Design and experimental validation of an annular dielectric elastomer actuator for active vibration isolation

Mechanical Systems and Signal Processing 134 (2019) 106367

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Design and experimental validation of an annular dielectric elastomer actuator for active vibration isolation Yunhua Zhao a, Qiwei Guo b, Song Wu b, Guang Meng a, Wenming Zhang a,⇑ a b

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Key Laboratory of Space Structure Institutions, Shanghai 201108, China

a r t i c l e

i n f o

Article history: Received 13 March 2019 Received in revised form 28 July 2019 Accepted 10 September 2019 Available online 16 September 2019 Keywords: Dielectric elastomer Annular actuators Active vibration isolation Adaptive control

a b s t r a c t This paper proposes annular dielectric elastomer actuators (ADEAs) for active vibration isolation. A theoretical model is developed to characterize the actuator and parametrized based on experimental data. The parametric dependence of the natural frequency of the ADEA on the actuator geometry, applied voltage, and the pre-stretch of the elastomer is analyzed. The electromechanical behavior of the ADEA is also characterized experimentally and estimated by finite-impulse-response (FIR) based adaptive filter. Vibration control with the ADEA as the active vibration isolator is implemented by employing the filtered-x leastmean-square (Fx-LMS) algorithm due to incorporating the effect of the secondary path function. The vibration isolation ratios at the 7 and 11 Hz harmonics with the amplitudes of 320 lm and 330 lm are 10.53 and 11.81 dB, respectively. For the isolated mass under the 9 Hz sinusoidal disturbance with peak-to-peak amplitude of 1400 lm, vibration attenuation of 10.74 dB is achieved. The results show that ADEAs have the potential for active vibration isolation systems even with large amplitude disturbance. It provides guidelines for the actuator design and promote the dielectric elastomers (DEs) for engineering applications. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction High-precision equipment is increasingly emerged and the accuracy of analysis instruments has improved over the past decades, raising the demands for environments with a minimum of vibration [1,2]. Various isolation techniques have been employed and can be categorized into three types: passive [3,4], active [5,6] and semi-active [7–9] methods. In many cases, passive structural provisions cannot meet these demands especially in low frequency and active systems have to be considered. Compared with traditional actuators such as hydraulic and electro-magnetic ones to achieve vibration control, smart material based actuators are particularly attractive owing to satisfied accuracy and direct energy conversion, resulting in compact structures. Among them, piezoelectric and magnetostrictive based actuators have been extensively investigated for active vibration isolation [10,11]. Small stroke characteristic, however, makes them unsatisfied for the applications where large-scale disturbances exist. Dielectric elastomers (DEs), possessing large deformation and fast response [12,13], have great potential for active vibration isolation in certain applications. Despite their promising properties DEs have rarely been investigated for vibration isolation systems. Great attention has been put on artificial muscles [14], loudspeakers [15] and soft robotics [16]. Deformation of DEs can be configured in many ways to produce actuation. Common configurations for dielectric elastomer actuator (DEA) are diaphragm, roll, tube, stack, stretched over a frame and laminated on a flexible

⇑ Corresponding author. E-mail address: [email protected] (W. Zhang). https://doi.org/10.1016/j.ymssp.2019.106367 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

