Available online at www.sciencedirect.com
Procedia Engineering 16 (2011) 685 – 694
International Workshop on Automobile, Power and Energy Engineering
Free-interface component mode synthesis method with link substructure as super-element Kai-Liang LUa*, Yuan LIUa, Wei-guo ZHANGa,b, Hui-qing QIUb, Wei-jian MIa a
Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China b College of Mechanical Engineering, Tongji University, Shanghai 201804, China
Abstract According to the definition of link substructure in component mode synthesis (CMS) method, link substructure is essentially a kind of super element whose master degree of freedoms (DOF) is the interface DOF. Based on this, a new method of free-interface CMS, compatible for both displacement and force on interfaces, was proposed, by transforming link substructure into super element with Guyan static condensation or dynamic condensation. The new method not only retains free-interface CMS’s advantages of reducing the system DOF efficiently and high accuracy , but also can deal with lumped damping reasonably, thus, it has a widespread application prospect in dynamic analysis of the structures with local non-linearity. Then, the application of the proposed technique was shown by modal and seismic response analysis of a truss bridge in which the girder and brace are lead rubber bearings (LRB) linked, in the automated container terminal. Regarding LRB as super element link substructure, the calculation accuracy and efficiency of Guyan and dynamic condensation methods are compared with finite element method (FEM) or direct integration method. Furthermore, inherent characteristic results of the truss bridge under different LRB disposition forms were obtained.
© 2010 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Automobile, Power and Energy Engineering Open access under CC BY-NC-ND license. Keywords: component mode synthesis; free interface; link substructure; super element method; lead rubber bearing
1. Introduction On the basis of the wide use of finite element method and computer science, component mode synthesis (CMS) method, which is used in dynamic analysis of complicated structures, has gradually
* Corresponding author. Tel.: +86-021-3828-2629; fax: +86-021-3828-2673. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.1142
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come into being. Among lots of different CMS methods, the free-interface CMS method has been the most widely used and constantly improved due to its advantages such as no interface DOF in synthesized equation, high computational efficient and convenient to validate with experimental modal analysis technology. Generally speaking, most CMS methods are direct connected, namely adjacent substructures share the same interface DOF. On indirect CMS method, many researchers did studies and put forward the concept of link substructure. Literature [1,2] first studied the problem of indirect connection between freeinterface substructures by Hou’s method, assembling system motion equation through coupling spring and damping force between substructures, and applied this method to the dynamic problem of weakcoupling and non-linear link between substructures. Literature [3], which aimed at some engineering structure with elastic links, such as the combined shafts of ships, the structure with vibration absorbers, and so on, suggested that elastic links should be treated as ‘soft substructure’ alone so that incompatible interface among substructure interfaces could be translated into the internal deformation of “soft substructure” to meet the compatibility conditions and dual-compatible - compatible for both displacement and force - free - interface CMS should be used to synthesize motion equation. Literature [4] further introduced the concept of link substructure, but the discussion of link substructure was still limited in elastic or rigid link. Literature [5,6] gave out the characterizing definition of link substructure in CMS, namely, only the interface DOF without internal DOF, and all the interface DOF shared with non-link substructures; and described three types of link substructures, namely, elastic, rigid and mixed. The author also suggested the with link substructure indirect-link mixed-interface CMS method; and discussed the two link mode of displacement coordination and dual coordination of displacement and force. In addition, others also carried out similar studies on indirect CMS method [7, 8]. According to the characterizing definition of link substructure in CMS, given by literature [5, 6], link substructure is essentially a kind of static or dynamic transformed super element. Based on this, this paper takes link substructure as super element to deduce the free-interface component mode synthesis technique with link substructure as super-element (referred to as super-element indirect CMS), and applies this technique to the modal and dynamic analysis of a truss bridge, whose girder and brace are lead rubber bearings (LRB) linked, in the automated container terminal (ACT). 2. Super-element indirect CMS by Guyan static condensation 2.1. Substructure’s Division Yet the general, discuss the case like a link substructure between two free-interface substructures, as shown in Fig. 1. All the interface DOF of substructure E are shared with adjacent substructures A and B. A u j , B u j are respectively the interface DOF of substructure A and B; E u jA , E u jB are respectively the communal interface DOF that substructure E shares with substructure A and B, apparently, T
um = ⎡⎣ E u jA E u jB ⎤⎦ . A f j , B f j are respectively the interface-force vectors that substructure A and B work on their respective interface DOF; E f jA , E f jB are respectively the interface-force vectors that E
substructure E work on interface DOF jA and jB.
