Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Piezoelectric optimal delayed feedback control for nonlinear vibration of beams:

Piezoelectric optimal delayed feedback control for nonlinear vibration of beams: An optimal delayed feedback control methodology is developed to mitigate the primary and super harmonic resonances of a flexible simply-simply supported beam with piezoelectric sensor and actuator. Stable vibratory regions of the feedback gains are obtained by using the stability conditions of eigenvalue equation. Attenuation ratio is used to evaluate the performance of vibration control by taking the proportion of peak amplitude of primary or super harmonic reson- ances for the suspension system with and without controllers. Optimal control parameters are obtained using an optimal method, which takes attenuation ratio as the objective function and the stable vibratory regions of the time delay and feedback gains as constraint conditions. The piezoelectric optimal controllers are designed to control the dynamic behaviour of the nonlinear dynamic system. It is found that the optimal feedback gains obtained by the optimal method result in a good control performance. Keywords Nonlinear vibration, optimal control, piezoelectricity Introduction In the past decades, considerable amount of research works have helped us better understand the effect of time delays on the behaviour of nonlinear dynamical systems controlled by linear or nonlinear feedback controllers for mitigating the vibration amplitude. Delayed vibration control is a technique of vibration suppression. The time delay can vary a fixed gain and change the range of saturation control, either widening or shrinking the effective frequency bandwidth. Since delayed control in nonlinear systems has an important role in vibration control engineering, the study of delayed vibration control of nonlinear systems is essential. Analytical methods for weakly nonlinear continuous systems include harmonic balance method, the Galerkin procedure and directed multiple scales. The first two approximate methods are easy and straightforward to apply. While the calculation of higher-order approximations may be complex. In the third approach, one calculates an approximate solution of the nonlinear differential equation by directly using multiple scales to expand the non- linear governing partial differential equation and boundary conditions continuous problem. The obtained mode shapes are in full agreement with those obtained by using discretization because the latter are performed by using a complete set of basic functions that satisfy the boundary conditions. This method is beneficial to the elimination of coupling terms and removal of solving multi-dimensional equations, and can be easily applied to analyse the dynamic response of higher modes. The technique of delayed-feedback control was introduced as an effective means of controlling a wide variety of mechanical systems. Olgac and Holm-Hansen introduced the concept of delayed resonators to control mechanical 3,4 systems. This concept was utilized to study time-delayed acceleration feedback control over continuous systems School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo, China Corresponding author: Canchang Liu, School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255049, China. Email: sdutlcch@163.com 26 Journal of Low Frequency Noise, Vibration and Active Control 35(1) 5,6 7 and time-delayed velocity feedback control over torsional mechanisms. Jalili and Olgac used time-delayed feedback resonators to control discrete multi-degree-of-freedom systems. Zhao and Xu used delayed feedback control to suppress the vibration of vertical displacement in a two-degree-of-freedom nonlinear system with external excitation. Plaut and Hsieh studied the effect of a damping time delay on nonlinear structural vibrations and analysed six resonance conditions. Sato et al. discussed the free and forced vibration of nonlinear systems with the time delays. Hu et al. studied the primary resonance, super harmonic resonance and stability of a nonlinear Duffing oscillator with time delays under harmonic excitations and gave out periodic solutions by using the method of modified target practice. Maccari dealt with the principal parametric resonance of a van der Pol oscillator with time-delay linear state feedback. Ji and Leung demonstrated that in parametrically excited Duffing systems stable region of the trivial solution could be broadened, a discontinuous bifurcation could be transformed into a continuous one and the jump phenomenon in the response could be removed, if an appropriate feedback control was used. Ji and Leung also studied the primary, super harmonic and subharmonic resonances of a harmonically excited nonlinear single degree of freedom system with two distinct time delays in the linear state feedback. The primary resonance of a cantilever beam under state feedback control with a time delay was 15 16 investigated. Qian and Tang discussed the primary resonance and the subharmonic resonances of a nonlinear beam under moving load based on time-delay feedback control. Daqaq et al. presented a comprehensive inves- tigation on the effect of feedback delays on the nonlinear vibrations of a piezoelectric actuated cantilever beam and analysed the effect of feedback delays on a beam subjected to a harmonic base excitations. Alhazza et al. investigated the effect of time delays on the stability, amplitude and frequency–response behaviour of a beam and found that even the minute amount of delays could completely alter the behaviour and stability of the parametrically excited beam and lead to unexpected behaviour and response. Liao et al. presented a feasible methodology by which could achieve good control performance of a dynamic beam structure system with time delay effect. A spring time-delay controller was designed to achieve good control performance. Liu et al. studied the primary, subharmonic and super harmonic resonances of an Euler–Bernoulli beam subjected to harmonic excitations with damping and spring delayed-feedback controllers. Gao and Chen combined cubic nonlinearity and time delay to improve the performance of vibration isolation. A time delayed nonlinear saturation controller was proposed to reduce the horizontal vibration of a magnetically levitated body described by a nonlinear dif- ferential equation subjected to both external and modulated forces. The time delay saturation-based controller was considered for active suppression of nonlinear beam vibrations, and the effects of time delays on the system behaviour were studied. The research works mentioned above are concerned with the selection of feedback gains and time delays that can enhance the control performance of nonlinear systems or change the position of the bifurcation point. However, all these works have failed to address how to choose optimized control parameters and time delays on the basis of keeping the system stable. In this paper, delayed piezoelectric feedback controllers are designed to change the beam’s nonlinear dynamical behaviour. The feedback gains and time delays of piezoelectric controllers are easy to set up and change by using active control facilities. The main purpose of this paper is to present an optimum control method for a nonlinear vibration system. The stable vibratory regions of time delay and feedback gains are given based on the analysis of stability conditions of eigenvalue equation. The control parameters are calculated with an optimal method, which takes attenuation ratio as the objective function and stable vibratory regions of time delays and feedback gains as constraint conditions. The multiple scales method is applied to obtain linear equations, by which it is easy to analyse the dynamic behaviour of high order modes of the beam. Time delay is taken as a control factor that can turn defective effect into favourable effect. Compared with damping force controllers, time delayed controllers have three parameters to be adjusted, two feedback gains and a time delay. Therefore, the space of design and adjustment is much wider. A closed-loop control system is designed to control the dynamic behaviour of the nonlinear dynamic system. Primary resonance analysis Programme formulation The beam is assumed to be inextensible, has uniform cross-sectional area and satisfies the Euler–Bernoulli beam theory. The simply-simply supported boundary conditions are considered (see as Figure 1). The uniform flexible beam with a piezoelectric sensor layer on its top surface and a piezoelectric drive layer on its low surface is considered. The piezoelectric sensor bends with deformation of the beam, which is excited by a Liu et al. 27 Delay x1 x2 Voltage Amplifier Figure 1. Schematic drawing of a piezoelectric actuated beam under harmonic excitations. harmonic excitation. The voltage of the piezoelectric sensor layer, which is induced by the deformation along the 24,25 beam direction, is given as 0  0 V ¼ K½w ðX , t Þ w ðX , t Þ ð1Þ 2 1 where K ¼ bhg =2C , b and h are width and height of the beam, respectively. g is piezoelectric parameter, C is 31 p 31 p capacitor of the piezoelectric sensor. w is the deflection of the beam. The prime of w is the first derivative with respect to the x-coordinates. X and t* are the coordinates of the axial direction and time, respectively. The voltage produced by the piezoelectric sensor can be delayed by the time-delay device and amplified by the amplifier. The bending moment produced by the delayed and amplified voltage on the beam can be written as MðX, t Þ¼ gRVðtÞ½HðX  X Þ HðX  X Þ ð2Þ 1 2 where g is feedback gain parameter g4 0, g5 0 and g ¼ 0 represent positive, negative and no feedback, respect- ively. H is the Heaviside unit doublet function, R ¼ bd E ðh þ  Þ, E is the elastic modulus,  is the height of 31 pe pe pe pe the piezoelectric sensor and d is the piezoelectric parameter. 