should be noted, however, that the characteristics are quite different. Since PPF controller takes the displacement as the input, the controller takes a higher input at lower frequencies. On the contrary, since AF controller takes the acceleration as the input, the controller takes a higher input at higher frequencies. Therefore, the stability of PPF controller is more affected by the low frequency response while that of AF controller is more affected by the high frequency response. The main advantage of AF control over the PPF control is the use of accelerometers as the feedback sensors. In this study, a multimode controllable SISO AF controller with the sensor/moment pair on the clamped–clamped beam configuration is investigated. In Section 2, the equation of motion of a clamped beam with lumped masses is explained. In Section 3, the AF control method is mathematically formulated, and the principle of the controller is explained. In Section 4, the influence of the design parameters of the AF controller on the stability and performance of the AF controllers is examined. In Section 5, simulations of the stability and performance of the multimode controllable SISO AF control system for clamped–clamped beams are presented. Experiments on the AF controller are presented in Section 6. Finally, our conclusions are given in Section 7. 2. Equation of motion of clamped beams with lumped masses Consider a clamped beam with lumped masses as shown in Fig. 1. The equation of motion for this system can be obtained as EIð1 þ jZÞ
! 2 2 q4 y q2 y qT s ðx,tÞ ð1Þ q y ð1Þ ð2Þ q y ð2Þ , þ r A þ m d ðxx Þ þm d ðxx Þ ¼ f p ðx,tÞ þ a a a a qx qx4 qt 2 qt 2 qt 2
(1)
where E is Young’s modulus, I is the moment of inertia of the beam and Z is the loss factor. r is the mass density, A is the ð2Þ cross sectional area, mð1Þ a is the mass of the force transducer on the beam, and ma is the total mass of the masses of the ð1Þ ð2Þ piezoceramic patch and the accelerometer on the beam. So ma ¼ mFT and ma ¼ macc þmPZT . f p ðx,tÞ is the external force and T s ðx,tÞ is the control moment pair. The general solution of Eq. (1) can be expressed as the sum of the normal modes. Hence, the displacement distribution of the beam is represented by Yðx, oÞ ¼ UðxÞpðoÞ,
(2)
T
where pðoÞ ¼ ½p1 p2 pN which is the modal displacement vector, U is the mode shape matrix defined by
U ¼ ½/1 /2 /N
(3)
and N is the number of modes superposed. Utilizing the mode orthogonality leads to the matrix equation as ½o2 Iþ j diagð2zn on oÞ þ diagðo2n Þp ¼ UT f,
(4)
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Fig. 1. Schematic configuration for active control system for a clamped beam with a sensor/moment pair actuator system.
where zn is the nth modal damping ratio, on is the nth natural frequency of the structure and f is the modal force vector, expressed by qfðxÞ , (5) f ¼ f p ffðxp ÞgT s qx x ¼ xs ð2Þ where the force and the moment pair are concentrated at xp ¼ xð1Þ a and xs ¼ xa 7 s, respectively.
3. Acceleration feedback control AF control is a recently proposed control method. Its best feature is that the feedback signal is acceleration, which can be directly measured by accelerometers [14]. The AF controller requires the modal acceleration, which can be extracted from the measured acceleration signal. The AF controller for the specific target mode generates a controller output signal using the modal acceleration as the input signal. The control signal goes to the piezoceramic actuator and the structure is then under control. Using Va in Fig. 1 and Eq. (4), the matrix form of the structure equation in the control system can be written as T qU Ba V a þ UT ðxp Þf p (6) ½o2 I þj diagð2zn on oÞ þdiagðo2n Þp ¼ qx and the equation of the AF controller is written of the form: ½o2 Iþ j diagð2zc oc Þ þ diagðo2c Þq ¼ g o2 Ec p,
(7)
where the control output vector, q, whose component represents the control signal for each target mode. It is used to generate the physical control signal, Va, as V a ¼ Bs UEa q,
(8)
where Bs is the participation matrix that indicates the locations of the accelerometers: Bs ¼ ½0 0 1 0 0 T
and Ea is the mode arrange matrix for the modal superposition. U primary force, fp. ½qU=qxT Ba V a in Eq. (6) denotes the modal force of qU qf1 qf2 ¼ qx qx qx
(9)
f p in Eq. (6) represents the modal force due to the the control action by the moment pair where qfN (10) qx
and Ba is the participation matrix for the moment pair action location of the actuators: Ba ¼ ½0 0 1 0 0 1 0 0T :
(11)
In Eq. (7), zc and oc are the damping ratio and natural frequency of the compensator, respectively. g is the controller gain, Ec is the mode extraction matrix for the control mode. According to Eq. (7), the compensator requires the structural modal acceleration, p. This structural modal acceleration can be obtained as o2 p ¼ ðBs UÞ þ V r , þ
(12)
where is the pseudo-inverse operator, and Vr is the voltage signal measured by the feedback accelerometer. The pseudoinverse operation in Eq. (12) poses unreliable control weights to modal coordinate unless the number of the feedback sensors are available enough to account the number of modes. For practical case of this study, only one accelerometer is used for the feedback. Thus, Bs U of the Eq. (12) is the real valued row vector of mode shape at the sensor location. Then, its pseudo-inverse, ðBs UÞ þ , is to be a real valued column vector. In other words, ðBs UÞ þ acts as just constant vector having
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real values. This means that the modal acceleration to be controlled is proportional to the physical acceleration measured by the feedback sensor. However, the error in the calculation by pseudo-inverse operator can be compensated by the characteristics of the AF controller—low pass filtering and amplifying at the tuning frequency. Although we estimated unreliable control weight the error can be compensated by the characteristics of the AF controller. The controller modal signal, q, as calculated by Eq. (7) should transform into the physical controller signal, Va, as in Eq. (6). Fig. 2 shows a block diagram for active control using the SISO AF controller. The plant response, GðxÞ, and the AF controller, HAF , in the physical coordinates are T qU Ba (13) GðoÞ ¼ o2 Bs U½o2 I þj diagð2zs os oÞ þ diagðo2s Þ1 qx and HAF ðoÞ ¼ Bs UEa diagðg i ÞHðoÞEc ðBs UÞ þ , where HðxÞ ¼ diag
! 1 : o2ci o2 þ j2zci oci
(14)
(15)
The index i specifies the mode order to be controlled. The sensor output, Vr, is the sum of the disturbance, Vp, and the structure response, Vs, due to the actuator action. Therefore, Vr can be expressed as V r ðoÞ ¼ V p ðoÞ þ V s ðoÞ ¼ o2 Bs Upp ðoÞ þGðoÞV a ðoÞ,
(16)
where pp is the modal displacement vector due to the primary force, which can be obtained by Eq. (7) with V a ¼ 0. Va in Eq. (16) is the input signal to the actuator or the output from the controller, i.e., V a ¼ HAF V r :
(17)
Substituting Eq. (17) into Eq. (16), the sensor output voltage and the control signal voltage for the actuator action become V r ðoÞ ¼ ½IGHAF 1 Bs Upp
(18)
V a ðoÞ ¼ HAF ½IGHAF 1 Bs Upp :
(19)
and
In this configuration, stability should be considered, because it is a feedback control system. The stability of the system can be estimated by the open loop transfer function (OLTF) OLTFðoÞ ¼ GHAF :
(20)
G is the acceleration response of the plant at 0.8L when the unit moment pair is applied to the structure. HAF is the transfer function of the AF controller as given in (14). Fig. 3 shows the plant response, GðoÞ, when the sensor/actuator pair is placed at 0.8L. In the Bode plot, the magnitude of the plant response increases as the frequency increases and the phase shifts. This means that an unmodeled response at high frequencies can cause the control system to be unstable. The phase remains between 01 and 1801 up to 17.3 kHz except for the first mode. The phase behavior at the first mode is directly affected by the location of the sensor/moment pair actuator. if this sensor/moment pair actuator is located within 0:2L of the structure boundary, then the plant becomes conditionally stable [16]. Because the sensor/moment pair actuator is on the 0:2L of the beam from the clamped boundary, the unstable phase response occurs in the first mode of the plant response. The Nyquist plot of the plant response shows that active damping cannot be obtained with simple proportional controllers. Fig. 4 shows the transfer function of the AF controller that is tuned to the fourth mode of the structural resonance at 302 Hz. The tuned frequency is the natural frequency of the target mode. This controller has two main characteristics:
Fig. 2. Block diagram for active feedback control using AF controller.
