Reduction of structural vibrations with the piezoelectric stacks ring

Abstract

It is known that piezoelectric material shunted with external circuits can convert mechanical energy to electrical energy, which is so called piezoelectric shunt damping technology. In this paper, a piezoelectric stacks ring (PSR) is designed for vibration control of beams and rotor systems. A relative simple electromechanical model of an Euler Bernoulli beam supported by two piezoelectric stacks shunted with resonant RL circuits is established. The equation of motion of such simplified system has been derived using Hamilton’s principle. A more realistic FEA model is developed. The numerical analysis is carried out using COMSOL® and the simulation results show a significant reduction of vibration amplitude at the specific natural frequencies. Using finite element method, the influence of circuit parameters on lateral vibration control is discussed. A preliminary experiment of a prototype PSR verifies the PSR’s vibration reduction effect.

Keywords

Piezoelectric shunt damping technology vibration control piezoelectric stacks ring

1. Introduction

Up to now, the lateral vibration control of flexible structures and rotor systems is still attracting researchers’ attention very much. Based on the piezoelectric effect, piezoelectric material can convert mechanical energy into electrical energy. For this reason, piezoelectric materials have been widely used as damp structures or transducers for structural vibration control [1–4] or mechanical energy harvesting [5,6]. In 1991, N.W. Hagood [7] investigated dissipating vibration energy using piezoelectric material. This method is also called piezoelectric shunt damping technology. Because this technology has many advantages, such as high robustness and do not need external power, it has been well developed. Jia Long Cao [8] studied the vibration control effect of PZT elements shunted with passive R-L circuits. C.H. Park [9] validated that the piezoelectric shunt damping effect generate an additional damping augmented to a system. Heiko Atzrodt [10] demonstrated the vibration control effect of piezoelectric multilayer actuators shunted with several classical circuits. However, it seems do not have many related researches focused on lateral vibration control of beams with piezoelectric patches.

PZT patches attached on the beam usually increases the complexity and weight of the structure. In this paper, a new ring-shaped damper, called piezoelectric stacks ring, is designed. The ring is installed directly on the support of shafts or beams, and can be inserted between driveline components, e.g. gears, bearings, shafts, etc. For this reason, this ring can also be used to suppress the vibration of the rotor system. As shown in Fig. 1, according to the cross-section size of the beams or shafts, 4–12 piezoelectric stacks shunted with resonant RL circuits are installed in the same plane, in order to dissipate vibration energy transmitted from the shafts or beams to the support. A relative simple model which composed by an Euler–Bernoulli beam and two piezo stacks is established. Due to the central symmetry of the PSR, it is applicable to use such simple model to indicate the vibration reduction potential of the PSR. The equations of motion and associated boundary conditions of that model are derived using Hamilton’s principle. A FEA model is developed to discuss the influence of circuit parameters on lateral vibration control. Furthermore, an experiment of a prototype PSR is carried out to show its control performance.

Fig. 1.

Piezoelectric stacks rings containing different numbers of piezoelectric stacks.

2. Dynamic model of a beam clamped by two shunted piezo stacks

The piezoelectric stacks ring can be installed at the support of various structures, such as beams, shafts, rotor systems, etc. In this paper, an Euler–Bernoulli beam supported by two piezo stacks shunted with resonant RL circuits is analyzed. The equations of motion are derived using Hamilton’s principle.

Before starting the analytical process, some assumptions should be given.

In the following analysis, the gyroscopic effect and coriolis effect of rotor system is neglected, that is, the rotating shaft is static.

According to the characteristics of piezoelectric stacks, only the motion along the stack’s poling axis (the 3-axis) is described. Each piezoelectric stack is modeled using the linear constitutive equations of piezoelectricity.

Due to the symmetry of the structure, only the vertical harmonic excitation and deformation on x–y plane are considered.