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Fig.1. Sketch map of substructure interface connection
2.2. The first coordinate transformation for substructures (1) Free-interface substructure Set the stiffness matrix and mass matrix of substructure A in physical coordinates as A ⎡ Az z ⎤ A z = ⎢ A ii A ij ⎥ , z = k , m (1) z jj ⎦⎥ ⎣⎢ z ji Where, subscript i denote substructure’s non-interface DOF and j denotes substructure’s interface DOF. Substructure B is also free-interface substructure. Corresponding matrix of substructure B can be obtained A B to . So we omit the left superscript in the following deduction. by changing left superscript Use the assumed branch mode set Φ , composed of first-k-order normal modes Φk and residual attachment modes Ψ d as the transformation matrix, thus the transformation relation between physical coordinate u and modal coordinate p is Φ Ψ pf Φp ⎪⎧ ui ⎪⎫ ⎡ ik ⎪ k ⎫⎪ id ⎤ ⎧ (2) u=⎨ ⎬=⎢ ⎥⎨ ⎬ = ⎪ j ⎭⎪ jd ⎦ ⎩ ⎩⎪u j ⎭⎪ ⎣ jk Where, f j is the interface-force vector that substructure works on interface DOF j. The calculation of residual attachment modes Ψ d with or without substructure’s rigid body mode can refer to literature [9]. Through the regularization of mode to mass matrix, the stiffness matrix and mass matrix of freeinterface substructure in modal coordinates can mΨ be derived as kΨ I ⎡ kk ⎤ ⎡ Λkk ⎤ m=⎢ (3a)(3b) ⎥ , k=⎢ ⎥ T T Ψ Ψ d d ⎦ d d ⎦ ⎣ ⎣ (2) Super-element link substructure Super element is characterized by contracting all of substructure’s DOF on interface DOF (the master DOF) E um . Set the stiffness and mass matrixes of link substructure under master coordinate E um and slave coordinate E us as E ⎡ Ez z ⎤ z = ⎢ E ss E sm ⎥ , z = m , k zmm ⎦ ⎣ zms By Guyan static condensation, the stiffness and matrixes only Ψ mass z Ψ z m k under interface DOF is E z = E cT E E c , = , E
in which, Ψ c is static transformation matrix.
(4)
(5)
E
Ψ c = ⎡⎣ −
E
Since E um = ⎡⎣ E u jA
E
k E
k −1 E ss
I sm
⎤⎦
T
(6)
T
u jB ⎤⎦ , Equation (5) can be written in block form: E
E ⎡ E z ⎤ z z = ⎢ E ( jA)( jA) E ( jA)( jB ) ⎥ , z = m , k ⎢⎣ z( jB )( jA) z( jB )( jB ) ⎥⎦
(7)
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2.3. Assemble system equation and the second coordinate transformation (1) Coordination of interface displacement After the first coordinate transformation, the system’s non-damping vibration equation, under the coordinate p = ⎡⎣ A pk Where,
A
fj
E
u jA
E
B
u jB
T
f j ⎤⎦ , is
+ kp = 0 mp
m Ψ
⎡ A I kk ⎢ A Ψ dT A ⎢ ⎢ m=⎢ ⎢ ⎢ ⎢ ⎣⎢
B
pk m Ψ
A d E E
m ( jA)( jA) E m ( jA)( jB ) m ( jB )( jA) E m ( jB )( jB ) B
I kk Ψ
B
T B d
B
(8) k Ψ
⎤ ⎡ A Λkk ⎥ ⎢ A Ψ dT A ⎥ ⎢ ⎥ ⎢ ⎥ k=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ d⎦ ⎣⎢
k Ψ
A d E E
k( jA)( jA) E k( jA)( jB ) k( jB )( jA) E k( jB )( jB ) B
Λkk Ψ dT B
B
B
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ d⎦
(9a)(9b)
