Low-frequency broadband multidirectional vibration isolation by piezoelectric smart platform with active control

Abstract

A high-precision equipment is very sensitive to vibration, even micro vibration. How to isolate the low-frequency broadband multidirectional vibration remains a challenge. As the main component of piezoelectric smart vibration isolation model, the piezoelectric stack consists of a piezoelectric actuator, a piezoelectric sensor, and a rubber layer. The numerical results obtained by the finite element method agree well with theoretical solutions. The vibration attenuates rapidly under displacement and velocity feedback control with negative gains. As the control gain decreases, the vibration isolation band is extended to lower frequency. A piezoelectric smart multidirectional vibration isolation platform model is further proposed by inclined installation of two piezoelectric stacks. A spring-like structure is designed to exert a preload pressure on these piezoelectric stacks. After optimization on the control gain, the platform can isolate vibration from 0 to 3000 Hz in multiple directions.

Keywords

Piezoelectric smart stack active control low-frequency broadband multidirectional vibration isolation

Introduction

The active smart system is an effective way to control structural vibration, which always consists of piezoelectric sensor and actuator. Vibration isolation performance of the system can be effectively regulated by the voltage of the piezoelectric actuator. Active control with piezoelectric sensor and actuator was proposed many years ago.¹ Extensive research has been conducted for three decades to suppress the vibrations in large space structure, civil structure, helicopter, aircraft, and computer hard disk driver.² The voltage relationship between sensor and actuator is determined by piezoelectric active control strategy, which usually plays a decisive role in vibration reduction. Therefore, many scholars have conducted in-depth research. He et al.³ controlled the vibration of functionally graded material plate by velocity feedback control strategy. Furthermore, many other structures, such as piezoelectric plate and shell,^4,5 were studied by this strategy. In addition, Moita et al.⁶ applied displacement feedback control strategy to control the vibration of the piezoelectric laminated sheet. The vibration of a flexible beam was controlled by angular velocity feedback strategy.⁷ Furthermore, multiple control strategies were usually applied simultaneously. Karagülle et al.⁸ studied displacement, velocity, and displacement integral feedback control strategies to control the vibration of cantilever beam by ANSYS. Liew et al.⁹ employed both velocity and displacement feedback control strategies to control the vibration of laminated composite plates. Moreover, active vibration control by multiple strategies was also investigated extensively in different types of functionally graded plates.^10–12 So far, velocity feedback control and displacement feedback control have become the basic means for active control.

It is also an effective vibration isolation method to adjust the frequency band gap by modifying the stiffness of periodic structures through piezoelectric active control.¹³ Li et al.¹⁴ adjusted the stiffness of periodic piezoelectric beam by active control, and isolated vibration propagation successfully. With the same method, the stiffness of periodic piezoelectric beam was modulated to widen its band gap ^15,16 Yi et al.¹⁷ designed an appropriate feedback law to control the bending stiffness of the beam, and then achieved tunable band gaps and non-reciprocal wave propagation.

To enhance the effect of active control, some optimizations were studied. In terms of structural optimization, the placement of sensors and actuators was optimized on carbon nanotube reinforced composite plates and flexible cantilever plate.^18,19 The topology optimization of flextensional piezoelectric actuators was carried on a cantilever model and a thin shell.^20,21 In terms of control strategies, active vibration control as well as an optimal time delay were developed and identified for a cantilever beam.²²

Active control can also be applied in many other aspects, such as acoustic cloaks,²³ optical mirrors,²⁴ aircraft wing,²⁵ and thin-walled wing engine system.²⁶ In recent years, the innovation of active control algorithm^27–31 and methods^32,33 and the improvement of control system stability have been research hotspots of active control.³⁴ Moreover, as an actuator, the piezoelectric stack was usually applied in vibration isolation. He et al.³⁵ suppressed the vibration of the high-speed bearing-rotor system by using piezoelectric stack.

However, effective isolation of vibration input from the ground has always been a difficult issue in precision manufacturing. Nelson³⁶ proposed an active control structure system by installing piezoelectric sensor as well as piezoelectric stacks under the equipment. Ryan³⁷ introduced the active control system into a high-precision optical platform, but the control law was not given and the mechanism was also not clear. The team at University of Twente carried out the integrated design on active control system.^38–41 Chen and Li⁴² designed a monolithic self-sensing precision stage to achieve vibration isolation. However, the underlying mechanism needs to be further explored beyond the experiments. It is necessary to conduct in-depth theoretical research to improve performance of active vibration control. In addition, one of the advantages of active control is the effective control of low-frequency vibration. But controlling vibrations at lower frequencies and wider frequency bands remains a challenge. Multidirectional vibration is a common phenomenon in engineering, and the control of multidirectional vibration is another challenge of active control.

