Application of Energy Finite Element Method in Active Vibration Control of Piezoelectric Intelligent Beam

Abstract

Based on the transmission and equilibrium relationship of vibration energy in beam-like structures, the Galerkin weighted residual method was applied to equation discretization. An equivalent transformation of feedback element was suggested to develop the Energy Finite Element model of a composite piezoelectric cantilever beam driven by harmonic excitation on lateral direction, with both systems with and without time delay being studied and the power input estimation of harmonic excitation was discussed for the resolution of Energy Finite Element function. Then the energy density solutions of the piezoelectric coupling beam through Energy Finite Element Method (EFEM) and classical wave theory were compared to verify the EFEM model, which presented a good accordance. Further investigation was undertaken about the influence of control parameters including the feedback gain and arrangement of piezoelectric patches on characteristics of system energy density distribution.

1. Introduction

The birth of intelligent structure system is due to the mutual penetration of information science and engineering and materials science. Piezoelectric intelligent structure, as one of intelligent structures, is interpreted as light damping system attached with piezoelectric materials on the surface. Based on various advantages of piezoelectric materials, such as easy measurement and control of input and output of the signals, wide range of frequency response, both use of sensor and actuator and so on, it has been largely applied in aerospace, robot, information technology, IT, materials science, and other high technologies. Many scholars have made deep researches and gained rich achievements, and as a result, active vibration control systems using piezoelectric materials as distributed sensors and actuators (S/As) are becoming more and more popular in recent years.

As the object of study, piezoelectric intelligent structure’s dynamic modeling and optimal design of structure parameters are the problems we concerned about most. External perturbation is always treated as pulse excitation to discuss the suppression problems of system vibration. Obtaining the system’s state-space equation through modal theory, the optimal locations of piezoelectric sensors and actuators were searched taking the smallest vibration energy as the objective function [1]. In [2], governing equation was established based on steady stimulation and then made an experimental research on frequency response characteristic. In [3], the boring bar vibration was controlled by using piezoelectric materials; simulation results suggest that the control effect was well.

The application of Finite Element Method (FEM) will be involved when it needs to solve the analysis and design problems of complex engineering structures. In recent years, the FEM analysis for vibration control of piezoelectric intelligent structures has become some research hotspot. A piezoelectric smart structure model had been discussed in [4], using the finite element theory.  The finite element dynamics equation of piezoelectric intelligent structure was deduced in [5]. However, the application of FEM is generally confined in relatively lower frequency domain, which is resulted from a high sensitivity of the numerical calculation to tiny alteration of structure parameters when the number of elements in the FEM model becomes very great. Although Statistical Energy Analysis (SEA) has become a mature method in energy flow analysis, and related commercial software such as AutoSEA has been in use, it would face the problem of insufficiency in modal density in relatively lower frequency band. In addition, there is not mature theory of structure in loss factor and coupling loss factor [6]. There exists some middle frequency band in which either FEM or SEA would not be credible. Compared with SEA, as an energy analysis method based on wave theory, EFEM does not have any request for structural modal density, and it is more rigorous in describing the energy of system balance. Thus, the EFEM would overcome both the disadvantages of FEM and SEA in solving middle frequency problems and will have a great development in resolving the structure vibration and acoustic radiation problems [7].

In this paper, based on the transmission and equilibrium relationship of vibration energy in beam-like structures, the Galerkin weighted residual method was applied to equation discretization. An equivalent transformation of feedback element was suggested to develop the Energy Finite Element model of a composite piezoelectric cantilever beam driven by harmonic excitation on lateral direction, both systems with and without time delay being studied. And the power input estimation of harmonic excitation was discussed for the resolution of Energy Finite Element function. Then the energy density solutions of the piezoelectric coupling beam through Energy Finite Element Method (EFEM) and classical wave theory were compared to verify the EFEM model, which presented a good accordance. Further investigation was undertaken about the influence of control parameters including the feedback gain and arrangement of piezoelectric patches on characteristics of system energy density distribution.

2. Energy Finite Element Model of Piezoelectric Cantilever System

2.1. System Model

Composite piezoelectric cantilever beam is one of the most typical piezoelectric intelligent structures, which is presented in Figure 1(a). In the demonstrated system, series of piezoelectric material patches are grouped in couples and attached symmetrically to a host beam, one patch of each couple being used as an actuator and the other a sensor. The host beam is excited by a harmonically varying force with magnitude 𝐹₀ which would cause a transverse bending vibration. ℎ_𝑝 and ℎ_𝑏 represent the thickness of piezoelectric patch and cantilever beam, respectively.

(a)

(b)

Figure 1 

Piezoelectric cantilever system model and energy flow balance.