substrate [17]. Among them, tubular and stack actuators are suitable and most commonly investigated for active vibration isolation because of load bearing capacity [18–20]. DEs were firstly investigated for potential use in vibration isolation applications in a Technology Investment Fund Project entitled dielectric polymer actuators for active/passive vibration isolation. The motivation was to combine their inherent passive isolation properties with an active functionality. However, this work did not exceed the fabrication ad characterization of rolled actuators. Afterwards, others developed a core free tubular actuator with no pre-strain [18], converting the biaxial expansion into the linear motion along the axis of the roll. In the experiments a 250 g mass was isolated from base vibration, both harmonic and random narrowband excitations provided by an electrodynamic shaker. Similarly, Berardi used a tubular PolyPower actuator for active vibration isolation in the low frequency range [21]. A theoretical model was developed by considering the actuator as a multi-element device, and experimental results were given only in the time domain, demonstrating the effective isolation. In addition, stacked DEAs are exploited for active vibration control applications. William shows the implement of an isolation system using a DEA with perforated metal electrodes, which allows both highly dynamic performance and operation with high static loads [19]. The active vibration isolation was implemented using two different signal feedback concepts, both of which can lead to a similar broadband attenuation of the disturbance. The first idea is to combine a positive absolute velocity feedback with a positive absolute acceleration feedback. While the latter uses both absolute and differential feedback signals. Specifically, it combines the positive absolute velocity feedback with a differential displacement feedback. Moreover, in Li’s experiment a loading mass (100 g) is used to represent the structure to be isolated and an attenuation of 34 dB could be achieved for harmonic disturbance with 17 Hz [20]. Nevertheless, both tubular and stack DEAs, despite many potential applications, suffer from complicated fabrication techniques which are vital for the performance of DEAs. For roll-based tubular DEAs it mainly involves pre-straining a roll structure, electrode application onto large areas and potential slip between layers of the rolled techniques. Moreover, the electrode design has proven to be one of the most challenging aspects in the development of DEAs and the novelty has largely been in the choice of electrodes. For example, silicone elastomer in conjunction with compliant metal electrode technology has been employed to produce linear tubular actuators in large quantities [18]. Specifically, a corrugated microstructure pattern is imprinted in the elastomer and then a metal layer spluttered on the surface. For stack DEAs depositing DE layers and depositing electrode layers are usually needed, and various electrode materials have been investigated, ranging from conductive polymers to graphite powder and carbon nanotubes [22,23]. Common method of depositing electrode layers is spraying graphite powder. But it is difficult to control the thickness of electrode layers accurately, which can affect greatly characteristics of the actuators to a certain extent. Moreover, casting process for stack DEAs is developed [20]. There are some preparations for actuation material and electrode material before casting or depositing process. Complicated procedure and special equipment make the fabrication of stacked and tubular actuators timeconsuming and high cost. Annular dielectric elastomer actuators (ADEAs), by comparison, exhibit simple fabrication and large strain and have been extensively investigated The aim of this work is to demonstrate an active vibration isolation system based on an annular dielectric elastomer actuator (ADEA). Although many researchers have focused on modeling and positioning control of DEA systems [24,25], and improving the performance of the DEA by changing biasing mechanism [26,27], its potential for active vibration isolation has not been demonstrated. In Section 2, we develop an analytical model of the considered actuator which relates geometric parameters and material properties to actuator displacement and natural frequency. Quasi-static and dynamic characteristics of the ADEA are described, and the effect of pre-stretch, size and applied voltage on the natural frequency of the ADEA is also studied in Section 3. Next, we give the principle of finite-impulse-response (FIR) based feed-forward control in Section 4. An active vibration isolation test is set up, the system is identified and vibration isolation results are presented in Section 5. Finally, the conclusion of this study is summarized in Section 6. 2. Working principle and modeling The structure considered in this paper is based on the annular DE membrane. The states that the DE goes through during the assembly are shown in Fig. 1. Fig. 1(a) displays a DE membrane of thickness T 0 and the membrane is pre-stretched with thickness t 0 , as shown in Fig. 1(b). Once completed, carbon grease as compliant electrodes are screen printed all over the annular area (radial length l0 ) of the polymeric film. The outer frame and the inner circular plate (radius r) are made of acrylic plate (in light purple). A mass is connected to the moving part of the membrane, constituting the actuator’s load. When a voltage is applied to the electrodes, Maxwell stress is generated which compresses the material in the thickness direction, producing a radial expansion, and the subsequent actuation motion is depicted in Fig. 2. The actuation range and dynamic characteristics of the actuator can be tuned by choosing different pre-stretch, voltage and size. An analytical model of the DEA is developed to establish the parametric dependence of the actuator displacement and natural frequency with the input voltage. It provides the basis for dynamic characteristics analysis and optimal design. 2.1. Material strain and geometrical description In the pre-stretch process, it satisfies T 0 ¼ t0 k2p where kp represents the in-plane pre-stretch of DE membrane. For the circular geometry, the deformation state of the membrane can be described by the radial, circumferential and thickness

Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

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Fig. 1. Schematic of the ADEA in three states: (a) Origin DE film; (b) Pre-stretched state; (c) Deformed configuration.

