Experiments on vibration control of a piezoelectric laminated paraboloidal shell

Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Experiments on vibration control of a piezoelectric laminated paraboloidal shell Honghao Yue a, Yifan Lu a,n, Zongquan Deng a, Hornsen Tzou b a b

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, Heilongjiang Province 150001, PR China College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

a r t i c l e i n f o

abstract

Article history: Received 23 December 2015 Received in revised form 10 May 2016 Accepted 16 May 2016

A paraboloidal shell plays a key role in aerospace and optical structural systems applied to large optical reflector, communications antenna, rocket fairing, missile radome, etc. Due to the complexity of analytical procedures, an experimental study of active vibration control of a piezoelectric laminated paraboloidal shell by positive position feedback is carried out. Sixteen PVDF patches are laminated inside and outside of the shell, in which eight of them are used as sensors and eight as actuators to control the vibration of the first two natural modes. Lower natural frequencies and vibration modes of the paraboloidal shell are obtained via the frequency response function analysis by Modal VIEW software. A mathematical model of the control system is formulated by means of parameter identification. The first shell mode is controlled as well as coupled the first and second modes based on the positive position feedback (PPF) algorithm. To minimize the control energy consumption in orbit, an adaptive modal control method is developed in this study by using the PPF in laboratory experiments. The control system collects vibration signals from the piezoelectric sensors to identify location(s) of the largest vibration amplitudes and then select the best two from eight PVDF actuators to apply control forces so that the modal vibration suppression could be accomplished adaptively and effectively. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Paraboloidal shell PVDF Vibration control Positive position feedback (PPF)

1. Introduction Large optical reflector, communications antenna, rocket fairing, missile radome and other aerospace structures are typically made into the shape of thin-walled paraboloidal shells. To reduce the weight of the system, these structures are usually made of ultra-light and ultra-thin materials and therefore have generally low modal frequencies, small damping ratio and large flexibility. Since these kind of structures mainly work in outer space with scarcely any air resistance, their vibration endures once excited and this would not only jeopardize relevant instruments and equipment, but also lead to system or structural failures. Thus, in order to improve the performance and precision of the spacecraft, adaptive and active vibration control to the key structures are needed. Since the piezoelectric effect was found by Jacques and Pierre Curie in 1880, piezoelectric materials have been used as sensors and actuators for plate and shell structures over the years. Bailey et al. [1] proposed the piezoelectric active damper in 1985. With distributed polyvinylidene fluoride (PVDF) piezoelectric film layers laminated on one side of a flexible

n

Corresponding author. E-mail address: [email protected] (Y. Lu).