Because of the PSR’s specific mechanical configuration, the piezoelectric stack can’t be pulled under tension. As shown in Fig. 2(a), each end of the beam is clamped by two piezoelectric stacks which are installed symmetrically in y direction. It can be found from Fig. 2(a) that the boundary condition of such beam is clamped-clamped. Because the piezo stack can only be compressed, the equilibrium digram of Fig. 2(a) can be simplified as shown in Fig. 2(b), where the equivalent stack supports can be subject to tension and pressure. Figure 2(b) shows an electromechanical model which composed by a simplified beam structure and two piezoelectric stacks shunted with resonant RL circuits.

Fig. 2.

Cross-sectional area of a beam (a) a beam with piezoelectric stacks supports and (b) simplified model.

Because the mass of piezoelectric stack is very small relative to the system, the stacks’ mass is neglected in following derivation. The kinetic energy due to tension and compression deformation of the piezoelectric stacks are ignored. The kinetic energy of the system is: $\begin{eqnarray}\displaystyle T=\frac{1}{2}\int _{0}^{l}{\rho}A{\dot{w}}^{2}dx & & \displaystyle\end{eqnarray}$ (1) Where 𝜌 and A are mass density and the cross-sectional area of the beam respectively; w (x, t) is the transverse displacement of the beam.

The potential energy of the system is: $\begin{eqnarray}\displaystyle U=\frac{1}{2}\int _{0}^{l}E_{b}Iw^{\prime \prime 2}dx+\frac{1}{2}\int _{V_{1}+V_{2}}(T_{3}S_{3}+DE)dV & & \displaystyle\end{eqnarray}$ (2) Where E_b is the elastic modulus of the beam, I is the diametrical moment of inertia of the beam about neutral axis; D and E are the electrical displacement and electrical field strength of PZT in “3” direction. V₁, V₂ are the volumes of the piezoelectric stacks. T₃ and S₃ is the stress and the strain of the piezoelectric stack.

The virtual work due to the distribution force and electrical displacement is: $\begin{eqnarray}\displaystyle W=\int _{0}^{l}p(x,t)dx\cdot w+\int _{V_{1}}EDdV+\int _{V_{2}}EDdV & & \displaystyle\end{eqnarray}$ (3) Where p (x, t) is the distributed harmonic excitation.

The piezoelectric constitutive equation for the ‘33’ mode piezoelectric stacks is given as $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}E\\ T_{3}\end{array}\right)=C_{0}\left(\begin{array}{@{}cc@{}}s^{E} & -d_{33}\\[3.0pt] -d_{33} & {\varepsilon}^{T}\end{array}\right)\left(\begin{array}{@{}c@{}}D\\ S_{3}\end{array}\right) & & \displaystyle\end{eqnarray}$ (4) Where C₀ = 1∕[s^Eϵ^T(1 − k²)] is a constant used to simplify the derivation; d₃₃ is the piezo constant of the PZT in “3-3” direction. s^E is the elastic compliance constant of PZT under constant electric field. ϵ^T is the permittivity constant of PZT under constant stress. k is the electromechanical coupling coefficient of PZT.

According to the Hamilton’s principle: $\begin{eqnarray}\displaystyle \int _{t_{1}}^{t_{2}}({\delta}T-{\delta}U+{\delta}W)dt=0. & & \displaystyle\end{eqnarray}$ (5)

Substituting Eqs (1)–(4) into Eq. (5) yields: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \int _{t_{1}}^{t_{2}}({\delta}T-{\delta}U+{\delta}W)dt\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\int _{t_{1}}^{t_{2}}\left[\int _{0}^{l}(-{\rho}Aw^{\prime \prime }-E_{b}Iw^{(4)}+p(x,t)){\delta}wdx-E_{b}Iw^{\prime \prime \prime }{\delta}w|_{0}^{l}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.\qquad -\left(\frac{k_{p}}{1-k^{2}}w(0,t)-\frac{{\theta}_{p}}{C_{p}^{s}}q_{1}\right){\delta}w(0,t)-\left(\frac{k_{p}}{1-k^{2}}w(l,t)-\frac{{\theta}_{p}}{C_{p}^{s}}q_{2}\right){\delta}w(l,t)\right]dt\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =0\end{eqnarray}$ (6)

Where q₁ and q₂ are the electrical charge of the piezoelectric stacks, k_p is the short circuit stiffness of the piezoelectric stack, θ_p = nd₃₃k_p is the equivalent piezoelectric coefficient, $C_{p}^{s}$ is the capacitance of the piezoelectric stack under constant strain.