26,27 The non-dimensional form of equation of motion and boundary conditions of the beam can be obtained as ð4Þ 00 02 0 0 w€ þ w ¼"w_ þ "w w dx þ "f ðxÞ cosðtÞþ "gg ½w ðx , t  Þ w ðx , t  Þ p 2 1 ð3Þ 00 00 ½H ðx  x Þ H ðx  x Þ 1 2 w ¼ w ¼ 0at x ¼ 0, 1 ð4Þ where g ¼ LRK/EI,  is the damping of the vibration system. L is the length of the beam, E and I are the elastic qffiffiffiffiffi pffiffiffiffiffiffiffiffi t EI modulus and moment of inertia of the beam, respectively. w ¼ w=,  ¼ I=A, x ¼ X/L. t ¼ ,  and A are L A the mass per unit length and the cross-section area of the beam, respectively.  is the time delay. The beam bending 28,29 vibration w can be expanded by order of " as wðx, t; "Þ¼ w ðx, T , T Þþ "w ðx, T , T Þ þ  ð5Þ 0 0 1 1 0 1 where T and T are the fast and slow time scales, respectively. Substituting expression (5) into the partial differ- 0 1 ential equation (3) and separating terms at orders of ", yields @ w 0 ð4Þ þ w ¼ 0 ð6Þ @T w ¼ w ¼ 0at x ¼ 0, 1 ð7Þ 2 2 1 @ w @ w @w 1 0 0 ð4Þ 00 02 þ w ¼2   þ w w dx þ f cosðT Þþ gg w  w 0 p 0 0 0 2 1 @T @T @T @T ð8Þ 0 1 0 0 00 00 ½H ðx  x Þ H ðx  x Þ 1 2 28 Journal of Low Frequency Noise, Vibration and Active Control 35(1) w ¼ w ¼ 0at x ¼ 0, 1 ð9Þ With the Galerkin approximation, w can be represented as a series of products of the spatial functions which only depends on time as 1 1 X X wðx, tÞ¼ w ðx, tÞ¼  ðxÞq ðtÞ ð10Þ n n n n¼1 n¼1 pffiffiffi where  is the nth eigen function of a linear uniform beam and  ¼ 2 sinðnxÞ. q is the nth generalized n n n time-dependent coordinates. The solution of linear equation (6) with the boundary conditions (4) can be written as i! T i! T k 0 k 0 w ¼  ðxÞ A ðT Þe þ A ðT Þe ð11Þ 0 k k 1 k 1 where A is the amplitude, A is the conjugate item of A . k k k The solvability condition requires that the eigen functions are orthogonal ð12Þ h ,  i¼  ðsÞ ðsÞds ¼ n m n m nm here  is the Kronecker delta. For the case of primary resonance, we have nm ¼ ! ð1 þ " Þð13Þ where is the detuning parameter. ! is the kth order circular frequency. Substituting equations (11), (12) and (13) 1 i into (8), applying the condition (9), and letting A ¼ a e , the equation of secular terms is k k 0 2 ið! T Þ i! k 1 k ð14Þ 2i! A þ i! A  3g A A  F e  2g A e ¼ 0 k k k kk k k k k k k DE hi R R 1 1 00 02 1 0 0 1 where g ð! Þ¼  ðxÞ,   dx ,  ¼ g ½ ðx Þ  ðx Þ , F ¼ f ðxÞdx kk k k k p 1 2 k k k k k k 0 2 2 0 Let ¼ ! T . Separating the real and imaginary parts of equation (14), yields k k 1 k a ¼ a þ sinð Þ ð15Þ k k k 0 3 ¼ a þ v a þ cosð Þ ð16Þ k k k k k k k g sin !  g cos !  3g 1 k k k k kk where ¼  þ , ¼ ! þ , v ¼ . k k K k 2 ! ! 8! k k 0 0 By letting a ¼ ¼ 0, the steady-state response in the system can be expressed as k k a þ sinð Þ¼ 0 ð17Þ k k k a þ v a þ cosð Þ¼ 0 ð18Þ k k k k Using modulation equations, the amplitude–frequency equation of the system can be obtained 2 3 ð19Þ ð a Þ þ a þ v a ¼ k k k k k k Liu et al. 29 The peak amplitude of the primary response, obtained from equation (19), is given by 2 k E ¼ a ¼ ð20Þ k max 2 2 k k For the purpose of comparison, the equation of motion for the nonlinear primary oscillator without control is ð4Þ 00 02 ð21Þ w€ þ "w_ þ w ¼ "w w dx þ "f ðxÞ cosðtÞ The corresponding peak amplitude for the nonlinear primary oscillator without control can be written as E ¼ a ¼ ð22Þ k max 2 2 k k where  ¼ =2. The performance of the vibration controllers on the reduction of nonlinear vibrations cannot be studied by using a similar procedure that discusses the ratio of response amplitude of the linear system because of the difficulty of finding the analytical solutions for a nonlinear system. Therefore, an attenuation ratio is utilized to evaluate the performance of the vibration controller by taking the proportion of vibration peaks of primary 30,31 resonances of the beam system with and without control. By this definition, the attenuation ratio can be written as 0 1 B C B C ð23Þ R ¼ @ A 2g sin ! k k 1 þ As defined by equation (23), a small value of the attenuation ratio R indicates a large reduction in the vibrations of the nonlinear primary system. From equations (17) and (18), it is known that the damping coefficients are functions of feedback gain and time delay. Small attenuation rate can be obtained by selecting proper parameters of feedback gain and time delay. Controller design for primary resonance For nonlinear systems, the nonlinear phenomena such as jump phenomenon and hysteresis may occur and the vibration of the system may become unstable. The vibration controllers should be designed to avoid the unstable vibration and reduce nonlinear vibration amplitude. The stability of the solutions depends on the eigenvalues of the corresponding Jacobian matrix of equations (17) and (18). The corresponding eigenvalues are the roots of equation 2 2 2 2 þ 2 þ þ þ v a þ 3v a ¼ 0 ð24Þ k k k k k k k k The sum of the two eigenvalues is 2 . The addition of the feedback gain and time delay alters the sum of the two eigenvalues. If < 0, it means at least one of the eigenvalues will always have a positive real part. The system will be unstable. If ¼ 0, it means a pair of purely imaginary eigenvalues and hence a Hopf bifurcation may occur. Based on the analyses mentioned above, unstable periodic solutions corresponding to a saddle is f ð Þ5 0 ð25Þ 2 2 2 2 4 where f ð Þ¼ þ 4v a þ þ 3v a . The value of f ð Þ is positive when there is no solution of equation k k k k k k k k k f ð Þ¼ 0 ð26Þ where a saddle-node bifurcation occurs. 30 Journal of Low Frequency Noise, Vibration and Active Control 35(1) If > 0, the sum of two eigenvalues is negative, and accordingly, at least one of the two eigenvalues will have a 16,32,33 negative real part. The sufficient conditions for guaranteeing the system stability are 2 2 2 2 4 f ð Þ¼ þ 4v a þ þ 3v a 4 0 4 0 ð27Þ k k k k k k k k k 2 2 4 2 4 The value of f ð Þ is positive when then no solution of critical equation f ð Þ¼ 0. Letting  v a  v a , k k k k k max k k there is 2 2 4 2 k v a ¼ v ð28Þ k k k max k 4 4 k k The region of stable vibration control parameters can be obtained sffiffiffiffiffiffiffiffiffiffiffiffi "# ! 1 jj v F k k ð29Þ g ¼ g sin !     þ 2 ! When there are two solutions of equation f ð Þ¼ 0, the solutions are 2 2 4 2 1=2 ¼2v a ðv a  Þ ð30Þ k k k k k As f ð Þ¼ 0 is a parabola whose mouth is opened upward, the inequality f ð Þ4 0 is satisfied when 5 k k k and 4 . In order to have a good control performance, is larger than  . Reducing or enlarging the roots of k k k the equation of f ð Þ¼ 0 and considering g < 0, we have k kk 1=2 2 4 2 ð31Þ ¼ v a k k max k k 2 2 4 2 1=2 þ ¼2v a þðv a   Þ  ð32Þ k k k max k k max k k Considering the formula of and equation (31), the region of stable vibration is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 4 v F   ! k k k k ! k ð33Þ g cos !    and 4 0 k k k K k Taking into account the formula (32), gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 6 4 2v F þ v F   ! ! k k k k k k ð34Þ g cos !   þ and 4 0 k k k k k Analysis of optimised controller design parameters The region of feedback parameter has been obtained based on the analysis of stability condition of the nonlinear vibration system, but it is difficult to obtain the optimal control parameters of the system. Taking the nonlinear vibration system with attenuation ratio as the objective function, the optimal feedback control parameters can be calculated with an optimal method. The optimal analysis is carried out by taking into account the two cases with and without solutions of the critical equation. Optimal design when the critical equation has no solution. 0 1 B C B C ð35Þ min R ¼ @ A 2g sin ! k k 1 þ k Liu et al. 31 sffiffiffiffiffiffiffiffiffiffiffiffi "# ! 1 jj v F k 3 k s:t: g sin !     þ  0, 4 0 k k 2 ! Optimal design when the critical equation has two solutions. 0 1 B C B C ð36Þ min R ¼ @ A 2g sin ! k k 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 6 4 v F   ! k k k k ! s:t: g cos !  þ þ  0 k k k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 6 4 2v F  v F   ! ! k k k k k or g cos !  