Discrete time transfer matrix method for dynamics of multibody system with flexible beams moving in space

Acta Mech. Sin. (2012) 28(2):490–504DOI 10.1007/s10409-012-0025-7

RESEARCH PAPER

Discrete time transfer matrix method for dynamics of multibody system withflexible beams moving in space

Xiao-Ting Rui · Edwin Kreuzer · Bao Rong · Bin He

Received: 6 April 2010 / Revised: 7 June 2010 / Accepted: 29 July 2010©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2012

Abstract In this paper, by defining new state vectors anddeveloping new transfer matrices of various elements mov-ing in space, the discrete time transfer matrix method ofmulti-rigid-flexible-body system is expanded to study thedynamics of multibody system with flexible beams movingin space. Formulations and numerical example of a rigid-flexible-body three pendulums system moving in space aregiven to validate the method. Using the new method tostudy the dynamics of multi-rigid-flexible-body system mov-ing in space, the global dynamics equations of system arenot needed, the orders of involved matrices of the systemare very low and the computational speed is high, irrespec-tive of the size of the system. The new method is simple,straightforward, practical, and provides a powerful tool formulti-rigid-flexible-body system dynamics.

Keywords Multi-rigid-flexible-body system · Spatial mo-tion · Dynamics · Discrete time transfer matrix method

1 Introduction

Multibody system dynamics (MSD) has become an impor-

The project was supported by the Natural Science Foundation ofChina Government (10902051), the Natural Science Foundation ofJiangsu Province (BK2008046), and the German Science Founda-tion.

X.-T. Rui · B. Rong · B. HeInstitute of Launch Dynamics,Nanjing University of Science & Technology,210094 Nanjing, Chinaemail: rongbao [email protected]

E. KreuzerMechanics and Ocean Enginnering of Hamburg Universityof Technology, Hamburg, Germany

tant theoretical tool for wide engineering problem analy-sis. Lots for MSD have been studied by many authors ontheoritical and computational methods [1–4]. With high-performance and high-precision demands of complex me-chanical systems such as lightweight robots, precision ma-chinery, aircrafts and vehicles, large scale, low stiffnessand flexibility have emerged as important trends in thesefields [1, 3]. The interaction of large motion and deforma-tional behavior of flexible body cause the dynamic equa-tions to be complicated and highly nonlinear. At present,the deformation of flexible bodies is mostly described byRayleigh–Rize, finite element or modal analysis methods.Then the global dynamic equations of multi-rigid-flexible-body systems can be obtained by using the vector mechanicsor analytical mechanics methods, such as Lagrange, Witten-burg, Schiehlen and Kane methods. The orders of the in-volved system matrices increase together with the numberof the system degrees of freedom; hence the orders of ma-trices involved in the global dynamic equations are ratherhigh for complex multi-rigid-flexible-body systems. Owingto the accurate, real-time, efficient demands of dynamic pre-diction and control of complex mechanical systems, findinga new method to study multi-rigid-flexible-body system dy-namics without the global dynamic equations and improvingthe computational efficiency has become a research focus inthe field of MSD [1, 3].

Classical transfer matrix method (TMM) has been de-veloped over seventy years and was used widely in struc-ture mechanics and rotor dynamics of linear time-invariantsystem [5, 6]. Rui et al. developed TMM of linear multi-body system (MS-TMM) for vibrations analysis of linearmultibody system by developing new transfer matrices andorthogonal property of multibody system [7]. Kumar andSankar developed discrete time TMM (DT-TMM) for struc-ture dynamics of time variant system by combining theTMM with the numerical integration procedure [8]. Rui et al.developed discrete time transfer matrix method of multibody

Discrete time transfer matrix method for dynamics of multibody system with flexible beams moving in space 491

system (MS-DT-TMM) [9, 10] to study general multi-rigid-body system dynamics by combining and expanding theTMM and the numerical integration procedure. Rui et al. de-veloped discrete time transfer matrix method of multi-rigid-flexible-body system (MRFS-DT-TMM) to study dynamicsof multi-rigid-flexible-body systems moving in plane [11,12]. Rong and Rui extended MRFS-DT-TMM to study thedynamics of planar controlled manipulator system [13, 14].Rong and Rui combine FEM and MS-DT-TMM to study dy-namics of general planar flexible multibody systems includ-ing flexible bodies with irregular shape [15]. When usingMRFS-DT-TMM, the global dynamics equations of systemare not needed and the orders of the involved system matricesare decreased greatly.

In this paper, by defining new state vectors and de-veloping new transfer matrices of various elements movingin space, the MRFS-DT-TMM is extended to study the dy-namics of multibody system with flexible beams moving inspace. Formulations of the proposed method as well as anumerical example of a rigid-flexible-body three pendulumssystem moving in space are given to validate the method.This paper is organized as follows: in Sect. 2, the generaltheorems and steps of the method are shown. In Sect. 3, thedynamics equations of the beam moving in space are devel-oped. In Sect. 4, the transfer matrices of typical elementsmoving in space are deduced. In Sect. 5, the numerical re-sults of a rigid-flexible-body three pendulums system mov-ing in space got by the proposed method and by ordinarymethod are given to validate the method. The conclusionsare presented in Sect. 6.

2 General theorems and steps of the proposed method

The general theorems and steps of this method are similar asthat used in Refs. [10, 11]. When using MRFS-DT-TMM tostudy the dynamics of multi-rigid-flexible-body systems, ac-cording to the natural attribute of bodies and hinges of thesystem, any complex multi-rigid-flexible-body system canbe divided into a certain number of subsystems, which canbe represented by various elements including bodies (rigidbodies, elastic bodies, lumped masses, and so on) and hinges(joints, ball-and-socket, pins, linear springs, rotary springs,linear dampers and rotary dampers, and so on). Linearizingthe dynamic equations of each element, the transfer equa-tions and transfer matrices of these bodies and hinges can bedeveloped respectively. By assembling these transfer matri-ces of elements at the required locations, the overall trans-fer equation and overall transfer matrix of any multi-rigid-flexible-body system can be obtained easily. Once the over-all transfer equation of a system is known, by applying sys-tem boundary conditions, multi-rigid-flexible-body systemdynamics can be calculated. In order to describe these stepsconveniently, the chain multi-rigid-flexible-body system istaken as an example in follows. However, the dynamic mod-eling method can be carried over straightforwardly to other

complex systems with complex topology structures.It should be pointed out that the positions of the bodies

and hinges are considered equivalent in transfer equationsand transfer matrices in this method. In the development pre-sented in the subsequent sections, the right-handed Cartesiancoordinate systems oxyz and O2x2y2z2 are used as an inertialreference system and a body-fixed reference system, respec-tively; the space-three-angles x-y-z defined in the Ref. [16],are used as the orientation angles of involved bodies.