Face to these two challenges, the paper aims to study low-frequency multidirectional vibration isolation. Firstly, the piezoelectric smart model is proposed by introducing the piezoelectric sensor and actuator. Different control laws of the active control are then implemented to adjust the active damping and stiffness. The vibration isolation performance under different control gains is studied in time domain and frequency domain, respectively. Finally, multidirectional vibration isolation platform is proposed to investigate vibration control in two directions.

Piezoelectric smart vibration isolation model and control laws

As shown in Figure 1, the proposed piezoelectric smart vibration isolation model consists of a piezoelectric stack and a high-precision equipment. The piezoelectric stack is composed of a piezoelectric actuator, a piezoelectric sensor, and a rubber layer. The high-precision equipment is simplified as a rigid body due to its large stiffness. As an insulation material, the rubber layer is introduced between the sensor and actuator. The resonance frequency is also reduced because of its low stiffness. As the sensor, a single piezoelectric patch is placed below the rigid body whose polarization direction is along the vertical direction. Upper and lower surfaces of the piezoelectric patch are electroded. The lower electrode is grounded. As the output voltage, potential difference between the upper and lower surfaces can be monitored by the voltage on the upper surface. The actuator is composed of N layers of piezoelectric patches, where N = 18. Half of the layers are polarized along the thickness direction, and the other half are polarized in the opposite direction. In order to produce the synchronous deformation, piezoelectric patches in two different polarization directions are arranged alternately.

Figure 1.

Piezoelectric smart vibration isolation model. (a) Structure and circuit connection. (b) Geometric dimensions and the boundary conditions.

The constitutive equation of piezoelectric materials can be written as

\begin{array}{l} T_{p} = c_{p q}^{E} S_{q} - e_{p i} E_{i} \\ D_{j} = e_{j p} S_{p} + ε_{j i}^{S} E_{i} \end{array}

(1)

where T_p and S_q represent the stress tensor and strain tensor, D_j and E_i denote electrical displacement vector and electric field vector, respectively.

c_{p q}^{E}

is stiffness constant under constant electric field, e_pi is piezoelectric constant, and

ε_{i j}^{S}

is dielectric constant under constant strain. These indices i, j = 1, 2, 3, and p, q = 1, 2, …, 6.

For one-dimensional thickness extension motion of a thin patch, equation (1) can be simplified as

\begin{array}{l} T_{3} = c_{33}^{E} S_{3} - e_{33} E_{3} \\ D_{3} = e_{33} S_{3} + ε_{33}^{S} E_{3} \end{array}

(2)

where D₃ is the electrical displacement component in the vertical direction, and T₃ and E₃ are the stress and electric field intensity components in the vertical direction, respectively.

For the piezoelectric sensor, the electric displacement is equal to 0 at the condition of open circuit. Hence, equation (2) becomes

\begin{array}{l} T_{3} = c_{33}^{E} S_{3} - e_{33} E_{3} \\ 0 = e_{33} S_{3} + ε_{33}^{S} E_{3} \end{array}

(3)

From equation (3), the electric field and voltage of the sensor is obtained as

E_{3} = - \frac{T_{3} e_{33}}{{\bar{c}}_{33} ε_{33}^{S}}

(4)

where

{\bar{c}}_{33} = c_{33}^{E} (1 + k_{33}^{2})

k_{33}^{2} = e_{33}^{2} / ε_{33}^{S} c_{33}^{E}

is electromechanical coupling coefficient. Then the sensor voltage is

V_{s} = - \frac{T_{3} e_{33} h_{s}}{{\bar{c}}_{33} ε_{33}^{S}}

(5)

where h_s is the thickness of the piezoelectric layer as the sensor. It can be known from equation (5) that the piezoelectric patch acts as a force sensor.

Displacement feedback control and velocity feedback control are basic control strategies.⁹ Base on these two control strategies, the relationships between voltages V_s and V_a of the sensor and the actuator can be written as

V_{a} = G_{d} V_{s}

(6)

V_{a} = G_{v} {\dot{V}}_{s}

(7)

where G_d and G_v denote displacement feedback control gain and velocity feedback control gain, respectively. After extracting the voltage of the sensor, the actuator voltage is determined by control strategy to achieve the feedback control. In practical engineering, these two control strategies can be realized by proportional and differential control circuits, respectively. From equations (5) and (6), the voltage of the actuator can be expressed as

V_{a} = - \frac{G_{d} T_{3} e_{33} h_{s}}{{\bar{c}}_{33} ε_{33}^{S}}

(8)

Similarly, the differential relation can be derived by equations (5) and (7).