þ þ  0, 4 0 k k k k k Super harmonic resonance analysis Super harmonic resonance For the case of super harmonic resonance, the harmonic excitation is assumed that pðxÞ¼ "f ðxÞð37Þ Substituting equations (5) and (37) into equation (3) and equating coefficients of like powers of ", the following equations are derived @ w ð4Þ þ w ¼ pðxÞ cosðTÞð38Þ @T w ¼ w ¼ 0at x ¼ 0, 1 ð39Þ 2 2 @ w @ w @w 1 ð4Þ 0 0 00 02 þ w ¼2   þ w w dx þ gg w p 0 1 0 0 @T @T @T @T 0 1 0 ð40Þ 00 00 ½H ðx  x Þ H ðx  x Þ 1 2 w ¼ w ¼ 0at x ¼ 0, 1 ð41Þ The solution of equation (38) is written as follows hi ! ðT þ T Þ k 0 1 i! T i k 0 w ¼  ðxÞ A ðT Þe þ B e þ cc ð42Þ 0 k k 1 k where B ¼ , F ¼  ðxÞ pðxÞdx. Substituting equation (42) into equation (40), the equation of secular k 2 2 k k 2ð!  Þ terms is written as 0 2 3 ið! T Þ i! k 1 k i2! A þ i! A  3g A A  6g A B B  g B e  2g A e ¼ 0 ð43Þ k k k kk k kk k k k kk k k k k k 32 Journal of Low Frequency Noise, Vibration and Active Control 35(1) 1 i Let A ¼ a e , ¼ ! T . Substituting them into equation (43) and separating the real and imaginary k k k k 1 k parts, yields g B kk 0 k ð44Þ a ¼ a þ sinð Þ k k k g B kk 0 3 ð45Þ ¼ a þ v a þ cosð Þ k sk k k k k k 3g B g cos !  kk k k k where ¼ ! þ þ . The fixed points of this system are given as sk k ! ! k k g B kk ð46Þ a  sinð Þ¼ 0 k k k g B kk 3 k ð47Þ a þ v a þ cosð Þ¼ 0 sk k k k By eliminating from equations (46) and (47), the equation of amplitude-frequency is obtained g B kk 2 3 k ð48Þ ð a Þ þ a þ v a ¼ k k sk k k The peak amplitude of the super harmonic resonance, which is obtained from equation (48), is given by 2 6 g B 2 kk k E ¼ a ¼ ð49Þ k max 2 2 k k For the purpose of comparison, the corresponding peak amplitude of the nonlinear super harmonic oscillator without control is 2 6 g B 2 kk k E ¼ a ¼ ð50Þ k max 2 2 k k The attenuation ratio of nonlinear super harmonic can be written as 0 1 B C B C R ¼ ð51Þ @ A 2g sin ! k k 1 þ The steady-state solutions of super harmonic resonance response depend on the eigenvalues of the characteristic equation, which are the roots of 2 2 2 2 þ 2 þ þ þ v a þ 3v a ¼ 0 ð52Þ k k k k k k k k Optimal design of controller parameters Taking the attenuation ratio of super harmonic resonance as the objective function, the optimal feedback control parameters can be worked out by the optimal method. When the critical equation has no solution. 0 1 B C B C ð53Þ min R ¼ @ A 2g sin ! k k 1 þ k Liu et al. 33 "# sffiffiffiffiffiffiffiffiffiffiffiffiffi jj ! 1 v F k 3 k s:t: g sin !     þ  0, 4 0 k k 2 ! where F ¼ g B . s kk When the critical equation has two solutions. 0 1 B C B C ð54Þ min R ¼ @ A 2g sin ! k k 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 6 4 2 2 v F   ! k s k k ! þ 3g B kk k k s:t: g cos !  þ þ  0 k k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 6 4 2 2 2v F  v F   ! s s k k k ! þ 3g B kk k k or g cos !  þ þ  0, 4 0 k k k k k Numerical simulation The nonlinear vibration control for a simply-simply supported beam is studied in the present paper. The length, width and height of beam are 500 mm, 50 mm, and 1 mm, respectively. The density and elastic modulus of the beam are 2700 kg m and 70 GPa, respectively. Both the length and width of piezoelectric sensor are 30 mm. The thickness of piezoelectric sensor is 0.2 mm, the piezoelectric constant is d ¼ 190 e  12 mV , the piezoelectric capacitance is Cs ¼ 15 nf, elastic modulus of the piezoelectric actuators E ¼ 63 GPa, and the constant of piezo- pe electric is g ¼ 1.9 C m . Other parameters are  ¼ 0.5,  ¼ 0.1, ¼ 0.01 and B ¼ 0.1, respectively. The control 31 1 analysis is carried out only for the first mode of the beam, and the control analysis of high modes is similar. Figure 2 shows the variation of damping coefficient of the nonlinear vibration for first order mode of the beam with feedback gain and time delay. The damping value varies with the alteration of time delay when the value of feedback gain is fixed. Usually, should have a larger positive value in order to improve the control performance. For the cases of three distinct feedback gains, the effective time delay region of vibration damping which is greater than is from 0 to 0.32 and 0.64 to 0.80. The damping coefficient reaches the peak of primary resonance when the time delay is 0.16. Figure 3 g =200 g =400 g =600 0.5 g =0 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 2. Variation of with time delay of the first mode primary resonance of the beam for different feedback control parameters. 1 34 Journal of Low Frequency Noise, Vibration and Active Control 35(1) displays the alteration of tuning coefficients of beam’s primary and super harmonic resonance of first-order mode with the time delay, respectively. It is found that the tuning coefficient varies with the alteration of feedback gain and time delay. Based on the analysis of Figures 2 and 3, it can be observed that stable vibration can be obtained by selecting proper feedback gain and time delay, and the peak amplitude of the resonance can be thus reduced. Figure 4 displays the variation of attenuation ratio R with time delay and feedback gain. As shown in the figure, for fixed value of the amplitude of excitation, a smaller value of the attenuation ratio R indicates a larger reduction in the primary resonances of the nonlinear system. A good selected value of time delay can relatively lead to a larger positive value of and a smaller attenuation ratio R . Therefore, the amplitude of vibration of the 1 1 nonlinear system can be reduced by properly selecting the feedback gain and the time delay. Taking the nonlinear vibration system with attenuation ratio as the objective function and the stable vibratory regions of the time delays and feedback gains as constraint conditions, the optimal feedback control parameters can be calculated by using the optimal method. The case that the characteristic equation has no solution is taken into account as an example to discuss the control parameters. As the value of  and  are all positive values in this example through calculation, the value of g should be positive in order to get better control performance shown in the formulation (34). A greater value of g can lead to a smaller attenuation rate R , and a better control perform- ance can be obtained. It is also found that g is the function of excitation amplitude F . When the value of F is 1 1 bigger, a larger value of control parameter is demanded to mitigate the vibration. The effect of the excitation amplitude on the stable minimum control parameters for three sets of time delays is shown in Figure 5. g=200 g=400 g=600 0.5 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 3. Variation of with time delay of the first mode primary resonance of the beam for different feedback control parameters. g =100 g =200 0.8 g =300 0.6 0.4 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Figure 4. Variation of R with time delay of the first mode primary resonance of the beam for different feedback control parameters. 1 Liu et al. 35 The stable minimum feedback gain varies from a small value to a big one with an increase in the value of excitation amplitude of F . The stable minimum feedback gain also changes with variation of the value of time delay In the region of 0-j/2! , a large value of time delay is prone to need a small feedback gain to control the . 1 vibration of the system. The optimal control time delay can be obtained when  ¼ j/2! , by which the smallest feedback gain g can be used to reduce the vibration. So the optimal control time delay can be taken as a control factor to improve the control performance as same as feedback gain. A good control performance can be obtained by selecting an optimal time delay. The amplitude of excitation F is 0.2. It can be worked out that the product of feedback gain g and sin(! )is 1 1 more than 40.3141 when the characteristic equation has no solution. Let  ¼j/3! . The optimal feedback gain can be calculated as g 2 [778.9,1581.2] for the case that there are two solutions for the characteristic equation. The calculation of optimal feedback gain and time delay of super harmonic resonance follows the same pro- cedure as the primary resonance. It can be calculated that the product of g and sin(! ) is more than 12.16 when the characteristic equation has no solution and B ¼ 0.12. Letting  ¼j/3! , the optimal feedback gain is calcu- 1 1 lated as g[184.8,473.1] when the characteristic equation has two solutions. Figures 6 and 7 show the primary and super harmonic response curves of first mode of the beam with three different sets of the time delay. There is no jump and hysteresis phenomenon when  ¼j/6! or  ¼j/3! . This 1 1 suggests that saddle node bifurcation and jump phenomenon can be eliminated by choosing certain values of the t=p/6w t=p/3w t=p/2w 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 5. Stable minimum control parameters of the first mode primary resonance of the beam for three sets of time delays. 