A vibration analysis of a deployable structure

Tao LIU1; Shaoze SHAO1

1 Tsinghua University, China

ABSTRACT

Deployable structures capable of being packaged into a small volume to be deployed on orbit to full size have

been widely utilized on aerospace vehicles. Owing to clearances, a nonlinear dynamic feature emerges. In

order to more accurately obtain the vibration properties of the solar arrays with clearances. In this paper,

vibrations of the structure are stimulated by means of hammer impact method. After that, the Constant-Q

non-stationary Gabor frame transform and Fourier transform are employed to analyze the corresponding

vibration properties. As the experimental results shown: (1) The damping will become greater along with the

vibration frequencies increasing; (2) The vibration of the outer panel tend to be more random, so the effect of

the clearance is stronger for the outer panel; (3) In contrast to deployed state, vibration energy in folded

situation is in more dispersed distribution, which indicates that the effect of the clearance is stronger in the

folded situation. The analysis may be valuable for the analysis of the nonlinear dynamics of the deployable

structure.

Keywords: Deployable structures; Nonstationary signals; Time–frequency analysis; Fourier transform;

Constant-Q nonstationary Gabor transform

I-INCE Classification of Subjects Number(s): 41.5

1. INTRODUCTION

Space deployable mechanisms have been widely employed in aerospace vehicles, duce to an ability

to improve space utility rates. As a type of deployable structure, solar cell arrays are an important part

of satellites. However, vibrations may occur during operations, and damping is minimal in space.

Therefore, suppressing the vibration caused by disturbances is difficult. Vibrations can lead to the

abnormal operation of satellites and may lead to accidents.

In order to obtain vibration properties of rigid arrays with clearances, a number of investigations

have been conducted for this topic [1]-[5], which reveals the importance of the vibration properties of

arrays for dynamic characteristics of astrovehicles. Therein, by comparing the flight data with

computer simulation results of the solar array for Hubble Space Telescope, Foster figured out that the

disturbance source was the thermally driven deformation of the solar arrays[1]. Many researches were

done by Thornton et al to obtain the thermal vibration of the Hubble Space Telescope solar array [2],

[3]. Baturkin surveyed current tendencies in thermal control of a micro-satellite [4]. Simplified

governing equations were proposed to study thermal vibrations by Tsai [5].

In 2011, P. Balazs and M. Dorfler made deep work to nonstationary Gabor transform (NSGT), and

literature [6], [7] reflected that not only the Constant-Q transform (CQT) can be done by the NSGT, the

computation time keeps low. And a prior research has been done by Yan by using the Constant -Q

nonstationary Gabor transform (CQ-NSGT) [8].Therefore, the CQ-NSGT is introduced in this paper.

In order to obtain as many resonant frequencies as possible, hammer impact method is used to

stimulate the natural frequencies of the arrays. And experiments are conducted to study the effect of

the clearance for the inner and outer panels and different working situations. After that, the CQ-NSGT

and Fourier transform (FT) is employed to analyze vibration signals to obtain vibration properties of

the deployable structure.

1 yansz@tsinghua.edu.cn

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2. CQ-NSGT

2.1 Frame

The short Fourier transform is a conventional method for the time-frequency analysis. However,

due to the application of the window function with a fixed length, the time resolution and frequency

resolution can not be promising at the same time, restricted by Heisenberg Uncertainty Principle.

Therefore, if variable length window functions are introduced in the transform process, the time

resolution and time resolution will be improved to some extent. And then, non-stationary frame should

be applied there. Besides, the constant-Q transform (CQT) provides a frequency resolution that

depends on geometrically spaced center frequencies of the analysis windows. The constant Q-factor

leads to a finer frequency resolution in low frequencies, whereas time resolution improves along with

increasing frequency, which can effectively reduce calculation amount. Just as mentioned above, the

CQ-NSGT, which is proposed by Nicki Holighaus, Gino Angelo Velasco, Monika Dorfler, et al, is a

high-efficiency algorithm to implement this algorithm [7]. So the algorithm is introduced in this paper.