V_{a} = - \frac{G_{v} {\dot{T}}_{3} e_{33} h_{s}}{{\bar{c}}_{33} ε_{33}^{S}}

(9)

As we can know, the driving force of the actuator is positively correlated with the actuator voltage V_a. Besides, T₃ is positively correlated with the acceleration of the rigid body. Thus, the relationship between the displacement of the rigid body and the driving force of the actuator can be regulated by G_d from equation (8). Similarly, the relationship between the velocity of the rigid body and the driving force of the actuator can be regulated by G_v from equation (9). Thus, the magnitude and direction of the actuator output force can be modulated by tuning G_d or G_v, which can be applied to generate damping or to enhance vibration.

In addition, the stiffness of the model and the vibration propagation characteristics can also be adjusted by displacement feedback control. Let the total deformation of the piezoelectric stack be Δx₃, and the strain of the piezoelectric stack can be assumed as

S_{3} = \frac{Δ x_{3}}{h_{a}}

(10)

where h_a denotes the thickness of the piezoelectric stack.

By ignoring the inertial force of the rubber, the stress T₃ on its upper and lower surfaces is equal. The displacement of the load is set as x₃. According to the constitutive relation of the rubber, the stress of the rubber is

T_{3} = \frac{(x_{3} - Δ x_{3}) E_{r}}{h_{r}}

(11)

where E_r and h_r denote Young’s modulus and thickness of the rubber, respectively.

Combine equations (5) and (6), the actuator voltage can be written as

V_{a} = - G_{d} \frac{e_{33} E_{r} h_{s} (x_{3} - Δ x_{3})}{{\bar{c}}_{33} ε_{33}^{S} h_{r}}

(12)

Thus, the electric field of the actuator is

E_{a} = - G_{d} \frac{n e_{33} E_{r} h_{s} (x_{3} - Δ x_{3})}{{\bar{c}}_{33} ε_{33}^{S} h_{a} h_{r}}

(13)

Combine equations (3), (10), (11), and (13), the stress of the actuator is

T_{3} = \frac{c_{33}^{E} E_{r} x_{3}}{E_{r} h_{a} + c_{33}^{E} h_{r} - n G_{d} h_{s} {\bar{k}}_{33}^{2} E_{r}}

(14)

where

{\bar{k}}_{33}^{2} = e_{33}^{2} / {\bar{c}}_{33} ε_{33}^{S}

. Then the output force, f_a, of the actuator is

\begin{array}{l} f_{a} = - K_{p} x_{3} \\ K_{p} = \frac{c_{33}^{E} E_{r} A}{E_{r} h_{a} + c_{33}^{E} h_{r} - n G_{d} h_{s} {\bar{k}}_{33}^{2} E_{r}} \end{array}

(15)

where A denotes the cross-sectional area of the actuator.

It can be seen from equation (15) that the stiffness can be modulated by controlling the gain G_d. For a positive G_d, the structure is hardened by the piezoelectric effects; while for a negative G_d, the structure is softened. Therefore, it can be further inferred from statics analysis that the stiffness and vibration propagation characteristic of the structure can be adjusted by displacement feedback control.

Piezoelectric smart vibration isolation model with active control

Time domain vibration attenuation analysis

The effect of active damping on structural vibration is investigated. Piezoelectric smart vibration isolation model and its geometric dimensions and boundary conditions are shown in Figure 1. The polarization directions are represented by arrows. The geometric parameters are listed in Table 1. For the rubber, the density is 1000 kg/m³, Young’s modulus is 1.4 MPa, and Poisson’s ratio is 0.49. The mass of the rigid body is 169 kg. PZT-5H is chosen as the piezoelectric material of the actuator and the sensor, and its material parameters are listed in Table 2.

Table 1.

Geometric parameters of the piezoelectric stack.

	Length (mm)	Height (mm)
Actuator (l, h_a)	650	27
Sensor (l, h_s)	650	3
Rubber (l, h_r)	650	3

Table 2.

Material parameters of PZT-5H.