Figure 1(b) shows the vibration energy inflow and outflow of cubic element, in which 𝑞 and 𝜋_diss represent power flow through cubic element and the energy dissipated by damping, respectively, and 𝑄_in is the energy input from external. At each point of the structure, the instantaneous energy balance equation is given by [8] 𝜕𝑒𝜕𝑡=−𝜕𝑞𝜕𝑥−𝜋_diss+𝑄_in, (1) where 𝜕𝑒/𝜕𝑡 is the variation ratio of energy density to time.

The time-averaged value of 𝑞, 𝜋_diss, and 𝑒, that is, ⟨𝑞⟩, ⟨𝜋_diss⟩, and ⟨𝑒⟩, could be related as 𝑞=𝑐²_𝑔⋅𝑑⟨𝑒⟩𝜂𝜔,𝜋𝑑𝑥^diss=𝜂𝜔⟨𝑒⟩. (2)

By substituting expression (2) into (1), the energy balance equation is written as 𝑐²_𝑔⋅𝑑𝜂𝜔²⟨𝑒⟩𝑑𝑥²+𝜂𝜔⟨𝑒⟩=𝑄_in, (3) where 𝑐_𝑔=2(𝜔²𝐸𝐼/𝜌𝐴) is the group velocity of bending wave and 𝜂 is the damping loss factor.

2.2. Energy Density Control Equations without Feedback

When the feedback does not exist, the harmonic excitation is the only energy source in the system. Equation (3) could be discretized by Galerkin weighted residual method: 𝐾^(𝑛)𝑒^(𝑛)=𝑃^(𝑛)+𝑄^(𝑛),𝐾^(𝑛)=^{𝑥𝑅(𝑛)}_{𝑥𝐿(𝑛)}−𝑐²_𝑔𝜔𝜂𝑑𝐍^𝑇𝑑𝑥𝑑𝐍𝑑𝑥+𝜔𝜂𝐍^𝑇𝐍𝐏𝑑𝑥,_(𝑛)=𝐍𝑥_𝑅^(𝑛)𝑞_{𝑥𝑅(𝑛)}−𝐍𝑥_𝐿^(𝑛)𝑞_{𝑥𝐿(𝑛)},𝐐^(𝑛)=^𝑥𝑗_𝑥𝑖𝐍𝑄_in𝑑𝑥, (4)(5)(6) where the superscript (𝑛) denotes the 𝑛th discretizednodes, 𝐊^(𝑛) is a coefficient matrix related to stiffness and mass, 𝐏^(𝑛) is the power flow at corresponding node, 𝐐^(𝑛) is the external energy input, and 𝐍(𝑥) is the lagrange linear interpolation basic function.

For solving expression (4), coupling relationship among elements needs to be solved. Suppose the numbers of two adjacent elements are 𝑖 and 𝑖+1 and 𝜏^{(𝑖,𝑖+1)} and 𝛾^{(𝑖,𝑖+1)} represent transmission and reflection coefficient between coupling nodes: 𝑞⁻_𝑅^(𝑖)=𝛾^{(𝑖,𝑖+1)}𝑞⁺_𝑅^(𝑖)+𝜏^{(𝑖,𝑖+1)}𝑞⁻_𝐿^(𝑖+1),𝑞⁺_𝐿^(𝑖+1)=𝜏^{(𝑖,,𝑖+1)}𝑞⁺_𝑅^(𝑖)+𝛾^{(𝑖,𝑖+1)}𝑞⁻_𝐿^(𝑖+1). (7)

The node input power flow for each element is 𝑞_𝑥𝑅^(𝑖)=𝜏^{(𝑖,𝑖+1)}𝑞⁻_𝐿^(𝑖+1)−1−𝛾^{(𝑖,𝑖+1)}𝑞⁺_𝑅^(𝑖),𝑞_𝑥𝐿^(𝑖+1)=𝜏^{(𝑖,𝑖+1)}𝑞⁺_𝑅^(𝑖)−1−𝛾^{(𝑖,𝑖+1)}𝑞⁻_𝐿^(𝑖+1). (8)

Combine expressions (7) and (8): 𝑞_𝑥𝑅^(𝑖)𝑞_𝑥𝐿^(𝑖+1)=𝛼^{(𝑖,𝑖+1)}𝑐_𝑔^(𝑖)−𝑐_𝑔^(𝑖+1)−𝑐_𝑔^(𝑖)𝑐_𝑔^(𝑖+1)⟨𝑒⟩_𝑅^(𝑖)⟨𝑒⟩_𝐿^(𝑖+1), (9) where 𝛼^{(𝑖,𝑖+1)}=0.5𝜏^{(𝑖,𝑖+1)}/𝛾^{(𝑖,𝑖+1)}.