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Fig. 4. The Bode and Nyquist plot of the transfer function of the AF controller response which is tuned to the fourth mode of the structural resonance.
phase shift and magnitude roll-off at high frequencies. The phase shift occurs at the target frequency from 1801 to 01. This phase shift gives the active damping to the system at the tuning frequency. The magnitude of the transfer function of the controller is rapidly reduced at high frequencies. Thus, the magnitude of the unmodeled plant response at high frequencies is reduced, and so the gain margin is significantly increased. Fig. 5 shows the OLTF with an AF controller that is tuned to the fourth mode of the structural resonance. The peak magnitude of this OLTF occurs at the fourth mode, and the phase is shifted to the corresponding mode. In the Nyquist plot, the response at the tuned mode has the largest magnitude in the right half-plane. This response of the OLTF at the tuned mode generates the active damping force. The magnitude of the OLTF is reduced as the frequency increases. This rapid roll-off phenomenon at high frequencies is due to the low pass filtering characteristic of the AF controller as shown in Fig. 4. The unstable condition is observed at the first mode. However, the magnitude difference between the first mode and the tuning mode is more than 50 dB. Therefore, a high gain margin of the control system can be achieved. It should be noted that the lower modes are not affected by the controller. In the Nyquist plot, the lower modes are on the positive imaginary axes. These correspond to the active stiffness, so the responses of the lower modes are partly in the unit circle about ð1,j0Þ, indicating enhancement at the frequencies in the Nyquist plot.
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Fig. 5. The Bode and Nyquist plot of the open loop transfer function with AF controller which is tuned to the fourth mode of the structural resonance.
4. Parametric study of AF controllers In this section, a series of design parametric studies in terms of the stability and performance in conducted to design the controller to have a larger gain margin and a high performance. It is essential to check the stability of the control system since it is tuned to the resonance frequency. The design parameters we investigated here are the tuning frequency, the damping ratio and gain. Fig. 6 shows the variation of the OLTF of the fourth mode tuned AF controller for the gains of 1:0 105 , 3:0 105 , 5:0 105 and 8:68 105 . The damping ratio of the AF controller is set to 5:0 102 . The magnitude of the OLTF is increased at all the frequencies as the gain increases, as shown in the Bode diagram of Fig. 6(a), while the phase of the OLTF is not changed. Fig. 6(b) shows the OLTF normalized by the corresponding feedback gains for the four gains. All the curves are perfectly overlapped. This means that the magnitude of the OLTF varies in proportion to the gain. There are the two Nyquist curves crossing the positive imaginary axis. These two responses correspond to modes lower than the tuned mode. This behavior is due to the low pass filter effect of the AF controller, as shown in Fig. 4. This effect leads to enhancement in the closed loop response at the some frequencies where the Nyquist curves of the two modes are in the unit circle about ð1,j0Þ. Fig. 7 shows the variation of the closed loop acceleration response at 0.8L of the structure with the uncontrolled response (solid line) and the controlled response for feedback gains of 1 105 (dashed line), 3 105 (dash-dotted line), 5 105 (dotted line) and 8:68 105 (thin dashed line) subjected to an AF controller tuned to the fourth mode of the structure. A damping ratio of 5 102 is used. The uncontrolled response is due to the disturbance, fp, at 0.2L. Two features can be observed. First, the closed loop acceleration response at the sensor location (xð2Þ a ¼ 0:8L) is decreased at the target mode as the gain is increased. This is illustrated by the increase of the magnitude of the OLTF as the gain increases, as shown in Fig. 6(a). Second, new peaks around the tuning frequency are also found. The difference between the frequencies and the new peaks increases as the gain increases, as if the mass ratio is increased in the dynamic absorbers [17]. In addition to the gain effects on the closed loop acceleration response around the tuning frequency, another significant behavior is found. The closed loop acceleration response around the tuning frequency is significantly changed in magnitude and structural resonance frequencies. As a result of the gain effects of the AF controller, the resonance frequencies of the second and third modes are significantly decreased. On the other hand, those from the fifth to thirteenth mode are slightly increased. The resonance frequencies of the second and third modes are decreased, as the gain increases, but, the resonance frequency of the first mode is not changed. For a gain of 8:68 105 , the peak of the second resonance for the gain of 5 105 is combined with that of the first mode. Hence, the magnitude of the first mode is rapidly amplified to 88.7 dB. This phenomenon can be explained by the response of the OLTF crossing the negative real axis around the frequency of the first mode in the Nyquist plot. These responses are also within the unit circle about the ð1,j0Þ point. The Nyquist curves at the second and third modes move toward the Nyquist point as the gain increases. This behavior leads to variation in the lower mode response in terms of the magnitude and the natural frequencies. The variation of the resonance frequency at the second and third modes can also be illustrated by the behavior of the transfer function of the AF controller shown in Fig. 4. The essential relevant feature of this transfer function is the phase behavior, in that there is no phase change at frequencies lower than the target frequency. Since the feedback signal is proportional to the acceleration at the actuator location, the AF controller plays a role of the stiffness at frequencies lower
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Fig. 6. Variation of the open loop transfer function of active vibration control systems at 0:8L with the feedback gain of AF controllers with a sensor/ actuator pair. The AF controllers are tuned to the fourth mode with damping ratio of 5:0 102 and gains of 1:0 105 (solid line), 3:0 105 (dashed line), 5:0 105 (dash-dotted line) and 8:68 105 (dotted line). Bode diagram (a) and normalized Nyquist diagram by the corresponding gain (b).
