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Dynamic response analysis of closed-loop control system for random intelligent truss structure under random forces
ARTICLE IN PRESS
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 18 (2004) 947–957 www.elsevier.com/locate/jnlabr/ymssp
Dynamic response analysis of closed-loop control system for random intelligent truss structure under random forces Wei Gaoa,*, Jianjun Chena, Yabin Zhoub, Mingtao Cuia a
School of Electronic Mechanical Engineering, Xidian University, Xi’an 710071, People’s Republic of China School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China
b
Received 11 September 2002; received in revised form 2 June 2003; accepted 28 July 2003
Abstract Considering the randomness of structural damping, physical parameters of structural materials, geometric dimensions of active bars and passive bars, applied loads and control forces simultaneously, the problems of dynamic response analysis of closed-loop control system based on probability for the random intelligent truss structures are studied in this paper. The computational expressions of numerical characteristics of structural dynamic response of closed-loop control system are derived by means of the mode superposition method. Through the engineering examples, the influences of the randomness of them on structural dynamic response are inspected and some significant conclusions are obtained. r 2003 Elsevier Ltd. All rights reserved. Keywords: Dynamic response analysis; Intelligent truss structures; Random factor method
1. Introduction Dynamic response analysis of closed-loop control system for intelligent structure is an important segment in the process of intelligent structural design and its shape and vibration control. The results of the dynamic response analysis are the important base of the determination of the active bars’ location. So far, however, almost of modelling on intelligent structural dynamic analysis basically belongs to the determinate models, that is, all structural parameters are regarded as determinate ones. Apparently, this kind of model cannot reflect the influence of the randomness of intelligent structure on the structural dynamic characteristic. As a matter of fact, in some situations the randomness of structure must be considered, especially in design stage. Such as, for one kind numerous or batch producing structures, their values of physical parameter of *Corresponding author. E-mail address: [email protected] (W. Gao). 0888-3270/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0888-3270(03)00100-6
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material has dispersivity, there is error on the geometric dimension of structural number in process of manufacture and assemblage, the support boundary condition is indeterminate, and so on. Therefore, studying the problem of random intelligent structural dynamic analysis is of much realistic engineering background and important theoretic signification. Structural dynamic analysis includes structural dynamic character and response analysis. Since 1970 decade, some research works about the eigenvalue problem of structure with random parameters has been published successively [1–4]. Their essential treatments are that the random factors in structure are described with small parameter, and the problem is solved by means of FEM or matrix perturbation method. Benaroya and Rehak [5] reviewed on some problems of finite element methods in probabilistic structural analysis. Zhu and Wu analysed the real eigenvalue problem by using the stochastic finite element method [6]. Recent years, Wu and Yang [7] took the randomness of structural parameter as the perturbation quantity nearby the mean value of random variables, and obtained the perturbing expression of structural dynamic characteristic by using the variation principle of Rayleigh’s quotient. Li presented a kind of modelling method with two-phase iterations for random parameter system based on the extendedorder method in random structural analysis [8]. Through the method of generalised sample subdomains, Wang and Mei [9] set-up the perturbation analysing formula of random eigenvalue for the cooling tower with random parameter, and discussed the effect of statistical characteristic of random parameter of the boundary support in tower ground on structural eigenvalue. Chen and Che [10] studied the structural dynamic characteristic, in which the randomness of both physical parameters (elastic module and mass density) of structural materials and geometric dimension (length) of bars were considered. In this paper, intelligent truss structures are taken as analysing objects and their structural dynamic modelling and analysing based on probability are studied. The computational expressions of numerical characteristics of structural dynamic response of closed-loop control system are derived by means of the mode superposition method, in which the randomness of structural damping (modal damping), physical parameters (elastic module and mass density) of structural materials, geometric dimensions (length and cross-section area) of active and passive bars, applied loads and control forces are considered simultaneously. Through engineering examples, the influences of them on structural dynamic response are inspected and some significant conclusions are obtained.
2. Structural dynamic response analysis of closed-loop control system Following the finite element formulation described in Ref. [11], the equation of motion for an intelligent structure is given by ½MfxðtÞg þ ½CfxðtÞg þ ½KfxðtÞg ¼ fF ðtÞg þ ½B1 fFP ðtÞg; . ’
ð1Þ
where ½M; ½C and ½K are the mass, damping and stiffness matrices, respectively. fxðtÞg; fxðtÞg ’ and fxðtÞg are structural displacement, velocity and acceleration vectors, respectively. fFðtÞg is . load force vector. The fFP ðtÞg term represents the initial strain effect of piezoelectric active bar. Since fFP ðtÞg appears on the right-hand side of the equation of motion, it is referred to as the
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apparent active bar force, that is, control force. The ½B1 matrix consists of the active bar’s direction cosines. By means of the mode superposition method, and the Wilson’s damping hypothesis will be adopted. The structural displacement response can be expressed as fxðtÞg ¼
n X
ffgi zi ðtÞ;
ð2Þ
i¼1
where the displacement response of jth degree of freedom is xj ðtÞ ¼
n X
fji zi ðtÞ;
j ¼ 1; 2; y; n;
ð3Þ
i¼1
1 zi ðtÞ ¼ o %i
Z
t 0
ðRi ðtÞ þ Hi ðtÞÞ exp½ xi oi ðt tÞ sin o % i ðt tÞ dt;
Ri ðtÞ ¼ ffgTi fF ðtÞg;
Hi ðtÞ ¼ ffgTi ð½B1 fFP ðtÞgÞ;
i ¼ 1; 2; y; n;
i ¼ 1; 2; y; n;
ð4Þ ð5Þ
where Ri ðtÞ; Hi ðtÞ and zi ðtÞ are load, control force and displacement response of ith degree of freedom in principal coordinates, respectively. oi ; ffgi and xi are ith order inherence frequency, modal shape and modal damping of structure, respectively. o %i ¼ % i ¼ oi ð1 x2i Þ1=2 : Since xi 51; o oi can be obtained. After the structural displacement response of closed-loop system obtained, the stress response of the eth element can be expressed as fsðtÞðeÞ g ¼ E ½B fxðtÞðeÞ g
ðe ¼ 1; 2; y; neÞ;
ð6Þ
where fxðtÞðeÞ g is the displacement response of the nodal point of the eth element, fsðtÞðeÞ g is the stress response of the eth element, ½B and E are the geometric matrix and elastic module of the eth element, respectively.