As an example, the piezoelectric beam is divided into four units as shown in Figure 2 according to the classification rule of energy finite element, in which unit 2 is a piezoelectric beam element, and others are base beam elements. Substitute expression (7) into (9):𝑘⁽¹⁾₁₁𝑘⁽¹⁾₁₂𝑘00000₍₁₎₂₁𝑘⁽¹⁾₂₂−𝛼^(1,2)𝑐_𝑔⁽¹⁾𝛼^(1,2)𝑐_𝑔⁽²⁾00000𝛼^(1,2)𝑐_𝑔⁽¹⁾𝑘⁽²⁾₁₁−𝛼^(1,2)𝑐_𝑔⁽²⁾𝑘⁽²⁾₁₂00000𝑘⁽²⁾₂₁𝑘⁽²⁾₂₂−𝛼^(2,3)𝑐_𝑔⁽²⁾𝛼^(2,3)𝑐_𝑔⁽³⁾00000𝛼^(2,3)𝑐_𝑔⁽²⁾𝑘⁽³⁾₁₁−𝛼^(2,3)𝑐_𝑔⁽³⁾𝑘⁽³⁾₁₂00000𝑘⁽³⁾₂₁𝑘⁽³⁾₂₂+𝑘⁽⁴⁾₁₁𝑘⁽⁴⁾₁₂00000𝑘⁽⁴⁾₂₁𝑘⁽⁴⁾₂₂𝑒₁𝑒₂𝑒₃𝑒₄𝑒₅𝑒₆𝑒₇=000000𝑄_in. (10)

Figure 2 

Element Finite Element model of cantilever beam system.

From expression (10), if the material parameters of the base beam and piezoelectric beam elements are known, the energy density of each node could be obtained after measuring or calculating the input power of system.

2.3. Transformation of Feedback Element

2.3.1. Ideal System without Time Delay

Equation (10) is deduced without consideration of the feedback element performed by the piezoelectric actuator and sensor. In the negative feedback control model, the piezoelectric actuator would exert a resistance moment to suppress the vibration of host beam according to the vibration response signal picked up by the piezoelectric sensor. In this situation, the active control moment would become a second energy source besides the harmonic exciting force. However, assuming an ideal system in which the transmission of feedback signal and the rapid reaction of the actuator were all instant, that is, there were not any time delay in the feedback control element; an equivalent transformation could be adopted to match the effect of feedback control with some simple modification of system parameters.

In most conditions, the attached piezoelectric patches should not be very long in size, and the electric charge 𝑄_𝑎(𝑥,𝑡) generated by the piezoelectric sensor under a bending deformation of the host beam could be expressed as 𝑄_𝑎(𝑡)=_𝑥^𝑥+𝑙𝑏𝑑₃₁𝐸_𝑝𝜀₁=1𝑑𝑥2𝑑₃₁𝐸_𝑝ℎ_𝑏𝑏𝑤_2,𝑥(𝑥+𝑙,𝑡)−𝑤_2,𝑥(≐1𝑥,𝑡)2𝑑₃₁𝐸_𝑝ℎ_𝑏𝑏𝑙𝑤_2,𝑥𝑥(𝑡), (11) where 𝑏, 𝑑₃₁,𝜀₁, and 𝐸_𝑝 represent width, piezoelectric strain constant, and elasticity modulus of piezoelectric element, 𝑥 and 𝑥+𝑙 are coordinates of the piezoelectric patch’s two ends, 𝑤_2,𝑥𝑥 is the curvature of the composite beam.

For a proportional feedback control, the bending moment 𝑚_𝑎(𝑡) supplied by the piezoelectric actuator is decided by 𝑚_𝑎(𝑡)=𝐾_𝑚𝑈_𝑎(𝑡)=𝐾_𝑚𝐶_𝑝𝑄_𝑎(𝑡)=𝐾𝑤_2,𝑥𝑥(𝑡), (12) where 𝐾_𝑚 is a transfer coefficient, 𝐶_𝑝 is the capacitance of the piezoelectric sensor, and 𝐾=0.5𝑑₃₁𝐸_𝑝ℎ_𝑏𝑏𝑙𝐾_𝑚/𝐶_𝑝 is total feedback gain.