Fig. 2. Actuation mechanism of the ADEA.

stretches, denoted as kl , kc and kt . In this work, the symbol ki;t represents the total stretch in the ith principal direction, and is equal to the product of the actuation stretch ki and the constant pre-stretch ki;p . So we have

kl;t ¼ kl;p kl ; kc;t ¼ kc;p kc ; kt;t ¼ kt;p kt

ð1Þ

Because of the incompressibility of the DE membrane, the following equality holds

kl;p kc;p kt;p ¼ kl kc kt ¼ kl;t kc;t kt;t ¼ 1

ð2Þ

Assuming that the membrane deformation follows a truncated cone-shaped geometry [28,29] shown in Fig. 2, the circumferential actuation stretch is constant and equal to one. Based on this assumption, it holds

l kl ¼ ¼ l0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 l0 þ d l0

; kc ¼ 1; kt ¼

t t0

ð3Þ

The volume of the DEA in the pre-stretched and deformed state satisfies

(

h i 2 V pre ¼ pt 0 ðr þ l0 Þ  r 2 V def ¼ pltð2r þ l0 Þ

:

ð4Þ

The relationship between variables d and t can be easily obtained by simple geometrical considerations:

t¼

t 0 l0 t 0 l0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi : l 2 2 l0 þ d0

ð5Þ

2.2. Dynamic equation In the analytical model, the voltage applied to the membrane U and the vertical displacement d are considered as the input and output, respectively. The vertical force equilibrium on the load mass can be expressed as follows:

€ þ Pðd; U; t Þsin/  mg ¼ 0; md

ð6Þ

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Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

where m is the total mass of the load, Pis the tensile force exerted by DE membrane, t is time, / is the deflection angle of the membrane, gis the gravitational acceleration. 2.3. Tensile force provided by DE Considering DE membrane coated with compliant electrodes as a parallel plate capacitor, the charge accumulating on the electrode can be calculated:

Q ¼e where

U T0

pl0 ðl0 þ 2rÞk2p k2l ;

ð7Þ

e is the permittivity of the DE film. kl is denoted as k and the variation of Q ¼ Q ðk; UÞ can be written as dQ ¼ e

1 U pl0 ðl0 þ 2rÞk2p k2 dU þ 2e pl0 ðl0 þ 2rÞk2p kdk: T0 T0

ð8Þ

When the charge on the electrodes varies by dQ , the applied voltage does work of UdQ . The total work done by damping   2 force is obtained as cl0 =2 ðdk=dtÞdk where c is viscous damping coefficient. For any arbitrary variation of the system, the variation in the free energy of the membrane should be equal to the work done jointly by the voltage, the mechanical load and damping force. So we have

1 2 dk V und dW ¼ UdQ þ Pl0 dk  cl0 dk: 2 dk

ð9Þ

The free energy density function W can be obtained as the sum of deformation and electric charge energies. For calculate the elastic energy density, the Gent model [30] is employed in this work to account for the behavior of limiting stretch of the DE membrane. Thus

W ¼

l 2

J m ln 1 

! k2l;tot þ k2c;tot þ k2t;tot  3 D2 þ Jm 2e

ð10Þ

where l is shear modulus, J m is a material constant related to the limiting stretch. The quantity D appearing in the second energetic term is the electrical displacement, namely the charge density over the electrodes surfaces assuming an ideal behavior of the DEA. Combining Eqs. (8) (9) and (10), we have

e

U2 t 20

k4p k þ

2 3 k2p k  k4 Pl0  12 cl0 dk p k dt : ¼l 2 2 2 4 2 k k þkp þkp k 3 pl0 t0 ð2r þ l0 Þ 1 p