than the target frequency. Note that the AF controller decreases the stiffness as the gain increases. Because of the phase change of 901 at the target frequency, the AF controller generates a damping force. Moreover, because of the phase change of 1801 at frequencies higher than the target frequency, the AF controller generates an inertia force. Fig. 8 shows the variation of the OLTF of the fourth mode tuned AF controller with the damping ratios of 5 103 , 5 102 , 1 101 , and 8 101 . The gain of 2 103 is used for this simulation. As shown in Fig. 8(a), the magnitude of the OLTF is decreased as the damping ratio is increased at the tuned mode, but it is not changed in the other modes. The phase around the tuned mode changes significantly when the damping ratio is increased. The phases of the OLTF are varied in a wide frequency range with increasing damping ratio. The phase behavior can be represented more clearly in the zoomed Nyquist diagram as the shown in Fig. 8(b). Two hollow marks are placed at the same frequency about 290 Hz, J for z ¼ 5 103 and n for z ¼ 1 101 . Two solid marks are also placed at the same frequency about 315 Hz, for z ¼ 5 103 and m for z ¼ 1 101 . Therefore, the bandwidth of the AF controller is not varied with the change of the damping ratio. The responses within the unit circle about ð1,j0Þ move away from the Nyquist point as the damping ratio increases. Thus, the control system is stabilized with the increase of the damping. Fig. 9 shows the variation of the closed loop acceleration response with the damping ratio of the AF controller when the AF controller is tuned to the fourth mode of the structure. The damping ratios for this simulation are 5 103 , 5 102 ,
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1:0 101 , and 8:0 101 . The feedback gain of the AF controller is set to 2 103 . The reduction in the structure response of the fourth mode is decreased as the damping ratio of the AF controller increases as shown in Fig. 9(a). Although the damping ratio is increased, the bandwidth of the AF controller from 290 Hz to 315 Hz is not remained, as shown in Fig. 9(b). Even if new peaks at the both side of the tuning frequency are emerged, their levels are decreased as the damping ratio is increased. The existence of the new peaks is explained by the enhancement related to the existence of the OLTF response in the unit circle about ð1,j0Þ.
5. Implementation of multimode AF controllers Based on the examination of the behavior of the AF controller, a SISO multimode control is examined in this section. To implement the SISO multimode AF controller, a control signal is obtained by summation of the signals from AF controllers that are tuned to each mode and connected in parallel. The parameters of the AF controllers are set based on the parametric study conducted in the previous section. The higher modes are designed first. This is to alleviate the effects of the higher modes on the responses of the lower modes. Fig. 10 show the schematic diagram of the multimode AF controller we have designed sequentially. The measured acceleration, Vr, is used to obtain the modal acceleration, pi, supplied to the AF controller of each mode. The AF controller generates the control signal for each mode, and then the control signal are superposed to make the control signal, Va, in order to actuate the PZT patch. Fig. 11 shows the open loop transfer function of the combined AF control system with the sensor/moment pair actuator at 0.8L of the clamped–clamped beam. The second mode tuned controller was designed with a gain of 3:3 102 and a damping ratio of 8 102 . The third mode tuned controller was designed with a gain of 7 102 and a damping ratio of 1 101 . The fourth mode tuned controller was designed with a gain of 1:1 103 and a damping ratio of 4 102 . It can be found from Fig. 11(a) that the magnitudes of the combined OLTF are quite high at the three tuned modes; they are almost the same. The phases of the three modes are in 7901. This means that the AF controller generates an active damping force. Hence, we can expect approximately the same performance. A maximum reduction in the acceleration response of about 12 dB at each tuning frequency can be expected in the combined AF controller. Fig. 11(b) is the Nyquist diagram of the combined OLTF. The multimode controller is stable with a high gain margin. The magnitude of the OLTF response within the unit circle about ð1,j0Þ is smaller than that outside this circle. Therefore, the magnitude enhancement is smaller than the reduction around the tuning frequency. Fig. 12 shows the closed loop acceleration response of the clamped–clamped beam at 0.8L, excited by a concentrated force at 0.2L and subjected to the control moment pair at 0.8L—without control (solid line) and with control (dashed line). The disturbances are reduced by about 12 dB at each tuned mode, as predicted in Fig. 11. The controller does not affect the response at modes other than the tuned modes.
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Real Fig. 8. Variation of the open loop transfer function of active vibration control systems at 0.8L with the damping ratio of AF controllers with a sensor/ actuator pair. The AF controllers are tuned at the fourth mode with gain of 2:0 103 and damping ratios of 5 103 (solid line), 5 102 (dashed line), 1:0 101 (dash-dotted line), and 8:0 101 (dotted line). Bode diagram (a) and zoomed Nyquist diagram (b).
6. Experiments Based on the analytic examination of the AF controller behavior, its performance of the AF controller is experimentally verified in this section. To implement HAF for a real structure, the transfer function of the AF controller shown in Eq. (15) can be rewritten in the s-domain function as ! 1 HAF ðsÞ ¼ diag 2 (21) s þ 2zci oci s þ o2ci and it is coded using Matlab/Simulink. The implementation of the transfer function, Eq. (21), is downloaded to the ROM of DSP (dSPACE DS1103). Then the DSP works as AF controller. Fig. 13 shows the experimental setup for the active control of the clamped beam with the AF controller. An aluminum beam is used for this experiment; its length is 0.5 m. Both edges of the beam are clamped by vises. The beam is disturbed by a mini-shaker (B&K 4810) at 0.2L. The disturbance force is measured by a force transducer (B&K 8200). An accelerometer (B&K 4393) is installed on the beam at 0.8L, and it measures the acceleration at the sensor location. The measured acceleration signal is fed to the controller (dSPACE DS1103), which generates the control signal. The power amplifier (AVL 790) amplifies the control signal to drive the piezoceramic actuator. The piezoceramic actuator is a
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Fig. 9. Variation of the closed loop acceleration response with damping ratio of the AF controller at 0:8L. The AF controllers are tuned to the fourth mode with gain of 2:0 103 . Uncontrolled response (solid line) and controlled response for damping ratios of 5 103 (dashed line), 5 102 (dash-dotted line), 1:0 101 (dotted line), and 8:0 101 (thin dashed line). Structural acceleration (a) and zoomed structural acceleration (b).
Fig. 10. Schematic diagram of the multimode AF controller.
rectangular shaped 1–3 mode PZT patch (25 mm 25 mm 1 T) whose material properties hold with PZT4 series, manufactured by Kyungwon Ferrite Ind. Co., Ltd (South Korea). Fig. 14 shows the experimental acceleration response of the clamped beam excited by a concentrated force at 0.2L (solid line), and when subjected to the AF controller with a sensor/moment pair actuator system at 0.8L (dashed line). The structural response is experimentally reduced to about 12 dB, as we predicted analytically; see Fig. 12. In this experimental examination, the controller does not affect the response at modes other than the tuned modes.