Taking into account the dynamic balance of moments, when a feedback control went into action, the internal bending moment 𝑀 in the composite beam would be 𝑀𝑗𝜔=𝑀_𝑎𝑗𝜔+𝐸𝐼1+𝑗𝜂𝑊_2,𝑥𝑥=𝑊𝑗𝜔𝐾+𝐸𝐼1+𝑗𝜂_2,𝑥𝑥𝑗𝜔. (13)

The above equation is expressed in frequency domain, where 𝑀_𝑎 and 𝑊_2,𝑥𝑥 are the Fourier transforms of 𝑚_𝑎 and 𝑤_2,𝑥𝑥, respectively.

The revelation of (13) is that an attendance of negative feedback control would be equivalent to an increase of the flexural rigidity of the composite beam by 𝐾.

2.3.2. Discussion of Systems with Time Delay

Time delay, which is caused by the transmission and processing of feedback signals and reaction inertia of actuators, is inevitable in all kinds of active control systems. Time delay would result in complicated dynamic characteristics. The following is a qualitative analysis of the influence of time delay to the modeling method investigated here.

When a time delay is taken into consideration, the expression (12) should be written in the following description: 𝑚_𝑎(𝑡)=𝐾𝑤_2,𝑥𝑥(𝑡−𝜏). (14)

Assuming the system is in a harmonic vibration style, making a Fourier transform to the above equation would result in 𝑀_𝑎𝑗𝜔=𝐾𝑒^{−𝑗𝜔𝜏}𝑊_2,𝑥𝑥(𝜔)=𝐾𝑊_2,𝑥𝑥(𝜔)cos(𝜔𝜏)−𝑗sin(𝜔𝜏). (15)

Substitute the above equation into (13), 𝑀=(()𝑗𝜔[𝐸𝐼+𝐾cos𝜔𝜏)]+𝐸𝐼⋅𝑗𝜂−𝐾sin𝜔𝜏𝐸𝐼×𝑊_2,𝑥𝑥𝑗𝜔. (16)

Comparing (16) with (13), one could conclude that(1)if 𝜔𝜏≪1, that means cos(𝜔𝜏)≈1 and sin(𝜔𝜏)≈0, (16) and (13) are actually the same and time delay could be neglected;(2)if 𝜔𝜏 is great enough to be counted, a feedback control accompanying time delay would be equal to a change of flexural rigidity by 𝐾cos(𝜔𝜏), and a change of damping loss factor by −𝐾sin(𝜔𝜏)/𝐸𝐼;(3)from the viewpoint of rigidity improvement, time delay would not be expected. The feedback control would more or less produce a positive effect when 0<𝜔𝜏<𝜋/2. But if 𝜔𝜏 is too great to exceed 𝜋/2, in the range of 𝜋/2<𝜔𝜏<3𝜋/2, the feedback control would produce a worsen effect than it comes into action before. And this might lead to a negative rigidity if the feedback gain 𝐾 was great enough, which would be the worst situation that the system would become unstable;(4)if there was any possibility that one might benefit from a time delay, the benefit might lie in an equivalent increase of damping loss factor when 𝜋<𝜔𝜏<2𝜋, which has been presented in the second part of (16), and this might be helpful to vibration damping. But there would also be a risk that negative damping might appear if improperly controlled, and the system stability would be destroyed.

2.4. Input Power of the Piezoelectric Cantilever Beam Using Two Different Calculation Methods

In order to solve expression (10), the input energy 𝑄_in should be determined at first. 𝑄_in here denotes the energy intensity input by the external excitation to the system and is specifically defined as the time-averaged power input in a vibration period.

For steady harmonic vibration, 𝑄_in can be expressed as 𝑄_in(𝜔)=𝜔2𝜋₀^2𝜋/𝜔(𝐹)(𝑉)=1Re⋅Re𝑑𝑡2|𝐹𝑉|1||cos𝜑=2|𝐹|²1Re𝑍=12|𝑉|²Re(𝑍), (17) where 𝐹=𝐹(𝑡)𝑒^𝑗𝜔𝑡, 𝑉=𝑉(𝑡)𝑒^{𝑗𝜔(𝑡+𝜑)}, and 𝑍 is the impedance of the structure at the excitation point.

The estimation of 𝑄_in could be performed by modal method or infinite beam method.

2.4.1. Modal Method

For beam structures as shown in Figure 1, the displacement response function is 𝑤(𝑥,𝑡)=𝐹₀𝑒^{∞𝑗𝜔𝑡}_𝑟=1𝑌_𝑟𝑥₀𝑌_𝑟(𝑥)𝑀_𝑟𝜔²_𝑟1+𝑗𝜂−𝜔², (18) where 𝑥₀ are the coordinates of the exciting force, and the beam end and 𝑌_𝑟(𝑥) and 𝑀_𝑟 are the modal shape function and modal mass of the beam, respectively.