3. Probabilistic structural dynamic response analysis of close-loop system Suppose that there are ne elements in the analysed intelligent truss structure. In the structure, any element can be taken as passive bar or active bar (the structure of active bar can be seen in Ref. [12]). Piezoelectric bar is utilised as active bar. In order to make use of the united form to express the structural stiffness and mass matrix, a kind of mixed element have been constructed. A Boolean algebra value named as y is introduced in the mixed element, when y ¼ 0; the mixed element is active element bar; when y ¼ 1; the mixed element is passive element bar. In the following, expressions of the stiffness matrix ½K and mass matrix ½M of intelligent truss structure in global coordinate will be developed by use of this kind of mixed element. (" # ) ðeÞ ðeÞ 2 ðeÞ ðeÞ ðeÞ Xne Xne c þ ðe Þ =e E A 33 33 ðeÞ ð7Þ ½K ðeÞ ¼ y m ðeÞ m þ ð1 yÞ 33 AP ½G ½K ¼ ðeÞ e¼1 e¼1 lm lP
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½M ¼
ne X e¼1
ne X 1 ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ½M ¼ ðyrm Am lm þ ð1 yÞrP AP lP Þ½I 2 e¼1 ðeÞ
ð8Þ
where ½K ðeÞ is the eth element’s stiffness matrix, ½M ðeÞ is the eth element’s mass matrix. ½I is a 6 ðeÞ ðeÞ order identity matrix. rðeÞ m ; Am and lm are the eth passive bars’ mass density, cross-section area ðeÞ ðeÞ and length, respectively. rP ; AP and lPðeÞ are the eth active bars’ mass density, cross-section area ðeÞ ðeÞ and length, respectively. EmðeÞ is the eth passive bars’ elastic module. cðeÞ 33 ; e33 and e33 are elastic module, piezoelectric force/electrical constant and dielectric constant, respectively. ½G is a 6 6 matrix, where g11 ¼ g44 ¼ 1; g14 ¼ g41 ¼ 1; other elements of ½G are all equal to 0. Let ðeÞ 2 ðeÞ ð9Þ EPðeÞ ¼ cðeÞ 33 þ ðe33 Þ =e33 : EPðeÞ just is the generalised elastic module of piezoelectric active bars while considering the mechanic–electronic coupling effect. Substituting Eq. (9) into Eq. (7), then (" # ) ne ne X X EmðeÞ Am EPðeÞ AP ðeÞ ½G : ð10Þ ½K ¼ y þ ð1 yÞ ½K ¼ lm lP e¼1 e¼1 ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ Considering the randomness of xi ; rðeÞ m ; rP ; Am ; AP ; lm ; lP ; Em ; c33 and fFðtÞg simultaneously, that is, they are all random variables. The amplitude of fF ðtÞg is obeyed a normal distribution. From Eq. (9), it can be obtained easily that EPðeÞ is random variable. In the closed-loop system, because the production and response process of fFP ðtÞg determined by fFðtÞg; fFP ðtÞg is random variable too, and these two variables are full positive correlation. In realistic engineering examples, normal distribution is a wide probabilistic distribution, and other probabilistic distributions can be equivalently transformed into normal distribution. Therefore, suppose that these random variables all obey normal distribution and also the amplitude of fF ðtÞg obeys a normal distribution. For linear structures, the structural displacement responses obey normal distribution too. The engineering background of this case is the stochastic loads and control forces act on the intelligent structure with random parameters. This case is the most general one of random model of intelligent structural dynamic response analysis. Suppose the length and cross-section areas of bars are two kinds of random variables and the dispersivity of each kind of random variables of each bar is equal, respectively. Then the length and cross-section area of the eth element can be written, respectively, as: l ðeÞ ¼ lðeÞ l; AðeÞ ¼ ZðeÞ Aðe ¼ 1; y; neÞ; where lðeÞ and ZðeÞ are the determinate quantities that denote the nominal length and nominal cross-section area of eth bar, respectively. l is the random variable factor of all bars’ length, its mean value is 1.0 and variance is v2l ; A is the random variable factor of all bars’ cross-section, its mean value is 1.0 and variance is v2A : If the physical parameters of each active element are equal, the physical parameters of each passive element are also equal, rm ¼ rðeÞ m ; rP ¼ ðeÞ ðeÞ ðeÞ rP ; Em ¼ Em ; EP ¼ EP can be obtained easily. Therefore, the matrices ½K and ½M can be expressed, respectively, as ! ðeÞ ne ne ðeÞ X X E A E A ð11Þ ½K ¼ ½K ðeÞ ¼ y m ½KmðeÞ # þ ð1 yÞ P ½KPðeÞ # ; l l e¼1 e¼1
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where ½KmðeÞ # and ½KPðeÞ # are determinate part in stiff matrices ½KmðeÞ and ½KPðeÞ ; respectively, just taking the parameters as lmðeÞ ¼ lðeÞ ; AðeÞ ¼ ZðeÞ ; EmðeÞ ¼ 1 and EPðeÞ ¼ 1: ne ne X X ðeÞ ðeÞ # ðeÞ # ½M ¼ ½M ðeÞ ¼ ðyrðeÞ ð12Þ m A l½Mm þ ð1 yÞrP A l½MP Þ; e¼1
e¼1
where and are determinate part in mass matrices ½MmðeÞ and ½MPðeÞ ; respectively, just ðeÞ taking the parameters as: lmðeÞ ¼ lðeÞ ; AðeÞ ¼ ZðeÞ ; rðeÞ m ¼ 1 and rP ¼ 1: If the physical parameters’ variation of active and passive elements are equal to each other, rP ¼ k1 rm ¼ k1 r and EP ¼ k2 Em ¼ k2 E (k1 and k2 are constant ratio factors) can be easily obtained. Therefore, the matrices ½K and ½M can be expressed, respectively, as ne ne X X EA ðeÞ # EA ðeÞ # ðeÞ ½Km þ ð1 yÞk2 ½Km ½K ¼ ½K ¼ y l l e¼1 e¼1 ½MmðeÞ #
½MPðeÞ #
EA ½y þ ð1 yÞk2 ½Km # ; l ne ne X X ½M ðeÞ ¼ ðyr A l ½MmðeÞ # þ ð1 yÞk1 r A l ½MmðeÞ # Þ ½M ¼ ¼
e¼1
ð13Þ
e¼1