2.1 State vectors

The state vectors of any point of rigid bodies moving in spaceare defined as

z = [x, y, z, θx, θy, θz, mx, my, mz, qx, qy, qz, 1]T. (1)

The state vectors of any point of flexible bodies moving inspace are defined as

z = [x, y, z, θx, θy, θz, mx, my, mz, qx, qy, qz, q1,

q2, · · · , qn, 1]T, (2)

where x, y, z, θx, θy and θz are the position coordinates of theconnection point with respect to oxyz and the correspondingorientation angles of local reference frame; mx, my, mz, qx,qy and qz are the corresponding interior torques and interiorforces defined with respect to oxyz, respectively. q1, q2, · · · ,qn are the generalized coordinates which describes the defor-mation of flexible body with modal method, the superscript“n” is the highest order of the modal considered.

For a chain multi-rigid-flexible-body system moving inspace, there are four kind of hinges: (1) both of the outboardbody and inboard body are flexible bodies; (2) both of theoutboard body and inboard body are rigid bodies; (3) its out-board body is flexible body, and its inboard body is rigidbody; (4) its outboard body is rigid body, and its inboardbody is flexible body. The state vectors of rigid body andflexible body moving in space have been given in Eqs. (1)and (2), respectively. So, the state vectors of any kind ofhinge can be defined easily by combing Eqs. (1) and (2), asshown in Sect. 4. The sketch map of the state vector of theinboard and outboard points of rigid body moving in spaceis shown as Fig. 1.

Fig. 1 State vector of the inboard and outboard points of rigid bodymoving in space

492 X.-T., Rui, et al.

2.2 Linearization of dynamics equations of elements

For developing transfer equations and transfer matrices ofelements, one needs to linearize the dynamic equations ofelements. The widely used linearization methods are intro-duced simply in this section, more details can be found inRefs. [10, 11].

According to the step by step time integration meth-ods (such as Newmark-βmethod, Wilson-θ method, Houboltmethod, and so on), the motion parameters ξ and ξ at timeinstant ti can be expressed as a linear function of ξ in form

ξ (ti) = χ1ξ (ti) + χ2,ξ, ξ (ti) = χ3ξ (ti) + χ4,ξ, (3)

where variable ξ may represent vector of position coordi-nates, x, y, z, orientation angles θx, θy, θz or generalized co-ordinates q1, q2, · · · , qn, respectively; ξ and ξ represent thefirst order and the second order derivatives of ξ with respectto time. The quantities χ1, χ2,ξ, χ3 and χ4,ξ will have dif-ferent expressions for different numerical integration meth-ods, as shown in Ref. [17]. For example, if Newmark-βmethod [18, 19] is used, then one can obtained

χ1 =1βΔT 2

I kxk,

χ2,ξ = − 1βΔT 2

[ξ (ti−1) + ΔT ξ (ti−1) +

(12− β)ΔT 2ξ (ti−1)

],

χ3 = γχ1ΔT,

χ4,ξ = ξ (ti−1) + ΔT [(1 − γ)ξ (ti−1) + γχ2,ξ],

where ΔT = ti − ti−1 is the time step, β and γ are the coef-ficients of Newmark-β method. Bold capital symbol I k×k isthe unit matrix, and its subscript “k” denotes the order of theunit matrix.

The trigonometric functions at time ti are expandedwith respect to ti−1 using the truncated Taylor series of or-der 3, that is

sin θ(ti) = sin[θ(ti−1) + Δθ] = s + o(ΔT 2),

cos θ(ti) = cos[θ(ti−1) + Δθ] = c + o(ΔT 2),(4)

where

s = sin θ(ti−1){1 − 1

2[θ(ti−1)ΔT ]2

}

+ cos θ(ti−1)[θ(ti−1)ΔT +

12θ(ti−1)ΔT 2

],

c = cos θ(ti−1){1 − 1

2[θ(ti−1)ΔT ]2

}

− sin θ(ti−1)[θ(ti−1)ΔT +

12θ(ti−1)ΔT 2

].

(5)

A multinomial in the dynamic equations can be approx-imated by

a(ti)b(ti) = a(ti−1)b(ti) + a(ti)b(ti−1)

−a(ti−1)b(ti−1) + a(ti−1)b(ti−1)ΔT 2. (6)

Using multiple Taylor expansion theorems, a directioncosine matrix A(ti) from the reference system O2x2y2z2 tooxyz can be approximately expressed with respect to timeti−1 by the truncated Taylor series of order 3, that is

A(ti) = A(ti−1)T 1(ti−1)θx(ti) + A(ti−1)T 2(ti−1)θy(ti)

+A(ti−1)T 3(ti−1)θz(ti) +Φ(ti−1), (7)

where T j ( j = 1, 2, 3) is a skew symmetric matrix of T j.

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

cycz sxcycz − cxsz cxsycz + sxcz

cysz sxsysz + cxcz cxsysz − sxcz

−sy sxcy cxcy

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

T 1 = [1, 0, 0]T,

T 2 = [0, cx, −sx]T,

T 3 = [−sy, sxcy, cxcy]T,

sr = sin θr, cr = cos θr, (r = x, y, z),

Φ(ti−1) = A(ti−1){I − T 1(ti−1)θx(ti−1) − T 2(ti−1)θy(ti−1)

−T 3(ti−1)θz(ti−1) +[12

T 21(ti−1)θ2x(ti−1)

+12

T 22(ti−1)θ2y (ti−1) +

12

T 23(ti−1)θ2z (ti−1)

+T 2(ti−1)T 1(ti−1)θy(ti−1)θx(ti−1)

+T 3(ti−1)T 2(ti−1)θz(ti−1)θy(ti−1)

+T 3(ti−1)T 1(ti−1)θz(ti−1)θx(ti−1)]ΔT 2}.

The motion quantities ξ (ti−1), ξ (ti−1), ξ (ti−1) at the pre-vious time instant are all known at time instant ti. Thus, thesequantities χ1, χ2,ξ, χ3, χ4,ξ, s and c etc. are all definable forany subsystem for the time interval (ti−ti−1), and hence aboveformulations are valid.

2.3 Transfer equation and transfer matrix of element

The dynamic equations of the j-th element that have beenlinearized using the linearization method can be rewritten asa single transfer equation

z j, j+1(ti) = U j(ti)z j, j−1(ti). (8)

The meaning of the subscripts of the state vectors z fol-lows the convention in Sect. 2.1 of Ref. [10]. The transferequation describes the mutual relationship between the statevectors at two ends of the j-th element. Here, U j(ti) is thetransfer matrix of the j-th element, it is the functions of themotion quantities (ξ (tk), ξ (tk) and ξ (tk), k = i− 1, i− 2, · · · )which are all known at time instant ti, and its order always is(13+n)× (13+n) for dynamics of chain multi-rigid-flexible-body systems moving in space.