0.5 t=p/6w Stable t=p/3w 0.4 t=0 0.3 Unstable 0.2 0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 Figure 6. Frequency–response curves for the first mode primary resonance of the beam for three sets of time delays. g 36 Journal of Low Frequency Noise, Vibration and Active Control 35(1) t=0 0.5 t=p/6w t=p/3w 0.4 Stable 0.3 Unstable 0.2 0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 Figure 7. Frequency–response curves for the first mode super harmonic resonance of the beam for three sets of time delays. Figure 8. Frequency–response curves for the first mode primary resonance of the beam for different excitations and time delays. time delay. Three solutions exist in a region of coexistence of  ¼ 0. The bending of the frequency response curves is the cause of a jump phenomenon. Moreover, the peak amplitude of the primary resonance response at  ¼ j/3! is the smallest one in the three cases. The vibration controllers can effectively suppress the amplitude oscillations of the nonlinear oscillator. Hence, by optimally choosing the feedback gain and time delay of the piezoelectric controllers, the primary and super harmonic resonance response of the nonlinear oscillator can be reduced. Figure 8 shows the primary response curves of first mode with different sets of the time delays and amplitudes of excitations. The feedback gain is also selected as 0.35. As shown in the figure the vibration of the system is unstable with  ¼ 0 and stable with  ¼ j/6! when the amplitude of the excitation is 0.25. The vibration of the system can be stable and the vibration displacement be mitigated when the amplitude of excitation changes from 0.25 to 0.45 and the time delay from j/6! to j/2! . Hence, the time delay can be used as a control factor to suppress the 1 1 resonance response of the nonlinear oscillator by choosing optimal value of the time delay. Figure 9 shows the critical curve of saddle bifurcation of the beam for different kinds of time delays. When f ð Þ¼ 0, the other eigenvalue is zero where a saddle-node bifurcation occurs. The formulas (26) are utilised to study the critical curves of the nonlinear vibration system. When f ð Þ4 0, the vibration system may be stable. However, when f ð Þ5 0, the vibration system is unstable. Stable and unstable regions are shown in the figure. s1 Liu et al. 37 Figure 9. Critical curve of saddle bifurcation of the beam for different time delays. With increase in the time delay, the stable regions of the nonlinear vibration system decrease. The value of the time delay can change the stable structure of the nonlinear vibration system. Conclusions The regions of time delay and feedback gains are obtained by enlarging or reducing the inequalities of the roots of eigenvalue equation. The primary and super harmonic resonance response of the nonlinear oscillator can be reduced by choosing stable feedback gains and time delay. The optimal control parameters of feedback gain and time delay are calculated by using the method of minimum optimal attenuation ratio, which takes the attenuation ratio as the objective function and stable regions of the time delay and feedback gain as constrained conditions. Optimal control performance can be achieved by using the optimal time delay and feedback gain controller. The value of the time delay can change the stable structure of the nonlinear vibration system. The intelligent control system attached piezoelectric elements has the characters of high efficiency and low energy consumption, therefore it can be widely used in the vibration control of the flexible structure. Test cases are designed and the simulation results are studied. Time-delay feedback controllers can be utilized to suppress the dynamic behaviours of nonlinear systems. Most of the vibration energy of the nonlinear oscillator is transferred to the controllers through piezoelectric controller. The optimal vibration controllers can effectively suppress the amplitude of oscillations of the nonlinear oscillator. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. References 1. Nayfeh AH and Nayfeh SA. On nonlinear modes of continuous systems. J Vibration Acoust 1994; 116: 129–136. 2. Olgac N and Holm-Hansen BT. A novel active vibration absorption technique: delayed resonator. J Sound Vibration 1994; 176: 93–104. 3. Olgac N, Elmali H, Hosek M, et al. Active vibration control of distributed systems using delayed resonator with acceleration feedback. J Dyn Syst Meas Control 1997; 119: 380–389. 4. Olgac N and Jalili N. Modal analysis of flexible beams with delayed resonator vibration absorber: theory and experiments. J Sound Vibration 1998; 218: 307–331. 38 Journal of Low Frequency Noise, Vibration and Active Control 35(1) 5. Hosek M, Elmali H and Olgac N. A tunable torsional vibration absorber: the centrifugal delayed resonator. J Sound Vibration 1997; 205: 151–165. 6. Filipovic D and Olgac N. Delayed resonator with speed feedback-design and performance analysis. Mechatronics 2002; 12: 393–413. 7. Jalili N and Olgac N. Multiple delayed resonator vibration absorbers for multi-degree-of-freedom mechanical structures. J Sound Vibration 1999; 223: 567–585. 8. Zhao YY and Xu J. Effects of delayed feedback control on nonlinear vibration absorber system. J Sound Vibration 2007; 308: 212–230. 9. Plaut RH and Hsieh JC. Non-linear structural vibrations involving a time delay in damping. J Sound Vibration 1987; 117: 497–510. 10. Sato K, Sumio Y and Okimura T. Dynamic motion of a nonlinear mechanical system with time delay: analysis of the self- excited vibration by an averaging method. Trans JSME Series C 1995; 685: 1861–1866. 11. Hu HY, Dowell EH and Virgin LN. Resonances of a harmonically forced Duffing oscillator with time delay state feed- back. Nonlinear Dyn 1998; 15: 311–327. 12. Maccari A. The response of a parametrically excited van der Pol oscillator to a time delay state feedback. Nonlinear Dyn 2001; 26: 105–119. 13. Ji JC and Leung AYT. Bifurcation control of parametrically excited Duffing system. Nonlinear Dyn 2002; 27: 411–417. 14. Ji JC and Leung AYT. Resonances of a non-linear s.d.o.f system with two time-delays in linear feedback control. J Sound Vibration 2002; 253: 985–1000. 15. Maccari A. Vibration control for the primary resonance of a cantilever beam by a time delay state feedback. J Sound Vibration 2003; 259: 241–251. 16. Qian CZ and Tang JS. A time delay control for a nonlinear dynamic beam under moving load. J Sound Vibration 2008; 309: 1–8. 17. Daqaq MF, Alhazza KA and Arafat HN. Non-linear vibrations of cantilever beams with feedback delays. Int J Non-Linear Mech 2008; 43: 962–978. 18. Alhazza KA, Daqaq MF, Nayfeh AH, et al. Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control. Int J Non-Linear Mech 2008; 43: 801–812. 19. Liao JJ, Wang AP, Ho CM, et al. A robust control of a dynamic beam structure with time delay effect. J Sound Vibration 2002; 252: 835–847. 20. Liu CC, Qiu JH, Sun HY, et al. Nonlinear vibrations of beams with spring and damping delayed feedback control. J Vibroeng 2013; 15: 340–354. 21. Gao X and Chen Q. Nonlinear analysis, design and vibration isolation for a bilinear system with time-delayed cubic velocity feedback. J Sound Vibration 2014; 333: 1562–1576. 22. Eissa M, Kandil A, El-Ganaini WA, et al. Vibration suppression of a nonlinear magnetic levitation system via time delayed nonlinear saturation controller. Int J Non-Linear Mech 2015; 72: 23–41. 23. Saeed NA, El-Ganini WA and Eissa M. Nonlinear time delay saturation-based controller for suppression of nonlinear beam vibrations. Appl Math Modell 2013; 37: 8846–8864. 24. Wang ZL, Lin QR and Liu ZX. Optimal design of vibration control for piezoelectric intelligent beams. Chin Q Mech 2000; 12: 454–461. (In Chinese). 25. Wang Y, Cao XS, Jing T, et al. Dynamic characteristics and stability of axially moving viscoelastic plate with piezoelectric layer. J Low Freq Noise Vibration Active Control 2014; 33: 341–356. 26. Nageswara Rao B. Large-amplitude free vibrations of simply supported uniform beams with immovable ends. J Sound Vibration 1992; 155: 523–527. 27. Mahmoodia SN, Jalilia N and Khadem SE. An experimental investigation of nonlinear vibration and frequency response analysis of cantilever viscoelastic beams. J Sound Vibration 2008; 311: 1409–1419. 28. Nayfeh AH. Perturbation methods. New York, NY: Wiley, 1973. 29. Nayfeh AH and Mook DT. Nonlinear oscillations. New York, NY: Wiley, 1979. 30. Ji JC and Zhang N. Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber. J Sound Vibration 2010; 329: 2044–2056. 31. Liu CC, Ren CB, Liu L, et al. Optimal control of nonlinear resonances for vehicle suspension using linear and nonlinear control. J Low Freq Noise Vibration Active Control 2013; 32: 335–346. 32. Li XY, Ji JC and Hansen CH. The response of a Duffing-van der Pol oscillator under delayed feedback control. J Sound Vibration 2006; 291: 644–655. 33. Malekshahi A, Mirzaei M and Aghasizade S. Non-Linear predictive control of multi-input multi-output vehicle suspension system. J Low Freq Noise Vibration Active Control 2015; 34: 87–106. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

Piezoelectric optimal delayed feedback control for nonlinear vibration of beams:

Download PDF
14 pages

Loading next page...