¼ r A l ½y þ ð1 yÞk1 ½Mm # :
ð14Þ
The randomness of physical parameters and geometric dimension will lead the structural inherence frequency o as well as randomness. Therefore, the randomness of the structural dynamic characteristics, damping, loads and control forces will lead the structural dynamic response (dynamic displacement and dynamic stress) as well as randomness. The statistical description of random variables represented by utilising its numerical characteristic. In the following, the expressions of numeral characteristics of dynamic response will be derived. 3.1. Numerical characteristics of dynamic displacement response of closed-loop system From Eq. (5), the mean value mRi ðtÞ and variance s2Ri ðtÞ of Ri ðtÞ and the mean value mHi ðtÞ and variance s2Hi ðtÞ of Hi ðtÞ in principal coordinates can be deduced by means of the algebra synthesis method: mRi ðtÞ ¼ fmf gTi fmF ðtÞ g;
ð15Þ
s2Ri ðtÞ ¼ fm2f gTi fm2F ðtÞ gðv2FðtÞ þ v2o þ v2FðtÞ v2o Þ;
ð16Þ
mHi ðtÞ ¼ fmf gTi fmFP ðtÞ g;
ð17Þ
s2Hi ðtÞ ¼ fm2f gTi fm2FP ðtÞ gðv2FP ðtÞ þ v2o þ v2FP ðtÞ v2o Þ;
ð18Þ
where voi ; vF ðtÞ and vFP ðtÞ are variation coefficient of oi ; fF ðtÞg and fFP ðtÞg; respectively. From Eq. (4), the mean value mZi ðtÞ and mean variance sZi ðtÞ of the displacement of ith degree of freedom in principal coordinate can be deduced by means of the random variable’s functional moment method: Z t 1 mzi ðtÞ ¼ ðm þ mHi ðtÞ Þ expð mx moi ðt tÞÞ sin moi ðt tÞ dt; ð19Þ moi 0 Ri ðtÞ
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szi ðtÞ
( Z t 1 2 ¼ vo ðm þ mHi ðtÞ Þ expð mx moi ðt tÞÞ sin moi ðt tÞ dt moi 0 Ri ðtÞ Z t mx ðmRi ðtÞ þ mHi ðtÞ Þðt tÞ expð mx moi ðt tÞÞ sinmoi ðt tÞ dt þ
0
Z
t 0
2 ðmRi ðtÞ þ mHi ðtÞ ðt tÞ expð mx moi ðt tÞÞ cos moi ðt tÞ dt
1 þ 2 fm2f ÞTi ½fm2FðtÞ gðn2FðtÞ þ n2oi þ n2F ðtÞ n2oi Þ moi þfm2FP ðtÞ gðn2FP ðtÞ þ n2oi þ n2FP ðtÞ n2oi Þ þ 2fmF ðtÞ gfmFP ðtÞ g ððn2F ðtÞ þ n2oi þ n2F ðtÞ n2oi Þðn2FP ðtÞ þ n2oi þ n2FP ðtÞ n2oi ÞÞ1=2
Z t 2 expð mx oi ðt tÞÞ sin moi ðt tÞ dt 0
þ
n2xi m2xi
1 moi
Z
t
0
2 )1=2 ðmRi ðtÞ þ mHi ðtÞ Þ expð mx moi ðt tÞÞ sin moi ðt tÞ dt ;
ð20Þ
where the symbol m; s and n denote random variable’s mean values, mean variances and variation coefficients, respectively. mFðtÞ ; vF ðtÞ ; mFP ðtÞ and vFP ðtÞ are given by the statistical information of the loads and control forces. The fmf gi term can be gained from the structural dynamic characteristic computation. moi ; s2oi and voi can be obtained from the following equations: !1=2 m E ½1 þ n2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r moi ¼ o#i mr mZ CEr ðn2A þ n2E þ n2A n2E Þ1=2 ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 2 1 ½2n2A þ n2E þ n2Z þ n2r þ n2A n2Z þ n2A n2E þ n2A n2r þ n2Z n2r þ n2A n2Z n2r 2
2CEr ðn2A
s2oi ¼ ðo#i Þ2
þ
n2E
þ
n2A n2E Þ1=2 ðn2A
þ
n2Z
þ
n2r
þ
n2A n2Z
þ
n2A n2r
þ
n2Z n2r
1=4
þ
n2A n2Z n2r Þ1=2
! mE n ½1 þ n2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r mr mZ
CEr ðn2A þ n2E þ n2A n2E Þ1=2 ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 f½1 þ n2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r CEr ðn2A þ n2E þ n2A n2E Þ1=2 ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 2 1 ½2n2A þ n2E þ n2Z þ n2r þ n2A n2Z þ n2A n2E þ n2A n2r þ n2Z n2r 2
; ð21Þ
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þ n2A n2Z n2r 2CEr ðn2A þ n2E þ n2A n2E Þ1=2
o ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 g1=2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4v2l þ 2v4l ; vz ¼ vl 2 ¼ 1 þ v2l so noi ¼ i ; moi
ð22Þ
ð23Þ ð24Þ
where vE and vr are variation coefficient of E and r; respectively. CEr is the correlation coefficient of variables E and r o#i can be obtained from the structural dynamic characteristic computation. From Eqs. (19) and (20), the variation coefficient of zi ðtÞ can be obtained as ð25Þ vzi ðtÞ ¼ szi ðtÞ =mzi ðtÞ : From Eq. (2), the mean value and mean variance of displacement response in nature coordinate can be deduced by means of the algebra synthesis method: n X fmf gi mzi ðtÞ ; ð26Þ fmxðtÞ g ¼ i¼1
fsxðtÞ % g¼
( n X
)1=2 ðfm2f gi m2zi ðtÞ ðv2zi ðtÞ þ v2oi þ v2oi v2zi ðtÞ ÞÞ
:
ð27Þ
i¼1
From above formulas, it can be easily seen that the randomness of the structural displacement response are dependent on not only the structural physical parameters xi ; E; r and the geometric % A% but also the randomness of loads and control forces. In addition, they are determined dimension L; by the values of the modal damping, inherence frequency, mode shape, loads and control forces. 3.2. Numeral characteristics of dynamic stress response of closed-loop system After the displacement responses of the element’s nodal are obtained, the mean value and mean variance of the eth element’s stress response can be obtained by means of the algebra synthesis method from equation ðeÞ fmðeÞ sðtÞ g ¼ ½BmE fmxðtÞ g
ðe ¼ 1; 2; y; ne Þ;
ðeÞ 2 2 2 2 1=2 fsðeÞ sðtÞ g ¼ ½BmE fmxðtÞ g½vfxðtÞðeÞ g þ vE þ vE vfxðtÞðeÞ g
ð28Þ ðe ¼ 1; 2; y; ne Þ:
ð29Þ
4. Examples In the following examples, for the intelligent structure’s active bars, a constant output velocity feedback control law is selected [12]. Active bar’s and passive bar’s materials and their parameters’ value are given in Table 1. Example 1. 20-bar planar truss structure (Fig. 1).