2.4 Overall transfer equation and transfer matrix of system

By using the same method as used in MS-DT-TMM [10],the overall transfer equation and transfer matrix of system ofa chain multi-rigid-flexible-body system, can be assembledand calculated as follows

zm,m+1 = U z1,0, (9)

U = U mUm−1 · · ·U2U 1. (10)

where the subscript “m” denotes the total number of the ele-ments of a chain multi-rigid-flexible-body system.

For a chain multi-rigid-flexible-body system, the orderof the overall transfer matrix of system is equal to the or-der of the transfer matrix of the element, and it does notincrease when the system degrees of freedom increase. Ir-respective of the size of a multi-rigid-flexible-body system,the highest order of the overall transfer matrix U is the samewith the order of transfer matrix of single element, that is(13+n)× (13+n) for dynamics of chain multi-rigid-flexible-body system moving in space. So, the matrices involved inthe proposed method are always small, which greatly reducesthe computational time and the memory storage requirement.

2.5 Solutions of multi-rigid-flexible-body system dynamics

Once the overall transfer equation of a multi-rigid-flexible-body system is known, the boundary conditions of the sys-tem can then be applied and the unknown quantities in theboundary state vectors can be computed. Now, knowingthe boundary state vectors completely, the state vectors andhence the motion quantities at each element at time ti canbe computed by the repeated use of corresponding transferequations of element similar to Eq. (8). The velocity, angu-lar velocity, acceleration and angular acceleration quantitiesat time ti are then obtained using Eq. (3), respectively. Thenentire procedure can be repeated for time ti+1 and so on. Theflow chart of algorithms for this method is shown as Fig.2.

It can be seen clearly from Eqs. (9) and (10) that theglobal dynamic equations of system are not needed if us-ing the proposed method to study the dynamics of multi-rigid-flexible-body systems moving in space. Irrespectiveof the size of a multi-rigid-flexible-body system, the ma-trices involved in the proposed method are always small,which greatly increases the computational speed and avoidsthe computing difficulties caused by too high matrix ordersfor complex multi-rigid-flexible-body systems. The overalltransfer equation of system can be assembled easily just us-ing the transfer equations of elements, and it is a set of alge-bra equations with respect to the system state variables. Sothis method simplifies the solving procedure of multi-rigid-flexible-body system dynamics greatly.

Fig. 2 Flow chart of algorithms for the proposed method

3 Dynamic equations of Euler–Bernoulli beam movingin space

As shown in Fig. 3a, the input end I of beam moving inspace is fixed on the original point O2 of the reference systemO2x2y2z2, and the axis O2x2 is along with the axis of beambefore deformation. rP is the position vector of any PointP with respect to oxyz, r I is the position vector of the inputpoint with respect to oxyz, l IP is the initial position vectorof any Point P with respect to O2x2y2z2 before deformation,uP is the deformation vector of any Point P with respect toO2x2y2z2. In the development presented in the subsequentsections, the beam satisfies the following assumptions: (1)the deformation obeys the linear elasticity assumption, andthe deformation is far smaller than the geometrical size ofbeam, namely the strain is far smaller than 1; (2) the beam isregarding as the Euler–Bernoulli beam.

Neglecting the axial deformation of the beam, the po-sition coordinates rP, velocity rP and acceleration rP of anyPoint P of the beam with respect to oxyz can be obtained

rP = r I + r IP = r I + A(l IP + u P), (11)

rP = r I + Aω(l IP + uP) + AuP, (12)

rP = r I + Aωω(l IP + uP) + A ˙ω(l IP + uP)

+2AωuP + AuP, (13)

where ω is the angular velocity of the body-fixed coordi-nate system, ω is the skew symmetric matrix of ω . r I =

[xI, yI, zI]T, rP = [x, y, z]T, r IP = Ar2,IP, r2,IP = [x2, v,w]T,


Fig. 3 Position description of any point on beam and model of beam element

l IP = [x2, 0, 0]T, uP = [0, v,w]T. v and w are the transversedeformation of any point on center line of the beam alongy2-axis and z2-axis with respect to O2x2y2z2, respectively.

According to the force and moment balance conditionsof beam element, one can obtain

q + f dx2 = q +∂q∂x2

dx2 + f ′dx2, (14)

m + dx2 × q +dx2

2× f dx2 = mdx2 + m +

∂m∂x2

dx2

+dx2

2× f ′dx2, (15)

where q , f , f ′, m and m are the interior force, distributedexterior force, distribution inertia force, interior moment anddistributed exterior moment of the element of the beam, re-spectively, f ′ = m(x2)rP, m(x2) is the linear density of beam.

According to the geometric relationship of the beam el-ement, one can obtain⎡⎢⎢⎢⎢⎢⎣α2,y

α2,z

⎤⎥⎥⎥⎥⎥⎦ = ∂∂x2

⎡⎢⎢⎢⎢⎢⎣−w

v

⎤⎥⎥⎥⎥⎥⎦ , (16)

where α2,y and α2,z are the angles caused by the deformationrotating about y2-axis and z2-axis, respectively.

According to material mechanics, one can obtain⎡⎢⎢⎢⎢⎢⎣

m2,y

m2,z

⎤⎥⎥⎥⎥⎥⎦ = EI∂

∂x2

⎡⎢⎢⎢⎢⎢⎣α2,y

α2,z

⎤⎥⎥⎥⎥⎥⎦ = EI∂2

∂x22

⎡⎢⎢⎢⎢⎢⎣−w

v

⎤⎥⎥⎥⎥⎥⎦ , (17)

where m2,y and m2,z are the interior moments along y2-axisand z2-axis with respect to O2x2y2z2, respectively, EI is thestiffness coefficient of the beam.

Substituting Eq. (17) to Eqs. (14) and (15), and omit-ting the high-order small parameters, the traversal vibrationequation of beam moving in space can been obtained

∂

∂x2

[∂

∂x2

(EI2,z

∂2

∂x22

v)+ m2,z

]

= f2,y − m(x2)[0, 1, 0]AT rP, (18)

∂

∂x2

[∂

∂x2

(EI2,y

∂2

∂x22

w)− m2,y

]

= f2,z − m(x2)[0, 0, 1]AT rP, (19)

where f2,y and f2,z are the distributing exterior forces alongy2-axis and z2-axis with respect to O2x2y2z2, m2,y and m2,z

are the distributing exterior moments along y2-axis and z2-axis with respect to O2x2y2z2, respectively.

According to the absolute angular momentum theoremof particle system to the motive Point I , one can obtain

dG I

dt+ mr IC × r I =

∑P

r IP × f P, (20)

where the symbol “C” denotes the point of mass center ofbeam, G I is the absolute moment of momentum of beam

moving in space with respect to Point I , G I =

∫ l

0m(x2)r IP×

r IPdx2.According to Eq. (20), the rotation equation of beam in

the inertia coordinate system can be obtained

A[ω

∫ l

0m(x2)J 1ωdx2 + ω

∫ l

0m(x2)r IuPdx2

+ddt

∫ l

0m(x2)J 1ωdx2 +

ddt

∫ l

0m(x2)r IuPdx2

]

+mr IC r I = A∑

P

r2,IP[ f2,x, f2,y, f2,z]TP, (21)

where l is the length of beam, r I is the skew symmetric ma-trix of vector r I, J 1 is inertia matrix, f2,x is the distributingexterior forces along x2-axis with respect to O2x2y2z2.