Property	Symbol	Value
Stiffness constant (GPa)	$c_{11}^{E}, c_{33}^{E}$	126, 117
	$c_{12}^{E}, c_{13}^{E}$	79.5, 84.1
	$c_{44}^{E}, c_{66}^{E}$	23, 23.25
Piezoelectric constant (C/m²)	e₃₁, e₃₂	−6.5
	e ₃₃	23.3
	e₂₄, e₁₅	17
Dielectric permittivity (F/m)	$ε_{11}^{S}, ε_{33}^{S}$	1700ε₀, 1470ε₀
Density (kg/m³)	ρ _p	7500

The piezoelectric smart vibration isolation model is calculated by software ANSYS with APDL language. As shown in Figure 1(b), an acceleration excitation along x₃-axis acts on the bottom of the model, which is set as excitation boundary (EB). The top surface of the rigid body is set as observation boundary (OB), whose vibration along x₃-axis is measured. The presets of structural damping α = β = 1 × 10⁻⁷. The time step is set as 0.05 ms. The acceleration excitation versus time is shown in Figure 2(a), here the excitation lasts 2 ms. The displacement of the observation boundary (OB) versus time without active control is shown in Figure 2(a). The vibration of the rigid body and excitation are synchronous within excitation time. After the excitation, the structure vibrates at a frequency, 840.3 Hz. The frequency is close to the natural frequency, 836.5 Hz, of the first thickness stretch mode, which is also plotted in Figure 2(b). The vibration further decays slowly under the effect of structural damping.

Figure 2.

Excitation and response versus time in 10 ms. (a) Acceleration excitation and displacement response at observation boundary (OB). (b) Modal shape of the first thickness stretch mode with natural frequency, 836.5 Hz.

The vibration of the rigid body is further investigated under the displacement and velocity feedback control. Figure 3(a) illustrates that the vibration of the rigid body gradually decreases at a negative G_d, while the vibration increases at a positive gain as shown in Figure 3(b). Similar results are obtained under velocity feedback control as shown in Figure 3(c) and (d). The vibration of the rigid body gradually decreases for the negative G_v but increases for the positive one. For the negative G_v, the force acting on the rigid body by the actuator is opposite to the velocity direction; thus, vibration attenuates rapidly. However, for the positive G_v, the situation is reversed.

Figure 3.

Displacement time history of OB at different feedback control gains. (a) G_d = −100, (b) G_d = 50, (c) G_v = −0.004, and (d) G_v = 0.004.

To qualitatively represent the attenuation coefficient of each strategy, attenuation coefficient, n, is defined as⁴³

n = \frac{2}{T} \ln \frac{| A_{1} | + | A_{2} |}{| A_{2} | + | A_{3} |}

(16)

where T is the vibration period of the structure; A₁, A₂, and A₃ are three consecutive peaks in the vibration curve. The attenuation coefficients and driving voltage under different control conditions are shown in Table 3. The velocity feedback control has the highest attenuation coefficient n = 327.41. Moreover, its driving voltage amplitude is lower than that of displacement feedback control. Therefore, the damping effect of velocity feedback control is more obvious.

Table 3.

Attenuation coefficient, n, and actuator voltage amplitude under different control conditions.

	Uncontrolled	G_v = −0.004	G_d = −100
n	137.18	327.41	251.46
V_a (V)	—	20.32	34.38

Since the output force of velocity feedback control has the same property as the damping force, it is obvious that velocity feedback control has good damping effect. However, displacement feedback control cannot form effective damping, and furthermore its damping mechanism is still not clear. Therefore, it is necessary to analyze the relationship between actuator voltage and load displacement. Figure 4(a) and (b) show actuator voltage and velocity of rigid body versus time under velocity feedback control and displacement feedback control, respectively. A certain time lag between the sensor and the actuator is equal to the time step, where 0.05 ms is selected in calculation. However, in displacement feedback control, the voltage and displacement of displacement feedback control are not completely synchronized because of the hysteresis between the actuator and the sensor. But piezoelectric damping can still be formed in the driving force of the actuator during the hysteresis period, which results in vibration attenuation.

Figure 4.

Actuator voltage and velocity of rigid body versus time at different control gains. (a) G_v = −0.004. (b) G_d = −100.

Figure 5 shows the displacement time history of the rigid body within 30 ms when G_v = −0.004. It can be noted that the vibration of the structure is attenuated to an equilibrium position after 20 ms. Under the continuous action of active control, the structure remains in this equilibrium position. If the structure is excited by vibration again, the amplitude of the sensor voltage generated by deformation will be greater, and according to the control strategy, the amplitude of the actuator voltage will also be recovered. No manual intervention is required, and the vibration suppression performance remains as effective as before.

Figure 5.

Displacement time history of OB over a long period of time when G_v = −0.004.