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The elastic module E; mass density r; bars’ length L% and bars’ cross-section area A% are all random variables and obeyed normal distribution. A step load along with the positive direction of x-axis is acted on the nodal 9 of the structure. The amplitude of the step load is a normal random variable. Its mean value is mF ðtÞ ¼ 5000ðNÞ: The gain of the closed-loop control system is g ¼ 50: Now, the first element and 20th element are utilised as piezoelectric active bar, respectively. The corresponding computational results of the structural dynamic displacement response and dynamic stress response are given in Table 2. In this table, the dynamic displacement response is that of the nodal 12, the dynamic stress response is that of the first element. In order to compare, two kinds of models, the determinate model and random model, are adopted, respectively, in computational process. In the determinate model, the mean values of all random variables are regarded as determinate quantity, and their variation coefficients are taken as 0. In the random model, three situations are considered, respectively: (1) the physical parameters and geometric dimensions of the structure are random variables, (2) the modal damping, loads and the control forces of the closed-loop system for the structure are random variables, (3) the physical parameters, geometric dimensions, modal damping, loads and control forces are simultaneously random variables. That is, random model (I), random model (II) and random model (III), respectively. It can be seen from Table 2 that under the conditions that the variation coefficients of physical parameters and geometric dimensions are equal to the variation coefficients of the damping, loads and control forces, the randomness of physical parameters and geometric dimensions will produce greater effect on the randomness of structural stress response than the one produced by the randomness of the damping, loads and control forces. However, the randomness of the damping, loads and control forces will produce greater effect on the randomness of structural displacement Table 1 Intelligent truss structural physical parameters Active bar (PZT-4) Mean value of mass density r Mean value of elastic module c33 Piezoelectric force/electric constant e33 Dielectric constant e33 Cross-section area A
7800 kg=m3 2:1 1011 N=m2 — — 3:0 10 4 m2
7600 kg=m 8:807 1010 N=m2 18:62 C=m2 5:92 10 9 C=V m 3:0 10 4 m2
•
1
Passive bar (steel)
3
•
•
3
•
2 • 250
•
4
•
5
• •
250
(unit: mm)
•
6
•
7
• •
250
•
8
•
9
• •
250
•
10
11
• •
250 12
Y
250 X
Fig. 1. 20-bar planar intelligent truss structure.
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Table 2 The computational results of dynamic response of 20-bar planar structure Model
Dynamic response Maximal displacement response (mm)
Maximal stress response (Mpa)
First element act as active bar
20th element act as active bar
First element act as active bar
20th act as active bar as active bar
Determinate model
5.046
7.397
50.729
81.301
Random model (I) vE ¼ vr ¼ nl ¼ nA ¼ 0:1 nxi ¼ vF ðtÞ ¼ nFP ðtÞ ¼ 0
mzmax ðtÞ ¼ 5:046 szmax ðtÞ ¼ 0:1797
mzmax ðtÞ ¼ 7:397 szmax ðtÞ ¼ 0:2635
msmax ðtÞ ¼ 50:729 ssmax ðtÞ ¼ 6:138
msmax ðtÞ ¼ 81:301 ssmax ðtÞ ¼ 9:837
Random model (II) vE ¼ vr ¼ nl ¼ nA ¼ 0 nxi ¼ vF ðtÞ ¼ nFP ðtÞ ¼ 0:1
mzmax ðtÞ ¼ 5:046 szmax ðtÞ ¼ 0:4087
mzmax ðtÞ ¼ 7:397 szmax ðtÞ ¼ 0:5992
msmax ðtÞ ¼ 50:729 ssmax ðtÞ ¼ 4:053
msmax ðtÞ ¼ 81:301 ssmax ðtÞ ¼ 6:496
Random model (III) vE ¼ vr ¼ nl ¼ nA ¼ nxi ¼ vF ðtÞ ¼ nFP ðtÞ ¼ 0:1
mzmax ðtÞ ¼ 5:046 szmax ðtÞ ¼ 0:4385
mzmax ðtÞ ¼ 7:397 szmax ðtÞ ¼ 0:6428
msmax ðtÞ ¼ 50:729 ssmax ðtÞ ¼ 8:421
msmax ðtÞ ¼ 81:301 ssmax ðtÞ ¼ 13:496
response than the one produced by physical parameters and geometric dimensions. In addition, comparing with the conditions that only the randomness of the physical parameters and geometric dimensions or the randomness of the damping, loads and control forces are considered, under the condition that their randomness are all considered the randomness of the structural displacement and stress response are greater. Example 2. 4-bar space truss structure (Fig. 2). A step load along with the negative direction of z-axis is acted on the nodal 5 of the structure. The amplitude of the step load is a normal random variable. Its mean value is mF ðtÞ ¼ 75 000ðNÞ: The gain of the closed-loop control system is g ¼ 100: Now, the 4th element is used as piezoelectric active bar. The computational results of the dynamic displacement response and dynamic stress response of the structure are given in Table 3. In Table 3, the dynamic displacement response is that of nodal 5, the dynamic stress response is that of first element. In order to investigate the effect of the dispersal degree of random variables E; r and dimension random variable factor l; A and xi ; loads, control forces on the structural dynamic characteristic, the values of variation coefficients of parameters E; r; l; A; xi ; F ; FP are, respectively, taken as two groups. Group I : nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:01: Group II: nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:1: In order to compare, two kinds of models, the determinate and random models, are adopted, respectively, in computational process too. It can be seen from Table 3 easily that changing the variation coefficients of physical parameters, geometric dimensions, damping, loads and control forces will produce considerable effect on the computational results of structural dynamic response. Along with the increase of the
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400
Z
•
2
350 Y
5 •
200
3
•
1
•
X
150 350
4
(unit: mm) Fig. 2. 4-bar space intelligent truss structure.
Table 3 The computational results of dynamic response of 4-bar space structure Model
Dynamic response Maximal displacement response (mm)
Maximal stress response (mm)
Determinate model
2.15
41.29
Random model (I) nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:01
mxmax ðtÞ ¼ 2:15 sxmax ðtÞ ¼ 0:1339
msmax ðtÞ ¼ 41:29 ssmax ðtÞ ¼ 0:4874
Random model (II) nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:1
mxmax ðtÞ ¼ 2:15 sxmax ðtÞ ¼ 0:1868
msmax ðtÞ ¼ 41:29 ssmax ðtÞ ¼ 6:854
variation coefficients of physical parameters, geometric dimensions, damping, loads and control forces, the dispersal degree of structure’s dynamic response will notably increase.
5. Conclusions (1) The examples show that the analysing results of intelligent structural dynamic response of the determinate model are different from the one of the random model. So that when one of the physical parameters, geometric dimensions, damping, loads and control forces is random variable, the conventional determinate analysis method of structural dynamic response will not reflect the effect of the randomness. It is only dependent on the structural dynamic response analysis method based on probability. (2) The examples show that the model and solving method of dynamic response analysis of intelligent truss structure based on probability presented in this paper are rational and feasible.
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References [1] C.J. Astill, M. Shinozuka, Random eigenvalue problems in structural analysis, AIAA Journal 10 (1972) 456–562. [2] R. Vaicaitis, Free vibrations of beams with random characteristics, Journal of Sound and Vibration 35 (1974) 13–21. [3] J.V. Scheidt, W. Purkert, Random Eigenvalue Problem, Elsevier, New York, 1983. [4] M. Kaminski, Perturbation based on stochastic finite element method homogenization of two-phase elastic composites, Computers and Structures 78 (6) (2000) 811–826. [5] H. Benaroya, M. Rehak, Finite element methods in probabilistic structural analysis: a selective review, Applied Mechanics Review ASME 41 (1988) 201–213. [6] W.Q. Zhu, W.Q. Wu, A stochastic finite element method for real eigenvalue problem, Problems in Engineering and Mechanics 118 (1992) 496–511. [7] X.F. Wu, G.S. Yang, Statistical analysis of the dynamic characteristics of structures including random parameters, Journal of Beijing Science and Technical University 16 (1996) 43–47. [8] J. Li, A research on modeling stochastic structures, Journal of Vibration Engineering 9 (1996) 46–53. [9] X.T. Wang, Z.X. Mei, Random eigenvalue analysis of cooling tower on the spline subdomain random perturbation methods, Computational Structural Mechanics and Application 14 (1997) 174–181. [10] J.J. Chen, J.W. Che, H.A. Sun, H.B. Ma, M.T. Cui, Probabilistic dynamic analysis of truss structures, Structural Engineering and Mechanics 13 (2002) 231–239. [11] Gun-Shing Chen, Robin J. Bruno, Moktar Salama, Optimal placement of active/passive members in structures using simulated annealing, AIAA Journal 29 (1991) 1327–1334. [12] Nie Runtu, Shao Chengxun, Zou Zhenzu, Dynamic modelling and vibration control of piezoelectric truss structures, Journal of Astronautics 19 (1998) 8–14.