 
/lp/sage/piezoelectric-optimal-delayed-feedback-control-for-nonlinear-vibration-gibf4xYTyU

References (34)

Publisher
SAGE
Copyright
Copyright © 2022 by SAGE Publications Ltd unless otherwise noted. Manuscript content on this site is licensed under Creative Commons Licenses
ISSN
0263-0923
eISSN
2048-4046
DOI
10.1177/0263092316628255
Publisher site
See Article on Publisher Site

Abstract

An optimal delayed feedback control methodology is developed to mitigate the primary and super harmonic resonances of a flexible simply-simply supported beam with piezoelectric sensor and actuator. Stable vibratory regions of the feedback gains are obtained by using the stability conditions of eigenvalue equation. Attenuation ratio is used to evaluate the performance of vibration control by taking the proportion of peak amplitude of primary or super harmonic reson- ances for the suspension system with and without controllers. Optimal control parameters are obtained using an optimal method, which takes attenuation ratio as the objective function and the stable vibratory regions of the time delay and feedback gains as constraint conditions. The piezoelectric optimal controllers are designed to control the dynamic behaviour of the nonlinear dynamic system. It is found that the optimal feedback gains obtained by the optimal method result in a good control performance. Keywords Nonlinear vibration, optimal control, piezoelectricity Introduction In the past decades, considerable amount of research works have helped us better understand the effect of time delays on the behaviour of nonlinear dynamical systems controlled by linear or nonlinear feedback controllers for mitigating the vibration amplitude. Delayed vibration control is a technique of vibration suppression. The time delay can vary a fixed gain and change the range of saturation control, either widening or shrinking the effective frequency bandwidth. Since delayed control in nonlinear systems has an important role in vibration control engineering, the study of delayed vibration control of nonlinear systems is essential. Analytical methods for weakly nonlinear continuous systems include harmonic balance method, the Galerkin procedure and directed multiple scales. The first two approximate methods are easy and straightforward to apply. While the calculation of higher-order approximations may be complex. In the third approach, one calculates an approximate solution of the nonlinear differential equation by directly using multiple scales to expand the non- linear governing partial differential equation and boundary conditions continuous problem. The obtained mode shapes are in full agreement with those obtained by using discretization because the latter are performed by using a complete set of basic functions that satisfy the boundary conditions. This method is beneficial to the elimination of coupling terms and removal of solving multi-dimensional equations, and can be easily applied to analyse the dynamic response of higher modes. The technique of delayed-feedback control was introduced as an effective means of controlling a wide variety of mechanical systems. Olgac and Holm-Hansen introduced the concept of delayed resonators to control mechanical 3,4 systems. This concept was utilized to study time-delayed acceleration feedback control over continuous systems School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo, China Corresponding author: Canchang Liu, School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255049, China. Email: sdutlcch@163.com 26 Journal of Low Frequency Noise, Vibration and Active Control 35(1) 5,6 7 and time-delayed velocity feedback control over torsional mechanisms. Jalili and Olgac used time-delayed feedback resonators to control discrete multi-degree-of-freedom systems. Zhao and Xu used delayed feedback control to suppress the vibration of vertical displacement in a two-degree-of-freedom nonlinear system with external excitation. Plaut and Hsieh studied the effect of a damping time delay on nonlinear structural vibrations and analysed six resonance conditions. Sato et al. discussed the free and forced vibration of nonlinear systems with the time delays. Hu et al. studied the primary resonance, super harmonic resonance and stability of a nonlinear Duffing oscillator with time delays under harmonic excitations and gave out periodic solutions by using the method of modified target practice. Maccari dealt with the principal parametric resonance of a van der Pol oscillator with time-delay linear state feedback. Ji and Leung demonstrated that in parametrically excited Duffing systems stable region of the trivial solution could be broadened, a discontinuous bifurcation could be transformed into a continuous one and the jump phenomenon in the response could be removed, if an appropriate feedback control was used. Ji and Leung also studied the primary, super harmonic and subharmonic resonances of a harmonically excited nonlinear single degree of freedom system with two distinct time delays in the linear state feedback. The primary resonance of a cantilever beam under state feedback control with a time delay was 15 16 investigated. Qian and Tang discussed the primary resonance and the subharmonic resonances of a nonlinear beam under moving load based on time-delay feedback control. Daqaq et al. presented a comprehensive inves- tigation on the effect of feedback delays on the nonlinear vibrations of a piezoelectric actuated cantilever beam and analysed the effect of feedback delays on a beam subjected to a harmonic base excitations. Alhazza et al. investigated the effect of time delays on the stability, amplitude and frequency–response behaviour of a beam and found that even the minute amount of delays could completely alter the behaviour and stability of the parametrically excited beam and lead to unexpected behaviour and response. Liao et al. presented a feasible methodology by which could achieve good control performance of a dynamic beam structure system with time delay effect. A spring time-delay controller was designed to achieve good control performance. Liu et al. studied the primary, subharmonic and super harmonic resonances of an Euler–Bernoulli beam subjected to harmonic excitations with damping and spring delayed-feedback controllers. Gao and Chen combined cubic nonlinearity and time delay to improve the performance of vibration isolation. A time delayed nonlinear saturation controller was proposed to reduce the horizontal vibration of a magnetically levitated body described by a nonlinear dif- ferential equation subjected to both external and modulated forces. The time delay saturation-based controller was considered for active suppression of nonlinear beam vibrations, and the effects of time delays on the system behaviour were studied. The research works mentioned above are concerned with the selection of feedback gains and time delays that can enhance the control performance of nonlinear systems or change the position of the bifurcation point. However, all these works have failed to address how to choose optimized control parameters and time delays on the basis of keeping the system stable. In this paper, delayed piezoelectric feedback controllers are designed to change the beam’s nonlinear dynamical behaviour. The feedback gains and time delays of piezoelectric controllers are easy to set up and change by using active control facilities. The main purpose of this paper is to present an optimum control method for a nonlinear vibration system. The stable vibratory regions of time delay and feedback gains are given based on the analysis of stability conditions of eigenvalue equation. The control parameters are calculated with an optimal method, which takes attenuation ratio as the objective function and stable vibratory regions of time delays and feedback gains as constraint conditions. The multiple scales method is applied to obtain linear equations, by which it is easy to analyse the dynamic behaviour of high order modes of the beam. Time delay is taken as a control factor that can turn defective effect into favourable effect. Compared with damping force controllers, time delayed controllers have three parameters to be adjusted, two feedback gains and a time delay. Therefore, the space of design and adjustment is much wider. A closed-loop control system is designed to control the dynamic behaviour of the nonlinear dynamic system. Primary resonance analysis Programme formulation The beam is assumed to be inextensible, has uniform cross-sectional area and satisfies the Euler–Bernoulli beam theory. The simply-simply supported boundary conditions are considered (see as Figure 1). The uniform flexible beam with a piezoelectric sensor layer on its top surface and a piezoelectric drive layer on its low surface is considered. The piezoelectric sensor bends with deformation of the beam, which is excited by a Liu et al. 27 Delay x1 x2 Voltage Amplifier Figure 1. Schematic drawing of a piezoelectric actuated beam under harmonic excitations. harmonic excitation. The voltage of the piezoelectric sensor layer, which is induced by the deformation along the 24,25 beam direction, is given as 0  0 V ¼ K½w ðX , t Þ w ðX , t Þ ð1Þ 2 1 where K ¼ bhg =2C , b and h are width and height of the beam, respectively. g is piezoelectric parameter, C is 31 p 31 p capacitor of the piezoelectric sensor. w is the deflection of the beam. The prime of w is the first derivative with respect to the x-coordinates. X and t* are the coordinates of the axial direction and time, respectively. The voltage produced by the piezoelectric sensor can be delayed by the time-delay device and amplified by the amplifier. The bending moment produced by the delayed and amplified voltage on the beam can be written as MðX, t Þ¼ gRVðtÞ½HðX  X Þ HðX  X Þ ð2Þ 1 2 where g is feedback gain parameter g4 0, g5 0 and g ¼ 0 represent positive, negative and no feedback, respect- ively. H is the Heaviside unit doublet function, R ¼ bd E ðh þ  Þ, E is the elastic modulus,  is the height of 31 pe pe pe pe the piezoelectric sensor and d is the piezoelectric parameter. 