Thus, the input power could be calculated by 𝑄_in=12|𝐹|²1Re𝑍=12|𝐹|²Re𝑗𝜔^∞_𝑟=1𝑌_𝑟𝑥₀𝑌_𝑟(𝑥)𝑀₀𝜔²_𝑟1+𝑗𝜂−𝜔². (19)

2.4.2. Infinite Beam Method

Considering a bilateral infinite beam excited by a harmonic force at point 𝑥₀=0, the displacement response in frequency domain could be obtained according to the wave theory: 𝑊_𝑖(𝑥,𝜔)=𝐶_𝑖1𝑒^{−𝑗𝜆𝑖𝑥}+𝐶_𝑖2𝑒^{−𝜆𝑖𝑥}+𝐶_𝑖3𝑒^{𝑗𝜆𝑖𝑥}+𝐶_𝑖4𝑒^𝜆𝑖𝑥, (20) where 𝐶₁, 𝐶₂, 𝐶₃, and 𝐶₄ are determined by the shear equilibrium conditions as below: 𝐹₀2𝜕−𝐸𝐼³𝑤𝜕𝑥³_𝑥=0=𝐹₀2+𝐸𝐼𝐾³_𝐵−𝑗𝐶₃+𝐶₄=0(𝑥=0⁻),𝐹₀2𝜕+𝐸𝐼³𝑤𝜕𝑥³_𝑥=0=𝐹₀2+𝐸𝐼𝐾³_𝐵−𝑗𝐶₁+𝐶₂=0𝑥=0⁺,𝐾_𝐵𝑗𝐶₃+𝐶₄=0,−𝐾_𝐵𝑗𝐶₁+𝐶₂=0(𝑥=0). (21)(22)(23)

Solving the simultaneous equations (21)∼(23), 𝐶₁=𝑗𝐶₂=𝐶₃=𝑗𝐶₄𝐹=−𝑗⁰4𝐸𝐼𝐾³_𝐵. (24)

Thus, the input impedance of bilateral infinite beam at the excitation point is 𝑍_𝑠=𝐹₀𝑒^𝑗𝜔𝑡𝜕𝑤(0,𝑡)/𝜕𝑡=2𝜌𝐴𝐶_𝐵1+𝑗≈2.67𝜌𝐴𝐶_𝑙ℎ𝑓1+𝑗. (25)

Similarly, the input impedance of unilateral infinite beam could also be deduced: 𝑍_𝑑=𝐹₀𝑒^𝑗𝜔𝑡𝜕𝑤(0,𝑡)/𝜕𝑡=2𝜌𝐴𝐶_𝐵1+𝑗≈0.67𝜌𝐴𝐶_𝑙ℎ𝑓1+𝑗. (26)

The input power of infinite beam could then be acquired by (17).

Figure 3 shows the results of modal method and infinite beam method. It could be found that the latter is the average value of modal solution over modal density. Figure 4 presents the influence of the damping loss factor on the input power. Increasing the damping will reduce the size of each peak in the input power curve, but the sizes of other regions will increase. Figure 5 compares input power curves under different elastic modulus. It can be observed that reducing elastic modulus will lead to the movement of peaks to lower frequency region. This is because the beam’s vibration modals have been excited earlier after the drop of the stiffness. Reducing the beam density would bring about a similar result as that of increasing the elastic modulus, which is shown in Figure 6.

(a)

(b)

Figure 3 

The input power of modal method and infinite beam method.

Figure 4 

Influence of damping on the input power.

Figure 5 

Influence of stiffness on the input power.

Figure 6 

Influence of density on the input power.

3. Numerical Analysis

A concrete example of the system in Figure 2 is investigated. The physical parameters are listed in Table 1.

Table 1 

Parameters of piezoelectric cantilever beam.

The EFEM model would be verified through a comparison with the result from a wave theory which has been deduced in the appendix. As a first stage, Table 2 gives a comparison between natural frequencies of the composite piezoelectric cantilever beam calculated by wave theory and ANSYS, respectively. It could also be found that the natural frequencies are little higher than those of the host beam after the path of piezoelectric elements.

Table 2 

Natural frequency of the system before and after the patch of piezoelectric elements.

3.1. Comparison of EFEM and Exact Solutions by Wave Theory

Figure 7 gives a comparison between the energy density calculated by EFEM and the wave theory in the appendix. The average value of wave solution over the beam length, or in other words, EFEM solution is a time-averaged and space-averaged energy density. Figure 7 also indicates an excellent identity in the results of EFEM and wave method. In Figure 8 different feedback gain is applied in the wave method and the effect of feedback control is evaluated through the variation of vibration amplitude. A careful comparison of Figures 8 and 9(a) could further verify the coincidence of EFEM and wave theory.