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 18 (2004) 947–957 www.elsevier.com/locate/jnlabr/ymssp
Dynamic response analysis of closed-loop control system for random intelligent truss structure under random forces Wei Gaoa,*, Jianjun Chena, Yabin Zhoub, Mingtao Cuia a
School of Electronic Mechanical Engineering, Xidian University, Xi’an 710071, People’s Republic of China School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China
b
Received 11 September 2002; received in revised form 2 June 2003; accepted 28 July 2003
Abstract Considering the randomness of structural damping, physical parameters of structural materials, geometric dimensions of active bars and passive bars, applied loads and control forces simultaneously, the problems of dynamic response analysis of closed-loop control system based on probability for the random intelligent truss structures are studied in this paper. The computational expressions of numerical characteristics of structural dynamic response of closed-loop control system are derived by means of the mode superposition method. Through the engineering examples, the influences of the randomness of them on structural dynamic response are inspected and some significant conclusions are obtained. r 2003 Elsevier Ltd. All rights reserved. Keywords: Dynamic response analysis; Intelligent truss structures; Random factor method
1. Introduction Dynamic response analysis of closed-loop control system for intelligent structure is an important segment in the process of intelligent structural design and its shape and vibration control. The results of the dynamic response analysis are the important base of the determination of the active bars’ location. So far, however, almost of modelling on intelligent structural dynamic analysis basically belongs to the determinate models, that is, all structural parameters are regarded as determinate ones. Apparently, this kind of model cannot reflect the influence of the randomness of intelligent structure on the structural dynamic characteristic. As a matter of fact, in some situations the randomness of structure must be considered, especially in design stage. Such as, for one kind numerous or batch producing structures, their values of physical parameter of *Corresponding author. E-mail address: [email protected] (W. Gao). 0888-3270/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0888-3270(03)00100-6
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material has dispersivity, there is error on the geometric dimension of structural number in process of manufacture and assemblage, the support boundary condition is indeterminate, and so on. Therefore, studying the problem of random intelligent structural dynamic analysis is of much realistic engineering background and important theoretic signification. Structural dynamic analysis includes structural dynamic character and response analysis. Since 1970 decade, some research works about the eigenvalue problem of structure with random parameters has been published successively [1–4]. Their essential treatments are that the random factors in structure are described with small parameter, and the problem is solved by means of FEM or matrix perturbation method. Benaroya and Rehak [5] reviewed on some problems of finite element methods in probabilistic structural analysis. Zhu and Wu analysed the real eigenvalue problem by using the stochastic finite element method [6]. Recent years, Wu and Yang [7] took the randomness of structural parameter as the perturbation quantity nearby the mean value of random variables, and obtained the perturbing expression of structural dynamic characteristic by using the variation principle of Rayleigh’s quotient. Li presented a kind of modelling method with two-phase iterations for random parameter system based on the extendedorder method in random structural analysis [8]. Through the method of generalised sample subdomains, Wang and Mei [9] set-up the perturbation analysing formula of random eigenvalue for the cooling tower with random parameter, and discussed the effect of statistical characteristic of random parameter of the boundary support in tower ground on structural eigenvalue. Chen and Che [10] studied the structural dynamic characteristic, in which the randomness of both physical parameters (elastic module and mass density) of structural materials and geometric dimension (length) of bars were considered. In this paper, intelligent truss structures are taken as analysing objects and their structural dynamic modelling and analysing based on probability are studied. The computational expressions of numerical characteristics of structural dynamic response of closed-loop control system are derived by means of the mode superposition method, in which the randomness of structural damping (modal damping), physical parameters (elastic module and mass density) of structural materials, geometric dimensions (length and cross-section area) of active and passive bars, applied loads and control forces are considered simultaneously. Through engineering examples, the influences of them on structural dynamic response are inspected and some significant conclusions are obtained.
2. Structural dynamic response analysis of closed-loop control system Following the finite element formulation described in Ref. [11], the equation of motion for an intelligent structure is given by ½MfxðtÞg þ ½CfxðtÞg þ ½KfxðtÞg ¼ fF ðtÞg þ ½B1 fFP ðtÞg; . ’
ð1Þ
where ½M; ½C and ½K are the mass, damping and stiffness matrices, respectively. fxðtÞg; fxðtÞg ’ and fxðtÞg are structural displacement, velocity and acceleration vectors, respectively. fFðtÞg is . load force vector. The fFP ðtÞg term represents the initial strain effect of piezoelectric active bar. Since fFP ðtÞg appears on the right-hand side of the equation of motion, it is referred to as the
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apparent active bar force, that is, control force. The ½B1 matrix consists of the active bar’s direction cosines. By means of the mode superposition method, and the Wilson’s damping hypothesis will be adopted. The structural displacement response can be expressed as fxðtÞg ¼
n X
ffgi zi ðtÞ;
ð2Þ
i¼1
where the displacement response of jth degree of freedom is xj ðtÞ ¼
n X
fji zi ðtÞ;
j ¼ 1; 2; y; n;
ð3Þ
i¼1
1 zi ðtÞ ¼ o %i
Z
t 0
ðRi ðtÞ þ Hi ðtÞÞ exp½ xi oi ðt tÞ sin o % i ðt tÞ dt;
Ri ðtÞ ¼ ffgTi fF ðtÞg;
Hi ðtÞ ¼ ffgTi ð½B1 fFP ðtÞgÞ;
i ¼ 1; 2; y; n;
i ¼ 1; 2; y; n;
ð4Þ ð5Þ
where Ri ðtÞ; Hi ðtÞ and zi ðtÞ are load, control force and displacement response of ith degree of freedom in principal coordinates, respectively. oi ; ffgi and xi are ith order inherence frequency, modal shape and modal damping of structure, respectively. o %i ¼ % i ¼ oi ð1 x2i Þ1=2 : Since xi 51; o oi can be obtained. After the structural displacement response of closed-loop system obtained, the stress response of the eth element can be expressed as fsðtÞðeÞ g ¼ E ½B fxðtÞðeÞ g
ðe ¼ 1; 2; y; neÞ;
ð6Þ
where fxðtÞðeÞ g is the displacement response of the nodal point of the eth element, fsðtÞðeÞ g is the stress response of the eth element, ½B and E are the geometric matrix and elastic module of the eth element, respectively.