http://dx.doi.org/10.1016/j.ymssp.2016.05.023 0888-3270/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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cantilever, the first modal damping ratio of the beam could be increased to 4.5 times. Also, it was investigated by Burke et al. [2] that vibration control could be achieved by changing the shape of PVDF actuators. A complete theory of distributed sensing and control of piezoelectric shells was presented by Tzou in 1988 [3]. Finite element method was employed combined with distributed sensing and control to analyze the dynamic performance of plate [4] and beam [5]. With piezoelectric lead zirconate titanate (PZT) ceramic and PVDF as patching on/ embedding in sensors and actuators, dynamic performance and theoretical model of piezoelectric structures were investigated and active damping control was achieved [6]. Tzou et al. carried out a multilayer actuator theory for distributed control of flexible shell structures [7]. With orthogonal winding piezoelectric sensors and actuators, Tzou developed a free modal control method for a flexible ring [8]. Relevant theories and applications of laminated composite cylindrical shells with sensor and actuator layers were discussed [9] and distributed piezoelectric elements were used to study the modal sensing performance of a rotational cylindrical shell [10]. A cylindrical shell controlled by completely/ partially distributed actuators and linear excitation was investigated [11,12]. Modal control force, excitation factor, feedback factor and control ratio of cylindrical shells with 4 distributed actuators under different deformation were studied [13]. Dynamic modeling and vibration control of spherical [14–16], composite spherical [17], conical [18] and toroidal [19,20] thin shell were also developed. Micro control and sensing action of smart structures was extended to nonlinear thick paraboloidal shell [21–23] by Tzou et al. The Newmark time integration method was used to calculate the dynamic response and negative velocity feedback control algorithm was used to control the dynamic response [24]. Dynamic modeling and active vibration control of a cylindrical shell with piezoelectric sensors and actuators was investigated [25]. The precision distributed control effectiveness of simple supported paraboloidal shells with laminated PVDF actuators was investigated [26]. Microscopic actuations of paraboloidal membrane shells and parabolic cylindrical shells were studied and optimal actuator locations were investigated [27,28]. Control methods and strategies are of importance to vibration control effectiveness. Although many modern control theories and strategies have been proposed in recent years, real-life engineering applications are still needed in practice. Positive position feedback (PPF) control, optimal control, fuzzy control, neural network control and adaptive filtering feedforward control are commonly used control strategies for current active vibration control. The PPF control method was introduced to control vibrations of large flexible space structures by Fanson and Caughey in 1985 [29]. It was applied by feeding the structural position coordinate directly to a compensator and then the product of the compensator and a scalar gain positively back to the structure. Neural network control was used in system identification and active control experiment. The result was compared with that of optimal control [30,31]. Adaptive neural network control was utilized in vibration control of flexible aerospace structure system where a back propagation (BP) algorithm and the stochastic optimization searching method were adopted [32]. An LMS (Least Mean Square) adaptive algorithm was applied to active vibration control of a broad band structure [33]. Directly adaptive algorithm [34], gain adjustment algorithm [35] and filtered-X-least mean square (LMS) algorithm [36] were studied respectively and employed in different structures. Nonlinear vibrations of laminated rectangular plates with free boundary conditions were discussed [37,38] and active control of a sandwich plate by collocated and the non-collocated PPF algorithm was developed respectively [39,40]. Due to the complexity of analytical procedures and unavailability of analytical solutions, this study focuses on an experimental study of the active modal vibration control of a flexible paraboloidal shell with free boundary conditions (BCs). Eight PVDF patches are laminated outside the shell as sensors and eight are inside as actuators to control the vibration of the first two shell modes. A parameter identification method is first used herein to establish the mathematical model of the control system. Controllers for the first order mode as well as the first and second order coupled mode of the paraboloidal shell are developed based on the positive position feedback (PPF) algorithm. Then, an experimental platform for adaptive modal control of precision paraboloidal shell is established and two from eight PVDF actuators could be selected by the control system to alleviate the oscillation caused by external excitations.

2. Positive position feedback control Positive position feedback (PPF) was firstly proposed by Fanson and Caughey in 1982 and has been applied extensively to the low frequency vibrations of thin walled structures [29]. In the case of a second-order single-degree-of-freedom system, the modal displacement is processed by a second-order low-pass filter and, then, positively fed-back into the structure. The modal equations of the structure and the controller are respectively represented by

ξ¨ + 2ζωξ ̇ + ω2ξ = γω2η

(1)

η¨ + 2ζc ωc η ̇ + ωc2 η = ωc2 ξ

(2)

where ξ is the coordinate of the structure, η is the coordinate of the controller, ω and ωc are the structural and controller natural frequencies, respectively. ζ and ζc are the structural and controller damping ratios, respectively. γ is a positive system scalar gain, i.e., γ 4 0. The transfer function of the controller C(s) becomes

C (s ) =

γc ωc2 s2 + 2ζc ωc s + ωc2

(3)

Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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where γc is the scalar gain of the controller. Eqs. (1) and (2) denote the decoupled model equations of the structure and the controller, respectively. From descriptions above, it indicates that PPF control is essentially a kind of algorithm based on the modal space and each parameter can be clearly analyzed in the modal space. To describe the working principle of the controller, assume that the modal vibration displacement of the paraboloidal shell is

ξ (t ) = αe jωt where

(4)

α is the vibration amplitude of the structure. The output of the controller is η (t ) = βe j (ωt − φ)

(5)

⎛ 2ζ ω /ω ⎞ α where β is the modal amplitude of the controller, β = , and φ is the phase angle, φ = tan−1⎜ c 2 c2 ⎟. ⎝ 1 −ω /ωc ⎠ (1 − ω2 / ωc2 )2 + (2ζc ω / ωc )2 Three cases, i.e. ω«ωc, ω E ωc and ω»ωc, could lead to different value of ϕ and thus change the original structure in different ways. In this study, only the second case, ω E ωc, is adopted to augment the damping of the system and derivations are presented below. When the natural frequency of the flexible paraboloidal shell ω is similar to that of the controller ωc, according to the expression of phase angle, it can be concluded that φ E π/2. Substituting Eq. (5) into Eq. (1) yields

⎛ γβ ⎞⎟ ̇ ξ¨ + 2 ⎜ ζ + ωξ + ω2ξ = 0 ⎝ 2α ⎠

(7)

In this situation, the system damping increases and the modal vibration of the structure can be suppressed.