26,27 The non-dimensional form of equation of motion and boundary conditions of the beam can be obtained as ð4Þ 00 02 0 0 w€ þ w ¼"w_ þ "w w dx þ "f ðxÞ cosðtÞþ "gg ½w ðx , t  Þ w ðx , t  Þ p 2 1 ð3Þ 00 00 ½H ðx  x Þ H ðx  x Þ 1 2 w ¼ w ¼ 0at x ¼ 0, 1 ð4Þ where g ¼ LRK/EI,  is the damping of the vibration system. L is the length of the beam, E and I are the elastic qffiffiffiffiffi pffiffiffiffiffiffiffiffi t EI modulus and moment of inertia of the beam, respectively. w ¼ w=,  ¼ I=A, x ¼ X/L. t ¼ ,  and A are L A the mass per unit length and the cross-section area of the beam, respectively.  is the time delay. The beam bending 28,29 vibration w can be expanded by order of " as wðx, t; "Þ¼ w ðx, T , T Þþ "w ðx, T , T Þ þ  ð5Þ 0 0 1 1 0 1 where T and T are the fast and slow time scales, respectively. Substituting expression (5) into the partial differ- 0 1 ential equation (3) and separating terms at orders of ", yields @ w 0 ð4Þ þ w ¼ 0 ð6Þ @T w ¼ w ¼ 0at x ¼ 0, 1 ð7Þ 2 2 1 @ w @ w @w 1 0 0 ð4Þ 00 02 þ w ¼2   þ w w dx þ f cosðT Þþ gg w  w 0 p 0 0 0 2 1 @T @T @T @T ð8Þ 0 1 0 0 00 00 ½H ðx  x Þ H ðx  x Þ 1 2 28 Journal of Low Frequency Noise, Vibration and Active Control 35(1) w ¼ w ¼ 0at x ¼ 0, 1 ð9Þ With the Galerkin approximation, w can be represented as a series of products of the spatial functions which only depends on time as 1 1 X X wðx, tÞ¼ w ðx, tÞ¼  ðxÞq ðtÞ ð10Þ n n n n¼1 n¼1 pffiffiffi where  is the nth eigen function of a linear uniform beam and  ¼ 2 sinðnxÞ. q is the nth generalized n n n time-dependent coordinates. The solution of linear equation (6) with the boundary conditions (4) can be written as i! T i! T k 0 k 0 w ¼  ðxÞ A ðT Þe þ A ðT Þe ð11Þ 0 k k 1 k 1 where A is the amplitude, A is the conjugate item of A . k k k The solvability condition requires that the eigen functions are orthogonal ð12Þ h ,  i¼  ðsÞ ðsÞds ¼ n m n m nm here  is the Kronecker delta. For the case of primary resonance, we have nm ¼ ! ð1 þ " Þð13Þ where is the detuning parameter. ! is the kth order circular frequency. Substituting equations (11), (12) and (13) 1 i into (8), applying the condition (9), and letting A ¼ a e , the equation of secular terms is k k 0 2 ið! T Þ i! k 1 k ð14Þ 2i! A þ i! A  3g A A  F e  2g A e ¼ 0 k k k kk k k k k k k DE hi R R 1 1 00 02 1 0 0 1 where g ð! Þ¼  ðxÞ,   dx ,  ¼ g ½ ðx Þ  ðx Þ , F ¼ f ðxÞdx kk k k k p 1 2 k k k k k k 0 2 2 0 Let ¼ ! T . Separating the real and imaginary parts of equation (14), yields k k 1 k a ¼ a þ sinð Þ ð15Þ k k k 0 3 ¼ a þ v a þ cosð Þ ð16Þ k k k k k k k g sin !  g cos !  3g 1 k k k k kk where ¼  þ , ¼ ! þ , v ¼ . k k K k 2 ! ! 8! k k 0 0 By letting a ¼ ¼ 0, the steady-state response in the system can be expressed as k k a þ sinð Þ¼ 0 ð17Þ k k k a þ v a þ cosð Þ¼ 0 ð18Þ k k k k Using modulation equations, the amplitude–frequency equation of the system can be obtained 2 3 ð19Þ ð a Þ þ a þ v a ¼ k k k k k k Liu et al. 29 The peak amplitude of the primary response, obtained from equation (19), is given by 2 k E ¼ a ¼ ð20Þ k max 2 2 k k For the purpose of comparison, the equation of motion for the nonlinear primary oscillator without control is ð4Þ 00 02 ð21Þ w€ þ "w_ þ w ¼ "w w dx þ "f ðxÞ cosðtÞ The corresponding peak amplitude for the nonlinear primary oscillator without control can be written as E ¼ a ¼ ð22Þ k max 2 2 k k where  ¼ =2. The performance of the vibration controllers on the reduction of nonlinear vibrations cannot be studied by using a similar procedure that discusses the ratio of response amplitude of the linear system because of the difficulty of finding the analytical solutions for a nonlinear system. Therefore, an attenuation ratio is utilized to evaluate the performance of the vibration controller by taking the proportion of vibration peaks of primary 30,31 resonances of the beam system with and without control. By this definition, the attenuation ratio can be written as 0 1 B C B C ð23Þ R ¼ @ A 2g sin ! k k 1 þ As defined by equation (23), a small value of the attenuation ratio R indicates a large reduction in the vibrations of the nonlinear primary system. From equations (17) and (18), it is known that the damping coefficients are functions of feedback gain and time delay. Small attenuation rate can be obtained by selecting proper parameters of feedback gain and time delay. Controller design for primary resonance For nonlinear systems, the nonlinear phenomena such as jump phenomenon and hysteresis may occur and the vibration of the system may become unstable. The vibration controllers should be designed to avoid the unstable vibration and reduce nonlinear vibration amplitude. The stability of the solutions depends on the eigenvalues of the corresponding Jacobian matrix of equations (17) and (18). The corresponding eigenvalues are the roots of equation 2 2 2 2 þ 2 þ þ þ v a þ 3v a ¼ 0 ð24Þ k k k k k k k k The sum of the two eigenvalues is 2 . The addition of the feedback gain and time delay alters the sum of the two eigenvalues. If < 0, it means at least one of the eigenvalues will always have a positive real part. The system will be unstable. If ¼ 0, it means a pair of purely imaginary eigenvalues and hence a Hopf bifurcation may occur. Based on the analyses mentioned above, unstable periodic solutions corresponding to a saddle is f ð Þ5 0 ð25Þ 2 2 2 2 4 where f ð Þ¼ þ 4v a þ þ 3v a . The value of f ð Þ is positive when there is no solution of equation k k k k k k k k k f ð Þ¼ 0 ð26Þ where a saddle-node bifurcation occurs. 30 Journal of Low Frequency Noise, Vibration and Active Control 35(1) If > 0, the sum of two eigenvalues is negative, and accordingly, at least one of the two eigenvalues will have a 16,32,33 negative real part. The sufficient conditions for guaranteeing the system stability are 2 2 2 2 4 f ð Þ¼ þ 4v a þ þ 3v a 4 0 4 0 ð27Þ k k k k k k k k k 2 2 4 2 4 The value of f ð Þ is positive when then no solution of critical equation f ð Þ¼ 0. Letting  v a  v a , k k k k k max k k there is 2 2 4 2 k v a ¼ v ð28Þ k k k max k 4 4 k k The region of stable vibration control parameters can be obtained sffiffiffiffiffiffiffiffiffiffiffiffi "# ! 1 jj v F k k ð29Þ g ¼ g sin !     þ 2 ! When there are two solutions of equation f ð Þ¼ 0, the solutions are 2 2 4 2 1=2 ¼2v a ðv a  Þ ð30Þ k k k k k As f ð Þ¼ 0 is a parabola whose mouth is opened upward, the inequality f ð Þ4 0 is satisfied when 5 k k k and 4 . In order to have a good control performance, is larger than  . Reducing or enlarging the roots of k k k the equation of f ð Þ¼ 0 and considering g < 0, we have k kk 1=2 2 4 2 ð31Þ ¼ v a k k max k k 2 2 4 2 1=2 þ ¼2v a þðv a   Þ  ð32Þ k k k max k k max k k Considering the formula of and equation (31), the region of stable vibration is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 4 v F   ! k k k k ! k ð33Þ g cos !    and 4 0 k k k K k Taking into account the formula (32), gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 6 4 2v F þ v F   ! ! k k k k k k ð34Þ g cos !   þ and 4 0 k k k k k Analysis of optimised controller design parameters The region of feedback parameter has been obtained based on the analysis of stability condition of the nonlinear vibration system, but it is difficult to obtain the optimal control parameters of the system. Taking the nonlinear vibration system with attenuation ratio as the objective function, the optimal feedback control parameters can be calculated with an optimal method. The optimal analysis is carried out by taking into account the two cases with and without solutions of the critical equation. Optimal design when the critical equation has no solution. 0 1 B C B C ð35Þ min R ¼ @ A 2g sin ! k k 1 þ k Liu et al. 31 sffiffiffiffiffiffiffiffiffiffiffiffi "# ! 1 jj v F k 3 k s:t: g sin !     þ  0, 4 0 k k 2 ! Optimal design when the critical equation has two solutions. 0 1 B C B C ð36Þ min R ¼ @ A 2g sin ! k k 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 6 4 v F   ! k k k k ! s:t: g cos !  þ þ  0 k k k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 6 4 2v F  v F   ! ! k k k k k or g cos !  