3. Probabilistic structural dynamic response analysis of close-loop system Suppose that there are ne elements in the analysed intelligent truss structure. In the structure, any element can be taken as passive bar or active bar (the structure of active bar can be seen in Ref. [12]). Piezoelectric bar is utilised as active bar. In order to make use of the united form to express the structural stiffness and mass matrix, a kind of mixed element have been constructed. A Boolean algebra value named as y is introduced in the mixed element, when y ¼ 0; the mixed element is active element bar; when y ¼ 1; the mixed element is passive element bar. In the following, expressions of the stiffness matrix ½K and mass matrix ½M of intelligent truss structure in global coordinate will be developed by use of this kind of mixed element. (" # ) ðeÞ ðeÞ 2 ðeÞ ðeÞ ðeÞ Xne Xne c þ ðe Þ =e E A 33 33 ðeÞ ð7Þ ½K ðeÞ ¼ y m ðeÞ m þ ð1 yÞ 33 AP ½G ½K ¼ ðeÞ e¼1 e¼1 lm lP
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½M ¼
ne X e¼1
ne X 1 ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ½M ¼ ðyrm Am lm þ ð1 yÞrP AP lP Þ½I 2 e¼1 ðeÞ
ð8Þ
where ½K ðeÞ is the eth element’s stiffness matrix, ½M ðeÞ is the eth element’s mass matrix. ½I is a 6 ðeÞ ðeÞ order identity matrix. rðeÞ m ; Am and lm are the eth passive bars’ mass density, cross-section area ðeÞ ðeÞ and length, respectively. rP ; AP and lPðeÞ are the eth active bars’ mass density, cross-section area ðeÞ ðeÞ and length, respectively. EmðeÞ is the eth passive bars’ elastic module. cðeÞ 33 ; e33 and e33 are elastic module, piezoelectric force/electrical constant and dielectric constant, respectively. ½G is a 6 6 matrix, where g11 ¼ g44 ¼ 1; g14 ¼ g41 ¼ 1; other elements of ½G are all equal to 0. Let ðeÞ 2 ðeÞ ð9Þ EPðeÞ ¼ cðeÞ 33 þ ðe33 Þ =e33 : EPðeÞ just is the generalised elastic module of piezoelectric active bars while considering the mechanic–electronic coupling effect. Substituting Eq. (9) into Eq. (7), then (" # ) ne ne X X EmðeÞ Am EPðeÞ AP ðeÞ ½G : ð10Þ ½K ¼ y þ ð1 yÞ ½K ¼ lm lP e¼1 e¼1 ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ ðeÞ Considering the randomness of xi ; rðeÞ m ; rP ; Am ; AP ; lm ; lP ; Em ; c33 and fFðtÞg simultaneously, that is, they are all random variables. The amplitude of fF ðtÞg is obeyed a normal distribution. From Eq. (9), it can be obtained easily that EPðeÞ is random variable. In the closed-loop system, because the production and response process of fFP ðtÞg determined by fFðtÞg; fFP ðtÞg is random variable too, and these two variables are full positive correlation. In realistic engineering examples, normal distribution is a wide probabilistic distribution, and other probabilistic distributions can be equivalently transformed into normal distribution. Therefore, suppose that these random variables all obey normal distribution and also the amplitude of fF ðtÞg obeys a normal distribution. For linear structures, the structural displacement responses obey normal distribution too. The engineering background of this case is the stochastic loads and control forces act on the intelligent structure with random parameters. This case is the most general one of random model of intelligent structural dynamic response analysis. Suppose the length and cross-section areas of bars are two kinds of random variables and the dispersivity of each kind of random variables of each bar is equal, respectively. Then the length and cross-section area of the eth element can be written, respectively, as: l ðeÞ ¼ lðeÞ l; AðeÞ ¼ ZðeÞ Aðe ¼ 1; y; neÞ; where lðeÞ and ZðeÞ are the determinate quantities that denote the nominal length and nominal cross-section area of eth bar, respectively. l is the random variable factor of all bars’ length, its mean value is 1.0 and variance is v2l ; A is the random variable factor of all bars’ cross-section, its mean value is 1.0 and variance is v2A : If the physical parameters of each active element are equal, the physical parameters of each passive element are also equal, rm ¼ rðeÞ m ; rP ¼ ðeÞ ðeÞ ðeÞ rP ; Em ¼ Em ; EP ¼ EP can be obtained easily. Therefore, the matrices ½K and ½M can be expressed, respectively, as ! ðeÞ ne ne ðeÞ X X E A E A ð11Þ ½K ¼ ½K ðeÞ ¼ y m ½KmðeÞ # þ ð1 yÞ P ½KPðeÞ # ; l l e¼1 e¼1
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where ½KmðeÞ # and ½KPðeÞ # are determinate part in stiff matrices ½KmðeÞ and ½KPðeÞ ; respectively, just taking the parameters as lmðeÞ ¼ lðeÞ ; AðeÞ ¼ ZðeÞ ; EmðeÞ ¼ 1 and EPðeÞ ¼ 1: ne ne X X ðeÞ ðeÞ # ðeÞ # ½M ¼ ½M ðeÞ ¼ ðyrðeÞ ð12Þ m A l½Mm þ ð1 yÞrP A l½MP Þ; e¼1
e¼1
where and are determinate part in mass matrices ½MmðeÞ and ½MPðeÞ ; respectively, just ðeÞ taking the parameters as: lmðeÞ ¼ lðeÞ ; AðeÞ ¼ ZðeÞ ; rðeÞ m ¼ 1 and rP ¼ 1: If the physical parameters’ variation of active and passive elements are equal to each other, rP ¼ k1 rm ¼ k1 r and EP ¼ k2 Em ¼ k2 E (k1 and k2 are constant ratio factors) can be easily obtained. Therefore, the matrices ½K and ½M can be expressed, respectively, as ne ne X X EA ðeÞ # EA ðeÞ # ðeÞ ½Km þ ð1 yÞk2 ½Km ½K ¼ ½K ¼ y l l e¼1 e¼1 ½MmðeÞ #
½MPðeÞ #
EA ½y þ ð1 yÞk2 ½Km # ; l ne ne X X ½M ðeÞ ¼ ðyr A l ½MmðeÞ # þ ð1 yÞk1 r A l ½MmðeÞ # Þ ½M ¼ ¼
e¼1
ð13Þ
e¼1