Figure 7 

Comparison between solutions of EFEM and wave theory.

Figure 8 

Vibration amplitude under different feedback gain solved by wave theory.

(a) Comparison of different feedback gains

(b) Comparison of different piezoelectric patch positions

(c) Comparison of different piezoelectric patch lengths

(d) Comparison of different piezoelectric patch pairs

Figure 9 

Influence of variation of system parameters on energy density distribution by EFEM.

3.2. The Influence of Variation of System Parameter on the Distribution of Energy Density

Figure 9 is an investigation about the influence of variation of system parameters on distribution of system energy density along the piezoelectric cantilever beam by FFEM solution. As can be seen in Figure 9(a), increasing the feedback gain can reduce the energy density value of the beam. By comprehensive comparison of Figure 8 with Figure 9(a), one can still conclude that the energy density and vibration amplitude are equivalent in describing vibration level. However, the energy density, on account of having been space averaged, is more recapitulative in vibration evaluation. And the EFEM could be utilized to deal with more complex engineering structures than classical wave method.

Figure 9(b) shows the energy density distribution when the piezoelectric patch is located at different positions. It seems that the active control would be more effective if the piezoelectric patches were fixed near the clamped end of the cantilever beam. Figure 9(c) shows the alteration of energy density distribution when the length of piezoelectric element is altered, and it presents that a bit longer piezoelectric patch might be more beneficial to vibration suppression. Figure 9(d) compares the situation that two pairs of piezoelectric patches are attached discretely to the host beam with only one pair of piezoelectric patches being used. Obviously, lower energy density could be achieved in the former situation. And it could be realized that the beam segments having been patched with piezoelectric elements have lower energy density than other unpatched beam segments, and a more satisfied control effect would be expected if more pairs of piezoelectric patches were applied.

4. Summary and Conclusions

The research in this paper is focused on the modeling and analysis of active vibration control system of the composite piezoelectric beam based on Energy Finite Element Method. It should be pointed that environment noise, signal transmission error, and other uncontrollable factors might cause more or less departure of theoretical model from the practice, which is difficult to forecast now. However, through the theoretical and numerical analysis in this work, it could be proved that the EFEM is of application value in dealing with similar systems and it could be concluded that(1)the EFEM gives a time-averaged and space-averaged value of energy density at the conjunction node between discretized beam elements. It could be equally efficient as classical wave method or other mature method such as FEM and SEA in vibration evaluation, but the method would be more suitable for dealing with complex engineering structures associating broad frequency band;(2)in modeling the piezoelectric intelligent beam, the feedback gain could be equivalent to an extra increase of the flexural rigidity of the composite beam segment; that is, greater feedback gain would provide greater reduction of system vibration level, whereas the introduction of time delay would bring about complicated situations. In general, both rigidity modification and damping modification should be taken into consideration, and attention should be paid to potential problems on negative rigidity and damping;(3)numerical analysis indicates that proper design of system configuration parameters would be essential in order to achieve the best control efficiency, and how to find the optimal system parameters would be of significant value to be explored further. For the piezoelectric cantilever beam as studied in this paper, the control efficiency would be raised by arranging the piezoelectric patches near-clamped-end in position, a bit longer in size and multipled in number.

Appendix

Wave Solution of Energy Density of Piezoelectric Beam System

Referring to Figure 2, investigate a host beam symmetrically attached with a pair of piezoelectric patches. As a general discussion, supposing the excitation force is exerted at node 6, the whole beam would consequently fall into 4 segments, which are partitioned at nodes 2, 4, and 6, and are serially numbered as 1~4.

Let 𝑤_𝑖(𝑥,𝑡)(𝑖=1,2,3,4)   represent the lateral displacement of each beam segment, and 𝐸_𝑏𝐼_𝑏,  𝜌_𝑏𝐴_𝑏,𝐸_𝑝𝐼_𝑝 and 𝜌_𝑝𝐴_𝑝 represent the bending rigidity and mass density per unit length of the host beam and the piezoelectric patch.