þ þ  0, 4 0 k k k k k Super harmonic resonance analysis Super harmonic resonance For the case of super harmonic resonance, the harmonic excitation is assumed that pðxÞ¼ "f ðxÞð37Þ Substituting equations (5) and (37) into equation (3) and equating coefficients of like powers of ", the following equations are derived @ w ð4Þ þ w ¼ pðxÞ cosðTÞð38Þ @T w ¼ w ¼ 0at x ¼ 0, 1 ð39Þ 2 2 @ w @ w @w 1 ð4Þ 0 0 00 02 þ w ¼2   þ w w dx þ gg w p 0 1 0 0 @T @T @T @T 0 1 0 ð40Þ 00 00 ½H ðx  x Þ H ðx  x Þ 1 2 w ¼ w ¼ 0at x ¼ 0, 1 ð41Þ The solution of equation (38) is written as follows hi ! ðT þ T Þ k 0 1 i! T i k 0 w ¼  ðxÞ A ðT Þe þ B e þ cc ð42Þ 0 k k 1 k where B ¼ , F ¼  ðxÞ pðxÞdx. Substituting equation (42) into equation (40), the equation of secular k 2 2 k k 2ð!  Þ terms is written as 0 2 3 ið! T Þ i! k 1 k i2! A þ i! A  3g A A  6g A B B  g B e  2g A e ¼ 0 ð43Þ k k k kk k kk k k k kk k k k k k 32 Journal of Low Frequency Noise, Vibration and Active Control 35(1) 1 i Let A ¼ a e , ¼ ! T . Substituting them into equation (43) and separating the real and imaginary k k k k 1 k parts, yields g B kk 0 k ð44Þ a ¼ a þ sinð Þ k k k g B kk 0 3 ð45Þ ¼ a þ v a þ cosð Þ k sk k k k k k 3g B g cos !  kk k k k where ¼ ! þ þ . The fixed points of this system are given as sk k ! ! k k g B kk ð46Þ a  sinð Þ¼ 0 k k k g B kk 3 k ð47Þ a þ v a þ cosð Þ¼ 0 sk k k k By eliminating from equations (46) and (47), the equation of amplitude-frequency is obtained g B kk 2 3 k ð48Þ ð a Þ þ a þ v a ¼ k k sk k k The peak amplitude of the super harmonic resonance, which is obtained from equation (48), is given by 2 6 g B 2 kk k E ¼ a ¼ ð49Þ k max 2 2 k k For the purpose of comparison, the corresponding peak amplitude of the nonlinear super harmonic oscillator without control is 2 6 g B 2 kk k E ¼ a ¼ ð50Þ k max 2 2 k k The attenuation ratio of nonlinear super harmonic can be written as 0 1 B C B C R ¼ ð51Þ @ A 2g sin ! k k 1 þ The steady-state solutions of super harmonic resonance response depend on the eigenvalues of the characteristic equation, which are the roots of 2 2 2 2 þ 2 þ þ þ v a þ 3v a ¼ 0 ð52Þ k k k k k k k k Optimal design of controller parameters Taking the attenuation ratio of super harmonic resonance as the objective function, the optimal feedback control parameters can be worked out by the optimal method. When the critical equation has no solution. 0 1 B C B C ð53Þ min R ¼ @ A 2g sin ! k k 1 þ k Liu et al. 33 "# sffiffiffiffiffiffiffiffiffiffiffiffiffi jj ! 1 v F k 3 k s:t: g sin !     þ  0, 4 0 k k 2 ! where F ¼ g B . s kk When the critical equation has two solutions. 0 1 B C B C ð54Þ min R ¼ @ A 2g sin ! k k 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 6 4 2 2 v F   ! k s k k ! þ 3g B kk k k s:t: g cos !  þ þ  0 k k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 6 4 2 2 2v F  v F   ! s s k k k ! þ 3g B kk k k or g cos !  þ þ  0, 4 0 k k k k k Numerical simulation The nonlinear vibration control for a simply-simply supported beam is studied in the present paper. The length, width and height of beam are 500 mm, 50 mm, and 1 mm, respectively. The density and elastic modulus of the beam are 2700 kg m and 70 GPa, respectively. Both the length and width of piezoelectric sensor are 30 mm. The thickness of piezoelectric sensor is 0.2 mm, the piezoelectric constant is d ¼ 190 e  12 mV , the piezoelectric capacitance is Cs ¼ 15 nf, elastic modulus of the piezoelectric actuators E ¼ 63 GPa, and the constant of piezo- pe electric is g ¼ 1.9 C m . Other parameters are  ¼ 0.5,  ¼ 0.1, ¼ 0.01 and B ¼ 0.1, respectively. The control 31 1 analysis is carried out only for the first mode of the beam, and the control analysis of high modes is similar. Figure 2 shows the variation of damping coefficient of the nonlinear vibration for first order mode of the beam with feedback gain and time delay. The damping value varies with the alteration of time delay when the value of feedback gain is fixed. Usually, should have a larger positive value in order to improve the control performance. For the cases of three distinct feedback gains, the effective time delay region of vibration damping which is greater than is from 0 to 0.32 and 0.64 to 0.80. The damping coefficient reaches the peak of primary resonance when the time delay is 0.16. Figure 3 g =200 g =400 g =600 0.5 g =0 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 2. Variation of with time delay of the first mode primary resonance of the beam for different feedback control parameters. 1 34 Journal of Low Frequency Noise, Vibration and Active Control 35(1) displays the alteration of tuning coefficients of beam’s primary and super harmonic resonance of first-order mode with the time delay, respectively. It is found that the tuning coefficient varies with the alteration of feedback gain and time delay. Based on the analysis of Figures 2 and 3, it can be observed that stable vibration can be obtained by selecting proper feedback gain and time delay, and the peak amplitude of the resonance can be thus reduced. Figure 4 displays the variation of attenuation ratio R with time delay and feedback gain. As shown in the figure, for fixed value of the amplitude of excitation, a smaller value of the attenuation ratio R indicates a larger reduction in the primary resonances of the nonlinear system. A good selected value of time delay can relatively lead to a larger positive value of and a smaller attenuation ratio R . Therefore, the amplitude of vibration of the 1 1 nonlinear system can be reduced by properly selecting the feedback gain and the time delay. Taking the nonlinear vibration system with attenuation ratio as the objective function and the stable vibratory regions of the time delays and feedback gains as constraint conditions, the optimal feedback control parameters can be calculated by using the optimal method. The case that the characteristic equation has no solution is taken into account as an example to discuss the control parameters. As the value of  and  are all positive values in this example through calculation, the value of g should be positive in order to get better control performance shown in the formulation (34). A greater value of g can lead to a smaller attenuation rate R , and a better control perform- ance can be obtained. It is also found that g is the function of excitation amplitude F . When the value of F is 1 1 bigger, a larger value of control parameter is demanded to mitigate the vibration. The effect of the excitation amplitude on the stable minimum control parameters for three sets of time delays is shown in Figure 5. g=200 g=400 g=600 0.5 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 3. Variation of with time delay of the first mode primary resonance of the beam for different feedback control parameters. g =100 g =200 0.8 g =300 0.6 0.4 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 Figure 4. Variation of R with time delay of the first mode primary resonance of the beam for different feedback control parameters. 1 Liu et al. 35 The stable minimum feedback gain varies from a small value to a big one with an increase in the value of excitation amplitude of F . The stable minimum feedback gain also changes with variation of the value of time delay In the region of 0-j/2! , a large value of time delay is prone to need a small feedback gain to control the . 1 vibration of the system. The optimal control time delay can be obtained when  ¼ j/2! , by which the smallest feedback gain g can be used to reduce the vibration. So the optimal control time delay can be taken as a control factor to improve the control performance as same as feedback gain. A good control performance can be obtained by selecting an optimal time delay. The amplitude of excitation F is 0.2. It can be worked out that the product of feedback gain g and sin(! )is 1 1 more than 40.3141 when the characteristic equation has no solution. Let  ¼j/3! . The optimal feedback gain can be calculated as g 2 [778.9,1581.2] for the case that there are two solutions for the characteristic equation. The calculation of optimal feedback gain and time delay of super harmonic resonance follows the same pro- cedure as the primary resonance. It can be calculated that the product of g and sin(! ) is more than 12.16 when the characteristic equation has no solution and B ¼ 0.12. Letting  ¼j/3! , the optimal feedback gain is calcu- 1 1 lated as g[184.8,473.1] when the characteristic equation has two solutions. Figures 6 and 7 show the primary and super harmonic response curves of first mode of the beam with three different sets of the time delay. There is no jump and hysteresis phenomenon when  ¼j/6! or  ¼j/3! . This 1 1 suggests that saddle node bifurcation and jump phenomenon can be eliminated by choosing certain values of the t=p/6w t=p/3w t=p/2w 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 5. Stable minimum control parameters of the first mode primary resonance of the beam for three sets of time delays. 