Since the super-element link substructure E doesn’t have non-interface DOF, and its interface DOF are shared with adjacent free-interface substructure A and B, the coordination conditions of interface displacement can be deduced as E u jA = A u j , E u jB = B u j (10a)(10b) On the basis of Equation (10), carry topf the secondpf line, we obtain Φout Equation Φ Ψ(2) according Ψ A ⎧ pk ⎫ ⎧ A pk ⎫ ⎪ ⎪ ⎪B ⎪ E A A 0 0 ⎤⎪B k ⎪ ⎪⎧ u jA ⎪⎫ ⎡ jk ⎪ k⎪ jd ⎥ ⎨ A ⎬ = T1 ⎨ A ⎬ ⎨E ⎬=⎢ B B 0 ⎥⎪ j ⎪ ⎪⎩ u jB ⎪⎭ ⎣⎢ 0 jk jd ⎦ ⎪ j⎪ ⎪ B fj ⎪ ⎪ B fj ⎪ ⎩ ⎭ ⎩ ⎭ Thus, can receive Ψ Ψ ⎧ A pk ⎫ ⎡ I 0 0 0 ⎤ ⎪A ⎪ ⎢ ⎧ A pk ⎫ I 0 0 ⎥⎥ ⎧ A pk ⎫ ⎪ fj ⎪ ⎢ 0 ⎪ ⎪ ⎪B ⎪ E B A ⎪⎪ u jA ⎪⎪ ⎢ AΦ jk 0 0 ⎥ ⎪ pk ⎪ ⎪ pk ⎪ jd T = = ⎥⎨ A ⎬ ⎨E ⎬ ⎢ ⎬ 2 ⎨ A B B Φ jk 0 jd ⎥ ⎪ f j ⎪ ⎪ u jB ⎪ ⎢ 0 ⎪ fj ⎪ ⎪ B pk ⎪ ⎢ 0 ⎪ B fj ⎪ I 0 0 ⎥ ⎪⎩ B f j ⎪⎭ ⎩ ⎭ ⎥ ⎪B ⎪ ⎢ I ⎦⎥ 0 0 ⎪⎩ f j ⎪⎭ ⎣⎢ 0 (2) Coordination of interface force The static equilibrium equation of link substructure E is given by ⎧⎪ E f jA ⎫⎪ ⎡ E k( jA)( jA) E k( jA)( jB ) ⎤ ⎧⎪ E u jA ⎫⎪ ⎥⎨ ⎨E ⎬ = ⎢E ⎬ E k( jB )( jB ) ⎦⎥ ⎩⎪ E u jB ⎭⎪ ⎩⎪ f jB ⎭⎪ ⎣⎢ k( jB )( jA) From Equation (11) and Equation (13), we can obtain ⎧⎪ E f jA ⎫⎪ ⎡ E k( jA)( jA) ⎨E ⎬ = ⎢E ⎩⎪ f jB ⎭⎪ ⎣⎢ k( jB )( jA)
E E
k( jA)( jB ) ⎤ A ⎥ T1 ⎡ pk k( jB )( jB ) ⎦⎥ ⎣
B
pk
A
fj
B
f j ⎤⎦
(11)
(12)
(13)
(14)
T
Then, according to the equilibrium conditions of interface force E f jA = − A f j , E f jB = − B f j
(15a)(15b)
And from Equation Φ k (14)Φand Equation kk ΨΨ(15), I we k kcan receive ΨΨ I ⎡ E k( jA)( jA) A ⎢E A ⎣⎢ k( jB )( jA)
E jk E jk
B ( jA )( jB ) B ( jB )( jB )
E jk jk
+
A ( jA )( jA ) jd E A ( jB )( jA ) jd
E E
( jA )( jB ) B ( jB )( jB )
B
⎤ A ⎥ × ⎡ pk + ⎦⎥ ⎣
jd jd
B
pk
A
fj
B
f j ⎤⎦ = [C kk T
⎪⎧ pk ⎪⎫ C dd ] ⎨ ⎬ = 0 ⎩⎪ f j ⎭⎪
(16)
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Then A T ⎡ I ⎤ ⎪⎧ A p ⎪⎫ ⎪⎧ p ⎪⎫ ⎡⎣ A pk B pk A f j B f j ⎤⎦ = ⎢ −1 ⎥ ⎨ B k ⎬ = T3 ⎨ B k ⎬ ⎣ −C dd C kk ⎦ ⎩⎪ pk ⎭⎪ ⎩⎪ pk ⎭⎪ Thus we can obtain the second coordinates Ψ transformation Ψ A ⎧ pk ⎫ ⎡ I 0 0 0 ⎤ ⎪A ⎪ ⎢ 0 0 ⎥⎥ I ⎪ fj ⎪ ⎢ 0 E A A A A 0 0 ⎥ ⎡ I ⎤ ⎪⎧ A pk ⎪⎫ ⎪⎪ u jA ⎪⎪ ⎢ Φ jk ⎪⎧ pk ⎪⎫ ⎪⎧ pk ⎪⎫ jd T T T = × = = ⎥ ⎢ −1 ⎥ ⎨ B ⎬ ⎨E ⎬ ⎢ ⎬ ⎨B ⎬ 2 3 ⎨B B B 0 Φ jk ⎪ pk ⎭⎪ jd ⎥ ⎣ −C dd C kk ⎦ ⎩ ⎩⎪ pk ⎭⎪ ⎩⎪ pk ⎭⎪ ⎪ u jB ⎪ ⎢ 0 ⎥ ⎪ B pk ⎪ ⎢ 0 0 0 I ⎥ ⎪B ⎪ ⎢ 0 0 I ⎦⎥ ⎪⎩ f j ⎪⎭ ⎣⎢ 0
(17)
(18)
Take Equation (18) into Equation (8), we can obtain system’s equation of free vibration under the approximate space of generalized coordinate q = ⎡⎣ A pk
B
Mq + Kq = 0
T
pk ⎤⎦ , which is given by (19)
Where, M = T T mT , K = T T kT
(20a)(20b)
2.4. Inverse transform to physical coordinates For free-interface substructure,
u = ⎡⎣ A u
B
T
u ⎤⎦ = T4T3 q
(21)
Where B A ⎡ AΦ T4 = ⎢ A ik B ik A jk id jd ⎣⎢ Ψ id For super-element link substructure, Ψ TT q E u= E c 1 3
B
⎤ ⎥ ⎥ jd ⎦
T
jk B
(22)
(23)
3. Super-element indirect CMS by dynamic condensation 3.1. Dynamic condensation for super-element link substructure Set the stiffness matrix, damping matrix and mass matrix of link substructure in master and slave DOF as