3. Laboratory experiments The experimental setup is composed by two systems, a vibrating paraboloidal shell with free edges and an active vibration controller. Note that the piezoelectric patches serve as monitoring transducers whose signals are used in feedback control. 3.1. Modal analysis of the paraboloidal shell The tested flexible paraboloidal shell model is made of DSM Somos-14120 resin, whose geometrical parameters are listed in Table 1. The mobile point excitation method is used for its dynamic testing and lower vibration natural frequencies and modes are obtained via the frequency response analysis by Modal VIEW software. The mass density of the shell is 1120 kg/m3 (25 °C), and its elastic modulus 2800 MPa. In a previous study of a free paraboloidal shell, with experienced “trail-and-error” and verification, a new transverse mode shape function of a paraboloidal membrane shell with free boundary condition has been derived [27]

U3k = Ak ( k + 1) cos ϕ sink ϕ cos kψ

(8)

where the mode number k ¼1,2,3…, Ak is the kth modal amplitude, ϕ is the meridional angle measured from the pole, and ψ is the circumferential angle. In correspondence with the theoretical analysis, the flexible paraboloidal shell is suspended in the pole with the boundary free. An impact hammer excitation is used as an external force which is monitored by an accelerometer. The experimental setup is shown in Fig. 1. Natural frequencies of the first three natural modes of the paraboloidal thin shell are extracted from experimental modal analysis data and they are listed in Table 2. By synthesizing the frequency response function (FRF) and analyzing the multi-reference least square complex frequency (LSCF) steady-state plots, location and shape of the selected modes are determined and the mode shapes can be finally obtained. Comparisons between the experimental and theoretical results regarding the first three natural modes of the laminated paraboloidal thin shell are shown in Figs. 2–4. Dotted line indicates the envelope of each mode shape in the top views. These figures suggest the physical model indeed behavior as theoretical predictions. (Note that these four modes resemble to corresponding ring modes.) With the validated modal information, locations of piezoelectric sensor and actuator patches can be determined and the closed-loop control can be performed. Table 1 Geometrical parameters of the flexible paraboloidal shell. Type

Height c (m)

Radial distance a (m)

Thickness h (m)

Paraboloidal shell

0.1

0.2

0.001

Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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5000N ICP hammer

Accelerometer (M353B18)

ModalVIEW interface Paraboloidal shell

DAQ (NI-PXI4472)

Fig. 1. Modal experiment setting of free paraboloidal thin shell.

Table 2 Natural frequency of paraboloidal thin shell model. Type

1st mode (Hz)

2nd mode (Hz)

3rd mode (Hz)

Paraboloidal shell

8.90

23.60

43.36

a) Experimental top view

b) Theoretical top view

c) Experimental left view

d) Theoretical left view

Fig. 2. Comparison for first mode of free paraboloidal thin shell.

3.2. Active vibration control setup Fig. 5 illustrates a system block diagram of a piezoelectric laminated paraboloidal shell structure in laboratory experiments. An A/D card (PXI-6284) and a D/A card (PXI-6723)(NI-DAQ) are employed in the experimental setup for data acquisition of the sensors and outputs of the control signals. A high-performance embedded microcontroller PXI-8106 supplied by NI company is selected. The geometry and physical properties of the tested paraboloidal thin shell were summarized in Table 1. Eight pieces of Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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a) Experimental top view

c) Experimental left view

5

b) Theoretical top view

d) Theoretical left view

Fig. 3. Comparison for second mode of free paraboloidal thin shell.

PVDF sensor patches are uniformly arranged on the outside of the flexible paraboloidal thin shell, with an interval of 22.5 degrees between each in the circumferential direction. Eight PVDF actuators are arranged inside the shell on the opposite sides of the sensors. In order to identify the location of vibration in the modal control experiments, eight sensor/actuator sets are numbered clockwise (see Fig. 6). The parameters of PVDF are listed in Appendix A (Table A1). The active control of piezoelectric laminated paraboloidal shell testing platform is shown in Fig. 7. The hardware consists of a piezoelectric laminated paraboloidal shell (PVDF sensors, actuators and flexible paraboloidal shell), a multi-channel sensing signal conditioning circuit, an A/D and D/A conversion module, an embedded microcontroller, a multi-channel voltage amplifying circuit and other parts. By identifying the parameters of major components of the system and establishing mathematical model of each part, one obtains the control model of the integrated paraboloidal shell system.