0.5 t=p/6w Stable t=p/3w 0.4 t=0 0.3 Unstable 0.2 0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 Figure 6. Frequency–response curves for the first mode primary resonance of the beam for three sets of time delays. g 36 Journal of Low Frequency Noise, Vibration and Active Control 35(1) t=0 0.5 t=p/6w t=p/3w 0.4 Stable 0.3 Unstable 0.2 0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 Figure 7. Frequency–response curves for the first mode super harmonic resonance of the beam for three sets of time delays. Figure 8. Frequency–response curves for the first mode primary resonance of the beam for different excitations and time delays. time delay. Three solutions exist in a region of coexistence of  ¼ 0. The bending of the frequency response curves is the cause of a jump phenomenon. Moreover, the peak amplitude of the primary resonance response at  ¼ j/3! is the smallest one in the three cases. The vibration controllers can effectively suppress the amplitude oscillations of the nonlinear oscillator. Hence, by optimally choosing the feedback gain and time delay of the piezoelectric controllers, the primary and super harmonic resonance response of the nonlinear oscillator can be reduced. Figure 8 shows the primary response curves of first mode with different sets of the time delays and amplitudes of excitations. The feedback gain is also selected as 0.35. As shown in the figure the vibration of the system is unstable with  ¼ 0 and stable with  ¼ j/6! when the amplitude of the excitation is 0.25. The vibration of the system can be stable and the vibration displacement be mitigated when the amplitude of excitation changes from 0.25 to 0.45 and the time delay from j/6! to j/2! . Hence, the time delay can be used as a control factor to suppress the 1 1 resonance response of the nonlinear oscillator by choosing optimal value of the time delay. Figure 9 shows the critical curve of saddle bifurcation of the beam for different kinds of time delays. When f ð Þ¼ 0, the other eigenvalue is zero where a saddle-node bifurcation occurs. The formulas (26) are utilised to study the critical curves of the nonlinear vibration system. When f ð Þ4 0, the vibration system may be stable. However, when f ð Þ5 0, the vibration system is unstable. Stable and unstable regions are shown in the figure. s1 Liu et al. 37 Figure 9. Critical curve of saddle bifurcation of the beam for different time delays. With increase in the time delay, the stable regions of the nonlinear vibration system decrease. The value of the time delay can change the stable structure of the nonlinear vibration system. Conclusions The regions of time delay and feedback gains are obtained by enlarging or reducing the inequalities of the roots of eigenvalue equation. The primary and super harmonic resonance response of the nonlinear oscillator can be reduced by choosing stable feedback gains and time delay. The optimal control parameters of feedback gain and time delay are calculated by using the method of minimum optimal attenuation ratio, which takes the attenuation ratio as the objective function and stable regions of the time delay and feedback gain as constrained conditions. Optimal control performance can be achieved by using the optimal time delay and feedback gain controller. The value of the time delay can change the stable structure of the nonlinear vibration system. The intelligent control system attached piezoelectric elements has the characters of high efficiency and low energy consumption, therefore it can be widely used in the vibration control of the flexible structure. Test cases are designed and the simulation results are studied. Time-delay feedback controllers can be utilized to suppress the dynamic behaviours of nonlinear systems. Most of the vibration energy of the nonlinear oscillator is transferred to the controllers through piezoelectric controller. The optimal vibration controllers can effectively suppress the amplitude of oscillations of the nonlinear oscillator. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. References 1. Nayfeh AH and Nayfeh SA. On nonlinear modes of continuous systems. J Vibration Acoust 1994; 116: 129–136. 2. Olgac N and Holm-Hansen BT. A novel active vibration absorption technique: delayed resonator. J Sound Vibration 1994; 176: 93–104. 3. Olgac N, Elmali H, Hosek M, et al. Active vibration control of distributed systems using delayed resonator with acceleration feedback. J Dyn Syst Meas Control 1997; 119: 380–389. 4. Olgac N and Jalili N. Modal analysis of flexible beams with delayed resonator vibration absorber: theory and experiments. J Sound Vibration 1998; 218: 307–331. 38 Journal of Low Frequency Noise, Vibration and Active Control 35(1) 5. Hosek M, Elmali H and Olgac N. A tunable torsional vibration absorber: the centrifugal delayed resonator. J Sound Vibration 1997; 205: 151–165. 6. Filipovic D and Olgac N. Delayed resonator with speed feedback-design and performance analysis. Mechatronics 2002; 12: 393–413. 7. Jalili N and Olgac N. Multiple delayed resonator vibration absorbers for multi-degree-of-freedom mechanical structures. J Sound Vibration 1999; 223: 567–585. 8. Zhao YY and Xu J. Effects of delayed feedback control on nonlinear vibration absorber system. J Sound Vibration 2007; 308: 212–230. 9. Plaut RH and Hsieh JC. Non-linear structural vibrations involving a time delay in damping. J Sound Vibration 1987; 117: 497–510. 10. Sato K, Sumio Y and Okimura T. Dynamic motion of a nonlinear mechanical system with time delay: analysis of the self- excited vibration by an averaging method. Trans JSME Series C 1995; 685: 1861–1866. 11. Hu HY, Dowell EH and Virgin LN. Resonances of a harmonically forced Duffing oscillator with time delay state feed- back. Nonlinear Dyn 1998; 15: 311–327. 12. Maccari A. The response of a parametrically excited van der Pol oscillator to a time delay state feedback. Nonlinear Dyn 2001; 26: 105–119. 13. Ji JC and Leung AYT. Bifurcation control of parametrically excited Duffing system. Nonlinear Dyn 2002; 27: 411–417. 14. Ji JC and Leung AYT. Resonances of a non-linear s.d.o.f system with two time-delays in linear feedback control. J Sound Vibration 2002; 253: 985–1000. 15. Maccari A. Vibration control for the primary resonance of a cantilever beam by a time delay state feedback. J Sound Vibration 2003; 259: 241–251. 16. Qian CZ and Tang JS. A time delay control for a nonlinear dynamic beam under moving load. J Sound Vibration 2008; 309: 1–8. 17. Daqaq MF, Alhazza KA and Arafat HN. Non-linear vibrations of cantilever beams with feedback delays. Int J Non-Linear Mech 2008; 43: 962–978. 18. Alhazza KA, Daqaq MF, Nayfeh AH, et al. Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control. Int J Non-Linear Mech 2008; 43: 801–812. 19. Liao JJ, Wang AP, Ho CM, et al. A robust control of a dynamic beam structure with time delay effect. J Sound Vibration 2002; 252: 835–847. 20. Liu CC, Qiu JH, Sun HY, et al. Nonlinear vibrations of beams with spring and damping delayed feedback control. J Vibroeng 2013; 15: 340–354. 21. Gao X and Chen Q. Nonlinear analysis, design and vibration isolation for a bilinear system with time-delayed cubic velocity feedback. J Sound Vibration 2014; 333: 1562–1576. 22. Eissa M, Kandil A, El-Ganaini WA, et al. Vibration suppression of a nonlinear magnetic levitation system via time delayed nonlinear saturation controller. Int J Non-Linear Mech 2015; 72: 23–41. 23. Saeed NA, El-Ganini WA and Eissa M. Nonlinear time delay saturation-based controller for suppression of nonlinear beam vibrations. Appl Math Modell 2013; 37: 8846–8864. 24. Wang ZL, Lin QR and Liu ZX. Optimal design of vibration control for piezoelectric intelligent beams. Chin Q Mech 2000; 12: 454–461. (In Chinese). 25. Wang Y, Cao XS, Jing T, et al. Dynamic characteristics and stability of axially moving viscoelastic plate with piezoelectric layer. J Low Freq Noise Vibration Active Control 2014; 33: 341–356. 26. Nageswara Rao B. Large-amplitude free vibrations of simply supported uniform beams with immovable ends. J Sound Vibration 1992; 155: 523–527. 27. Mahmoodia SN, Jalilia N and Khadem SE. An experimental investigation of nonlinear vibration and frequency response analysis of cantilever viscoelastic beams. J Sound Vibration 2008; 311: 1409–1419. 28. Nayfeh AH. Perturbation methods. New York, NY: Wiley, 1973. 29. Nayfeh AH and Mook DT. Nonlinear oscillations. New York, NY: Wiley, 1979. 30. Ji JC and Zhang N. Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber. J Sound Vibration 2010; 329: 2044–2056. 31. Liu CC, Ren CB, Liu L, et al. Optimal control of nonlinear resonances for vehicle suspension using linear and nonlinear control. J Low Freq Noise Vibration Active Control 2013; 32: 335–346. 32. Li XY, Ji JC and Hansen CH. The response of a Duffing-van der Pol oscillator under delayed feedback control. J Sound Vibration 2006; 291: 644–655. 33. Malekshahi A, Mirzaei M and Aghasizade S. Non-Linear predictive control of multi-input multi-output vehicle suspension system. J Low Freq Noise Vibration Active Control 2015; 34: 87–106.

Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Feb 25, 2016

Keywords: Nonlinear vibration; optimal control; piezoelectricity

There are no references for this article.