⎡ Ez Ez ⎤ z = ⎢ E ss E sm ⎥ , z = k , c, m ⎣ zms zmm ⎦ By dynamic condensation, equation (24)Ψbecomes z Ψ z k c m T E z = E d ( ) E E d ( ) , = , , E
In which, Ψ d ( ) denotes dynamic condensation matrix and denotes system’s natural frequency E
which is still unknown.
(24)
(25)
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D Ψ d ( ) = ⎡ − ⎣
E
E
D
( )
−1 E ss
I
( )sm
⎤ ⎦
T
(26)
E
D ( )ss = E kss + j E css − 2 E mss
(27)
E
D ( ) sm = E k sm + j E csm − 2 E msm
(28)
Equation (25) can be written in block form E ⎡ E z ⎤ z E z = ⎢ E ( jA)( jA) E ( jA)( jB ) ⎥ , z = k , c, m (29) ⎢⎣ z( jB )( jA) z( jB )( jB ) ⎥⎦ Now the first coordinate transformation for link substructure is completed. The transformation for freeinterface substructure is the same as in section 2.2. 3.2. Coordination of interface force The dynamic equilibrium equation of link substructure E is given by E E D ( )( jA)( jB ) ⎤ ⎪⎧ E u jA ⎪⎫ ⎧⎪ E f jA ⎫⎪ ⎡ D ( )( jA)( jA) ⎥⎨E ⎨E ⎬ = ⎢E ⎬ E D ( )( jB )( jB ) ⎥ ⎪⎩ u jB ⎪⎭ ⎪⎩ f jB ⎪⎭ ⎣⎢ D ( )( jB )( jA) ⎦ In which E D ( ) = E k + j E c − 2 E m
(30)
(31)
The coordination of interface displacement is as same as that of static condensation. Note that Equation (30) has the same form with Equation (13), so Equation (13) is obtained by replacing k with D ( ) in Equation (13). Therefore, we can replace k with D ( ) in Equation (16) to obtain the second coordinates transformation in the dynamic condensation. As coordinates transformation matrix is a function of requested system’s natural frequency in the dynamic condensation, it will lead to non-linear eigenvalue problem which requires iterative calculation with resulting computational efficiency’s fall. 3.3. Summary From Equations (27), (28) and (30) we can see, when damping terms and inertia terms are neglected, that is, = 0 , the dynamic condensation turns into static condensation, so the latter is the zero-order approximation of the former. Whether in static condensation or dynamic condensation, the super-element indirect CMS deduced by this paper will eventually solve system’s dynamic problem in the approximate space which is set out by the normal mode coordinates of every free-interface substructure, which reduces system’s DOF enormously. Because the generalized coordinates of link substructure don’t take part in constructing approximate solving space. In that case, when there are non-linear elements in link substructure, we only need to modify the coordinate transformation matrix to modify system’s generalized k, c and m matrix. Therefore, the computational efficiency improves. On the other hand, the introduction of super-element link substructure can handle components’ structural damping and connectors’ lumped damping separately and then couple into the entire system, which is better than Raleigh damping, especially for some devices with concentrated damping such as rubber bearings, dampers and so on.