Frequency domain vibration isolation analysis

One-dimensional laminate theoretical model is proposed by taking the vertical vibration of piezoelectric damping structure into consideration. The frequency response of the structure is calculated by transfer matrix method (TMM). The thickness of each layer of the actuator is set as 2h_p and the density is ρ_p. The thickness of elastic rubber layer is set as h_r = 2h_e, and the thickness of piezoelectric sensor as h_s = 4h_p. h_p = 0.75 mm, h_e = 1.5 mm, and total layer number of the actuator N = 18 are fixed in the calculation.

Only considering the vibration in the thickness direction, the displacement u₃ and potential φ of the piezoelectric layer can be written as

\begin{array}{l} u_{3} = u_{3} (x_{3}) e^{i ω t} \\ ϕ = ϕ (x_{3}) e^{i ω t} \end{array}

(17)

where i and ω denote imaginary unit and angular frequency, respectively. Then the strain and electric field are

S_{3} = u_{3,3}, E = - ϕ_{, 3} .

(18)

Submitting into equation (2) yields

\begin{array}{l} T_{3} = c_{33}^{E} u_{3,3} + e_{33} ϕ_{, 3}, \\ D_{3} = e_{33} u_{3,3} - ε_{33}^{S} ϕ_{, 3} . \end{array}

(19)

The equation of motion and the charge equation require that

\begin{array}{l} T_{3,3} = ρ_{p} {\ddot{u}}_{3} \\ D_{3,3} = 0 \end{array}

(20)

Equations (19) and (20) yield

\begin{array}{l} c_{33}^{E} u_{3,33} + e_{33} ϕ_{, 33} = - ρ_{p} ω^{2} u_{3} \\ e_{33} u_{3,33} - ε_{33}^{S} ϕ_{, 33} = 0 \end{array}

(21)

where the time-harmonic factor has been dropped. Equation (21) can be integrated to yield

ϕ = \frac{e_{33}}{ε_{33}^{S}} u_{3} + B_{1} x_{3} + B_{2},

(22)

where B₁ and B₂ are integral constants. Substituting equation (22) into equation (19), we obtain

T_{33} = {\bar{c}}_{33} u_{3,3} + B_{1} e_{33}

(23)

D_{3} = - B_{1} ε_{33}^{S}

(24)

By substituting equation (22) into equation (21), the equation of motion can be simplified as

{\bar{c}}_{33} u_{3,33} = - ρ_{p} ω^{2} u_{3}

(25)

The general solution of displacement and voltage can be obtained as

\begin{array}{l} u_{3} = A_{1} \sin (α x_{3}) + A_{2} \cos (α x_{3}) \\ ϕ = \frac{e_{33}}{ε_{33}^{S}} [A_{1} \sin (α x_{3}) + A_{2} \cos (α x_{3})] + B_{1} x_{3} + B_{2} \end{array}

(26)

where

α^{2} = ρ_{p} ω^{2} / {\bar{c}}_{33}

. A₁ and A₂ are integral constants. Then the stress can be written as

T_{3} = {\bar{c}}_{33} [A_{1} α \cos (α x_{3}) - A_{2} α \sin (α x_{3})] + B_{1} e_{33}

(27)

For the piezoelectric sensor, current and electric displacement are 0 because of its open circuit condition. So the displacement, potential, and stress of the sensor can be simplified as

\begin{array}{l} u_{3} = A_{1} \sin (α x_{3}) + A_{2} \cos (α x_{3}) \\ ϕ = \frac{e_{33}}{ε_{33}^{S}} [A_{1} \sin (α x_{3}) + A_{2} \cos (α x_{3})] + B_{2} \\ T_{3} = {\bar{c}}_{33} [A_{1} α \cos (α x_{3}) - A_{2} α \sin (α x_{3})] \end{array}

(28)

For the elastic rubber layer, the constitutive equation and equilibrium equation are

T_{3} = (λ + 2 G) u_{3,3}

(29)

where

λ = E_{r} ν / [(1 + ν) (1 - 2 ν)]

and

G = E_{r} / [2 (1 + ν)]

T_{3,3} = ρ_{r} {\ddot{u}}_{3}

(30)

where ρ_r represents density of the rubber. By combining the above three equations, the governing equation of the elastic layer can be written as

(λ + 2 G) u_{3,33} = - ρ_{r} ω^{2} u_{3}

(31)

Its general solution is

\begin{array}{l} u_{3} = C_{1} \sin (β x_{3}) + C_{2} \cos (β x_{3}) \\ T_{33} = C_{1} (λ + 2 G) β \cos (β x_{3}) - C_{2} (λ + 2 G) β \sin (β x_{3}) \end{array}

(32)

where

β^{2} = ρ_{r} ω^{2} / (λ + 2 G)

. C₁ and C₂ are undetermined coefficients.