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Fig. 11. Final designed open loop transfer function of active control system with AF controller at 0.8L. Bode diagram (a) and Nyquist diagram (b).
Fig. 12. Analytic acceleration response of clamped beam excited by a concentrated force at 0.2L (solid line), and when subjected to the AF controller with sensor/moment pair actuator system at 0.8L (dashed line).
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Fig. 13. Experimental setup for active control of clamped beam with AF controller.
Accelerance (dB, ref. 1m/s /N)
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7. Conclusion In this study, the active vibration control of clamped beams using AF controllers with a non-collocated sensor/moment pair actuator configuration is investigated. The moment pair actuator is used as the control actuator and the accelerometer is used as the feedback sensor located at the opposite center of the moment pair. This non-collocated configuration is the main source of instability problems in control systems. To overcome the instability problem due to the non-collocated sensor/moment pair actuator configuration, an AF controller with a sensor/moment pair actuator for a clamped beam with lumped masses has been designed, implemented and experimentally verified. A parametric study of the design parameters of the AF controller was conducted to characterize their effects on the stability and performance of the controller. These parameters were the gain, damping ratio and tuning frequency. The magnitude of the OLTF was increased as the gain increased, while the characteristics of the phase were retained. Thus, the gain did not change the stability but changed the performance. Furthermore, the difference between the frequencies of the new peaks produced by the AF controller increased as the gain increased. The effective bandwidth for the reduction broadened as the gain increased at the tuned modes. The magnitude of the OLTF was decreased at the tuned frequency, and the phase of the OLTF was affected over a wider range of the tuned mode as the damping ratio increased. Increasing the damping ratio resulted in an increase in the stability of control system, but the control system required a higher gain to achieve the same performance. Note that the difference between the frequencies of the new peaks produced by the AF controller did not vary as the damping ratio increased. Finally, when the AF controller was tuned at a higher mode, the effects of the gain and the damping were the same as those of the controller tuned to a lower mode; the only difference
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was that the AF controller significantly affected the response in the lower modes. The natural frequency of the combined system (structure and AF controller) was decreased at the lower modes. A multimode (the second, third and fourth modes) controllable AF controller was implemented, and it was verified that the controller is stable and shows high performance. The control signal was obtained by summation of the signals from each AF controller, which was tuned to each mode and connected in parallel. The vibration levels at the tuned modes were reduced by about 12 dB, as determined numerically and experimentally.
Acknowledgement This work was partly supported by the 2011 Research Fund of Ulsan College. References [1] M. Serrand, S.J. Elliott, Multichannel feedback control for the isolation of base-excited vibration, Journal of Sound and Vibration 234 (2000) 681–704. [2] S.J. Elliott, P. Gardonio, T.C. Soros, M.J. Brennan, Active vibroacoustic control with multiple local feedback loops, Journal of the Acoustic Society of America 111 (2002) 908–915. [3] M.J. Balas, Direct velocity feedback control of large space structure, Journal of Guidance, Control and Dynamics 2 (1979) 252–253. [4] R.H. Cannon Jr., D.E. Rosenthal, Experiments on control of flexible structure with noncollocated sensors and actuators, Journal of Guidance, Control and Dynamics 7 (1984) 546–553. [5] Q. Zhang, S. Shelley, R.J. Allemang, Active damping design of flexible structures based on SISO and SIMO noncollocated sensor/actuator velocity feedback, Journal of Dynamic systems Measurement and Control—Transactions of the ASME 113 (1991) 259–266. [6] G. Gatti, M.J. Brennan, P. Gardonio, Active damping of a beam using a physically collocated accelerometer and piezoelectric patch actuator, Journal of Sound and Vibration 303 (2007) 798–813. [7] P. Gardonio, E. Bianchi, S.J. Elliott, Smart panel with multiple decentralized units for the control of sound transmission. Part II: design of the decentralized control units, Journal of Sound and Vibration 274 (2004) 193–213. [8] C. Hong, S.J. Elliott, Local feedback control of light honeycomb panels, Journal of Acoustical Society of America 121 (2006) 222–233. [9] C.J. Goh, T.K. Caughey, On the stability problem caused by finite actuator dynamics in the collocated control of large space structures, International Journal of Control 41 (1985) 787–802. [10] G. Song, S.P. Schmidt, B.N. Agrawal, Experimental robustness study of positive position feedback control for active vibration suppression, Journal of Guidance 25 (2002) 179–182. [11] Z.-c. Qiu, X.-m. Zhang, H.-X. Wu, H.-h. Zhang, Optimal placement and active vibration control for piezoelectric smart flexible cantilever plate, Journal of Sound and Vibration 301 (2007) 521–543. [12] M.K. Kwak, S. Heo, Active vibration control of smart grid structure by multiinput and multioutput positive position feedback controller, Journal of Sound and Vibration 304 (2007) 230–245. [13] J.N. Juang, M. Phan, Robust controller design for second-order dynamic systems: a virtual passive approach, Journal of Guidance, Control and Dynamics 15 (1992) 1192–1198. [14] A. Preumont, N. Loix, Active damping of a stiff beam-like structure with acceleration feedback, Experimental Mechanics 34 (1) (1994) 23–26. [15] E. Sim, S.W. Lee, Active vibration control of flexible structures with acceleration feedback, Journal of Guidance, Control and Dynamics 16 (2) (1993) 413–415. [16] C. Hong, S.J. Elliott, Active control of resiliently-mounted beams with a moment pair actuator, Smart Materials and Structures 14 (2005) 727–738. [17] C.M. Harris, A.G. Piersol, Harris’ Shock and Vibration Handbook, fifth Ed., McGraw-Hill, 2002.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Active vibration control of beam structures using acceleration feedback control with piezoceramic actuators Changjoo Shin a, Chinsuk Hong b,n, Weui Bong Jeong a a b
School of Mechanical Engineering, Pusan National University, Busan, 609-735, South Korea School of Mechanical Engineering, Ulsan College, Ulsan, 680-749, South Korea
a r t i c l e i n f o
abstract
Article history: Received 25 January 2011 Received in revised form 25 October 2011 Accepted 4 November 2011 Handling Editor: D.J. Wagg Available online 23 November 2011
In this study, the active vibration control of clamped–clamped beams using the acceleration feedback (AF) controller with a sensor/moment pair actuator configuration is investigated. The sensor/moment pair actuator is a non-collocated configuration, and it is the main source of instability in the direct velocity feedback control system. First, the AF controller with non-collocated sensor/moment pair actuator is numerically implemented for a clamped–clamped beam. Then, to characterize and solve the instability problem of the AF controller, a parametric study is conducted. The design parameters (gain and damping ratio) are found to have significant effects on the stability and performance of the AF controller. Next, based on the characteristics of AF controllers, a multimode controllable single-input single-output (SISO) AF controller is considered. Three AF controllers are connected in parallel with the SISO architecture. Each controller is tuned to a different mode (in this case, the second, third and fourth modes). The design parameters are determined on the basis of the parametric study. The multimode AF controller with the selected design parameters has good stability and a high gain margin. Moreover, it reduces the vibration significantly. The vibration levels at the tuned modes are reduced by about 12 dB. Finally, the performance of the AF controller is verified by conducting an experiment. The vibration level of each controlled mode can be reduced by about 12 dB and this value is almost same as the theoretical result. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Active vibration control methods are used for reduction of low frequency vibrations that are the main cause of structural weakness and damage. When any reference signal is not available, the vibration can be controlled by feedback control strategies including direct velocity feedback (DVFB), positive position feedback (PPF), and acceleration feedback (AF). DVFB uses structural velocities for feedback signals to actuators. It is a simple proportional controller. The controller needs only a signal amplifier, so it can be implemented at a low cost. DVFB generates a skyhook damping [1], hence, this controller can reduce vibrations at all frequencies. The collocated sensor/actuator pair in which both the sensor and force actuator are placed at the same point makes the control system unconditionally stable. This means that a control system with an infinite gain margin can be obtained. Elliott [2] and Balas [3] proposed a robust AVC system based on the use of DVFB with a collocated sensor/force actuator pair. However, when a moment pair actuator such as the piezoceramic patch is applied and the feedback velocity is acquired at the center of the moment pair, DVFB shows severe instability problem [4]. This is because of the non-collocated sensor/
n
Corresponding author. Tel.: þ82 52 279 3134; fax: þ82 52 279 3137. E-mail addresses: [email protected], [email protected] (C. Hong).