The governing equation of the beam can thus be given as $\begin{eqnarray}\displaystyle {\rho}Aw^{\prime \prime }+E_{b}Iw^{(4)}=p(x,t) & & \displaystyle\end{eqnarray}$ (7)

With boundary conditions: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle E_{b}Iw^{\prime \prime \prime }(l,t)=\frac{k_{p}}{1-k^{2}}w(l,t)-\frac{{\theta}_{p}}{C_{p}^{s}}q_{2},\quad w^{\prime }(l,t)=0\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle E_{b}Iw^{\prime \prime \prime }(0,t)=\frac{-k_{p}}{1-k^{2}}w(0,t)+\frac{{\theta}_{p}}{C_{p}^{s}}q_{1},\quad w^{\prime }(0,t)=0.\end{eqnarray}$ (9)

From the definition of E in Eq. (4), the electrical displacement D is written as: $\begin{eqnarray}\displaystyle D={\varepsilon}^{T}(1-k^{2})E+\frac{d_{33}}{s^{E}}S. & & \displaystyle\end{eqnarray}$ (10)

Substituting D = 1∕(nA_p) into Eq. (10) we have: $\begin{eqnarray}\displaystyle \frac{q_{1}}{nA_{p}}={\varepsilon}^{T}(1-k^{2})\frac{nU}{l_{p}}+\frac{d_{33}}{s^{E}}\frac{w(0,t)}{l_{p}} & & \displaystyle\end{eqnarray}$ (11)

Where n is the number of layers of piezoelectric sheets in a piezoelectric stack, A_p, l_p and U are the cross-sectional area, length and voltage of the stacks respectively.

Simplified Eq. (11) and the following equation can be obtained: $\begin{eqnarray}\displaystyle q_{1}=C_{p}^{s}U_{1}+{\theta}_{p}w(0,t). & & \displaystyle\end{eqnarray}$ (12)

A series resonant RL shunt circuit is connected across the electrodes of piezoelectric stack as shown in Fig. 3. Based on the Kirchhoff’s current law, we have: $\begin{eqnarray}\displaystyle U_{1}=-(L\ddot{q}_{1}+R\dot{q}_{1}). & & \displaystyle\end{eqnarray}$ (13)

Substituting Eq. (12) into Eqs (8) and (9), the boundary conditions can be simplified as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L\ddot{q}_{1}+R\dot{q}_{1}+\frac{1}{C_{p}^{s}}q_{1}=\frac{{\theta}_{p}}{C_{p}^{s}}w(0,t)\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L\ddot{q}_{2}+R\dot{q}_{2}+\frac{1}{C_{p}^{s}}q_{2}=\frac{{\theta}_{p}}{C_{p}^{s}}w(l,t)\end{eqnarray}$ (15)

3. The vibration control results of the finite element analysis

In order to verify the vibration reduction performance of the PSR intuitively, based on the electromechanical model shown in Fig. 2(b), a FEA model is developed as shown in Fig. 4(a). A beam is supported by two piezoelectric stacks. Piezoelectric stacks only deforms vertically. The harmonic excitation in vertical direction is applied at the midpoint of the beam. The material of the beam is structural steel. The material of the stack is lead zirconate titanate (PZT-5A). The number of layers of piezoelectric sheets in a piezoelectric stack is 100.

Fig. 3.

Schematic of a series resonant RL shunt circuit.

To show the effect of vibration suppression, for simplicity, each piezoelectric stack is shunted with series resonant RL circuits. Different circuit parameters will have different influences on the effect of vibration suppression. The inductance value is ranging from 0 to 30 H and the resistance value is ranging from 0 to 10 kΩ. After dividing these two continuous interval into two discrete arrays, these two one-dimensional arrays can be used to construct a two-dimensional array. The finite element analysis is namely carried out using the circuit parameter coordinates which come from the two-dimensional array. Several sets of simulated data about the amplitude of force transmissibility (defined by input force versus output force) near first modal frequency were selected, as shown in Fig. 4(b).