In frame theory, redundancies are introduced, which makes it be a generalization of the basic

function. Therefore, much more flexibility in design of the signal representation can be obtained, and

can be tailored to specific applications. Generally speaking, in transforming process, signals will be

represented by a linear combination of atoms i , with iZ , and be shown as Eq. (1):

i i

i

s (1)

therein, i

is corresponding coefficients of atom i . There, concept of frames for a Hilbert space H is

introduced. In a continuous setting, one may think of 2( )H L , whereas

LH , L is the length of

signal in this paper. { , }ii is a set of vectors, which belong to Hilbert space H , to signals s H , if

there are constants A and B , which can meet Eq. (2),

2 2 2

,i

i

A s s B s (2)

{ , }ii is a frame, and A and B are the boundaries of this frame.

We then define the frame operator S by Eq. (3),

,i i

i

x xS (3)

To discrete frame, frame operators can be calculated by Eq. (4)

S ΦΦ (4)

In the sequel, we can define a dual frame by Eq. (5)

1

i i

S (5)

where, { , }ii is the dual frame of { , }

ii . And then, synthesis of signal can be represented as a

linear combination of coefficients ,i i

x and the corresponding dual frame i ,

1 1

,i i i i

i i

s s s

S S S (6)

2.2 Constant-Q transform

The CQT ( , )CQ

X k n of a discrete time-domain signal is ( )x n defined by [9]

/2

/2

( , ) ( ) ( / 2)k

k

n N

CQ

k k

j n N

X k n x j j n N

(7)

where, k indexes the frequency bins of the CQT; denotes rounding towards negative infinity;

kN is the length of windows; ( )k n denotes the complex conjugate of ( )k n ; ( )k n are the basis

functions, and are complex-valued waveforms, and are defined by

( 2 )1

( ) ( )k

s

fi n

f

k

k k

nn e

N N

(8)

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where kf is the center frequency of bin k ; sf denotes the sampling rate; ( )t denotes a continuous

window function, for example, Hann or Blackman window, sampled at points determined by t . The

window length kN in Eq. (7) and Eq. (8) are inversely proportional to

kf in order to obtain the same

Q -factor for all bins k . Note that in Eq. (7) the windows are centered at all the sample n of the input

signal.

In the CQT considered here, the center frequencies kf obey

1

( )

12k

Bkf f

(9)

where 1f is the center frequency of the lowest-frequency bin; B determines the number of bins per octave.

The Q -factor of bin k is given by

def .k k k

k

k s

f N fQ

f f

(10)

where kf denotes the -3dB bandwidth of the frequency response of the atom ( )k n ; is the -3dB

bandwidth of the spectrum of the window function ( )t . Typically, Q is desirably made as large as

possible, so as to make the bandwidth kf of the each bin as small as possible, therefore, minimal

frequency smearing is introduced. the value of Q that introduces minimal frequency smearing bet still

allows signal reconstruction is

1

(2 1)B

qQ

(11)

where q ( 0 1q ) is a scaling factor, and typically 1q .

Values of q smaller than 1 can be used to improve the time resolution at the cost of degrading the

frequency resolution.

Eq(11) is substituted in Eq(10), and kN is obtained,

1

(2 1)

sk

Bk

qfN

f

(12)

By employing Eq(8)-Eq(12), the sequence of decomposition atoms can be obtained. Based on this,

frame operators S can be got with Eq. (4). Furthermore, the sequence of reconstruction atoms can be

obtained with Eq. (5). To decrease the computation intensity, FFT-based processing is applied, so the

computation time keeps low, and more detail can be found in Ref. [7].