0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.11.004
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moment pair actuator configuration. The non-collocated sensor/moment pair actuator induces phase shifts, and this phenomenon leads to instability of the control system, particularly at high frequencies [5,6]. Since the non-collocated configuration of the sensor/actuator holds stable behavior at low frequencies, this configuration is referred to as nearcollocated or quasi-collocated configuration. The configuration with the near-collocated configuration leads, however, to instability. To reduce this instability of the control system with the sensor/moment pairs, Gardonio proposed phase lag compensators to his multichannel smart panel [7]. Hong [8] implemented a multichannel active control system for honeycomb structures. He found that the honeycomb structure works as a low pass filter, so the control system can overcome the instability problem in the high frequencies. PPF controller is an effective active vibration control method that helps to overcome the instability problem of the control system. The PPF controller was developed to suppress the vibration of large flexible structures such as space structures [9,10]. The PPF controller is an electronically implemented dynamic absorber. Hence, the controller is tuned to the structural resonance to be controlled. The PPF control can make the system more stable and so obtained a high performance at the target frequency, because the controller generates the control signal at its resonance. In addition, the instability problem of the sensor/moment pair configuration can be overcome by the roll-off characteristic of the PPF controller: this characteristic results in rapid reduction of the magnitude of the open loop transfer function. Its behavior is similar to that of the low pass filter [11]. However, the PPF controller requires the modal displacement of the structure for the feedback signal and this displacement cannot be measured. Kwak [12] suggested that the modal displacement can be obtained by the block-inverse technique using the measured displacement. The displacement measurement is not easy because the sensors are large and expensive. On the other hand, accelerometers are general equipments that are small, light, and cheap. Juang [13] proposed AF for the second-order compensator. Preumont applied AF to beam structures and investigated its characteristics [14]. Sim and Lee [15] proved the stability of the AF controller in their theoretical study. The sensor and actuator in their study is the collocated configuration. For non-collocated sensor and actuator configuration as in this paper, the proved condition in stability does not hold. The stability of AF controller using the non-collocated configuration needs to be clarified. It is interesting that PPF and AF have the same transfer function, but have different inputs. Input to the PPF is displacement while the input to the AF is acceleration. Thus, AF controller could be defined as the PPF controller with the input of the accelerometer signal. It should be noted, however, that the characteristics are quite different. Since PPF controller takes the displacement as the input, the controller takes a higher input at lower frequencies. On the contrary, since AF controller takes the acceleration as the input, the controller takes a higher input at higher frequencies. Therefore, the stability of PPF controller is more affected by the low frequency response while that of AF controller is more affected by the high frequency response. The main advantage of AF control over the PPF control is the use of accelerometers as the feedback sensors. In this study, a multimode controllable SISO AF controller with the sensor/moment pair on the clamped–clamped beam configuration is investigated. In Section 2, the equation of motion of a clamped beam with lumped masses is explained. In Section 3, the AF control method is mathematically formulated, and the principle of the controller is explained. In Section 4, the influence of the design parameters of the AF controller on the stability and performance of the AF controllers is examined. In Section 5, simulations of the stability and performance of the multimode controllable SISO AF control system for clamped–clamped beams are presented. Experiments on the AF controller are presented in Section 6. Finally, our conclusions are given in Section 7. 2. Equation of motion of clamped beams with lumped masses Consider a clamped beam with lumped masses as shown in Fig. 1. The equation of motion for this system can be obtained as EIð1 þ jZÞ
! 2 2 q4 y q2 y qT s ðx,tÞ ð1Þ q y ð1Þ ð2Þ q y ð2Þ , þ r A þ m d ðxx Þ þm d ðxx Þ ¼ f p ðx,tÞ þ a a a a qx qx4 qt 2 qt 2 qt 2
(1)
where E is Young’s modulus, I is the moment of inertia of the beam and Z is the loss factor. r is the mass density, A is the ð2Þ cross sectional area, mð1Þ a is the mass of the force transducer on the beam, and ma is the total mass of the masses of the ð1Þ ð2Þ piezoceramic patch and the accelerometer on the beam. So ma ¼ mFT and ma ¼ macc þmPZT . f p ðx,tÞ is the external force and T s ðx,tÞ is the control moment pair. The general solution of Eq. (1) can be expressed as the sum of the normal modes. Hence, the displacement distribution of the beam is represented by Yðx, oÞ ¼ UðxÞpðoÞ,
(2)
T
where pðoÞ ¼ ½p1 p2 pN which is the modal displacement vector, U is the mode shape matrix defined by
U ¼ ½/1 /2 /N
(3)
and N is the number of modes superposed. Utilizing the mode orthogonality leads to the matrix equation as ½o2 Iþ j diagð2zn on oÞ þ diagðo2n Þp ¼ UT f,
(4)
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Fig. 1. Schematic configuration for active control system for a clamped beam with a sensor/moment pair actuator system.
where zn is the nth modal damping ratio, on is the nth natural frequency of the structure and f is the modal force vector, expressed by qfðxÞ , (5) f ¼ f p ffðxp ÞgT s qx x ¼ xs ð2Þ where the force and the moment pair are concentrated at xp ¼ xð1Þ a and xs ¼ xa 7 s, respectively.