ð11Þ

Jm

Substituting Eq. (11) into (6) we obtain the dynamic equation to accurately characterize the behavior of the actuator. 3. Actuator characteristics In this section both static behavior and natural frequency of the ADEA are investigated. The static behavior provides insight into the actuation range of the devices while the dynamic characteristics provide insight in the operation speed as well as providing an indication of the working frequency range(s) where active vibration isolation might be successful. 3.1. Static behavior The DEA may reach a state of equilibrium under the application of static voltage. The stretch ratio can be derived from the equation of motion by neglecting the inertia and damping effect. The relationship between stretch and applied voltage becomes 3 k2p k  k4 eU2 4 mg 1 p k k k þ : ¼ 2 p k 3 lT 20 p sin/ lpt0 ð2r þ l0 Þ 1  k2p k2 þk2p þk4

ð12Þ

Jm

Theoretically the model is capable of predicting actuation strains with reasonable accuracy. In practice, however, maximum stretch of the actuator is limited by the maximal allowed voltage corresponding to electrical breakdown strength. The actuator parameters used in this study are listed in Table 1. The experimental displacement-voltage data and the simulation results are compared in Fig. 3. They are in good agreement and both show that the actuator displacement increases progressively with higher voltage. Also, it is noted that the actuation range of the ADEA is almost one magnitude larger than magnetostrictive actuators or piezoelectric actuators with comparable size [10]. Thus the DE can be an effective supplement to the smart material family and annular DE sheet is an alternative geometry in applications especially requiring long range.

Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

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Table 1 Material and geometric parameters of ADEAs. Symbol

Description

Value

Unit

r l0 T0 kp m

Radius Radial length Thickness Pre-stretch Mass Shear modulus Limiting stretch Permittivity

10 20 1.0 3.0 50 7.6 100

mm mm mm Dimensionless g kPa Dimensionless F/m

l Jm

e

1:8  1011

Fig. 3. Analytical and experimental voltage-displacement relation of ADEAs.

The discrepancy between experimental and theoretical results is mainly brought by the employed material model without considering the effect of viscoelastic behavior. 3.2. Natural frequency The natural frequency of the actuator is a significant indicator especially when applied in vibration control. Analyzing the effect of various factors on the natural frequency can provide guidance on designing the actuators. Rewrite governing equation of the ADEA as

€ þ c d_ þ hðd; UÞ sin /  g ¼ 0; d m

ð13Þ

where

hðd; UÞ ¼ l

3 k2p k  k4 p k

1

2 k2p k2 þk2p þk4 p k 3 Jm

pt0 ð2r þ l0 Þ  e

U2 t0

kpð2r þ l0 Þ:

ð14Þ

The natural frequency of the actuator takes the form

w2n ¼

 1 @ ½hðd; UÞsin/ ;  m @d d¼deq

ð15Þ

where the partial derivative @ ðhsin/Þ=@d is evaluated at the equilibrium stretch keq . The natural frequency and the dynamic response of the ADEA can be tuned by the voltage U, the pre-stretch kp and geometry size r and l0 . Fig. 4 presents the voltage and pre-stretch dependence of the natural frequency of the ADEA when geometry size takes the value of r ¼ 10mm and l0 ¼ 20mm. Fig. 4(b) depicts the natural frequency as a function of the pre-stretch with different applied voltages. It shows that the natural frequency decreases at first within a narrow pre-stretch range and then increases. This is because a large pre-stretch makes the DE membrane tense, improving the equivalent stiffness. While Fig. 4(c) presents the natural frequency-voltage relation for different pre-stretches. The opposite tendency that the natural frequency declines with increasing voltage is because a higher voltage softens the membrane. The voltage dependence of the natural frequency increases with increasing pre-stretch, as shown in Fig. 4. Furthermore, geometry size has an influence on the natural frequency of the ADEA. The variation of the natural frequency under different radiuses and radial lengths is presented in Fig. 5 when kp ¼ 3 and U ¼ 1:5kV. It is obvious that greater natural

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Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

Fig. 4. (a) The voltage and pre-stretch dependence of the natural frequency of the ADEA; (b) The natural frequency as a function of the pre-stretch kp with the indicated voltage; (c) The natural frequency as a function of the voltage for the indicated pre-stretch.

Fig. 5. The effect of radius r and radial length l0 on the natural frequency of ADEAs.

frequency of the actuator can be achieved with increasing radial length l0 and decreasing radius r. Furthermore, the natural frequency is more sensitive to radial length l0 .