Kai-Liang LU et al. / Procedia Engineering 16 (2011) 685 – 694
4. Application: Inherent characteristic analysis of a truss bridge in ACT 4.1. Brief introduction to the truss bridge In existing ACT, it usually uses automatic guided vehicles (AGV) or container trucks to transport containers to and from between container cranes and yard. But both of them have the same problems: high cost and low efficiency. As shown in Fig. 2, a creative plan of constructing special truss bridge between container cranes and the yard to transport containers by electric vehicles through this truss bridge [10] has been proposed to solve the problem.
Fig.2. Virtual Reality simulation for the automated container terminal experimentation
The truss bridge (Fig. 3) mainly consists of trussed girder, braces, and bind members etc. And, the girder and the braces either have rigid connection or are LRB linked. LRB is a common device for seismic isolation and energy consumption, and widely used in bridges and constructions for its merits of simple structure, easy manufacture, and convenient installation and so on. Because of LRB’s complicated non-linear characteristic, researchers always adopt the equivalent linear model or the bilinear model in current analysis and design.
Fig.3. Structure sketch map of the truss bridge
According to the equivalent linear model, mechanical characteristic parameters of three kinds of LRB are listed in Table 1. Table 1. Characteristic parameters of LRB
Type
Mass
Vertical stiffness
Equivalent stiffness
Equivalent damping ratio
m (kg)
K v (N/m)
k B (N/m)
B (%)
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GZY300
56
1.188 × 109
1.13 × 106
21.9
GZY400
126
1.85 × 109
1.69 × 106
23.0
GZY500
228
1.972 × 109
1.91 × 106
22.4
4.2. Comparison between static condensation and dynamic condensation With the application of static condensation and dynamic condensation super-element indirect CMS, treating LRB as link substructures while trussed girder, braces and bind members as free-interface substructures (take the first-30-order substructures’ modes into use), solved the natural characteristic of the truss bridge LRB linked. Taking the FEM solution results as exact solution, the results are shown in Table 2; values in brackets are errors (%) compared with FEM solution. Table 2. Natural frequency (Hz ) results comparison of truss bridge LRB linked Mode order 1 5 10 15 20
FEM f 1.1197 1.6164 4.3933 6.9069 10.261
GZY300 Static Dynamic f (e) f (e) 1.1213(0.14) 1.1203(0.05) 1.6222(0.36) 1.6192(0.17) 4.4438(1.15) 4.4194(0.59) 7.0173(1.59) 6.9668(0.86) 0.445(1.79) 10.357(0.94)
FEM f 1.2973 1.8548 4.7391 7.0339 10.269
GZY400 Static f (e) 1.3024 (0.39) 1.8738(1.02) 4.8348(2.02) 7.1927(2.26) 10.516(2.41)
Dynamic f (e) 1.2990(0.13) 1.8602(0.29) 4.7855(0.98) 7.1107(1.09) 10.387(1.15)
FEM f 1.3497 1.9201 4.8372 7.0407 10.204
GZY500 Static f (e) 1.3596(0.73) 1.9480(1.45) 4.9605(2.55) 7.2524(3.01) 10.539(3.43)
Dynamic f (e) 1.3535(0.28) 1.9311(0.57) 4.8955(1.21) 7.1378(1.38) 10.357(1.50)
It shows in Tab.2 that: (1) Apparently, as the vertical stiffness and horizontal stiffness of LRB increases, the natural frequency of the entire structure improves; (2) Both the Super-element indirect CMS by Guyan condensation and by dynamic condensation have high accuracy, compared with the FEM results, natural frequency of the maximum error is 3.43% and 1.50%. The error between the two is small, which shows the contribution of LRB (link substructure) to the truss bridge (entire structure)’s inherent characteristics is close to the static component. 4.3. Natural characteristic comparison of the truss bridge in different support conditions Natural frequency results calculated with Super-element indirect CMS by static condensation of the truss bridge in different support conditions are shown in Table 3. Table 3. Natural frequency (Hz) results comparison of truss bridge in different support conditions Mode order 1 5 10 15 20
Condition 1
Condition 2
Condition 3
Condition 4