4. System parameter identification The experimental setup of the active vibration control of paraboloidal shell system has been established. As discussed previously, sixteen PVDF patches are laminated respectively on the inner and outer surfaces of the paraboloidal shell. Eight inside patches are used as actuators and eight outside patches are used as sensors. The PVDF sensors measure the vibration amplitude of the shell based on the direct piezoelectric effect. Due to the characteristics of piezoelectric thin film, sensor output signals need to be filtered and amplified. Signal filtering and amplifying circuits are contained in the control system. In this section, the transfer function model of each part of the hardware system is identified. It is worth noting that the mathematical models established by parameter identification could not only truly mirror the characteristics of the system but also have practical significance in updating model parameters and implementing dynamic adjustment in practical engineering applications. 4.1. PVDF sensor/actuator The sensing principle of PVDF is based on the direct piezoelectric effect of piezoelectric materials and the actuating principle is based on the converse piezoelectric effect. Assuming that distributed piezoelectric sensors and actuators are perfectly bonded on the shell structure, the voltage generated by the sensor is proportional to the mechanical strain and the Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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a) Experimental top view

b) Theoretical top view

c) Experimental left view

d) Theoretical left view

Fig. 4. Comparison for third mode of free paraboloidal thin shell.

Distributed sensing signal module

Sensor patches

Multi-channel signal conditioning circuit

A/D controller

Signal processing

Controller system microcontroller Paraboloidal shell Actuator patches

Multi-channel high voltage power amplifying circuit

D/A controller

Distributed high voltage excitation module Fig. 5. Block diagram of a piezoelectric laminated paraboloidal shell smart structure system.

actuation strain induced by the actuator is proportional to the excitation electric signal. According to the linear properties of piezoelectric thin film, the distributed sensor and actuator could be regarded as a proportional component and the transfer function becomes

GYD (s ) = Kc

(9)

where Kc is a scale factor. Note that the sensing signals need to be amplified before entering the A/D controller and the actuating signals also need to be amplified after coming out of the D/A controller in order to drive the PVDF actuators. In this experiment, appropriate gains for the PVDF sensor and actuator are selected as Ksen ¼ 20 and Kact ¼50, respectively. Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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1 8

2

3

7

6

4 5

Fig. 6. A paraboloidal shell model for adaptive vibration control experiment.

Paraboloidal shell

oscilloscope

Power supply of voltage

NI PXI- 8016 controller

amplifying circuit

Power supply of conditioning circuit

Multi-channel signal conditioning circuit A/D and D/A converter Pendulum ball Multi-channel voltage

Signal acquisition, controlling, processing interface

amplifying circuit

Sensing sinal(V)

Fig. 7. Experimental setup for active control system of a piezoelectric laminated paraboloidal thin shell.

Times(s) Fig. 8. Time responses of shell after tapping on the edge.

4.2. Flexible paraboloidal shell Two model identification methods are used herein to test the shell model, i.e., the impulse response method and the frequency response method. Firstly, the impulse response method is selected to identify the model of the vibration. Tapping Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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Sensing sinal(V)

8

Frequency(Hz) Fig. 9. Frequency responses of shell after tapping on the edge.

on the edge of the shell, the time response and frequency response are shown in Figs. 8 and 9. Fig. 9 reveals that the shell vibration frequencies mainly concentrate in the vicinity of 8.9 Hz and 23.6 Hz, as suggested in the modal testing, Table 2. These two resonant frequencies are the first and second natural frequencies of the paraboloidal shell. The mathematical model of the shell's first two modes can be respectively approximated by a second order transfer function as follows.