3.3. Dynamic response The dynamic properties of ADEAs are investigated by applying a time dependent voltage U ¼ U0 þ Ua sinð2pft Þ, where U0 is the bias voltage, Ua is the amplitude of the input voltage and f is the excitation frequency. Unless otherwise specified, we take the values of r ¼ 10mm, l0 ¼ 20mm and k1p ¼ k2p ¼ 3. The steady-state time response and frequency spectrum of the ADEA under sinusoidal periodic excitation with different bias voltages are shown in Fig. 6 when Ua ¼ 1:0 kV and f ¼ 5 Hz. It is noticeable that the bias and the amplitude of the output displacement increase with increasing bias voltage, indicating that the actuation performance of the actuator can be improved with larger bias voltage. This is because increasing bias voltage can lead to higher driving voltage, inducing larger Maxwell stress. Moreover, it is worth noting that multiple frequency components of the output displacement are negligibly small for U0 ¼ 1:0 kV and U0 ¼ 2:0 kV. While for no bias voltage, the dual frequency component becomes dominant as illustrated in Fig. 6(b), because U2 ¼ U2a ð1  cosð4pft ÞÞ=2 in Eq. (14) when U0 ¼ 0 V. Therefore, applying bias voltage can reduce the multiple frequency components, weakening the output nonlinearity of the actuator. Fig. 7 presents the response of the ADEA subject to voltages with different amplitudes when

Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

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Fig. 6. (a) Steady state time response and (b) Frequency spectrum of the ADEA under periodic excitation with the indicated bias voltage.

Fig. 7. Displacement response of the ADEA in (a) time domain; (b) frequency domain when subject to periodic excitation with the indicated amplitude of input voltage.

U0 ¼ 2:0 kV and f ¼ 5 Hz. It shows that larger bias displacement and actuation range can be achieved with increasing amplitude of the input voltage, the same for the dual frequency component. The dynamic response of the actuator in time and frequency domain under excitation of different frequencies when U0 ¼ 1:5 kV and Ua ¼ 1:5 kV are shown in Fig. 8. It demonstrates that the output displacement shows a clear dependence on the excitation frequency. Together, the dependence of the static and dynamic characteristics on applied voltage and geometric factors are analyzed. When the DEAs are used in the specific applications with requirements for loading conditions and operating frequency, basic rules are provided for determining the actuator parameters in designing compact actuator devices. 4. Controller design Passive approaches to isolation using devices like rubber mounts and/or springs are usually effective for high frequency vibration only. These passive isolators act as a rigid link when the vibration frequency is below the resonance frequency of the isolator. While active vibration control is an approach which can be employed to achieve isolation for low frequency vibrations. Fig. 9 depicts the overall control scheme represented in the discrete-time domain. The signals uðkÞ and eðkÞ respectively denote the vibration from the exciting source and the displacement of the isolated object, and f ðkÞ is the feedforward control signal. The control objective is to design a controller W to output signal f ðkÞ, in turn minimizing the power of eðkÞ. Gp ðzÞ and Gs ðzÞ represent transfer functions of the primary and secondary path, respectively. In this work, both the estimated dynamics ^ s ðzÞ and the feedforward controller W are constructed by adaptive filters. of the secondary path G

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Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

Fig. 8. The output displacement of the ADEA in (a) time domain; (b) frequency domain under excitation of different frequencies.

Fig. 9. Schematic diagram of adaptive active vibration control scheme.

An adaptive filter is a linear system whose transfer function is determined by variable parameters and those parameters can be adjusted according to certain update strategy. Finite-impulse-response (FIR) filter, a delayed adaptive filter, has gained substantial popularity, especially in active noise and vibration control owing to its simplicity and ease of implemen^ s ðzÞ and the structure of the feedforward controller W tation [31,32]. Here the estimated dynamics of the secondary path G are both constructed by FIR filters respectively as

^ s ðzÞ ¼ G

N 1 X

v i zi ; W ¼

i¼0

N 1 X

wj ðkÞzj

ð16Þ

j¼0

where N refers to the order of both filters, v i are the coefficients of the FIR filter corresponding to the estimated dynamics ^ s ðzÞ and wj ðkÞ represents the jth weight at kth time of the FIR filter corresponding to the estimated of the secondary path G ^ s ðzÞ. Since the filter output of a given input signal is determined by the filter weights, dynamics of the secondary path G adjusting

the

weights

to

obtain

the

desired

output

is

the

primary

task.