4 Transfer matrices of typical elements

By creating a library of transfer matrices for typical elementsand by assembling these elements at the required locations,various configurations of multi-rigid-flexible-body systemscan be modeled easily. The transfer matrices of elements canbe developed easily from the dynamic equations of elements


by using the linearization method. The transfer matrices ofsome elements have been developed in Refs. [10–12]. Thetransfer matrices of other elements, such as Euler–Bernoullibeam moving in space, smooth ball-and-socket hinge con-nected with beam moving in space, fixed hinge connectedwith beam moving in space, elastic hinge connected withbeam moving in space, are developed in follows.

4.1 Transfer matrix of Euler–Bernoulli beam moving inspace

Considering Euler–Bernoulli beam moving in space with thesmall transverse deformation, the state vector can be de-fined as Eq. (2). For describing conveniently, only the uni-form material constant section beam is considered, namelym(x2) = m. The mass of beam is m = ml.

Considering Eq. (11), integrating Eq. (14) along thebeam axis in [0, l] and projecting it to the inertia referencesystem, then one can obtain

qO = q I + A∫ l

0[ f2,x, f2,y, f2,z]

TPdx2 −

∫ l

0mrPdx2, (22)

where q I and qO are the interior forces of the input andoutput ends of beam in the inertia reference system, respec-tively, q I = [qx, qy, qz]T

I , qO = [qx, qy, qz]TO.

Using modal method, the transverse deformation v andw of a beam can be expressed as

v(x2, t) =n∑

k=1

Yk(x2)qk(t), w(x2, t) =n∑

k=1

Zk(x2)qk(t), (23)

where Yk(x2) and Zk(x2) are the k-th eigenvectors of thebeam in the y2- and z2-directions.

Submitting Eq. (23) to Eq. (22), and using the lin-earization formulae (3) and (7) to linearize this equation,then one can obtain

[qx, qy, qz]TO = U41[x, y, z]T

I + U42[θx, θy, θz]T + [qx, qy, qz]TI

+U45[q1, q2, · · · , qn]T + U46, (24)

where

U41 = −mχ1I3×3,

U42 = −mχ1 A ti−1

[T 1

∫ l

0r2,IPdx2, T 2

∫ l

0r2,IPdx2,

T 3

∫ l

0r2,IPdx2

]ti−1

,

U45 = −mχ1 A(ti−1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∫ l

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Y1(x2)

Z1(x2)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dx2,

∫ l

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Y2(x2)

Z2(x2)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dx2, · · · ,

∫ l

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Yn(x2)

Zn(x2)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dx2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

U46 = A∫ l

0[ f2,x, f2,y, f2,z]

TPdx2 − mχ2,rI − mχ1Φ(ti−1)

×∫ l

0r2,IP(ti−1)dx2 + mχ1 A(ti−1)

∫ l

0uP(ti−1)dx2

−m∫ l

0χ2,rIP dx2.

Let

h2 =

∫ l

0mJ 1ωdx2,

h3 =

∫ l

0mr I[0, v, w]Tdx2,

h4 = mr IC r I.

(25)

If considering the effect of distributed exterior force f anddistributed exterior moment m, and using Eq. (20), then ro-tation equation of beam in inertial reference system can bewritten as

mO =ddt

[A(h2 + h3)] + h4 −∫ l

0mdx2 + r IOqO

−A∫ l

0

∑j

r2,IP[ f2,x, f2,y, f2,z]TPdx2 + mI, (26)

where mI and mO are the interior moments of the input andoutput ends of beam in the inertia reference system, respec-tively, mI = [mx, my, mz]T

I , mO = [mx, my, mz]TO.

Linearizing h2, h3, h4, respectively, one can obtain

h2 = H 22[θx, θy, θz]Tti + H 25[q1, q2, · · · , qn]T

ti + H 26,

h3 = H 35[q1, q2, · · · , qn]Tti + H 36,

h4 = H 41[xI, yI, zI]Tti + H 45[q1, q2, · · · , qn]T

ti + H 46,

(27)

where

H 22 = χ3m∫ l

0J 1(ti−1)Hdx2,

H 26 =

∫ l

0mJ 1(ti−1)Hχ4,θdx2−

∫ l

0m(E2νti−1 +E3ω ti−1 )dx2,

H 25 =

[∫ l

0m(E2Y1 + E3Z1)dx2,

∫ l

0m(E2Y2 + E3Z2)dx2, · · · ,

∫ l

0m(E2Yn + E3Zn)dx2

],

H 35 =

[∫ l

0m(E4Y1 + E5Z1)dx2,

∫ l

0m(E4Y2 + E5Z2)dx2, · · · ,

∫ l

0m(E4Yn + E5Zn)dx2

],


H 36 =

∫ l

0mE6dx2,

H 41 = mχ1 r IC(ti−1),

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 −sy

0 cx sxcy

0 −sx cxcy

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

E2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2v −x2 0

−x2 0 −w

0 −w 2v

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ti−1

ω ti−1 ,

H 45 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣∫ l

0E7

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Y1

Z1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dx2,

∫ l

0E7

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Y2

Z2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dx2,

· · · ,∫ l

0E7

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Yn

Zn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dx2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

H 46 = −ml2

˜r I(ti−1)A

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

0

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+ mr IC(ti−1)[χ2,rI − r I(ti−1)],

E3 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2w 0 −x2

0 2w −v

−x2 −v 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ti−1

ω ti−1 ,

E4 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

wti−1 − χ3wti−1

0

χ3x2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

E5 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−vti−1 + χ3vti−1

−χ3x2

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

E6 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

vti−1 (χ4,w − wti−1 ) + wti−1 (vti−1 − χ4,v)

−x2χ4,w

x2χ4,v

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

E7 = −ml

˜r I(ti−1)A = −ml

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 −zI −yI

zI 0 −xI

−yI xI 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ti−1

A .

Substituting Eq. (27) to Eq. (26), one can obtain⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

mx

my

mz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦O

= U31

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

xI

yI

zI

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+ U 32

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

θx

θy

θz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

mx

my

mz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦I

+U34

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

qx

qy

qz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦I

+ U 35

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q1

q2

...

qn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+ U 36, (28)

where

U31 = H 41 + r IOU41,

U32 = χ3 AH22 + r IOU42,

U34 = r IO,

U35 = χ3 A(H 25 + H 35) + H 45 + r IOU45,

U36 = χ3 A(H 26 + H 36) + χ4,G I+ H 46 + r IOU46

−∫ l

0mdx2 − A

∫ l

0

∑j

r2,IP[ f2,x, f2,y, f2,z]TPdx2.