3. Acceleration feedback control AF control is a recently proposed control method. Its best feature is that the feedback signal is acceleration, which can be directly measured by accelerometers [14]. The AF controller requires the modal acceleration, which can be extracted from the measured acceleration signal. The AF controller for the specific target mode generates a controller output signal using the modal acceleration as the input signal. The control signal goes to the piezoceramic actuator and the structure is then under control. Using Va in Fig. 1 and Eq. (4), the matrix form of the structure equation in the control system can be written as T qU Ba V a þ UT ðxp Þf p (6) ½o2 I þj diagð2zn on oÞ þdiagðo2n Þp ¼ qx and the equation of the AF controller is written of the form: ½o2 Iþ j diagð2zc oc Þ þ diagðo2c Þq ¼ g o2 Ec p,
(7)
where the control output vector, q, whose component represents the control signal for each target mode. It is used to generate the physical control signal, Va, as V a ¼ Bs UEa q,
(8)
where Bs is the participation matrix that indicates the locations of the accelerometers: Bs ¼ ½0 0 1 0 0 T
and Ea is the mode arrange matrix for the modal superposition. U primary force, fp. ½qU=qxT Ba V a in Eq. (6) denotes the modal force of qU qf1 qf2 ¼ qx qx qx
(9)
f p in Eq. (6) represents the modal force due to the the control action by the moment pair where qfN (10) qx
and Ba is the participation matrix for the moment pair action location of the actuators: Ba ¼ ½0 0 1 0 0 1 0 0T :
(11)
In Eq. (7), zc and oc are the damping ratio and natural frequency of the compensator, respectively. g is the controller gain, Ec is the mode extraction matrix for the control mode. According to Eq. (7), the compensator requires the structural modal acceleration, p. This structural modal acceleration can be obtained as o2 p ¼ ðBs UÞ þ V r , þ
(12)
where is the pseudo-inverse operator, and Vr is the voltage signal measured by the feedback accelerometer. The pseudoinverse operation in Eq. (12) poses unreliable control weights to modal coordinate unless the number of the feedback sensors are available enough to account the number of modes. For practical case of this study, only one accelerometer is used for the feedback. Thus, Bs U of the Eq. (12) is the real valued row vector of mode shape at the sensor location. Then, its pseudo-inverse, ðBs UÞ þ , is to be a real valued column vector. In other words, ðBs UÞ þ acts as just constant vector having
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real values. This means that the modal acceleration to be controlled is proportional to the physical acceleration measured by the feedback sensor. However, the error in the calculation by pseudo-inverse operator can be compensated by the characteristics of the AF controller—low pass filtering and amplifying at the tuning frequency. Although we estimated unreliable control weight the error can be compensated by the characteristics of the AF controller. The controller modal signal, q, as calculated by Eq. (7) should transform into the physical controller signal, Va, as in Eq. (6). Fig. 2 shows a block diagram for active control using the SISO AF controller. The plant response, GðxÞ, and the AF controller, HAF , in the physical coordinates are T qU Ba (13) GðoÞ ¼ o2 Bs U½o2 I þj diagð2zs os oÞ þ diagðo2s Þ1 qx and HAF ðoÞ ¼ Bs UEa diagðg i ÞHðoÞEc ðBs UÞ þ , where HðxÞ ¼ diag
! 1 : o2ci o2 þ j2zci oci
(14)
(15)
The index i specifies the mode order to be controlled. The sensor output, Vr, is the sum of the disturbance, Vp, and the structure response, Vs, due to the actuator action. Therefore, Vr can be expressed as V r ðoÞ ¼ V p ðoÞ þ V s ðoÞ ¼ o2 Bs Upp ðoÞ þGðoÞV a ðoÞ,
(16)
where pp is the modal displacement vector due to the primary force, which can be obtained by Eq. (7) with V a ¼ 0. Va in Eq. (16) is the input signal to the actuator or the output from the controller, i.e., V a ¼ HAF V r :
(17)
Substituting Eq. (17) into Eq. (16), the sensor output voltage and the control signal voltage for the actuator action become V r ðoÞ ¼ ½IGHAF 1 Bs Upp
(18)
V a ðoÞ ¼ HAF ½IGHAF 1 Bs Upp :
(19)
and
In this configuration, stability should be considered, because it is a feedback control system. The stability of the system can be estimated by the open loop transfer function (OLTF) OLTFðoÞ ¼ GHAF :
(20)
G is the acceleration response of the plant at 0.8L when the unit moment pair is applied to the structure. HAF is the transfer function of the AF controller as given in (14). Fig. 3 shows the plant response, GðoÞ, when the sensor/actuator pair is placed at 0.8L. In the Bode plot, the magnitude of the plant response increases as the frequency increases and the phase shifts. This means that an unmodeled response at high frequencies can cause the control system to be unstable. The phase remains between 01 and 1801 up to 17.3 kHz except for the first mode. The phase behavior at the first mode is directly affected by the location of the sensor/moment pair actuator. if this sensor/moment pair actuator is located within 0:2L of the structure boundary, then the plant becomes conditionally stable [16]. Because the sensor/moment pair actuator is on the 0:2L of the beam from the clamped boundary, the unstable phase response occurs in the first mode of the plant response. The Nyquist plot of the plant response shows that active damping cannot be obtained with simple proportional controllers. Fig. 4 shows the transfer function of the AF controller that is tuned to the fourth mode of the structural resonance at 302 Hz. The tuned frequency is the natural frequency of the target mode. This controller has two main characteristics:
Fig. 2. Block diagram for active feedback control using AF controller.
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Fig. 4. The Bode and Nyquist plot of the transfer function of the AF controller response which is tuned to the fourth mode of the structural resonance.
phase shift and magnitude roll-off at high frequencies. The phase shift occurs at the target frequency from 1801 to 01. This phase shift gives the active damping to the system at the tuning frequency. The magnitude of the transfer function of the controller is rapidly reduced at high frequencies. Thus, the magnitude of the unmodeled plant response at high frequencies is reduced, and so the gain margin is significantly increased. Fig. 5 shows the OLTF with an AF controller that is tuned to the fourth mode of the structural resonance. The peak magnitude of this OLTF occurs at the fourth mode, and the phase is shifted to the corresponding mode. In the Nyquist plot, the response at the tuned mode has the largest magnitude in the right half-plane. This response of the OLTF at the tuned mode generates the active damping force. The magnitude of the OLTF is reduced as the frequency increases. This rapid roll-off phenomenon at high frequencies is due to the low pass filtering characteristic of the AF controller as shown in Fig. 4. The unstable condition is observed at the first mode. However, the magnitude difference between the first mode and the tuning mode is more than 50 dB. Therefore, a high gain margin of the control system can be achieved. It should be noted that the lower modes are not affected by the controller. In the Nyquist plot, the lower modes are on the positive imaginary axes. These correspond to the active stiffness, so the responses of the lower modes are partly in the unit circle about ð1,j0Þ, indicating enhancement at the frequencies in the Nyquist plot.