The

coefficient

vector

T

wðkÞ ¼ ½w0 ðkÞ; w1 ðkÞ;    ; wN1 ðkÞ can be updated according to an optimization algorithm. Among adaptive filter algorithms, Filtered-X least mean square (LMS) algorithm compensating for the effect of the secondary path function performs well in the vibration control systems [32,33]. The weights are updated by minimizing the expectation value of the squared error e2 ðkÞ, that is

n o    2 min E e2 ðkÞ ¼ min E½ys ðkÞ  dðkÞ ;

ð17Þ

where ys ðkÞ and dðkÞ are the transmitted vibration at the output terminal through the primary path and secondary path, respectively. The LMS algorithm is an iterative procedure which makes successive corrections to the weights of the filter in the negative gradient direction [34]. The adaptation algorithm then becomes

Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

wj ðkÞ ¼ wj ðk  1Þ þ 2lg ðk  jÞeðkÞ;

9

ð18Þ

where wj ðkÞ and wj ðk  1Þ represent the updated and present controller coefficients respectively. g ðkÞ is the excited vibration

^ s ðzÞ. The convergence coefficient l determines the convergence speed toward the optimal values of wj . Also, it is filtered by G worth noting that, the feedforward controller W is auto-tuning online but the secondary path Gs ðzÞ is estimated offline in advance. Moreover, the adaptive filtering control which the filtered-x least-mean-square (Fx-LMS) belongs to is stable and robust to the external and internal disturbance according to [35,36]. 5. Experiment 5.1. Experiment setup

The schematic of the active vibration isolation based on the ADEA is shown in Fig. 10. A loading mass (50 g) was utilized to represent the structure to be isolated and the aim is to isolate the mass from low frequency base vibration. The actuator was mounted between the mass and electrodynamic shaker (Bruel and Kjaer, 4810) with the input voltage being controlled to reduce the movement of the mass. In this work, LABVIEW software was used to implement the control algorithms. A multifunction I/O device (NI, USB-6212) hosted by the personal computer generated the disturbance signal and corresponding control signal. Then the vibration signal was fed to the shaker via the power amplifier (Bruel and Kjaer, 2706) and the control signal was amplified by a high voltage amplifier (TREK MODEL 10/10B-HS). The real-time displacement of the mass was measured by a laser displacement sensor (KEYENCE, LK-G80). The sampling frequency was set as 1 kHz. 5.2. Experiment implementation To implement the active vibration isolation, the FIR-based transfer function of the secondary path was estimated. Specifically, a white noise signal, with the bandwidth of 100 Hz, was used to drive the DEA via a high voltage amplifier to acquire ^ s ðzÞ and the iterthe transfer function of the secondary path Gs ðzÞ. LMS algorithm was utilized to determine the weighs of G ation coefficient l was experimentally determined. Moreover, the order of the FIR filter was experimentally chosen based on the trade-off between the approximation error and the filter complexity while keeping the approximation error small. Next, Fx-LMS based active vibration isolation experiments were carried out. Single tone (sine) and sweep frequency signal were selected as the disturbances. The disturbance signal generated by LABVIEW via the power amplifier was fed to the shaker. Control voltage signal f ðkÞ was calculated after determining the weight coefficients of the controller W, whose order and ^ s ðzÞ. The weights of the filter Wwere updated in real-time based on convergence coefficient follow the same procedure with G the input disturbance signal uðkÞ and the output displacement signal eðkÞ by LMS algorithm. 5.3. Results and discussion A 32-order FIR model was applied to the estimation of the secondary path. Fig. 11 shows the comparison between experimentally measured response of the secondary path and the estimated results using a FIR-based adaptive filter. It is clear that they are generally in agreement, meaning that the obtained FIR filter can capture the secondary path dynamics, though the

Fig. 10. (a) Schematic of the active vibration isolation test. (b) An assembled annular DEA.