For beam segments without piezoelectric patches, that is, segments 1, 3, and 4, we have the bending vibration equations: 𝜌_𝑏𝐴_𝑏𝑤_{𝑖,𝑡𝑡}+𝐸_𝑏𝐼_𝑏𝑤_{𝑖,𝑥𝑥𝑥𝑥}=0,(𝑖=1,3,4). (A.1)

For segment 2, in the feedback control mode, the governing equation is 𝜌_𝑏𝐴_𝑏+2𝜌_𝑝𝐴_𝑝𝑤_2,𝑡𝑡+𝐸_𝑏𝐼_𝑏+2𝐸_𝑝𝐼_𝑝𝑤_{2,𝑥𝑥𝑥𝑥}−𝑚_{𝑎,𝑥𝑥}=0. (A.2)

Letting 𝜆⁴₁=𝜆⁴₃=𝜆⁴₄=𝜔²𝜌_𝑏𝐴_𝑏/𝐸_𝑏𝐼_𝑏 and 𝜆⁴₂=𝜔²(𝜌_𝑏𝐴_𝑏+2𝜌_𝑝𝐴_𝑝)/(𝐸_𝑏𝐼_𝑏+2𝐸_𝑝𝐼_𝑝), after Fourier transformation, (A.1) and (A.2) are rewritten as 𝑊_{𝑖,𝑥𝑥𝑥𝑥}(𝑥,𝜔)+𝜆⁴_𝑖𝑊_𝑖(𝑥,𝜔)=0,(𝑖=1,2,3,4). (A.3)

The general expression of 𝑊_𝑖(𝑥,𝜔) would be 𝑊_𝑖(𝑥,𝜔)=𝐶_𝑖1𝑒^{−𝑗𝜆𝑖𝑥}+𝐶_𝑖2𝑒^{−𝑗𝜆𝑖𝑥}+𝐶_𝑖3𝑒^{−𝑗𝜆𝑖𝑥}+𝐶_𝑖4𝑒^{−𝑗𝜆𝑖𝑥},(𝑖=1,2,3,4). (A.4)

Substituting the boundary conditions of each beam segments into the above equation and defining 𝐶_𝑖⁺=𝐶_𝑖1,𝐶_𝑖2^𝑇,𝐶_𝑖⁺=𝐶_𝑖1,𝐶_𝑖2^𝑇,𝐶_𝑖⁻=𝐶_𝑖3,𝐶_𝑖4^𝑇,𝐞⁺_𝑖(𝑥)=𝑒^{−𝑗𝜆𝑖𝑥}00𝑒^{−𝜆𝑖𝑥},𝐞⁻_𝑖(𝑥)=𝑒^{𝑗𝜆𝑖𝑥}00𝑒^𝜆𝑖𝑥,𝝀⁺_𝑊𝑖=11−𝑗𝜆_𝑖−𝜆_𝑖,𝝀⁻_𝑊𝑖=11𝑗𝜆_𝑖𝜆_𝑖,𝝀⁺_𝐹𝑖=𝐸_𝑖𝐼_𝑖𝜆²_𝑖−11𝑗𝜆_𝑖−𝜆_𝑖,𝝀⁻_𝐹𝑖=𝐸_𝑖𝐼_𝑖𝜆²_𝑖−11−𝑗𝜆_𝑖𝜆_𝑖. (A.5)