According to equation (28), we get

ϕ = \frac{e_{33}}{ε_{33}^{S}} [A_{1 s} \sin (α x_{3}) + A_{2 s} \cos (α x_{3})] + B_{2 s}

(33)

where A_1s, A_2s, and B_2s are the undetermined coefficients of the sensor. As the output voltage of the sensor, the voltage difference between its upper and lower surfaces is

V_{s} = ϕ (2 h_{p}) - ϕ (- 2 h_{p}) = 2 \frac{e_{33}}{ε_{33}^{S}} A_{1 s} \sin (2 α h_{p})

(34)

According to equation (6), the actuator voltage is

V_{a} = 2 G_{d} \frac{e_{33}}{ε_{33}^{S}} A_{1 s} \sin (2 α h_{p})

(35)

According to equation (26), the electric potential of the actuator is

ϕ = \frac{e_{33}}{ε_{33}^{S}} [A_{1 a} \sin (α x_{3}) + A_{2 a} \cos (α x_{3})] + B_{1 a} x_{3} + B_{2 a}

(36)

The electric potentials of the upper and lower surfaces of the odd-layer actuator are

\begin{array}{l} ϕ (- h_{p}) = 0 \\ ϕ (h_{p}) = V_{a} \end{array}

(37)

Combining equations (36) and (37), B_1a can be expressed as

B_{1 a} = \frac{1}{2 h_{p}} [V_{a} - 2 A_{1 a} \frac{e_{33}}{ε_{33}} \sin (α h_{p})]

(38)

Similarly, for even-layer actuators

B_{1 a} = - \frac{1}{2 h_{p}} [V_{a} - 2 A_{1 a} \frac{e_{33}}{ε_{33}} \sin (α h_{p})]

(39)

After determining B_1a and V_a, only two coefficients are undetermined for each layer, A_1s and A_2s for sensing layer, A_1a and A_2a for actuator layer, and C₁ and C₂ for the rubber layer. These undetermined coefficients of the structure can be derived by the transfer matrix method.

At the interface between the rubber and the sensor, the continuity conditions of stress and displacement are

\begin{array}{l} u_{3 s} (- 2 h_{p}) = u_{3 r} (h_{e}) \\ T_{3 s} (- 2 h_{p}) = T_{3 r} (h_{e}) \end{array}

(40)

Then the transfer matrix between the rubber and the sensor can be written as follows

\begin{array}{l} [\begin{array}{c} \sin (- 2 α h_{p}) & \cos (- 2 α h_{p}) \\ {\bar{c}}_{33} α \cos (- 2 α h_{p}) & - {\bar{c}}_{33} α \sin (- 2 α h_{p}) \end{array}] [\begin{array}{c} A_{1 s} \\ A_{2 s} \end{array}] \\ = [\begin{array}{c} \sin (β h_{e}) & \cos (β h_{e}) \\ (λ + 2 G) β \cos (β h_{e}) & - (λ + 2 G) β \sin (β h_{e}) \end{array}] [\begin{array}{c} C_{1} \\ C_{2} \end{array}] \end{array}

(41)

Similarly, the continuity conditions at the interface between the rubber layer and the actuator are

\begin{array}{l} u_{3 r} (- h_{e}) = u_{3 a}^{(N)} (h_{p}) \\ T_{3 r} (- h_{e}) = T_{3 a}^{(N)} (h_{p}) \end{array}

(42)

where N is the total layer number of the actuator. Combining equations (35) and (39), we get

\begin{array}{l} [\begin{array}{c} \sin (- β h_{e}) & \cos (- β h_{e}) \\ (λ + 2 G) β \cos (- β h_{e}) & - (λ + 2 G) β \sin (- β h_{e}) \end{array}] [\begin{array}{c} C_{1} \\ C_{2} \end{array}] - [\begin{array}{c} 0 \\ \frac{G_{d} {\bar{c}}_{33} {\bar{k}}_{33}^{2}}{h_{p}} \sin (2 α h_{p}) \end{array}] \\ = [\begin{array}{c} \sin (α h_{p}) & \cos (α h_{p}) \\ {\bar{c}}_{33} α \cos (α h_{p}) - \frac{{\bar{c}}_{33} {\bar{k}}_{33}^{2}}{h_{p}} \sin (α h_{p}) & - {\bar{c}}_{33} α \sin (α h_{p}) \end{array}] [\begin{array}{c} A_{1 a}^{(N)} \\ A_{2 a}^{(N)} \end{array}] \end{array}