According to Eq. (11), the position coordinates of theoutput Point O of beam with respect to oxyz can be obtainedas

rO = r I + r IO. (29)

Substituting Eqs. (6) and (7) into Eq. (29), one canobtain

rO = r I + U12[θx, θy, θz]Tti + U 15[q1, q2, · · · , qn]T

ti + U16, (30)

where

U12 = A(ti−1)[T 1(ti−1)l IO, T 2(ti−1)l IO, T 3(ti−1)l IO],

U16 = Φ(ti−1)l IO,

U15 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣A

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Y1(l)

Z1(l)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, A

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Y2(l)

Z2(l)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, · · · , A

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

Yn(l)

Zn(l)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

As the orientation angle of the body-fixed referencesystem are equal at the output end and input end of the beam,then one can obtain

[θx, θy, θz]TO = [O3×3 I3×3 O3×3 O3×3 O3×n O3×1]z I. (31)

As the generalized coordinates are the same to anypoint on the beam, so

[q1, q2, · · · , qn]TO

= [On×3 On×3 On×3 On×3 In×n On×1]z I. (32)

From Eqs. (24), (28), (30)–(32), the transfer equationand transfer matrix of Euler–Bernoulli beam moving in spacecan be obtained

zO = Uz I, (33)


U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I3×3 U12 O3×3 O3×3 U15 U16

O3×3 I3×3 O3×3 O3×3 O3×n O3×1

U31 U32 I3×3 U 34 U35 U36

U41 U42 U12 I3×3 U45 U46

On×3 On×3 On×3 On×3 In×n On×1

O1×3 O1×3 O1×3 O1×3 O1×n 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (34)

4.2 Transfer matrix of smooth ball-and-socket hinge con-nected with beam moving in space

For a smooth ball-and-socket hinge, its mass and size havebeen neglected, the position coordinates and interior forcesare equal and interior moments equal to zero on its two ends.

4.2.1 Smooth ball-and-socket hinge whose inboard body isrigid body and outboard body is beam

Submitting Eq. (23) to the transverse vibration Eqs. (18) and(19) of Euler–Bernoulli beam moving in space, multiplyingY j(x2) ( j = 1, 2, · · · , n) on the two sides of Eq. (18), multi-plying Z j(x2) ( j = 1, 2, · · · , n) on the two sides of Eq. (19),integrating Eqs. (18) and (19) along x2 and linearizing, thenone can obtain⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q1

q2

...

qn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦O

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P11

P12

...

P1n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦I

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

θx

θy

θz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦O

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P51

P52

...

P5n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (35)

where

P1 j =−mχ1

(χ1 + Ω2j )dj

∫ l

0[0, Y j, Z j]AT(ti−1)dx2,

P2 j =F j

(χ1 + Ω2j )dj,

P5 j =1

(χ1 + Ω2j )dj

∫ l

0[Y j, Z j]

×[

f2y − ∂m2z

∂x2, f2z +

∂m2y

∂x2

]Tdx2

+mχ1

(χ1 + Ω2j )dj

∫ l

0[0, Y j, Z j]AT(ti−1)[xI, yI, zI]

Tti−1

dx2

+E j

(χ1 + Ω2j )dj,

E j = −m∫ l

0[0, Y j, Z j]ΦT

ti−1(χ2,rIP + χ1[xI, yI, zI]

Tti−1

)dx2,

F j = m∫ l

0[0, Y j, Z j]

×[T 1,ti−1 ATti−1

(χ2,r IP + χ1[xI, yI, zI]Tti−1

),

T 2,ti−1 ATti−1


),

T 3,ti−1 ATti−1


)]dx2,

∫ l

0m

⟨⎡⎢⎢⎢⎢⎢⎣Y j(x2)

Z j(x2)

⎤⎥⎥⎥⎥⎥⎦ ,⎡⎢⎢⎢⎢⎢⎣

Yk(x2)

Zk(x2)

⎤⎥⎥⎥⎥⎥⎦⟩

dx2

=

⎧⎪⎪⎨⎪⎪⎩0, j � k,

dj, j = k,∫ l

0EI

⟨⎡⎢⎢⎢⎢⎢⎣Y j(2)(x2)

Z j(2)(x2)

⎤⎥⎥⎥⎥⎥⎦ ,⎡⎢⎢⎢⎢⎢⎣

Yk(2)(x2)

Zk(2)(x2)

⎤⎥⎥⎥⎥⎥⎦⟩

dx2

=

⎧⎪⎪⎨⎪⎪⎩0, j � k,

Ω2j d j, j = k.

(36)

If the output end of outboard beam is also connectedwith a smooth ball-and-socket hinge, according to the inte-rior moments are all zero in the two ends of the outboardbeam, using the transfer equation of the outboard beam, thenone can obtain

03×1 = U31[x, y, z]TI + U 32[θx, θy, θz]T

O + U34[qx, qy, qz]TI

+U35[q1, q2, · · · , qn]TO + U 36, (37)

where the submatrices U i j are the elements of the transfermatrix of outboard beam.

Using Eqs. (35) and (37), one can obtain

[θx, θy, θz]TO = [−U−1

32 U 31, O3×3, O3×3,

−U−132 U34, O3×3, −U−1

32 U 36]z I,

[q1, q2, · · · , qn]TO = [U−1

55 U 51, O3×3, O3×3,

−U−155 U54, O3×3, U−1

55 U56]z I,

(38)

where

U31 = U 31 + U35

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P11

P12

...

P1n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

U32 = U 32 + U35

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

U36 = U 35

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P51

P52

...

P5n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+ U 36,


U55 = I +

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦U−1

32 U 35,

U51 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P11

P12

...

P1n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦−

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦U−1

32 U31,

U54 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦U−1

32 U 34,

U56 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P51

P52

...

P5n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦−

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦U−1

32 U 36.

The state vectors of the input end and output end of thesmooth ball-and-socket hinge whose inboard body is rigidbody and outboard body is beam are defined, respectively, as

z I = [x, y, z, θx, θy, θz,mx,my,mz, qx, qy, qz, 1]T,

zO = [x, y, z, θx, θy, θz,mx,my,mz, qx, qy, qz, q1,

q2, · · · , qn, 1]T.

(39)

From Eq. (38), the transfer equation and transfer ma-trix of smooth ball-and-socket hinge whose inboard body isrigid body and outboard body is beam can be obtained

zO = Uz I, (40)

U =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I3×3 O3×3 O3×3 O3×3 O3×1

−U−132 U31 O3×3 O3×3 −U−1

32 U 34 −U−132 U36

O3×3 O3×3 O3×3 O3×3 O3×1

O3×3 O3×3 O3×3 I3×3 O3×1

U−155 U51 On×3 On×3 −U−1

55 U 54 U−155 U56

O1×3 O1×3 O1×3 O1×3 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (41)

4.2.2 Smooth ball-and-socket hinge whose inboard body andoutboard body are beams

The state vectors of the input end and output end of thesmooth ball-and-socket hinge whose inboard body and out-

board body are beams are defined, respectively, as

z I = [x, y, z, θx, θy, θz,mx,my,mz, qx, qy, qz, q1,

q2, · · · , qn, 1]T,


q2, · · · , qn, 1]T.