G1 (s ) =

G2 (s ) =

ks ω12 3125ks = 2 s + 5.59s + 3125 s2 + 2ζ1ω1 + ω12

s2

(10)

ks ω22 21963ks = 2 s + 8.892s + 21963 + 2ζ2 ω2 + ω22

(11)

whereω1,ω2are the natural frequencies, ω1 = 2π × 8.9 = 55.9 (rad/s), ω2 = 2π × 23.6 = 148.2 (rad/s), ζ1,ζ2 are the first and second modal damping ratios respectively, i.e., ζ1 = 0.05, ζ2 = 0.03. ks is the static gain of the shell. The static gain is calculated via the frequency response method below. By inputting a sinusoidal wave to the PVDF actuator and measuring the output signal of the PVDF sensor in the corresponding position, one obtains the typical time response curve of the shell, shown in Fig. 10. The corresponding amplitude and frequency of the input excitation signal are 2 V and 10 Hz, respectively. The static gain of the flexible shell is calculated as

ks =

Kresp 0.7 = = 0.0007 Ksen Kact 20 × 50

(12)

Sensing sinal(V)

where Kresp is the amplitude ratio of the output and input sinusoidal waves. Recall that Ksen and Kact are the gains of the PVDF sensor and the actuator, respectively. Note that the amplitude ratio of the output signal from the sensor and the input signal from the actuator differs slightly with the change of the input amplitude. This indicates that there may be some uncertainties between the transducers and

Time(s) Fig. 10. Time responses of the shell under sinusoidal excitation (2 V, 10 Hz).

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the paraboloidal shell, such as the effect of the bonding layer. Since the change of the amplitude ratio is not significant, it can be regarded approximately as a constant. 4.3. Signal conditioning circuit Since the response of PVDF film shows some degree of nonlinearity when the vibration frequency is below 1 Hz, a highpass filter is used in the circuit system. In order to suppress the high frequency interference, a low-pass filter is also imposed. Besides, power line interference is introduced by the 50 Hz/220 V power supply system. Therefore, three notch filters are included in the conditioning circuit to filter out 50 Hz, 100 Hz and 150 Hz interference signals. The transfer function of the high-pass filter is

Ghp ( s ) =

0.0225s2 0.0225s2 + 0.15s + 1

(13)

The transfer function of the low-pass filter is

Glp ( s ) =

3.843 × 106 s2 + 3921s + 3.843 × 106

(14)

The transfer functions of three notch filters are

s2 + (2π × 50)2 s2 + 98700 = 2 s2 + 2 × (2π × 50) s + (2π × 50)2 s + 628.3s + 98700 s2 + (2π × 100)2 s2 + 394800 G100 (s ) = 2 = 2 2 s + 2 × (2π × 100) s + (2π × 100) s + 1257s + 394800 s2 + (2π × 150)2 s2 + 888300 = 2 G150 (s ) = 2 2 s + 2 × (2π × 150) s + (2π × 150) s + 1885s + 888300 G50 (s ) =

(15) (16) (17)

The transfer function model of the whole signal conditioning circuit becomes

Gfilter ( s ) = Ksen⋅Ghp ( s )⋅G50 ( s )⋅G100 ( s )⋅G150 ( s )⋅Glp ( s )

(18)

The Bode diagram of Gfilter (s ) is shown in Fig. 11. By comparing with the actually tested amplitude-frequency and phase-frequency characteristics of the circuit, one could find that this model can approximate the hardware system with a relatively high degree of accuracy. The final transfer function model of the experimental paraboloidal shell (including the first two shell natural modes) system becomes

G ( s ) = ⎡⎣ G1 ( s ) + G2 ( s ) ⎤⎦⋅Gfilter ( s )⋅Kact ⎛ ⎞ 2.19 15.37 0.0225s2 ⎟ × 20 × =⎜ 2 + 2 ⎝ s + 5.59 s + 3125 s + 8.892s + 21963 ⎠ 0.0225s2 + 0.15s + 1 s2 + 98700 s2 + 394800 s2 + 888300 × 2 × 2 s + 1257s + 394800 s + 1885s + 888300 + 628.3s + 98700 3.843 × 106 × 2 × 50 s + 3921 s + 3.843 × 106 ×

s2

(19)

Phase(deg)

Amplitude(dB)

This transfer function is used in conjunction with control algorithms to evaluate shell's vibration control effects.

Frequency(Hz)

Frequency(Hz)

(a) Amplitude-frequency characteristics

b) Phase-frequency characteristics

Fig. 11. Bode diagram of the signal conditioning circuit.

Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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Uncontrolled

Sensing sinal(V)

Controlled

Time(s) Fig. 12. Time responses of independent modal controller.

