In this paper, an attempt is made to determine the electric potential that would be generated in the piezoelectric vibration absorber using finite element piezoelectric analysis to determine optimal locations for damping of the first mode. Optimal placement of piezoelectric vibration absorber for passive vibration control application of a cantilever composite plate is investigated. Finite element piezoelectric modal analysis is performed. Models based on placing piezoelectric vibration absorbers at five different locations on the surface of the plate and incorporating piezoelectric properties are built. Modal analysis is used to find the electric potential developed in the piezoelectric vibration absorber. The location that yields the highest amount of electric potential would naturally be the best location for the vibration absorber. First bending mode of the cantilever composite plate is aimed for damping. Results of the analysis are verified with an experimental testing of the composite plate with piezoelectric vibration absorber firmly attached to the plate on the most effective location. A good agreement is found between the analytical and experimental results. Further, a resistive shunt circuit is designed for the passive damping of the first mode and attached to the vibration absorber in which the electric potential developed would be dissipated as heat to obtain passive vibration compensation. The experiment also demonstrates that a damping of 6 percent is obtained in the first mode of vibration and a great amount of damping is achieved in the second and third modes as well.
A piezoelectric vibration absorber is a patch of piezoelectric actuator placed on the surface of the plate to be damped, which is shunted with resistive, inductive, or both types of circuits designed to damp one or more modes. The mechanism of damping is based on the energy dissipation in the shunt circuit. The key factor in effective damping is the voltage developed in the piezoelectric vibration absorber as a result of energy conversion in the piezoelectric material. The higher the voltage, more is the damping amount of vibration control. Conventionally, the passive vibration absorbers are placed at the locations where the strain energy in the structure is at the highest value. These locations are determined by conducting stress analysis of the structure. However, several optimization methods have been proposed and many more are under study with the use of piezoelectric actuators and sensors in smart structures. Optimization of the locations of the piezoelectric sensors and actuators placement using generic algorithms has been proposed by Han and Lee [
In order to understand the vibration characteristics of the cantilever plate, modal analyses are performed with and without the piezoelectric vibration absorbers. The cantilever composite plate that was considered has a pair of PZT actuators near and parallel to the fixed edge on both sides and a sensor on one side.
Initially, a finite element model of the plate with these actuators and sensor was built and modal analysis was performed to obtain the natural frequencies, mode shapes and strain energy distributions for different modes. The mass and stiffness of the piezoelectric materials were also considered. However, the piezoelectric effect was not taken into account, as it is not required.
Then, five locations are chosen randomly for the placement of the passive piezoelectric vibration absorber (
A finite element model was built for each location considered and modal analysis was carried out to determine the natural frequencies, mode shapes, strain energy distributions and the electric potential developed in the vibration absorber. The first five natural frequencies of the plate with and without the passive vibration transducers are given in
Once the location for the piezoelectric transducer is finalized, the shunt circuit in which the energy would be dissipated is to be designed. The electrical energy produced in the piezoelectric vibration absorber is forced to flow as an electric current in the external circuit and be dissipated by joule heating through the shunt circuit. The elements of the shunt circuit can be of different kinds. It may consist of resistance only [
(a) Location # 1
(b) Location # 2
(c) Location # 3
(d) Location # 4
(e) Location # 5
in series [
At a non-dimensional frequency of
where is the electromechanical coupling coefficient of the piezoelectric, is the shunt resistance, is the piezoelectric capacitance across the electrodes when the piezoelectric vibration absorber is bonded to the structure and w is the frequency. The subscripts i and j represent the direction of electric current and the direction of strain, respectively. Thus by an appropriate choice of resistor, the peak of the loss factor curve can be manipulated and moved to the desired frequency.
The above equations are good when strain is in onedirection only. But, in the case of plates, we have twodimensional or planar strains. To incorporate this change we have to use a different set of equations for calculating the shunt resistance. Also, Equation (2) does not include any energy losses that may occur during the transfer of energy from one form to another. The more accurate method would be to use the generalized electromechanical coupling coefficient (K), which is defined as the square root of the ratio of electrical energy and mechanical energy. That is,
where w is the natural frequency of the structure without piezoelectric transducer and wE is the natural frequency of the structure with piezoelectric vibration absorber attached to it but shortened. kp is the planar electromechanical coupling coefficient of the piezoelectric given by the manufacturer. Equation (3) thus, incorporates the external factors in the form of change in frequencies and also the planar electromechanical coupling coefficient is used which is more appropriate for plates. The expression for shunt resistance, given in Wu and Bicos’s work [
where, is the piezoelectric capacitance across the electrodes when the piezoelectric vibration absorber is bonded to the structure, which is given as
where is the specified capacitance of the piezoelectric vibration absorber given by the manufacturer.
The manufacturer specified values of the constants are as follows,
.
The first five natural frequencies of the laminate composite plate, with and without piezoelectric vibration absorber are obtained and given in
.
The experimental setup for the vibration testing of the prototype plate, with actuators, sensors and piezoelectric passive piezoelectric vibration absorber attached to it at location # 4 (
gradually from 0 Hz to 500 Hz. The frequencies can be adjusted to three decimals to obtain precise values as shown in
Alternately, the modal response of the structure can be obtained by giving a sine wave input signal. The output signal from the sensor was captured with the help of dSPACE. The responses of the plate to a sweep sine input signal, without the shunt circuit attached to it, is shown in
The second step is to test the amount of damping obtained. This is obtained by analyzing the response of the structure to small tapping at its free end with and without the shunt circuit attached to the piezoelectric transducer. Data obtained from sensor is captured and the “psd” plot of frequencies with and without the vibration absorber is given in
The amount of electric potential developed in the passive vibration absorber is obtained by using it as a sensor and connecting it to the oscilloscope and by exciting the plate at different modes using the actuators.
tion absorber for Modes 2 and 4 are lower due to the fact that these modes are twisting modes and they cannot be excited well with the current location of actuators. Also the location of the vibration absorber is at a place where the strain energy in the plate is very low. However, the values of the electric potentials agree fairly well for the first, third and fifth modes which are the bending modes and could be excited well with the current location of the actuators. In this case, the location of the vibration absorber also contributes to the high voltage.
From the results obtained it can be concluded that piezoelectric modal analysis serve as a very good tool in determining the location of the passive vibration absorber without much effort. This method, compared to others, is more effective which enables us to predict the electric potential that would be generated in the vibration absorber on which the amount of damping depends directly. This method can also be used in optimizing not only the location, but also the size and shape of the passive vibration absorber to obtain maximum amount of damping. This can be achieved by simply changing the dimensions and shape of the piezoelectric vibration absorber in the finite element model on an iterative basis to find the configuration that gives maximum electric potential. Piezoelectric modal analysis can also be used in optimization of the location of the actuators and sensors for various applications to minimize the actuation effort and to maximize the sensing capability, respectively. In addition, considering the mass and stiffness of the piezoelectric material, analysis can be brought much closer to reality and hence the results predicted would be highly accurate.
This project was supported in part by OAI CCRP program. The authors gratefully acknowledged the agency and research funding awarded.