¼ r A l ½y þ ð1 yÞk1 ½Mm # :
ð14Þ
The randomness of physical parameters and geometric dimension will lead the structural inherence frequency o as well as randomness. Therefore, the randomness of the structural dynamic characteristics, damping, loads and control forces will lead the structural dynamic response (dynamic displacement and dynamic stress) as well as randomness. The statistical description of random variables represented by utilising its numerical characteristic. In the following, the expressions of numeral characteristics of dynamic response will be derived. 3.1. Numerical characteristics of dynamic displacement response of closed-loop system From Eq. (5), the mean value mRi ðtÞ and variance s2Ri ðtÞ of Ri ðtÞ and the mean value mHi ðtÞ and variance s2Hi ðtÞ of Hi ðtÞ in principal coordinates can be deduced by means of the algebra synthesis method: mRi ðtÞ ¼ fmf gTi fmF ðtÞ g;
ð15Þ
s2Ri ðtÞ ¼ fm2f gTi fm2F ðtÞ gðv2FðtÞ þ v2o þ v2FðtÞ v2o Þ;
ð16Þ
mHi ðtÞ ¼ fmf gTi fmFP ðtÞ g;
ð17Þ
s2Hi ðtÞ ¼ fm2f gTi fm2FP ðtÞ gðv2FP ðtÞ þ v2o þ v2FP ðtÞ v2o Þ;
ð18Þ
where voi ; vF ðtÞ and vFP ðtÞ are variation coefficient of oi ; fF ðtÞg and fFP ðtÞg; respectively. From Eq. (4), the mean value mZi ðtÞ and mean variance sZi ðtÞ of the displacement of ith degree of freedom in principal coordinate can be deduced by means of the random variable’s functional moment method: Z t 1 mzi ðtÞ ¼ ðm þ mHi ðtÞ Þ expð mx moi ðt tÞÞ sin moi ðt tÞ dt; ð19Þ moi 0 Ri ðtÞ
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szi ðtÞ
( Z t 1 2 ¼ vo ðm þ mHi ðtÞ Þ expð mx moi ðt tÞÞ sin moi ðt tÞ dt moi 0 Ri ðtÞ Z t mx ðmRi ðtÞ þ mHi ðtÞ Þðt tÞ expð mx moi ðt tÞÞ sinmoi ðt tÞ dt þ
0
Z
t 0
2 ðmRi ðtÞ þ mHi ðtÞ ðt tÞ expð mx moi ðt tÞÞ cos moi ðt tÞ dt
1 þ 2 fm2f ÞTi ½fm2FðtÞ gðn2FðtÞ þ n2oi þ n2F ðtÞ n2oi Þ moi þfm2FP ðtÞ gðn2FP ðtÞ þ n2oi þ n2FP ðtÞ n2oi Þ þ 2fmF ðtÞ gfmFP ðtÞ g ððn2F ðtÞ þ n2oi þ n2F ðtÞ n2oi Þðn2FP ðtÞ þ n2oi þ n2FP ðtÞ n2oi ÞÞ1=2
Z t 2 expð mx oi ðt tÞÞ sin moi ðt tÞ dt 0
þ
n2xi m2xi
1 moi
Z
t
0
2 )1=2 ðmRi ðtÞ þ mHi ðtÞ Þ expð mx moi ðt tÞÞ sin moi ðt tÞ dt ;
ð20Þ
where the symbol m; s and n denote random variable’s mean values, mean variances and variation coefficients, respectively. mFðtÞ ; vF ðtÞ ; mFP ðtÞ and vFP ðtÞ are given by the statistical information of the loads and control forces. The fmf gi term can be gained from the structural dynamic characteristic computation. moi ; s2oi and voi can be obtained from the following equations: !1=2 m E ½1 þ n2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r moi ¼ o#i mr mZ CEr ðn2A þ n2E þ n2A n2E Þ1=2 ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 2 1 ½2n2A þ n2E þ n2Z þ n2r þ n2A n2Z þ n2A n2E þ n2A n2r þ n2Z n2r þ n2A n2Z n2r 2
2CEr ðn2A
s2oi ¼ ðo#i Þ2
þ
n2E
þ
n2A n2E Þ1=2 ðn2A
þ
n2Z
þ
n2r
þ
n2A n2Z
þ
n2A n2r
þ
n2Z n2r
1=4
þ
n2A n2Z n2r Þ1=2
! mE n ½1 þ n2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r mr mZ
CEr ðn2A þ n2E þ n2A n2E Þ1=2 ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 f½1 þ n2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r CEr ðn2A þ n2E þ n2A n2E Þ1=2 ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 2 1 ½2n2A þ n2E þ n2Z þ n2r þ n2A n2Z þ n2A n2E þ n2A n2r þ n2Z n2r 2
; ð21Þ
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þ n2A n2Z n2r 2CEr ðn2A þ n2E þ n2A n2E Þ1=2
o ðn2A þ n2Z þ n2r þ n2A n2Z þ n2A n2r þ n2Z n2r þ n2A n2Z n2r Þ1=2 g1=2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4v2l þ 2v4l ; vz ¼ vl 2 ¼ 1 þ v2l so noi ¼ i ; moi
ð22Þ
ð23Þ ð24Þ
where vE and vr are variation coefficient of E and r; respectively. CEr is the correlation coefficient of variables E and r o#i can be obtained from the structural dynamic characteristic computation. From Eqs. (19) and (20), the variation coefficient of zi ðtÞ can be obtained as ð25Þ vzi ðtÞ ¼ szi ðtÞ =mzi ðtÞ : From Eq. (2), the mean value and mean variance of displacement response in nature coordinate can be deduced by means of the algebra synthesis method: n X fmf gi mzi ðtÞ ; ð26Þ fmxðtÞ g ¼ i¼1
fsxðtÞ % g¼
( n X
)1=2 ðfm2f gi m2zi ðtÞ ðv2zi ðtÞ þ v2oi þ v2oi v2zi ðtÞ ÞÞ
:
ð27Þ
i¼1
From above formulas, it can be easily seen that the randomness of the structural displacement response are dependent on not only the structural physical parameters xi ; E; r and the geometric % A% but also the randomness of loads and control forces. In addition, they are determined dimension L; by the values of the modal damping, inherence frequency, mode shape, loads and control forces. 3.2. Numeral characteristics of dynamic stress response of closed-loop system After the displacement responses of the element’s nodal are obtained, the mean value and mean variance of the eth element’s stress response can be obtained by means of the algebra synthesis method from equation ðeÞ fmðeÞ sðtÞ g ¼ ½BmE fmxðtÞ g
ðe ¼ 1; 2; y; ne Þ;
ðeÞ 2 2 2 2 1=2 fsðeÞ sðtÞ g ¼ ½BmE fmxðtÞ g½vfxðtÞðeÞ g þ vE þ vE vfxðtÞðeÞ g
ð28Þ ðe ¼ 1; 2; y; ne Þ:
ð29Þ
4. Examples In the following examples, for the intelligent structure’s active bars, a constant output velocity feedback control law is selected [12]. Active bar’s and passive bar’s materials and their parameters’ value are given in Table 1. Example 1. 20-bar planar truss structure (Fig. 1).