(42)

Inserting n columns zero elements behind column 12 inthe transfer matrix of smooth ball-and-socket hinge whoseinboard body is rigid body and outboard body is beam de-scribed in Eq. (41), n is the highest order of the modal whichdescribes the deformation of inboard beam. The transferequation and transfer matrix of smooth ball-and-socket hingewhose inboard body and outboard body are beams can be ob-tained

zO = Uz I, (43)

U =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I 3×3 O3×3 O3×3 O3×3 O3×n O3×1

−U−132 U 31 O3×3 O3×3 −U−1

32 U 34 O3×n −U−132 U 36

O3×3 O3×3 O3×3 O3×3 O3×n O3×1

O3×3 O3×3 O3×3 I 3×3 O3×n O3×1

U−155 U 51 On×3 On×3 −U−1

55 U 54 On×n U−155 U 56

O1×3 O1×3 O1×3 O1×3 O1×n 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.(44)

4.2.3 Smooth ball-and-socket hinge whose inboard body isbeam and outboard body is rigid body

The state vectors of the input end and output end of thesmooth ball-and-socket hinge whose inboard body is beamand outboard body is rigid body are defined, respectively, as


q2, · · · , qn, 1]T,

zO = [x, y, z, θx, θy, θz,mx,my,mz, qx, qy, qz, 1]T.

(45)

Inserting n columns zero elements behind column 12 inthe transfer matrix of smooth ball-and-socket hinge whoseinboard body and outboard body are rigid body describedwith Eq. (40) in Ref. [10], the transfer equation and transfermatrix of smooth ball-and-socket hinge whose inboard bodyis beam and outboard body is rigid body can be obtained

zO = Uz I, (46)

U =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I3×3 O3×3 O3×3 O3×3 O3×n O3×1

−U−132 U 31 O3×3 O3×3 −U−1

32 U 34 O3×n −U−132 U 35

O3×3 O3×3 O3×3 O3×3 O3×n O3×1

O3×3 O3×3 O3×3 I3×3 O3×n O3×1

O1×3 O1×3 O1×3 O1×3 O1×n 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.(47)


4.3 Transfer matrix of fixed hinge connected with beammoving in space

For a fixed hinge, its mass and size have been neglected, theposition coordinates, interior forces and interior moments areequal on its two ends.

4.3.1 Fixed hinge whose inboard body is rigid body and out-board body is beam

Using the similar method as Sect. 4.2.1, the transfer rela-tion between the generalized coordinates which describes theoutboard beam deformation and the state vector of input endof the fixed hinge can be obtained, as described by Eq. (35).The state vectors of the input end and output end of the fixedhinge whose inboard body is rigid body and outboard bodyis beam are defined, respectively, as



q2, · · · , qn, 1]T.

(48)

According to Eq. (35), then the transfer equation andthe transfer matrix of fixed hinge whose inboard body is rigidbody and outboard body is beam can be obtained

zO = Uz I, (49)

U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I3×3 O3×3 O3×3 O3×3 O3×1

O3×3 I3×3 O3×3 O3×3 O3×1

O3×3 O3×3 I3×3 O3×3 O3×1

O3×3 O3×3 O3×3 I3×3 O3×1

U51 U52 On×3 On×3 U55

O1×3 O1×3 O1×3 O1×3 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (50)

where

U51 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P11

P12

...

P1n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, U52 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, U55 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P51

P52

...

P5n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (51)

P1 j, P2 j and P5 j have the same means with Eq. (36).

4.3.2 Fixed hinge whose inboard body and outboard bodyare beams

For the fixed hinge whose inboard body and outboard bodyare beams, the direction cosine matrices of its inboard beamand outboard beam have the following relation

AO = A I Ad, (52)

where

Ad =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 −Δθz Δθy

Δθz 1 0

−Δθy 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎣Δθy

Δθz

⎤⎥⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− ∂w∂x2

∂ν

∂x2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=

n∑k=1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−dZk(l)dx2

dYk(l)dx2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦qk(t),

Yk, Zk, x2, l, qk, w, v are the parameters of its inboard beam,AO and A I are the direction cosine matrices of the outboardbeam and inboard beam, respectively.

According to Eq. (52), one can obtain

[θx, θy, θz]TO = [θx, θy, θz]T

I + U25[q1, q2, · · · , qn]TI , (53)

where

U25 = [P51, P52, · · · , P5n],

P5k = H−1[0, −dZk(l)

dx2,

dYk(l)dx2

]T,

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 −sy

0 cx sxcy

0 −sx cxcx

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦I

.

Substituting Eq. (53) into Eq. (35), one can obtain

[q1, q2, · · · , qn]TO = U 51[x, y, z]T

I + U52[θx, θy, θz]TI

+U52U25[q1, q2, · · · , qn]TI + U56. (54)

The state vectors of the input end and output end ofthe fixed hinge whose inboard body and outboard body arebeams are defined, respectively, as


q2, · · · , qn, 1]T,


q2, · · · , qn, 1]T.

(55)

According to Eq. (54), then the transfer equation andthe transfer matrix of fixed hinge whose inboard body andoutboard body are beams can be obtained.

zO = Uz I, (56)

U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I 3×3 O3×3 O3×3 O3×3 O3×n O3×1

O3×3 I3×3 O3×3 O3×3 U25 O3×1

O3×3 O3×3 I3×3 O3×3 O3×n O3×1

O3×3 O3×3 O3×3 I3×3 O3×n O3×1

U 51 U52 On×3 On×3 U52U25 U56

O1×3 O1×3 O1×3 O1×3 O1×n 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (57)


4.3.3 Fixed hinge whose inboard body is beam and outboardbody is rigid body

The state vectors of the input end and output end of the fixedhinge whose inboard body is beam and outboard body isrigid body are defined, respectively, as


q2, · · · , qn, 1]T,


(58)

Deleting n rows elements which describes the outboardbeam deformation behind Row 12 in the transfer matrixof fixed hinge whose inboard body and outboard body arebeams described with Eq. (57), the transfer equation andtransfer matrix of fixed hinge whose inboard body is beamand outboard body is rigid body can be obtained

zO = Uz I, (59)

U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I3×3 O3×3 O3×3 O3×3 O3×n O3×1

O3×3 I3×3 O3×3 O3×3 U25 O3×1

O3×3 O3×3 I3×3 O3×3 O3×n O3×1

O3×3 O3×3 O3×3 I3×3 O3×n O3×1

O1×3 O1×3 O1×3 O1×3 O1×n 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (60)

4.4 Transfer matrix of elastic hinge connected with beammoving in space

Here, the elastic hinge including linear spring and rotaryspring, and the mass of the spring are zero.