5. Implementation and experimental results Independent and coupled modal space controls are tested and PPF controllers are designed respectively in laboratory experiments of the flexible paraboloidal shell. An adaptive modal control method based on the PPF algorithm is proposed to alleviate the first modal vibration. 5.1. Independent modal control effect Since the PPF requires the controlled system to a second order system, for the first shell mode of 8.9 Hz, the system model could be simplified to

G (s ) =

2.19 × 20 × 50 s2 + 5.59 s + 3125

(20)

The PPF controller is designed as in Eq. (3). The frequency, damping ratio and scalar gain of the controller are chosen as ωc = 1.1ω = 61.5 (rad/s), ζc = 0.1, γc = 1. The transfer function of the controller becomes

C (s ) =

3782 s2 + 12.3s + 3782

(21)

Time and frequency responses of the first mode with and without control are shown in Figs. 12 and 13. Partially enlarged detail of the time response curve in the first centisecond is plotted in the bottom-right corner of Fig. 12. Figs. 12 and 13 indicate that damping of the first shell mode is increased by the controller and that this independent modal controller can restrain the vibration amplitude and shorten the stability time observably. The amplitude of the first mode decreases by approximately 62.3%, while the rest modes of the shell structure remain almost unchanged, which demonstrates that it is feasible to design the PPF controller for an independent mode without influencing of the others. Also, it can be found in Fig. 13 that a 13 Hz mode is generated by the controller and its amplitude is even larger than that of the first mode under suppression. The reason of this phenomenon could be complicated. The PPF algorithm chosen here is based on the linear control theory. But the paraboloidal shell, the PVDF actuators/sensors and the hardware circuit could involve

Uncontrolled

Sensing sinal(V)

Controlled

Frequency(Hz) Fig. 13. Frequency responses of independent modal controller.

Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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Uncontrolled

Sensing sinal(V)

Controlled

Time(s) Fig. 14. Time responses of independent modal controller after adjusting.

unpredicted nonlinear behaviors, system uncertainties, model imperfections, etc. Also, the whole system is approximated and modeled as a second order system and it is surely over simplified. These might lead to the generation of the spillover secondary mode of 13 Hz in Fig. 13. Based on the theoretical analysis of the PPF algorithm, reducing the controller gain may diminish the generated mode. Set the gain to 0.5, the transfer function turns into

C (s ) =

1936 s2 + 12.3s + 3872

(22)

Repeat the experiment with the new controller and results are shown in Figs. 14 and 15. From Fig. 15, one can find that the generated mode amplitude is decreased in this manner, but the vibration reduction effect is weaken as well, which is consistent with the theoretical analysis. Control effect of the PPF controller of shell's first mode after parameter adjustment is summarized in Table 3. Note that the first-mode oscillation amplitude is reduced by 51.6%. 5.2. Coupled modal control effect According to the PPF theory, multiple PPF controllers can be designed, aiming at controlling different modal resonant frequencies to satisfy the requirement of the coupled modal control. In this study, a coupled modal controller is designed by adding the independent modal controller together for controlling of the first (8.9 Hz) and second (23.6 Hz) coupled mode vibration. The transfer function is expressed as

C (s ) = C1 (s ) + C2 (s ) =

γc1ωc21 s2

+ 2ζc1ωc1s +

ωc21

+

γc2 ωc22 s2

+ 2ζc2 ωc2 s + ωc22

(23)

Uncontrolled

Sensing sinal(V)

Controlled

Frequency(Hz) Fig. 15. Frequency responses of independent modal controller after adjusting.

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Table 3 Control effect of PPF independent modal controller. Vibration mode

Amplitude without control (V)

Amplitude with control(V)

Suppression ratio

1st mode

0.411

0.199

51.6%

where C1 (s ) and C2 (s ) are the transfer functions of PPF controllers designed for the first and second shell modes, respectively. C1 (s ) is identical to Eq. (21) of the 1st-mode independent modal control and the parameters in C2 (s ) are defined as ωc2 = 1.1ω2 = 165.9 (rad/s ), ξc2 = 0.25, γc2 = 0.4 . The transfer function model of the controller is

C (s ) =

3869 22020 + 2 s2 + 12.44s + 3869 s + 82.95s + 27520

(24)