Because this paper is interested in minimizing the reaction force, the minimization of force transmissibility is used as an objective function, where the resistance R and inductance L are the wait-optimize parameters. Comparing the results in different circuit parameter situations, the amplitude of force transmissibility is reduced most significantly about 84% in the case of R = 5 kΩ and L = 17.62 H. The result of this optimal parameter combination is shown in Fig. 4(c).

Fig. 4.

A beam with two piezoelectric stack supports (a) a FEA model and (b) the amplitudes of force transmissibility near the first modal frequency and (c) the simulated result with optimal circuit parameters.

Moreover, in order to further demonstrate the PSR’s control performance, a numerical simulation using COMSOL is also carried out. Where a finite element model of a simplified rotor system is developed. Assume that the rotor is not rotating, the harmonic excitation acting on the midpoint of shaft is considered, as shown in Fig. 5(a). To have an intuitive sense of the rotor’s first mode, the rotor’s first mode shape is shown in Fig. 5(b), where the cloud pattern represent the deformation in z direction.

Fig. 5.

A simplified rotor system with a piezoelectric stacks ring (a) the finite element model and (b) the first mode of nonrotating rotor.

The simulated results are shown in Fig. 6. It can be seen from Fig. 6(a) that when the piezo stack is shunted with resonant RL circuits, the two-peak phenomenon appears, which is caused by introducing the additional degree of freedom, i.e., the electrical charge. One can also find that when the resistance R and inductance L are 29 kΩ and 26.3 H, respectively, the rotor’s amplitude of force transmissibility is reduced maximally about 94%, as shown in Fig. 6(b).

Fig. 6.

The simulation results (a) the amplitudes of force transmissibility near the first modal frequency and (b) the simulated result with optimal circuit parameters.

4. Preliminary experiment of a prototype PSR

In this section, a preliminary experiment of a prototype PSR was conducted to verify the PSR’s vibration control performance. As shown in Fig. 7, the PSR was excited by a shaker in the vertical direction. The relevant parameters of the piezoelectric stack are shown in Table 1.

Fig. 7.

The experimental platform.

Table 1

Parameters of the piezoelectric stack

Geometric size/mm	C/μF	E/(N/m²)
3.5 × 4.5 × 10	0.15	4.4 × 10¹⁰
d₃₃ /(10⁻¹² C/N)	$s_{33}^{E}$ /(10⁻¹² m²/N)	𝜌/(g/cm³)
606	22.7	4.44

Each piezoelectric stack was connected with a series resonant RL circuit in which case the resistance value of 870 Ω and inductance value of 1.2 H was used. Input force and output force were measured by two force sensors and the excitation frequency of input force is 370 Hz (near the resonant frequencies of the system). The experiment was performed in two cases, that is, the piezo stack was shunted with and without the resonant RL circuit respectively. As shown in Fig. 8, the amplitude of the output force was significantly reduced after connecting the circuit. The result validates that the piezoelectric stacks ring shunted with the resonant RL circuit can significantly suppress the vibration near the resonant frequencies of the system.

Fig. 8.

Time domain experiment result.

5. Conclusion

In this work, the piezoelectric stacks ring used for vibration control of beams, shafts and rotor systems is designed, which may replace the traditional damper. The equations of motion of a simplified beam model clamped by two piezoelectric stacks has been derived using the Hamilton’s principle. The simulated and experimental results have shown that the proposed damper can significantly suppress the structure’s vibration at the specific natural frequency. When the piezoelectric stacks are shunted with the resonant RL circuit, the amplitude of force transmissibility is reduced greatly. The analytical and simulated results suggest that the piezoelectric stacks ring provides an effective means for vibration control.

Footnotes

Acknowledgements

This work is supported by the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant No. MCMS-I-0118G01).

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