(43)

The continuity conditions at the interface between n-1 and n layer of the actuator are

\begin{array}{l} u_{3 a}^{(n)} (- h_{p}) = u_{3 a}^{(n - 1)} (h_{p}) \\ T_{3 a}^{(n)} (- h_{p}) = T_{3 a}^{(n - 1)} (h_{p}) \end{array}

(44)

Then the transfer matrix of the coefficients is

\begin{array}{l} [\begin{array}{c} - \sin (α h_{p}) & \cos (α h_{p}) \\ α \cos (α h_{p}) - \frac{{\bar{k}}_{33}^{2}}{h_{p}} \sin (α h_{p}) & α \sin (α h_{p}) \end{array}] [\begin{array}{c} A_{1 a}^{(n)} \\ A_{2 a}^{(n)} \end{array}] \\ = [\begin{array}{c} \sin (α h_{p}) & \cos (α h_{p}) \\ α \cos (α h_{p}) - \frac{{\bar{k}}_{33}^{2}}{h_{p}} \sin (α h_{p}) & - α \sin (α h_{p}) \end{array}] [\begin{array}{c} A_{1 a}^{(n - 1)} \\ A_{2 a}^{(n - 1)} \end{array}] \end{array}

(45)

For the rigid body, the equilibrium condition satisfies

m_{0} {\ddot{u}}_{3 s} (2 h_{p}) = - A T_{3 s} (2 h_{p})

(46)

where m₀ is the mass of the rigid body. A is the cross-sectional area of the laminates. Assuming the displacement input at the bottom is U_in, we have

u_{3 a}^{(1)} (- h_{p}) = U_{i n}

(47)

The frequency response relation of the structure can be written as

T_{r} {= 20 \log}_{10} | \frac{U_{o u t}}{U_{i n}} | {= 20 \log}_{10} | \frac{u_{3 s} (2 h_{p})}{u_{3 a}^{(1)} (- h_{p})} |

(48)

The effect of active control on vibration isolation is calculated by the finite element method (FEM) on software COMSOL Multiphysics. As illustrated in Figure 1, the bottom is excited by the vibration with displacement amplitude, U_in, along the x₃ axis. The output displacement amplitude, U_out, along the x₃ axis at the top of the model is then measured. Damping is introduced using complex elastic constants. c₃₃ is replaced by c₃₃ (1 + iQ⁻¹) in the theoretical calculation. For ceramics, Q is of the order of 100–1000. Q = 100 is used.^44,45

In order to verify these methods, harmonic response analyses on the model in Figure 1 are made by theoretical calculation and FEM. Figure 6(a) shows the frequency response curve obtained by TMM as theoretical calculation, and FEM as numerical calculation, where G_d = −1000 is fixed. Figure 6(b) shows the characteristic frequency versus control gain calculated by these two methods. The results agree well with each other, which verify the accuracy of the two calculation methods. Figure 7 shows the vibration displacement amplitude nephograms at 200 Hz without active control and with active control, where the control gain G_d = −2 × 10⁴. It can be seen from Figure 7(a) that the vibration amplitude of the rigid body is greater than that of the bottom. Hence, the vibration is enhanced when without active control. On the contrary, with active control, the vibration amplitude is weakened layer by layer along the piezoelectric actuator, as shown in Figure 7(b).

Figure 6.

Comparison of calculation results obtained from theoretical solution (TMM) and finite element calculation (FEM). (a) Frequency response curves when G_d = −1000. (b) The characteristic frequency versus control gain.

Figure 7.

Displacement distribution of the piezoelectric smart vibration isolation model under excitation of 200 Hz. (a) Without active control. (b)With active control and the control gain G_d = −2 × 10⁴.

Figure 8 shows the frequency response curves of the model under different displacement feedback control gains. It is worth noting that the model is softened by the negative G_d. It leads to a decrease of the natural frequency of stretch mode. Comparing to the free vibration without control, the frequency response curve shifts to the low-frequency band. As G_d decreases, the vibration isolation band is extended to lower frequency. On the contrary, the model is hardened by the positive G_d. The frequency corresponding to the peak increases with the increase of the gain, as shown in Figure 8(b). Therefore, from Figures 3 and 7, each control strategy is less sensitive to parameter errors but is more sensitive to the positive and negative signs of their gain coefficients.

Figure 8.

Frequency response curves of the displacement at different control gains. (a) G_d < 0; (b) G_d > 0.