4.4.1 Elastic hinge whose inboard body and outboard bodyare beams

According to the equilibrium of forces and torques at PointO or I, the following equations can be obtained.

q I + K (rO − r I) = 0, mI = K ′(θ ′O − θ ′I), (61)

where θ ′I and θ ′O are the orientated angles of the input endand output end of the elastic hinge, respectively.

K = diag (Kx,Ky,Kz), K ′= diag (K′x,K′y,K′z)

Kx, Ky, Kz and K′x, K′y, K′z are the stiffness coefficients oflinear springs and rotary springs, respectively

The direction cosine matrices of the input end of elastichinge and its inboard beam have the following relation

A ′I = A I A Iin, (62)

where A ′I and A I are the direction cosine matrices of theinput end of elastic hinge and its inboard beam, respectively.

A Iin =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 −ΔθI,z ΔθI,y

ΔθI,z 1 0

−ΔθI,y 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎢⎢⎣ΔθI,y

ΔθI,z

⎤⎥⎥⎥⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−∂wI

∂x2

∂νI∂x2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=

n∑k=1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−dZkI (lI)

dx2

dYkI (lI)

dx2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦qk

I (t),

YkI , Zk

I , lI, qkI , wI, vI are the parameters of the inboard beam.

The direction cosine matrices of the output end of elas-tic hinge and its outboard beam have the following relation

A ′O = AO AOout, (63)

where A ′O and AO are the direction cosine matrices of theoutput end of elastic hinge and its outboard beam, respec-tively.

AOout =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 −ΔθO,z ΔθO,y

ΔθO,z 1 0

−ΔθO,y 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎣ΔθO,y

ΔθO,z

⎤⎥⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−∂wO

∂x2

∂νO∂x2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=

n∑k=1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−dZkO(0)

dx2

dYkO(0)

dx2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦qk

O(t),

YkO, Zk

O, lO, qkO, wO, vO are the parameters of the inboard

beam.

Linearizing Eqs. (62) and (63), then

θ ′O = θO + H−1O [0,ΔθO,y,ΔθO,z]T,

θ ′I = θ I + H−1I [0,ΔθI,y,ΔθI,z]T,

(64)

where

H I =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 −sy

0 cx cy

0 −sx cxcy

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦I

, H O =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 −sy

0 cx cy

0 −sx cxcy

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦O

.

Substituting Eq. (64) into Eq. (61), one can obtain

θO + LO[q1, q2, · · · , qn]TO

= θ I + K′−1mI + L I[q1, q2, · · · , qn]TI , (65)

where

LO = H−1O

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 · · · 0

−dZ1O(0)

dx2−dZ2

O(0)

dx2· · · −dZn

O(0)

dx2

dY1O(0)

dx2

dY2O(0)

dx2· · · dYn

O(0)

dx2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,


L I = H−1I

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 · · · 0

−dZ1I (lI)

dx2−dZ2

I (lI)

dx2· · · −dZn

I (lI)

dx2

dY1I (lI)

dx2

dY2I (lI)

dx2· · · dYn

I (lI)

dx2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

Using the similar method as Sect. 4.2.1, the followingtransfer relation can be obtained⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q1

q2

...

qn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦O

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P11

P12

...

P1n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦I

− K−1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

qx

qy

qz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦I

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

θx

θy

θz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦O

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P51

P52

...

P5n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (66)

where P1 j, P2 j and P5 j have the same means with Eq. (36).Using Eqs. (65) and (66), one can obtain

[θx, θy, θz]TO = [U 21, U 22, U 23, U24, U25, U26]z I,

[q1, q2, · · · , qn]TO

= [U 51, U52, U 53, U54, U55, U 56]z I

(67)

where

U51 = U 5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P11

P12

...

P1n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, U 52 = U 5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

U53 = U 5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦K ′−1, U 54 = −U5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P11

P12

...

P1n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦K−1,

U55 = U 5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦L I, U 56 = U 5

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P51

P52

...

P5n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

U5 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝I +

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P21

P22

...

P2n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦LO

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

−1

,

U 21 = −LOU51, U22 = I − LOU52,

U 23 = K ′−1 − LOU53, U24 = −LOU54,

U 25 = L I − LOU55, U26 = −LOU56.

The state vectors of the input end and output end ofthe elastic hinge whose inboard body and outboard body arebeams moving in space are defined, respectively, as

z I = [x, y, z, θx, θy, θz,mx,my,mz, qx, qy, qz, q1, q2,

· · · , qn, 1]T,

zO = [x, y, z, θx, θy, θz,mx,my,mz, qx, qy, qz, q1, q2,

· · · , qn, 1]T.

(68)

From Eq. (67), the transfer equation and transfer ma-trix of elastic hinge whose inboard body and outboard bodyare beams moving in space can be obtained.

zO = Uz I, (69)

U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I 3×3 O3×3 O3×3 U14 O3×n O3×1

U 21 U22 U23 U24 U 25 U 26

O3×3 O3×3 I3×3 O3×3 O3×n O3×1

O3×3 O3×3 O3×3 I3×3 O3×n O3×1

U 51 U52 U53 U54 U 55 U 56

O1×3 O1×3 O1×3 O1×3 O1×n 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (70)

where U14 = −K−1.

4.4.2 Elastic hinge whose inboard body is rigid body andoutboard body is beam

The state vectors of the input end and output end of the elas-tic hinge whose inboard body is rigid body and outboardbody is beam are defined, respectively, as



q2, · · · , qn, 1]T.

(71)

Deleting n columns elements which describes the in-board beam deformation behind column 12 in the transfermatrix of elastic hinge whose inboard body and outboardbody are beams described with Eq. (70), the transfer equa-tion and transfer matrix of elastic hinge whose inboard bodyis rigid body and outboard body is beam can be obtained.

zO = Uz I, (72)


U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I3×3 O3×3 O3×3 U 14 O3×1

U21 U22 U23 U 24 U26

O3×3 O3×3 I3×3 O3×3 O3×1

O3×3 O3×3 O3×3 I3×3 O3×1

U51 U52 U53 U 54 U56

O1×3 O1×3 O1×3 O1×3 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (73)

4.4.3 Elastic hinge whose inboard body is beam and out-board body is rigid body

The state vectors of the input end and output end of the elas-tic hinge whose inboard body is beam and outboard body isrigid body are defined, respectively, as


q2, · · · , qn, 1]T,


(74)

For the elastic hinge whose inboard body is beam andoutboard body is rigid body, Eq. (65) can be simplified asfollowing.