The coefficients 𝐶_𝑖𝑗(𝑖,𝑗=1,2,3,4) could be resolved through the following equation: 𝐀₁₁𝐀₁₂𝐀𝐎𝐎𝐎𝐎𝐎𝐎₂₁𝐀₂₂𝐀₂₃𝐀₂₄𝐀𝐎𝐎𝐎𝐎₃₁𝐀₃₂𝐀₃₃𝐀₃₄𝐎𝐎𝐎𝐎𝐎𝐎𝐀₄₃𝐀₄₄𝐀₄₅𝐀₄₆𝐎𝐎𝐎𝐎𝐀₅₃𝐀₅₄𝐀₅₅𝐀₅₆𝐎𝐎𝐎𝐎𝐎𝐎𝐀₆₅𝐀₆₆𝐀₆₇𝐀₆₈𝐎𝐎𝐎𝐎𝐀₇₅𝐀₇₆𝐀₇₇𝐀₇₈𝐎𝐎𝐎𝐎𝐎𝐎𝐀₈₇𝐀₈₈𝐂⁺₁𝐂⁻₁𝐂⁺₂𝐂⁻₂𝐂⁺₃𝐂⁻₃𝐂⁺₄𝐂⁻₄=𝟎𝟎𝟎𝟎𝟎𝟎𝐅𝟎, (A.6) where 𝐎 is a 2 × 2 null matrix, 𝟎 a is 2 × 1 null vector, 𝐅=[^0𝐹₀]^𝑇, and 𝐀₁₁=𝝀⁺_𝑊1,𝐀₁₂=𝝀⁻_𝑊1,𝐀₂₁=𝝀⁺_𝑊1𝐞⁺₁𝑙₁,𝐀₂₂=𝝀⁻_𝑊1𝐞⁻₁𝑙₁𝐀₂₃=−𝝀⁺_𝑊2𝐞⁺₂𝑙₁,𝐀₂₄=−𝝀⁻_𝑊2𝐞⁻₂𝑙₁,𝐀₃₁=𝝀⁺_𝐹1𝐞⁺₁𝑙₁,𝐀₃₂=𝝀⁻_𝐹1𝐞⁻₁𝑙₁,𝐀₃₃=𝐊𝝀⁺_𝑊2𝐞⁺₂𝑙₁−𝝀⁺_𝑊2𝐞⁺₂𝑙₁+𝑙₂−𝝀⁺_𝐹2𝐞⁺₂𝑙₁,𝐀₃₄=𝐊𝝀⁻_𝑊2𝐞⁻₂𝑙₁−𝝀⁻_𝑊2𝐞⁻₂𝑙₁+𝑙₂−𝝀⁻_𝐹2𝐞⁻₂𝑙₁,𝐀₄₃=𝝀⁺_𝑊2𝐞⁺₂𝑙₁+𝑙₂,𝐀₄₄=𝝀⁻_𝑊2𝐞⁻₂𝑙₁+𝑙₂,𝐀₄₅=−𝝀⁺_𝑊3𝐞⁺₃𝑙₁+𝑙₂,𝐀₄₆=−𝝀⁻_𝑊3𝐞⁻₃𝑙₁+𝑙₂,𝐀₅₃=𝐊𝝀⁺_𝑊2𝐞⁺₂𝑙₁−𝝀⁺_𝑊2𝐞⁺₂𝑙₁+𝑙₂−𝝀⁺_𝐹2𝐞⁺₂𝑙₁+𝑙₂,𝐀₅₅=𝝀⁺_𝐹3𝐞⁺₃𝑙₁+𝑙₂,𝐀₅₄=𝐊𝝀⁻_𝑊2𝐞⁻₂𝑙₁−𝝀⁻_𝑊2𝐞⁻₂𝑙₁+𝑙₂−𝝀⁻_𝐹2𝐞⁻₂𝑙₁+𝑙₂,𝐀₅₆=𝝀⁻_𝐹3𝐞⁻₃𝑙₁+𝑙₂,𝐀₆₅=𝝀⁺_𝑊3𝐞⁺₃𝑙₁+𝑙₂+𝑙₃,𝐀₆₆=𝝀⁻_𝑊3𝐞⁻₃𝑙₁+𝑙₂+𝑙₃,𝐀₆₇=−𝝀⁺_𝑊4𝐞⁺₄𝑙₁+𝑙₂+𝑙₃,𝐀₆₈=−𝝀⁻_𝑊4𝐞⁻₄𝑙₁+𝑙₂+𝑙₃,𝐀₇₅=𝝀⁺_𝐹3𝐞⁺₃𝑙₁+𝑙₂+𝑙₃,𝐀₇₆=𝝀⁻_𝐹3𝐞⁻₃𝑙₁+𝑙₂+𝑙₃,𝐀₇₇=−𝝀⁺_𝐹4𝐞⁺₄𝑙₁+𝑙₂+𝑙₃,𝐀₇₈=−𝝀⁻_𝐹2𝐞⁻₄𝑙₁+𝑙₂+𝑙₃,𝐀₈₇=𝝀⁺_𝐹4𝐞⁺₄𝑙₁+𝑙₂+𝑙₃+𝑙₄,𝐀₈₈=𝝀⁻_𝐹4𝐞⁻₄𝑙₁+𝑙₂+𝑙₃+𝑙₄, (A.7) where 𝑙_𝑖(𝑖=1,2,3,4) is the length of the 𝑖th beam segment.

Finally, after every 𝐶_𝑖𝑗(𝑖,𝑗=1,2,3,4) is determined, the energy density of the 𝑖th beam segment is given by 𝑒_𝑖=𝜌_𝑖𝐴_𝑖𝜔²2×𝐶_𝑖1²𝑒^2𝑘2𝑥+𝐶_𝑖2²𝑒^{−2𝑘1𝑥}+𝐶_𝑖3²𝑒^{−2𝑘2𝑥}+𝐶_𝑖4²𝑒^2𝑘1𝑥+2Re𝐶_𝑖1𝐶^∗_𝑖3cos2𝑘₁𝑥+Im𝐶_𝑖1𝐶^∗_𝑖3sin2𝑘₁𝑥+2Re𝐶_𝑖2𝐶^∗_𝑖4cos2𝑘₂𝑥+Im𝐶_𝑖2𝐶^∗_𝑖4sin2𝑘₂𝑥, (A.8) where 𝑘₁ and 𝑘₂ represent the real and imaginary parts of the complex wavenumber of the corresponding beam segment.

The demonstrated method could be easily extended to systems with multiple piezoelectric patches.