3. THE EXPERIMENT DEVICES

3.1 Experimental device

In order to obtain the vibration properties of a certain type of deployable structure, a small-scale

model array, loosely based on a type of retractable rigid array, is manufactured to conduct the testing

experiment, and it is shown in Figure 1. Besides, the size diagram of the model array is shown in

Figure 2. It consists of inner and outer steel panel, connected by a pair of cylinder hinge, and the shaft

sleeves of the revolute pairs are made of brass. The size of the two panels are approximately

300×150×2.5mm. The material of the steering king pin 45#steel, with diameters 7.96mm，7.8mm and

7.2 mm, so the radius of the corresponding clearances are 0.02mm, 0.1 mm and 0.40 mm, and the

standard tolerance is 0.02mm. And the whole model is fixed in foundation bed by a cylinder hinge.

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Unfolded

(a) (b)

(c)

FoldedCylinder hinge

Figure 1 – The experimental model: (a) Cylinder hinge, (b) Folded, (c) Unfolded

56

150

150

56

131

300290126

加速度传感器

Y

Z

X

FED

J2

CBA

J1

Acceleration sensor

Outer panelInner panel

Figure 2 – The size diagram of the model array

3.2 The signal collection devices

The schematic diagram of the testing system is illustrated in Figure 3. To obtain the vibration signal

of the folded and deployed model array, a rubber hammer is used to knock on the array, and then two

pressure acceleration senores are employed to collect the acceleration of the vibration of the inner

panel and the outer panel, while the impact force produced by the rubber is collected by a piezoelectric

pressure sensor. After that two signals are filtered and amplified by a charge amplifier, and are, finally,

transmitted to a digital collection device.

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Pressure

acceleration

sensor

Piezoelectric

pressure sensor

Charge

amplifier

Digital

collection

device

Model array

Rubber

hammer

Figure 3 –The schematic diagram of the testing system

4. EXPERIMENT RESULTS AND ANALYSIS

To guarantee to fully obtain the vibration behavior of the model, the experiment consist of two

sections, and they are as follows: (1) impact forces are acted on inner panel or outer panel; (2) the

deployable structure are in different working situation (folded and unfolded). The experiment results

and corresponding analysis will be presented in following chapter.

4.1 Vibration collection positions and knocking positions in the inner panel or the outer

panel

Vibration characters of the inner panel and outer panels are different. To catch the change, the

experiments are designed as follows: when the model is unfolded, drub the model array in the point D,

and collect vibration signal in the points B and E (the point B is the mass center of inner panel, its

vibration characteristic is very representative for the inner panel, and the same to the point E and the

outer panel.). And the drubbing direction is along Y axis (as shown in Figure 3), and vibration

accelerations along Y axis are collected by the acceleration sensor. The clearance radius is 0.02mm.

The vibration acceleration signals in time domain, the corresponding FFT spectrums and

time-frequency distribution pictures are presented in Figures 4-6.

As can be seen in Figures 5 and 6, the energy of the signal collected from the point B concentrate

on few frequencies. Therein, the energy of 39.7 and 140.1 Hz (The frequencies are 40.3 and 140.5

Hz in the CQ-NSGT representation) are the relatively stronger than the others. It seems that the two

frequencies are the inherent frequency of the vibration mode, and 13.7 Hz is the first natural

frequency, and 140.1 Hz is a frequency doubling along Y axis. This phenomenon seems to suggest

that the effect of the clearance for the vibration model of the inner panel is small, at least is less than

that for the outer panel, which can be seen in Figures 5(b) and 6(b). The vibration energy of the outer

panel is in a more dispersive distribution. For a quantitative analysis, we calculate the standard

deviation of the two Fourier spectrums show in Figure 6, and the value of the inner panel is 8.37×10-7，

and the value of the outer panel is 5.43×10-7

, which also prove this phenomenon. Therefore, a greater

change occur in the vibration model of the outer panel.