θO = θ I + K ′−1mI + L I[q1, q2, · · · , qn]TI . (75)

According to Eq. (75), the transfer equation and trans-fer matrix of the elastic hinge whose inboard body is beamand outboard body is rigid body can be obtained

zO = Uz I, (76)

U =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

I3×3 O3×3 O3×3 U 14 O3×n O3×1

O3×3 I3×3 K ′−1 O3×3 L j O3×1

O3×3 O3×3 I3×3 O3×3 O3×n O3×1

O3×3 O3×3 O3×3 I3×3 O3×n O3×1

O1×3 O1×3 O1×3 O1×3 O1×n 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (77)

5 Numerical example

The dynamic model of a rigid-flexible-body three pendulumssystem moving in space under the effect of the gravity isshown in Fig. 4. The Elements 2 and 6 are rigid bodies, andthe Element 4 is Euler–Bernoulli beam. These three bod-ies (2, 4, 6) are connected by three smooth ball-and-sockethinges (1, 3, 5) successively. The parameters of system areas follows

m2 = m6 = 1 kg, m4 = 1.32 kg,

E4 = 2.5 × 1010 N/m2, ρ4 = 3.3 × 103 kg/m3,

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦2,C

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦4,C

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦6,C

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0.5

0.0

0.0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦m,

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦2,O

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦4,O

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x

y

z

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦6,O

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1.0

0.0

0.0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦m,

J 2 = J 6 = diag(16,

512,

512

)kg ·m2,

J 4 = diag (8.8 × 10−5, 0.44, 0.44) kg ·m2.

The initial orientation angles of the three bodies and the ini-tial angular velocities of bodies 2 and 4 are all zeros; theinitial angular velocities of body 6 are [0, 0.1, 0]T rad/s.

Fig. 4 Model of rigid-flexible-body three pendulums system mov-ing in space

Using the proposed method, the transfer equations ofthe elements can be obtained

z2,3 = U2 z2,1, z4,3 = U3 z2,3,

z4,5 = U4 z4,3, z6,5 = U5 z4,5,

z6,7 = U6 z6,5,

(78)

where U2 and U6 are transfer matrices of rigid bodies, ex-pressed with Eq. (31) in Ref. [10]; U4 is the transfer matrixof beam moving in space, expressed with Eq. (34); U 3 isthe transfer matrix of smooth ball-and-socket hinge whoseinboard body is rigid body and outboard body is beam, ex-pressed with Eq. (41); U5 is the transfer matrix of smoothball-and-socket hinge whose inboard body is beam and out-board body is rigid body, expressed with Eq. (47).

According to Eq. (78), the overall transfer equation andtransfer matrix of the rigid-flexible-body three pendulumssystem moving in space can be assembled

z6,7 = Uz2,1 = U6U5U 4U 3U2 z2,1. (79)

The boundary conditions of the system are

z2,1 = [0, 0, 0, θx, θy, θz, 0, 0, 0, qx, qy, qz, 1]T2,1,

z6,7 = [x, y, z, θx, θy, θz, 0, 0, 0, 0, 0, 0, 1]T6,7.

(80)

The system dynamics can be obtained by direct inte-grating the global dynamic equations of the system deducedby Newton–Euler method using R–K method and by solvingthe overall transfer equation of system using the proposedmethod respectively. The time histories of the deformationin the mid-point of beam 4 in y2- and z2-directionS got by


the two methods are shown in Figs. 5 and 6, respectively.The time histories of the orientation angles of rigid body 2and beam 4 got by the two methods are shown in Figs. 7and 8, respectively. It can be seen from Figs. 5–8 that theresults got by the two methods have good agreements, whichvalidate the proposed method.

Under the same microcomputer system and simulationcondition, the computational time required for the abovedynamics simulation by using the proposed method andNewton–Euler method has been obtained, respectively, asshown in Table 1. It can be seen clearly from Table 1 thatthe computational time reduces very largely if the proposedmethod is used.

Table 1 Computational time compared

Method Newton–Euler method The proposed method

Time/s 7.563 0.75

Fig. 5 Time history of the transverse deformation in y2 directionon mid-point of beam 4

Fig. 6 Time history of the transverse deformation in z2 directionon mid-point of beam 4

Fig. 7 Time history of the orientation angles of rigid body 2

Fig. 8 Time history of the orientation angles of beam 4

6 Conclusions

In this paper, by defining new state vectors and develop-ing new transfer matrices of elements moving in space, thediscrete time transfer matrix method of multi-rigid-flexible-body system is expanded to study the dynamics of multi-rigid-flexible-body system moving in space. The formula-tions and numerical example of a three pendulums systemmoving in space are given to validate the method. The trans-fer matrices of various typical elements are developed, whatcan be used directly to assemble many multi-rigid-flexible-body systems easily. So, this method provides flexibilityfor dynamics modeling of multi-rigid-flexible-body systems.That is, by creating a library of transfer matrices for typicalelements and by assembling them at the required locations,various configurations of multi-rigid-flexible-body systemscan be modeled easily. The proposed method avoids theglobal dynamic equations of system. It has very high com-putational efficiency and smaller memory storage require-


ment. The method provides a powerful tool for the multi-rigid-flexible-body system dynamics.

In the proposed method, we linearize the nonlinear dy-namic equations of each element to form its transfer equa-tion according to the step by step time integration methodin a proper time interval. The round off errors in TMM andtruncation errors caused by the linearization methods intro-duced in Sect. 2.2 are the main accumulated errors of thismethod involved in the computed solution. In fact, there is nogeneral method for exactly estimating these errors in differ-ent multibody systems. In author’s experience with the pro-posed method in solving multibody system dynamics, withtime step properly chosen, the truncation errors accumula-tion was not significant. On the other hand, the round offerrors accumulation is a function of the word length of thecomputing system as well as the number of elements to bemodeled, and is very significant for the proposed method.The numerical difficulties will arise when analyzing verylarge systems on small computing systems by using transfermatrices. Sometimes the multiplication of transfer matricesof different elements in a large system will lead to that sev-eral very large numbers are multiplied and several very smallnumbers are multiplied successively in nature sequence. Asthe word length of computing system is limited, if the ab-solute values of these numbers are too large or too small, itwill cause the solution process to probably eventually col-lapses. To avoid the problem of numerical difficulties arisingin the multiplication of transfer matrices, some methods canbe chosen, such as the Delta-matrix method [5], Modifiedtransfer-matrix method [5], Riccati TMM [17], and so on. Inauthor’s experience, when introducing Riccati transform asdescribed in Ref. [17], the proposed method is also valid fora very large system.

Acknowledgements When the first author worked as a guest Pro-fessor in Hamburg University of Technology, Germany, invited byProfessor Edwin Kreuzer. We owe special thanks to Professor JorgeA.C. Ambrosio (Instituto Superior Tecnico, Lisbon, Portugal) andProfessor Werner Schiehlen (University of Stuttgart, Germany) fortheir heuristic suggestions.

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Discrete time transfer matrix method for dynamics of multibody system with flexible beams moving in space