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Fig. 5. The Bode and Nyquist plot of the open loop transfer function with AF controller which is tuned to the fourth mode of the structural resonance.
4. Parametric study of AF controllers In this section, a series of design parametric studies in terms of the stability and performance in conducted to design the controller to have a larger gain margin and a high performance. It is essential to check the stability of the control system since it is tuned to the resonance frequency. The design parameters we investigated here are the tuning frequency, the damping ratio and gain. Fig. 6 shows the variation of the OLTF of the fourth mode tuned AF controller for the gains of 1:0 105 , 3:0 105 , 5:0 105 and 8:68 105 . The damping ratio of the AF controller is set to 5:0 102 . The magnitude of the OLTF is increased at all the frequencies as the gain increases, as shown in the Bode diagram of Fig. 6(a), while the phase of the OLTF is not changed. Fig. 6(b) shows the OLTF normalized by the corresponding feedback gains for the four gains. All the curves are perfectly overlapped. This means that the magnitude of the OLTF varies in proportion to the gain. There are the two Nyquist curves crossing the positive imaginary axis. These two responses correspond to modes lower than the tuned mode. This behavior is due to the low pass filter effect of the AF controller, as shown in Fig. 4. This effect leads to enhancement in the closed loop response at the some frequencies where the Nyquist curves of the two modes are in the unit circle about ð1,j0Þ. Fig. 7 shows the variation of the closed loop acceleration response at 0.8L of the structure with the uncontrolled response (solid line) and the controlled response for feedback gains of 1 105 (dashed line), 3 105 (dash-dotted line), 5 105 (dotted line) and 8:68 105 (thin dashed line) subjected to an AF controller tuned to the fourth mode of the structure. A damping ratio of 5 102 is used. The uncontrolled response is due to the disturbance, fp, at 0.2L. Two features can be observed. First, the closed loop acceleration response at the sensor location (xð2Þ a ¼ 0:8L) is decreased at the target mode as the gain is increased. This is illustrated by the increase of the magnitude of the OLTF as the gain increases, as shown in Fig. 6(a). Second, new peaks around the tuning frequency are also found. The difference between the frequencies and the new peaks increases as the gain increases, as if the mass ratio is increased in the dynamic absorbers [17]. In addition to the gain effects on the closed loop acceleration response around the tuning frequency, another significant behavior is found. The closed loop acceleration response around the tuning frequency is significantly changed in magnitude and structural resonance frequencies. As a result of the gain effects of the AF controller, the resonance frequencies of the second and third modes are significantly decreased. On the other hand, those from the fifth to thirteenth mode are slightly increased. The resonance frequencies of the second and third modes are decreased, as the gain increases, but, the resonance frequency of the first mode is not changed. For a gain of 8:68 105 , the peak of the second resonance for the gain of 5 105 is combined with that of the first mode. Hence, the magnitude of the first mode is rapidly amplified to 88.7 dB. This phenomenon can be explained by the response of the OLTF crossing the negative real axis around the frequency of the first mode in the Nyquist plot. These responses are also within the unit circle about the ð1,j0Þ point. The Nyquist curves at the second and third modes move toward the Nyquist point as the gain increases. This behavior leads to variation in the lower mode response in terms of the magnitude and the natural frequencies. The variation of the resonance frequency at the second and third modes can also be illustrated by the behavior of the transfer function of the AF controller shown in Fig. 4. The essential relevant feature of this transfer function is the phase behavior, in that there is no phase change at frequencies lower than the target frequency. Since the feedback signal is proportional to the acceleration at the actuator location, the AF controller plays a role of the stiffness at frequencies lower
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Fig. 6. Variation of the open loop transfer function of active vibration control systems at 0:8L with the feedback gain of AF controllers with a sensor/ actuator pair. The AF controllers are tuned to the fourth mode with damping ratio of 5:0 102 and gains of 1:0 105 (solid line), 3:0 105 (dashed line), 5:0 105 (dash-dotted line) and 8:68 105 (dotted line). Bode diagram (a) and normalized Nyquist diagram by the corresponding gain (b).
than the target frequency. Note that the AF controller decreases the stiffness as the gain increases. Because of the phase change of 901 at the target frequency, the AF controller generates a damping force. Moreover, because of the phase change of 1801 at frequencies higher than the target frequency, the AF controller generates an inertia force. Fig. 8 shows the variation of the OLTF of the fourth mode tuned AF controller with the damping ratios of 5 103 , 5 102 , 1 101 , and 8 101 . The gain of 2 103 is used for this simulation. As shown in Fig. 8(a), the magnitude of the OLTF is decreased as the damping ratio is increased at the tuned mode, but it is not changed in the other modes. The phase around the tuned mode changes significantly when the damping ratio is increased. The phases of the OLTF are varied in a wide frequency range with increasing damping ratio. The phase behavior can be represented more clearly in the zoomed Nyquist diagram as the shown in Fig. 8(b). Two hollow marks are placed at the same frequency about 290 Hz, J for z ¼ 5 103 and n for z ¼ 1 101 . Two solid marks are also placed at the same frequency about 315 Hz, for z ¼ 5 103 and m for z ¼ 1 101 . Therefore, the bandwidth of the AF controller is not varied with the change of the damping ratio. The responses within the unit circle about ð1,j0Þ move away from the Nyquist point as the damping ratio increases. Thus, the control system is stabilized with the increase of the damping. Fig. 9 shows the variation of the closed loop acceleration response with the damping ratio of the AF controller when the AF controller is tuned to the fourth mode of the structure. The damping ratios for this simulation are 5 103 , 5 102 ,
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Frequency (Hz) Fig. 7. Variation of the closed loop acceleration response with the gain of the AF controller at 0.8L. The AF controllers are tuned to the fourth mode with damping ratio of 5:0 102 . Uncontrolled response (solid line) and controlled response for gains of 1 105 (dashed line), 3 105 (dash-dotted line), 5 105 (dotted line) and 8:68 105 (thin dashed line).
1:0 101 , and 8:0 101 . The feedback gain of the AF controller is set to 2 103 . The reduction in the structure response of the fourth mode is decreased as the damping ratio of the AF controller increases as shown in Fig. 9(a). Although the damping ratio is increased, the bandwidth of the AF controller from 290 Hz to 315 Hz is not remained, as shown in Fig. 9(b). Even if new peaks at the both side of the tuning frequency are emerged, their levels are decreased as the damping ratio is increased. The existence of the new peaks is explained by the enhancement related to the existence of the OLTF response in the unit circle about ð1,j0Þ.