2.3634 4.8043 7.4432 8.7363 11.395
1.3024 1.8738 4.8348 7.1927 10.516
1.8796 3.2708 5.5838 7.1963 10.610
1.4939 3.7351 6.8161 8.5903 11.117
(NOTE: Condition 1 means that the girder and the braces have rigid connection; Condition 2 means that the girder and the braces are LRB linked; Condition 3 means that the end of the girder and the braces rigid connection, the middle of the girder and the braces LRB linked; Condition 4 is opposite to condition 3.)
Kai-Liang LU et al. / Procedia Engineering 16 (2011) 685 – 694
It is known from Tab.3 that the more the LRB arranged, the smaller the natural frequency is; the fifth mode in Condition 1 (rigid connected) is about 2.6 times than that in Condition 2 (LRB linked). Thus, LRB has notable effect on low natural frequency modulation of the entire structure. 4.4. Comparison of seismic response analysis Input the El Centro seismic wave (maximum acceleration: 0.34g, duration: 54s, data point: every 0.2s) horizontally along z direction in Fig. 3, the displacement response of a node in the middle span of the truss bridge in support condition 2, respectively solved by direct integration Newmark- method and super-element indirect CMS method based on ANSYS10.0 and Matlab6.5, on a Dell PC with basic configuration of Intel Pentium Dual CPU 1.60GHz, 1GB memory, are compared and shown in Fig. 4.
Fig.4. Displacement response time history curves of a node in the middle span with different solution methods
The peak and waveform of the response curves obtained by indirect CMS methods are in good agreement with that by direct integration method. The result by static condensation is almost the same as that by dynamic condensation, the peak value of the response is respectively 0.075m and 0.077m, and compared with the direct integration method’s 0.072m, the error is 4.2% and 6.9%, which indicates the indirect CMS methods also have high computational accuracy in dynamic response calculation. In the aspect of computation time, the time-consuming of the three methods are respectively 5h48min 2h5min7h12min, among which, the most efficient method is the indirect CMS method by static condensation. 5. Conclusions By transforming link substructure into super element with Guyan static condensation or dynamic condensation, a new method of free-interface CMS, both displacement and force on interfaces coordinated, was proposed in this paper. The new method not only retains free-interface CMS’s advantages of reducing the system DOF efficiently and high accuracy, but also has a widespread application prospect in dynamic analysis of the structures with local non-linearity or lumped damping. The proposed method was applied to modal and seismic response analysis of a truss bridge in which the girder and brace are lead rubber bearings (LRB) linked, in ACT. Comparing the results of indirect
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CMS methods by static and dynamic condensation with FEM or direct integration method, the following conclusions can be drawn: (1) Both the Super-element indirect CMS methods by Guyan condensation or by dynamic condensation have high accuracy for inherent characteristics and dynamic response calculation. The latter has better precision than the former, but with poor efficiency. The static-condensation indirect method is even better than Newmark- method in dynamic response calculation. (2) LRB has significant effect on frequency modulation for the truss bridge, through changing LRB’s type (i.e., stiffness, mass, damping) or disposition form.
Acknowledgements This work is sponsored by National 863 plans projects(2009AA043001), this paper is supported by Science & Technology Program of Shanghai Maritime University(20110041), also supported by Shanghai Education Committee Projects (J50604 and 09ZZ163) , Ministry of Communications Research Project(2009-329-810-020&2009-353-312-190), and Shanghai Science & Technology Committee Research Project (08DZ2210103 and 09DZ2250400).
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