The mathematical model of the paraboloidal shell is then simplified to

⎛ ⎞ 1.92 13.6 ⎟ × 20 × 50 G (s ) = ⎜ 2 + 2 ⎝ s + 5.655s + 3198 s + 9.048s + 22740 ⎠

(25)

Add Eq. (24) into the testing system after discretization and experiment. Time and frequency responses of the shell measured by the distributed PVDF sensors are shown in Figs. 16 and 17. Control effect of the PPF coupled modal controller is summarized in Table 4. Table 4 indicates that the suppression ratio of the first order mode is more significant than that of the second mode. This is qualitatively consistent with the theoretical results obtained in a previous study that the actuator control effect gradually decreases at higher modes since the inherent membrane effect diminishes as the mode increases [27]. Furthermore, since the sensor and actuator chosen in this experiment is near the excitation position, according to Figs. 2 and 3, vibration amplitude of the first mode is larger than that of the second mode at that point. This could also contribute to the suppression difference between the two modes. Also, the suppression ratio could be influenced by the selection of the control parameters and the complexity of unpredicted nonlinear behaviors, system uncertainties, model imperfections, etc. of the paraboloidal shell system. 5.3. Adaptive modal control A flexible precision paraboloidal shell is mainly used in the spacecraft system as high gain antenna, optical mirrors, etc. Since the energy in outer-space mainly comes from solar panels and spaceborne batteries, the energy provided by the system for active vibration control is therefore quite limited. To control the vibration of an intelligent structronic system consumes such large energy that the energy utilization efficiency has to be improved. In this section, based on the PPF algorithm, an adaptive vibration control method is proposed by identifying the area(s) or region(s) with the largest vibration amplitudes and choosing the best two actuators to control the vibration. In order to ensure the real-time property of the control system, the algorithm comparing the average vibration amplitude of the shell based on the first 256 sampling points is adopted to identify the best location of the actuator. The first shell mode is chosen as the major control mode and the algorithm is based on the PPF independent modal control demonstrated earlier. Firstly, an arbitrary location on the paraboloidal shell is selected to exert the excitation. The excitation position chosen in

Sensing sinal(V)

Uncontrolled Controlled

Time(s) Fig. 16. Time responses of coupled modal controller.

Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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Sensing sinal(V)

Uncontrolled Controlled

Frequency(Hz) Fig. 17. Frequency responses of coupled modal controller.

Table 4 Control effect of PPF coupled modal controller. Vibration mode

Amplitude without control (V)

Amplitude with control(V)

Suppression ratio

1st order 2nd order

0.3925 0.1419

0.1417 0.1177

63.9% 17%

this experiment is below No. 5 PVDF sensor near the edge of the shell (see Fig. 6). The absolute value of the first 256 response data of the 8 PVDF sensors are acquired by the data acquisition system and are averaged, respectively, which is shown in Table 5. According to the absolute average value in Table 5, the controller makes the judgment that No.5 and No.7 areas are the best control positions for the first shell mode control. It is clear that the response of No.5 and No.7 are the largest followed by No. 1 and No. 3 and all response of the even-number areas are relatively weak. In fact, the actual excitation position of the pendulum ball is near the No. 5 area. Thus, theoretically, the largest response of the first shell vibration mode excited by the pendulum ball should be in the contacting point and the two axial symmetrical orthogonal positions, i.e., No. 1, No. 3, No. 5 and No. 7 areas, whose value are relatively large in Table 5. Thus it can be seen that by calculating the first 256 absolute average value of the 8-channel sensors, the largest response area can be correctly identified. When achieving the best actuating position, No. 5 and No. 7 actuators are chosen by the system program. The preset PPF independent modal control algorithm is called and control commands are outputted by a D/A converter and actuate the 2 actuators after voltage amplification. The response of 8 sensors with active control on No. 5 and No. 7 areas are plotted in Fig. 18. Fig. 18 indicates that the initial vibration amplitudes in odd areas are larger than those in even areas and the timedomain responses of No. 5 and No. 7 with active control decay more quickly than those of No.1 and No.3. Furthermore, due to the 90-degree phase difference between the odd area and the excitation location, it is obvious that sensing signals of odd areas contain more high-frequency components. For further analysis of the adaptive control effect, responses of No. 5 and No. 7 areas are extracted and filtered. The first order mode time-domain responses of the two areas with and without control effect are shown in Fig. 19, from which one can find that the first modal vibration of the two areas are controlled effectively. Meanwhile, it is clear that the effective control of the two actuators starts from about 0.125 s (256 sampling points at 2000 Hz sampling frequency) after the dynamic oscillation, which indicates that the controller has very strong real-time performance and this is of great importance in practical applications.