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The elastic module E; mass density r; bars’ length L% and bars’ cross-section area A% are all random variables and obeyed normal distribution. A step load along with the positive direction of x-axis is acted on the nodal 9 of the structure. The amplitude of the step load is a normal random variable. Its mean value is mF ðtÞ ¼ 5000ðNÞ: The gain of the closed-loop control system is g ¼ 50: Now, the first element and 20th element are utilised as piezoelectric active bar, respectively. The corresponding computational results of the structural dynamic displacement response and dynamic stress response are given in Table 2. In this table, the dynamic displacement response is that of the nodal 12, the dynamic stress response is that of the first element. In order to compare, two kinds of models, the determinate model and random model, are adopted, respectively, in computational process. In the determinate model, the mean values of all random variables are regarded as determinate quantity, and their variation coefficients are taken as 0. In the random model, three situations are considered, respectively: (1) the physical parameters and geometric dimensions of the structure are random variables, (2) the modal damping, loads and the control forces of the closed-loop system for the structure are random variables, (3) the physical parameters, geometric dimensions, modal damping, loads and control forces are simultaneously random variables. That is, random model (I), random model (II) and random model (III), respectively. It can be seen from Table 2 that under the conditions that the variation coefficients of physical parameters and geometric dimensions are equal to the variation coefficients of the damping, loads and control forces, the randomness of physical parameters and geometric dimensions will produce greater effect on the randomness of structural stress response than the one produced by the randomness of the damping, loads and control forces. However, the randomness of the damping, loads and control forces will produce greater effect on the randomness of structural displacement Table 1 Intelligent truss structural physical parameters Active bar (PZT-4) Mean value of mass density r Mean value of elastic module c33 Piezoelectric force/electric constant e33 Dielectric constant e33 Cross-section area A
7800 kg=m3 2:1 1011 N=m2 — — 3:0 10 4 m2
7600 kg=m 8:807 1010 N=m2 18:62 C=m2 5:92 10 9 C=V m 3:0 10 4 m2
•
1
Passive bar (steel)
3
•
•
3
•
2 • 250
•
4
•
5
• •
250
(unit: mm)
•
6
•
7
• •
250
•
8
•
9
• •
250
•
10
11
• •
250 12
Y
250 X
Fig. 1. 20-bar planar intelligent truss structure.
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Table 2 The computational results of dynamic response of 20-bar planar structure Model
Dynamic response Maximal displacement response (mm)
Maximal stress response (Mpa)
First element act as active bar
20th element act as active bar
First element act as active bar
20th act as active bar as active bar
Determinate model
5.046
7.397
50.729
81.301
Random model (I) vE ¼ vr ¼ nl ¼ nA ¼ 0:1 nxi ¼ vF ðtÞ ¼ nFP ðtÞ ¼ 0
mzmax ðtÞ ¼ 5:046 szmax ðtÞ ¼ 0:1797
mzmax ðtÞ ¼ 7:397 szmax ðtÞ ¼ 0:2635
msmax ðtÞ ¼ 50:729 ssmax ðtÞ ¼ 6:138
msmax ðtÞ ¼ 81:301 ssmax ðtÞ ¼ 9:837
Random model (II) vE ¼ vr ¼ nl ¼ nA ¼ 0 nxi ¼ vF ðtÞ ¼ nFP ðtÞ ¼ 0:1
mzmax ðtÞ ¼ 5:046 szmax ðtÞ ¼ 0:4087
mzmax ðtÞ ¼ 7:397 szmax ðtÞ ¼ 0:5992
msmax ðtÞ ¼ 50:729 ssmax ðtÞ ¼ 4:053
msmax ðtÞ ¼ 81:301 ssmax ðtÞ ¼ 6:496
Random model (III) vE ¼ vr ¼ nl ¼ nA ¼ nxi ¼ vF ðtÞ ¼ nFP ðtÞ ¼ 0:1
mzmax ðtÞ ¼ 5:046 szmax ðtÞ ¼ 0:4385
mzmax ðtÞ ¼ 7:397 szmax ðtÞ ¼ 0:6428
msmax ðtÞ ¼ 50:729 ssmax ðtÞ ¼ 8:421
msmax ðtÞ ¼ 81:301 ssmax ðtÞ ¼ 13:496
response than the one produced by physical parameters and geometric dimensions. In addition, comparing with the conditions that only the randomness of the physical parameters and geometric dimensions or the randomness of the damping, loads and control forces are considered, under the condition that their randomness are all considered the randomness of the structural displacement and stress response are greater. Example 2. 4-bar space truss structure (Fig. 2). A step load along with the negative direction of z-axis is acted on the nodal 5 of the structure. The amplitude of the step load is a normal random variable. Its mean value is mF ðtÞ ¼ 75 000ðNÞ: The gain of the closed-loop control system is g ¼ 100: Now, the 4th element is used as piezoelectric active bar. The computational results of the dynamic displacement response and dynamic stress response of the structure are given in Table 3. In Table 3, the dynamic displacement response is that of nodal 5, the dynamic stress response is that of first element. In order to investigate the effect of the dispersal degree of random variables E; r and dimension random variable factor l; A and xi ; loads, control forces on the structural dynamic characteristic, the values of variation coefficients of parameters E; r; l; A; xi ; F ; FP are, respectively, taken as two groups. Group I : nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:01: Group II: nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:1: In order to compare, two kinds of models, the determinate and random models, are adopted, respectively, in computational process too. It can be seen from Table 3 easily that changing the variation coefficients of physical parameters, geometric dimensions, damping, loads and control forces will produce considerable effect on the computational results of structural dynamic response. Along with the increase of the
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400
Z
•
2
350 Y
5 •
200
3
•
1
•
X
150 350
4
(unit: mm) Fig. 2. 4-bar space intelligent truss structure.
Table 3 The computational results of dynamic response of 4-bar space structure Model
Dynamic response Maximal displacement response (mm)
Maximal stress response (mm)
Determinate model
2.15
41.29
Random model (I) nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:01
mxmax ðtÞ ¼ 2:15 sxmax ðtÞ ¼ 0:1339
msmax ðtÞ ¼ 41:29 ssmax ðtÞ ¼ 0:4874
Random model (II) nE ¼ vr ¼ vl ¼ nA ¼ nxi ¼ nF ¼ nFP ¼ 0:1
mxmax ðtÞ ¼ 2:15 sxmax ðtÞ ¼ 0:1868
msmax ðtÞ ¼ 41:29 ssmax ðtÞ ¼ 6:854
variation coefficients of physical parameters, geometric dimensions, damping, loads and control forces, the dispersal degree of structure’s dynamic response will notably increase.
5. Conclusions (1) The examples show that the analysing results of intelligent structural dynamic response of the determinate model are different from the one of the random model. So that when one of the physical parameters, geometric dimensions, damping, loads and control forces is random variable, the conventional determinate analysis method of structural dynamic response will not reflect the effect of the randomness. It is only dependent on the structural dynamic response analysis method based on probability. (2) The examples show that the model and solving method of dynamic response analysis of intelligent truss structure based on probability presented in this paper are rational and feasible.
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957
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