Figure 7 shows the amplitude variations at 13.9, 40.3 and 183.8 Hz along with time of the CQ-NSGT

representations of the experimental signals collected from the point B. For a convenient contrast, a

conduction of normalization is done for the amplitude variations, and the equation is

0

( , )( , )

max( ( , ) )

CQ

freCQ

fre CQ

fre

X k nX k n

X k n (13)

where, 0 ( , )CQ

freX k n is the normalized amplitude variation at a certain frequency, and ( , )CQ

freX k n is

the CQ-NSGT representation of the experimental signal at a certain frequency. As revealed in Figure 7,

the amplitude at a high frequency decrease at a relatively higher speed, which indicates that a high

vibration frequency generally leads to a great attenuation rate.

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(a) (b) Figure 4 –The vibration signal in time domain: (a) point B and (b) point E. Points B and E are the mass

center of the inner panel and outer panel.

141.4 Hz

13.4 Hz

39.4 Hz

60.8 Hz

82.3 Hz

183.8 Hz

367.1 Hz

425.3 Hz

13.9 Hz

183.8 Hz

24.3 Hz

40.3 Hz

140.5 Hz

61.5 Hz

(a) (b)

Figure 5 –The CQ-NSGT representations of the experimental signals: (a) point B and (b) point E.

13.7

39.7

140.1

184107.760.424.1

(a)

39.0

13.3

60.3

81.9

141.0

183.8

366.5 424.5

(b) Figure 6 –The Fourier spectrums of the experimental signals: (a) point B and (b) point E.

Figure 7 –The amplitude variations of 13.9, 40.3 and 183.8 Hz along with time of the CQ-NSGT

representations of the experimental signals collected from the point B.

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4.2 Different working situations (unfolded and folded)

To obtain the vibration characteristics of the deployable structure at different work situation , a

targeted test is taken as an example to carried out, and as follows: The signal collection position is on

E point, and knocking position is on D point, which is conducted as the model panel is folded and

unfolded. The knocking direction and the vibration direction are also along the Y axis. The FFT

spectrum of the signal are shown in Figure 8. As can be seen in Figure 8, in contrast to deployed state,

vibration energy in the folded situation is in a more dispersed distribution, which indicates that a more

serious wide-frequency phenomenon occur in the folded situation than at the unfolded situation, which

reveals that the effect of the clearance is stronger in the folded situation. For a quantitative analysis, we

calculate the standard deviation of the two Fourier spectrums show in Figure 8, and the value of the

inner panel is 5.18×10-7， and the value of the outer panel is 2.11×10

-7, which also prove this

phenomenon.

(a) (b) Figure 8 –The Fourier spectrums of the experimental signals collected from Point E at different work

situation: (a) folded and (b) unfolded.

5. CONCLUSIONS

In order to investigate vibration properties of the rigid deployable structure, a model array is made,

loosely based on a certain type of solar cell array, to conduct hammer impact test. The CQT based on

nonstationary Gabor frame is introduced to study the time-frequency distribution characteristics of the

vibrations, and Fourier transform is used to analyze the frequency distribution characteristics. The

experiment is conducted. In the experiment, the signal collection position and the working state for the

time-frequency properties of the vibration signals are analyzed, and the experiment results seem to

suggest following conclusions: (1) The damping will become greater along with the vibration

frequencies increasing, which is proved by the experiment results that the greater vibration frequency

is, the shorter the corresponding vibration time is. (2) The vibration of the outer panel tend to be more

random, which indicate that the effect of the clearance is stronger for the outer panel. (3) In contrast to

deployed state, vibration energy in folded situation is in more dispersed distribution , which indicate

that the effect of the clearance is stronger in the folded situation. The conclusions above may be

helpful to study the dynamic performance of this type of deployable structures.

In this paper, qualitative researches are done for the vibration properties of the deployable structure.

Estimation of dynamic parameters of the model array such as damping, stiffness and other dynamic

parameters, are further researching focuses.

ACKNOWLEDGEMENTS

This work is supported by National Science Foundation of China under Contract No. 11272171,

Education Ministry Doctoral Fund of China under Contract No. 20120002110070, and Beijing Natural

Science Foundation under Contract No. 3132030.

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