To control vibration of a piezoelectric smart structure, a controller is usually designed based on a reduced order model (ROM) of the system. When such a ROM based controller operates in closed loop with the actual structure, spillover phenomenon occurs because the unmodeled dynamics, which are not included in ROM, will be excited. In this paper, a new approach aiming at investigating spillover effects in ANSYS software is presented. By using the ANSYS parametric design language (APDL), the ROM based controller is integrated into finite element model to provide an accurate representation of what will happen when the controller is connected to the real plant. Therefore, the issues of spillover effects can be addressed in the closed loop simulation. Numerical examples are presented for investigating spillover effects of a cantilever piezoelectric plate subjected to various types of loading. The importance of considering spillover effects in closed loop simulation of piezoelectric smart structures is demonstrated. Moreover, the present study may provide an efficient method especially beneficial for preliminary design of piezoelectric smart structure to evaluate the performance of candidate control laws in finite element environment considering spillover effects.
In the recent years, the active vibration control of piezoelectric smart structures has been intensively investigated due to their potential benefits in shape control and vibration suppression of lightweight flexible structures. The piezoelectric materials, such as lead zirconate titanate (PZT), have the property to generate electrical charge under mechanical load or deformation and the reverse; applying an electrical field to the material results in mechanical strains or stresses. Bonding or embedding piezoelectric patches in a structure can act as sensors to monitor or as actuators to control the response of the structure. It has been observed from the open literature that current analytical and numerical investigations focus on vibration control of thin components such as beams, plates, and shells with embedded or surface bonded piezoelectric patches which are used as sensors or actuators.
To control vibration of piezoelectric smart structures, both structural dynamics and control theory need to be considered. It is widely accepted that finite element (FE) method is an efficient tool for analyzing piezoelectric smart structures even with complicated boundary conditions. Comprehensive surveys on FE modeling of piezoelectric smart structures can be found in [
However, conventional controller is usually designed from a reduced order model (ROM) of the original system, whereas FE models inevitably have a large number of degrees of freedom (DOFs). FE models are appropriate for dynamic analysis but cannot be used directly as a basis for controller design. There is a gap between dynamic analysis and control design of a piezoelectric smart structure. To this end, some methods or combination of methods for creating a ROM must be employed to compensate the gap. The most widely used model reduction method to try on a large scale FE model is the mode superposition technique [
The present study is a continuation of the above mentioned works [
The constitutive equations of a deformable piezoelectric material in Voigt notation, coupling the elastic, and the electric fields can be expressed as
By applying element discretization and using the Hamilton’s principle, the global dynamic system equation for a piezoelectric smart structure is derived as
A rectangular aluminum cantilever plate which consists of a host aluminum plate (700 mm × 150 mm × 1.2 mm) and 3 pairs of PZT patches (70 mm × 15 mm × 0.5 mm), as shown in Figure
Material properties.
PZT  Host plate  


7600.0  2700 

63.0  69.0 

54.0  69.0 

24.6  27.0 

30.6  27.0 

0.3  0.32 

0.4  0.32 

−5.4, −5.4, 15.8, 12.3  / 

1.53, 1.53, 1.50  / 
Configuration of a cantilever plate with PZT S/A pairs.
FE model is appropriate for dynamic analysis but cannot be used directly as a basis for controller design. Considering that FE simulations are analogous to performing experimental investigations where the only direct outputs are the time histories, a socalled subspace identification method is employed to develop a state space ROM for the piezoelectric smart structure. In this way, the multipleinput multipleoutput state space model is identified, which is convenient for the following controller design procedure.
Suppose that the FE model of a piezoelectric smart structure can be represented by the following state space model:
The inputoutput relations of the linear control system described by (
The key idea of subspace identification method is to estimate the extended observability matrix
Band limited white noise signals in a frequency range varying from 0.5 Hz to 150 Hz, which cover 8 lowfrequency modes of the plate, are applied to the 3 PZT actuators as input data and the corresponding voltage output data of the 3 PZT sensors are obtained by performing open loop FE analysis using the software package ANSYS. Transient simulations are performed for 6 s with a time step of 0.0025 s. The subspace identification method is then used to create a state space ROM. The order of the state space model is determined upon inspecting the singular values of the matrix
Natural frequencies of the piezoelectric plate (Hz).
FEM  Model 1  Model 2  

Mode 1  2.178  2.180  2.182 
Mode 2  13.041  13.089  13.089 
Mode 3  20.392  20.427  35.795 
Mode 4  35.421  35.787  
Mode 5  61.081 
The distribution of singular values.
The LQR based optimal controller is designed for reducing the vibration of the piezoelectric plate. A state feedback is adopted to minimize the cost function such that the requisite design criteria are achieved. The cost function given in a quadratic form is defined as
The LQR control law expressed by (
In general, higher values for
In this section, we begin our discussion on implementing the LQR controller coupled with a Luenberger state estimator into ANSYS. The block diagram of the analysis is shown in Figure
Block diagram of the closed loop simulation in ANSYS.
To implement the controller into ANSYS, a macro that involves utilizing the commands for elementary matrix operations and basic program flow controls has to be developed by using APDL. Supposing that the ROM expressed by (
The spillover effects on the controller performance will be investigated through numerical simulations. For all the cases considered following, the plate is subjected to time dependent mechanical loads applied at the free end and the tip displacement responses are used to illustrate the spillover effects. For the purpose of comparison, ROM based controllers of order 8 and 6, respectively, are integrated into the same FE model to perform the closed loop simulations.
The plate is subjected to a harmonic load
Tip displacement PSD under a sinusoidal load of 2.2 Hz.
Next, a harmonic load
Tip displacement PSD under a sinusoidal load of 20.1 Hz.
A final investigation of spillover effects on the controller performance is conducted by applying a mechanical load of the form
Tip displacement PSD under a multisinusoidal load.
In this paper, a preliminary research is presented for investigation of the spillover effects on the vibration control of piezoelectric smart structure in FE environment, and therefore an efficient method especially beneficial for preliminary design of piezoelectric smart structure is provided to evaluate the performance of candidate control laws in FE environment considering spillover effects. The basic idea of this paper is that the FE model is more accurate than ROM for modeling of physical systems such that the unmodeled dynamics can be included in FE model. By using APDL, a ROM based controller is integrated into the ANSYS software package to provide an accurate representation of what will happen when the controller is connected to the real plant. In this way, the actuators pump energy into the FE model and unmodeled dynamics, which are not included in ROM, and can be excited so that the issues of spillover effects can be addressed in the closed loop FE environment. The ROM based controller is designed to tailor the output of the ROM modes to meet performance specifications; however, there is no prior guarantee that a ROM based controller satisfying the control requirements still works well in closed loop with the FE model because of spillover effects. The spillover effects are illustrated via the PSD of the tip displacement response. It is demonstrated that by using a higher order controller the spillover problem can be prevented. Future research calls for investigation of spillover induced instability in vibration control of smart structures using the proposed method.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research was supported by National Science Fund for Distinguished Young Scholars (Grant no. 11125209) and Natural Science Foundation of China (Grant nos. 11322215 and 10702039).