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Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

Fig. 11. Comparison of the dynamic response of the secondary path between experimental data and the estimated results by the FIR filter.

secondary path shows much more complicated dynamic characteristics. We also note that the estimated error is mainly for the failure of capturing a few ‘‘dips” at the very low frequency. In this case, increasing the order of the filter can attain higher estimation accuracy but at the expense of computation complexity. Firstly, the ability to isolate the mass from sinusoidal disturbance at both 7 Hz and 11 Hz was investigated. The order of the FIR filter as a feedforward controller W was determined as 32. Fig. 12 shows the experimental results of the isolated mass vibration without active control and with the ADEA isolator using the Fx-LMS algorithm in (a) time domain; (b) frequency domain. The comparisons were carried out after the parameters of the feedforward controller W having converged. Intuitively it can be seen from Fig. 12(a) and (c) that a massive vibration reduction is achieved. We define

 rms2 rms1

a,  20lg

ð19Þ

as the indicator to describe vibration attenuation objectively, where rms1 and rms2 are the root-mean-square (RMS) values of the vibration displacement before and after active isolation in steady state. A higher value of a indicates better performance of vibration suppression, and a negative number means that the disturbance is amplified. Here, the attenuation of 10.53 and 11.81 dB are achieved for active vibration suppression with the DE annular actuator. From the prospective of frequency domain, in Fig. 12(b) and (d), it demonstrates that the vibration at the reference harmonic frequency (7 Hz and 11 Hz) was attenuated by almost 57.71 and 58.38 dB, respectively. Moreover, the vibration amplitudes at other frequencies were reduced to some extent, for example, the attenuation of 6.30 dB and 5.75 dB at 14 Hz and 22 Hz, respectively. We also see that low peaks at 14 Hz (second harmonics) and 21 Hz (third harmonics) exist and the same holds true for vibration isolation at objective frequency 11 Hz. It is mainly due to the weak nonlinearity of the electrodynamic shaker output. Nonetheless, the potential of the ADEA for active vibration isolation and the efficiency of the employed Fx-LMS algorithm are validated. The convergence coefficient l used in this experiment was chosen as 0.01, less than the value required to drive the LMS algorithm unstable. In addition, the applied voltage of the ADEA when used for active vibration isolation under the 7 Hz sinusoidal disturbance is presented in Fig. 13(a), which shows that the voltage amplitude and bias voltage are approximately 0.70 kV and 1.63 kV, respectively. The corresponding steady state response of the actuator are analyzed theoretically and given in both time domain (Fig. 13(b)) and frequency domain (Fig. 13(c)). Compared to the dominant frequency (7 Hz), multiple frequency components of the output displacement are negligibly small, indicating weak output nonlinearity of the ADEA. To some extent, it provides corroborative evidence that the FIR based adaptive filter, a linear system, can be effective in this vibration isolation experiment. To further demonstrate the potency of the ADEA for large amplitude vibration isolation, the experiment of isolating the mass under the 9 Hz sinusoidal disturbance with peak-to-peak amplitude of 1400 lm was carried out. 32 coefficients were used in the feedforward controller W and the iteration coefficient l used in this experiment was taken as 0.02. The response of the ADEA isolator with and without the converged adaptive controller was compared in both time domain (Fig. 14(a)) and frequency domain (Fig. 14(b)). It is clear from Fig. 14(a) that the mass vibration amplitude was reduced distinctly, and the vibration attenuation rate a can reach 10.74 dB. More specifically, the vibration at the reference frequency 9 Hz and the second harmonic frequency 18 Hz has been attenuated by almost 56.16 dB and 8.12 dB, respectively.

Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

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Fig. 12. Active vibration isolation with the ADEA under a single frequency excitation. Measured mass displacements in (a) time domain; (b) frequency domain under the 7 Hz sinusoidal disturbance; (c) time domain; (d) frequency domain under the 11 Hz sinusoidal disturbance. Blue solid curve and red dashed curve represent the mass position signal without and with active control, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. (a) Applied voltage of the ADEA for active vibration isolation under the 7 Hz sinusoidal disturbance. Theoretical output displacement of the ADEA in (b) time domain; (c) frequency domain with the voltage shown in (a).