Multidirectional vibration isolation platform

The tensile strength of piezoelectric ceramics is usually poor due to their easy fracture under tensile stress. When the control gain is large, the structure may become unsafe. To avoid the problem, a pressure is usually preloaded on the piezoelectric stack. Holterman³⁹ designed a piezoelectric vibration isolation platform by considering the strength of a piezoelectric actuator. Based on the idea, a piezoelectric smart multidirectional vibration isolation platform model is designed. As a symmetric structure, only the main view is displayed in Figure 9, where l₀ = 80 mm, h₀ = 30 mm, and the width of each gap is 0.5 mm. The model consists of a support platform and two piezoelectric stacks. Two cavities are reserved in the platform for mounting two piezoelectric stacks. The platform is carved with multiple gaps to form a spring-like structure. A preload pressure is exerted on the piezoelectric stacks by the spring-like structure. In addition, due to the inclined installation of the piezoelectric stacks, the movement of the platform in the horizontal and vertical directions can be controlled at the same time, and multidirectional vibration isolation is then achieved.

Figure 9.

Piezoelectric smart multidirectional vibration isolation platform.

The piezoelectric stack is adopted with similar structure as shown in Figure 1. The only difference is that the rubber is replaced by aluminum. The length l = 10 mm, the thickness of each piezoelectric layer is 1 mm, and the thickness of the aluminum layer is 1.5 mm. The piezoelectric material in the platform is still PZT-5H. For the aluminum layer, the density is 2700 kg/m³, the Young’s modulus is 79 GPa, and the Poisson’s ratio is 0.33.

Assuming a rigid body, a precision equipment with a mass of 24 kg is placed on a vibration isolation platform to evaluate its multidirectional vibration isolation performance. Displacement excitations in the horizontal and vertical directions are applied to the excitation boundary (EB) of the vibration isolation platform. The response of the rigid body in the horizontal and vertical directions is then measured. The harmonic response analysis of the model is operated by software COMSOL.

In order to display the multidirectional vibration isolation performance of the vibration isolation platform, Figure 10(a) and (b) are plotted to show the displacements of the platform along x₁ and x₃ axis at 1000 Hz without active control. The displacement of the OB is greater than that of the EB in either x₁ or x₃ direction. Therefore, vibration at 1000 Hz is not isolated by the platform without control. However, Figure 10(c) and (d) illustrate the vibration amplitudes of the OB in two directions are both smaller than those of the EB, where G_d = −100. Hence, the vibration is weakened by the platform with active control. The platform has excellent vibration isolation performance even when the vibration along x₁ and x₃ axis excites at the same time.

Figure 10.

Displacement of piezoelectric vibration isolation platform at 1000 Hz. (a) and (b) Without active control, displacements along x₁ axis and x₃ axis, respectively. (c) and (d) Active control with G_d = −100, displacements along x₁ axis and x₃ axis, respectively.

Figure 11 further demonstrates frequency response curves of the vibration along x₁ and x₃ axis, respectively. With the reduction of G_d, the frequency corresponding to the peak of the curve decreases, so the vibration isolation frequency band moves to the lower frequency. It is worth pointing out that when G_d = −30, vibration along x₁ axis is isolated in the range of 0–3000 Hz, but the vibration along x₃ axis is amplified. Nevertheless, when G_d = −100, the isolation vibration along both x₁ and x₃ axes can be achieved in the same frequency range. Accordingly, the piezoelectric vibration isolation platform can isolate the low-frequency and broadband vibration in two directions by tuning the control gain G_d.

Figure 11.

Frequency response curves of the piezoelectric vibration isolation platform. (a) Along x₁ axis. (b) Along x₃ axis.

Conclusions

The piezoelectric smart vibration isolation platform model with thickness of 30 mm was proposed, which can isolate low-frequency broadband multidirectional vibration. Dependence of vibration isolation performance upon control gain is studied based on displacement and velocity feedback control strategies of piezoelectric active control. Some conclusions are drawn as follows.

1. The vibration transmission can be adjusted by velocity and displacement feedback control strategies. A negative control gain weakens the vibration, while a positive control gain has opposite effect.

2. The stiffness of the structure can be modulated by displacement feedback control strategy. A negative control gain softens the structure. Hence, the lower frequency vibration can be isolated effectively.

3. The control gain of the vibration isolation platform has been optimized to isolate low-frequency and broadband vibration in multiple directions.

Footnotes

Acknowledgments

The computation is completed in the HPC Platform of Huazhong University of Science and Technology.

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 authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (NSFC) (11872186, 12232007, and 11972164), and Fundamental Research Funds for the Central Universities (HUST: 2016JCTD114).

ORCID iD

Hongping Hu

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