Table 5 The absolute average of first 256 response data of 8 PVDF sensors. Sensor No.

1

2

3

4

5

6

7

8

256 absolute average value (V)

1.5872

1.3067

1.5746

1.0668

1.6620

1.1153

1.6071

0.9527

Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

Sensing sinals(V)

Sensing sinals(V)

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Time(s)

a) No.1

b) No.2

Sensing sinals(V)

Sensing sinals(V)

Time(s)

Time(s)

Time(s)

d) No.4

Sensing sinals(V)

Sensing sinals(V)

c) No.3

Time(s)

Time(s)

f) No.6

Sensing sinals(V)

Sensing sinals(V)

e) No.5

Time(s)

Time(s)

g) No.7

h) No.8

Fig. 18. Time responses of eight sensors with adaptive modal control.

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Fig. 19. Dominating mode control effect of selected actuators.

In summary, the experimental results demonstrate that this approach could identify the input location of the external excitation and select the best actuators to exert control forces to the shell structure. Thus, the shell modal vibration can be controlled effectively.

6. Conclusion A flexible precision paraboloidal shell is often used as reflectors, high gain antennas, optical mirrors, etc. Due to unavailability of analytical solutions, an experimental platform for vibration control of a flexible paraboloidal shell with free boundary condition was set up and its vibration control performance was tested and evaluated in this study. A flexible paraboloidal shell laminated with 16 piezoelectric PVDF patches (i.e., eight as sensors and eight as actuators), signal conditioning circuits, data acquisition systems, AD/DA converters, controller (with embedded PPF and adaptive control algorithms), etc. were all connected on an isolation table as a real-time vibration control system. By parameter identifications, a mathematical model of the shell control system was established. Lower shell natural frequencies and normal modes of the paraboloidal shell were obtained via the frequency response function analysis by the Modal VIEW software. The first three mode shapes were compared with the theoretical results very well. Independent and coupled modal vibrations of the shell were controlled based on the PPF algorithm. Considering of the uncertainties of oscillation and the limitation of control energies in outer-space, an adaptive modal control strategy was proposed. Based on the PPF algorithm, an independent modal controller was developed and an adaptive vibration control of the shell was also achieved through eight distributed PVDF sensors and eight PVDF actuators laminated on the opposite sides of the flexible shell's surfaces. The effectiveness to shell vibration control was demonstrated in laboratory experiments. The experiment demonstrates that this control system could identify the area(s) with the largest vibration amplitude and automatically select the best two actuators to exert control forces and moments to the structure so that the modal vibration can be controlled effectively. In addition, this experimental setup proves to have a good real-time performance, which would be of great significance to practical application in space smart structures and structronic systems.

Acknowledgment This research is supported by National Natural Science Foundation of China (Grant no. 51175103) and Self-Planned Task of State Key Laboratory of Robotics and System (HIT) (Grant no. SKLRS201301B). Prof. Tzou would like to thank the “Ph.D. Supervisor Program” and the “111 Project” (B07018) at the Harbin Institute of Technology sponsored by the Chinese Ministry of Education.

Appendix A See Table A1 Please cite this article as: H. Yue, et al., Experiments on vibration control of a piezoelectric laminated paraboloidal shell, Mech. Syst. Signal Process. (2016), http://dx.doi.org/10.1016/j.ymssp.2016.05.023i

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Table A1 Physical properties of PVDF film. Parameter

Value

Unit

Young’s modulus (Y) Tensile strength (sb) Mass density (ρ) Piezoelectric strain constant (d31) Piezoelectric strain constant (d33) Piezoelectric voltage coefficient (g31) Piezoelectric voltage coefficient (g33) Dielectric constant (εr) Electromechanical coupling coefficient (K33) Pyroelectric coefficient (P) Surface resistance (R) Use of temperature (T)

2500 200 1.78  103 21 23 220–260  10  3

MPa MPa kg/m3 pC/N pC/N V m/N

330  10  3

10  3 V m/N

9.5 7 1.0 10–14% 40 r3  40 to 80

c/cm2 K Ω °C

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