5. Implementation of multimode AF controllers Based on the examination of the behavior of the AF controller, a SISO multimode control is examined in this section. To implement the SISO multimode AF controller, a control signal is obtained by summation of the signals from AF controllers that are tuned to each mode and connected in parallel. The parameters of the AF controllers are set based on the parametric study conducted in the previous section. The higher modes are designed first. This is to alleviate the effects of the higher modes on the responses of the lower modes. Fig. 10 show the schematic diagram of the multimode AF controller we have designed sequentially. The measured acceleration, Vr, is used to obtain the modal acceleration, pi, supplied to the AF controller of each mode. The AF controller generates the control signal for each mode, and then the control signal are superposed to make the control signal, Va, in order to actuate the PZT patch. Fig. 11 shows the open loop transfer function of the combined AF control system with the sensor/moment pair actuator at 0.8L of the clamped–clamped beam. The second mode tuned controller was designed with a gain of 3:3 102 and a damping ratio of 8 102 . The third mode tuned controller was designed with a gain of 7 102 and a damping ratio of 1 101 . The fourth mode tuned controller was designed with a gain of 1:1 103 and a damping ratio of 4 102 . It can be found from Fig. 11(a) that the magnitudes of the combined OLTF are quite high at the three tuned modes; they are almost the same. The phases of the three modes are in 7901. This means that the AF controller generates an active damping force. Hence, we can expect approximately the same performance. A maximum reduction in the acceleration response of about 12 dB at each tuning frequency can be expected in the combined AF controller. Fig. 11(b) is the Nyquist diagram of the combined OLTF. The multimode controller is stable with a high gain margin. The magnitude of the OLTF response within the unit circle about ð1,j0Þ is smaller than that outside this circle. Therefore, the magnitude enhancement is smaller than the reduction around the tuning frequency. Fig. 12 shows the closed loop acceleration response of the clamped–clamped beam at 0.8L, excited by a concentrated force at 0.2L and subjected to the control moment pair at 0.8L—without control (solid line) and with control (dashed line). The disturbances are reduced by about 12 dB at each tuned mode, as predicted in Fig. 11. The controller does not affect the response at modes other than the tuned modes.
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Real Fig. 8. Variation of the open loop transfer function of active vibration control systems at 0.8L with the damping ratio of AF controllers with a sensor/ actuator pair. The AF controllers are tuned at the fourth mode with gain of 2:0 103 and damping ratios of 5 103 (solid line), 5 102 (dashed line), 1:0 101 (dash-dotted line), and 8:0 101 (dotted line). Bode diagram (a) and zoomed Nyquist diagram (b).
6. Experiments Based on the analytic examination of the AF controller behavior, its performance of the AF controller is experimentally verified in this section. To implement HAF for a real structure, the transfer function of the AF controller shown in Eq. (15) can be rewritten in the s-domain function as ! 1 HAF ðsÞ ¼ diag 2 (21) s þ 2zci oci s þ o2ci and it is coded using Matlab/Simulink. The implementation of the transfer function, Eq. (21), is downloaded to the ROM of DSP (dSPACE DS1103). Then the DSP works as AF controller. Fig. 13 shows the experimental setup for the active control of the clamped beam with the AF controller. An aluminum beam is used for this experiment; its length is 0.5 m. Both edges of the beam are clamped by vises. The beam is disturbed by a mini-shaker (B&K 4810) at 0.2L. The disturbance force is measured by a force transducer (B&K 8200). An accelerometer (B&K 4393) is installed on the beam at 0.8L, and it measures the acceleration at the sensor location. The measured acceleration signal is fed to the controller (dSPACE DS1103), which generates the control signal. The power amplifier (AVL 790) amplifies the control signal to drive the piezoceramic actuator. The piezoceramic actuator is a
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Fig. 9. Variation of the closed loop acceleration response with damping ratio of the AF controller at 0:8L. The AF controllers are tuned to the fourth mode with gain of 2:0 103 . Uncontrolled response (solid line) and controlled response for damping ratios of 5 103 (dashed line), 5 102 (dash-dotted line), 1:0 101 (dotted line), and 8:0 101 (thin dashed line). Structural acceleration (a) and zoomed structural acceleration (b).
Fig. 10. Schematic diagram of the multimode AF controller.
rectangular shaped 1–3 mode PZT patch (25 mm 25 mm 1 T) whose material properties hold with PZT4 series, manufactured by Kyungwon Ferrite Ind. Co., Ltd (South Korea). Fig. 14 shows the experimental acceleration response of the clamped beam excited by a concentrated force at 0.2L (solid line), and when subjected to the AF controller with a sensor/moment pair actuator system at 0.8L (dashed line). The structural response is experimentally reduced to about 12 dB, as we predicted analytically; see Fig. 12. In this experimental examination, the controller does not affect the response at modes other than the tuned modes.
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Fig. 11. Final designed open loop transfer function of active control system with AF controller at 0.8L. Bode diagram (a) and Nyquist diagram (b).
Fig. 12. Analytic acceleration response of clamped beam excited by a concentrated force at 0.2L (solid line), and when subjected to the AF controller with sensor/moment pair actuator system at 0.8L (dashed line).
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Fig. 13. Experimental setup for active control of clamped beam with AF controller.
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7. Conclusion In this study, the active vibration control of clamped beams using AF controllers with a non-collocated sensor/moment pair actuator configuration is investigated. The moment pair actuator is used as the control actuator and the accelerometer is used as the feedback sensor located at the opposite center of the moment pair. This non-collocated configuration is the main source of instability problems in control systems. To overcome the instability problem due to the non-collocated sensor/moment pair actuator configuration, an AF controller with a sensor/moment pair actuator for a clamped beam with lumped masses has been designed, implemented and experimentally verified. A parametric study of the design parameters of the AF controller was conducted to characterize their effects on the stability and performance of the controller. These parameters were the gain, damping ratio and tuning frequency. The magnitude of the OLTF was increased as the gain increased, while the characteristics of the phase were retained. Thus, the gain did not change the stability but changed the performance. Furthermore, the difference between the frequencies of the new peaks produced by the AF controller increased as the gain increased. The effective bandwidth for the reduction broadened as the gain increased at the tuned modes. The magnitude of the OLTF was decreased at the tuned frequency, and the phase of the OLTF was affected over a wider range of the tuned mode as the damping ratio increased. Increasing the damping ratio resulted in an increase in the stability of control system, but the control system required a higher gain to achieve the same performance. Note that the difference between the frequencies of the new peaks produced by the AF controller did not vary as the damping ratio increased. Finally, when the AF controller was tuned at a higher mode, the effects of the gain and the damping were the same as those of the controller tuned to a lower mode; the only difference
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was that the AF controller significantly affected the response in the lower modes. The natural frequency of the combined system (structure and AF controller) was decreased at the lower modes. A multimode (the second, third and fourth modes) controllable AF controller was implemented, and it was verified that the controller is stable and shows high performance. The control signal was obtained by summation of the signals from each AF controller, which was tuned to each mode and connected in parallel. The vibration levels at the tuned modes were reduced by about 12 dB, as determined numerically and experimentally.
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