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Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367

Fig. 14. Comparison of active vibration isolation with the ADEA without and with active control by (a) time series; (b) frequency spectrum under the 9 Hz sinusoidal disturbance with the peak-to-peak amplitude of 1400 lm.

In terms of more complicated excitation signal, the results of active vibration suppression are compared in Fig. 15 where a 6–10 Hz swept frequency waveform (sweep speed 0.04 Hz/s) with the peak-to-peak amplitude of almost 330 lm was applied as the disturbances. The excited vibration is alleviated effectively both from the views of time domain and frequency domain, which echoes with the result that the vibration isolation ratio a ¼ 10:79dB. The vibration attenuation of 36.51 dB and 5.72 dB has been achieved at the frequency 8 Hz and corresponding second harmonic frequency 16 Hz. In addition, it can be noted from Fig. 14(a) that the output of the electrodynamic shaker is a distorted sinusoidal wave which is also verified in Fig. 14(b) because the second harmonics show comparable amplitude with the reference frequency. Normally, the distorting phenomenon could be brought from the power amplifier when its output saturates, and the aging shaker should be also considered. In this experiment, the practical disturbance and the reference signal were not in perfect coherence, which may reduce the anticipated performance of active vibration control. Table 2 comparatively evaluates the experimental results of the active vibration isolation ratio with the ADEA in steady state. Not only the amplitude attenuation at the reference frequency and the second harmonics are summarized, but the values of the vibration attenuation indicator, the ratio of the root-mean-square (RMS) values of the excitation and the response are given. Both of them cogently demonstrate that annular DE actuators can be employed for active vibration suppression even with large-amplitude disturbance.

Fig. 15. The response of the isolated mass with and without active controller in (a) time domain; (b) frequency domain where a 6–10 Hz swept frequency waveform (sweep speed 0.04 Hz/s) used as the excitation.

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Y. Zhao et al. / Mechanical Systems and Signal Processing 134 (2019) 106367 Table 2 Comparison of steady-state vibration isolation with ADEAs under different disturbances. Disturbance

Frequency f (Hz)

Peak-to-peak value (lm)

Attenuation at f (dB)

Attenuation at 2f (dB)

Vibration isolation ratio a (dB)

Sinusoidal Signal Sinusoidal Signal Sinusoidal Signal Sweep signal

7.0 11.0 9.0 6.0 to 10.0 (0.04 Hz/s)

320 330 1400 330

57.71 58.38 56.16 36.51

6.30 5.75 8.12 5.72

10.53 11.81 10.74 10.79

6. Conclusion In this paper, we investigate and particularize the potential of ADEAs for active vibration isolation. Annular DEAs are used for active isolators due to their lightness, compactness and flexibility, the more so as they enjoy the merits of simple fabrication and low cost compared to stack DEAs and tubular DEAs. An electromechanical model is developed and its analytical results show good consistency with experimental data in static mode. For the purpose of vibration control, the analysis of the natural frequency of the DEA is of primary importance. The parametric dependence of the natural frequency on the applied voltage, the actuator geometry and the pre-stretch of DE membrane is analyzed. Specially, larger natural frequency could be achieved with increasing radial length and decreasing radius. A higher applied voltage softens the membrane, declining the natural frequency of the DE actuator, while the pre-stretch of the DE membrane is the opposite. The loading mass supported by the actuator could also change the natural frequency, and geometric sizes of ADEAs should be determined for the specific applications with requirements for loading conditions and operating frequency. Furthermore, active vibration isolation with the ADEA was implemented employing the FIR-based adaptive controller. In this experiment, classical Fx-LMS algorithm, incorporating the effect of the secondary path function, was employed to tackle the vibration problem. With the DE annular actuator using the FIR-based adaptive controller, different types and amplitudes of excited disturbances were suppressed. Although the output disturbance is not in good accordance with the programmed reference signal, the experimental results still convincingly demonstrate that ADEAs could be used for active vibration isolation systems even with large amplitude disturbance. 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