Flexible Joints in Structural and Multibody Dynamics∗

Olivier A. Bauchau and Shilei HanUniversity of Michigan-Shanghai Jiao Tong University Joint Institute

Shanghai, 200240, China

August 13, 2015

Abstract

Flexible joints, sometimes called bushing elements or force elements, are found in all struc-tural and multibody dynamics codes. In their simplest form, flexible joints simply consist ofsets of three linear and three torsional springs placed between two nodes of the model. Forinfinitesimal deformations, the selection of the lumped spring constants is an easy task, whichcan be based on a numerical simulation of the joint or on experimental measurements. Ifthe joint undergoes finite deformations, identification of its stiffness characteristics is not sosimple, specially if the joint is itself a complex system. When finite deformations occur, thedefinition of deformation measures becomes a critical issue. This paper proposes a family oftensorial deformation measures suitable for elastic bodies of finite dimension. These familiesare generated by two parameters that can be used to modify the constitutive behavior of thejoint, while maintaining the tensorial nature of the deformation measures. Numerical resultsdemonstrate the objectivity of the deformations measures, a feature that is not shared by thedeformations measures presently used in the literature. The impact of the choice of the twoparameters on the constitutive behavior of the flexible joint is also investigated.

1 Introduction

Flexible joints, sometimes called bushing elements or force elements, are found in all multibodydynamics codes. In their simplest form, flexible joints simply consist of sets of three linear and threetorsional springs placed between two nodes of a multibody system. For infinitesimal deformations,the selection of the lumped spring constants is an easy task, which can be based on a numericalsimulation of the joint or on experimental measurements.

If the joint undergoes finite deformations, identification of its stiffness characteristics is notso simple, specially if the joint is itself a complex system. When finite deformations occur, thedefinition of the deformation measures becomes a critical issue. Indeed, for finite deformation, theobserved nonlinear behavior of materials is partly due to material characteristics, and partly dueto kinematics.

For instance, Anand [1, 2] has shown that the classical strain energy function for infinitesimalisotropic elasticity is in good agreement with experiment for a wide class of materials for moderatelylarge deformations, provided the infinitesimal strain measure used in the strain energy function isreplaced by the Hencky or logarithmic measure of finite strain. This means that the behaviorof materials for moderate deformations can be captured accurately using linear constitutive laws,

∗Mechanical Sciences, 4(1): pp 65-77, 2013

1

provided that the infinitesimal strain measures are replaced by finite deformation measures that arenonlinear functions of displacement.

These nonlinear deformation measures capture the observed nonlinear behavior associated withthe nonlinear kinematics of the problem. Degener et al. [3] also reported similar findings for thetorsional behavior of beams subjected to large axial elongation.

Much attention has been devoted to the problem of synthesizing accurate constitutive propertiesfor the modeling of flexible bushings presenting complex, time-dependent rheological behavior, [4,5]. It is worth stressing, however, that the literature seldom addresses three-dimensional jointdeformations.

Much like multibody codes, most FE codes also support the modeling of lumped structuralelements. While linear analysis is easily implemented, problems are encountered when dealing withfinite displacements and rotations, as pointed out by [6]. Structural analysis codes, either specificallyintended for multibody dynamics analysis, like MSC/ADAMS, or for nonlinear FEA with multibodycapabilities, like Abaqus/Standard, allow arbitrarily large absolute displacements and rotations ofthe nodes and correctly describe their rigid-body motion. When lumped deformable joints are used,relative displacements and rotations are often required to remain moderate, although not necessarilyinfinitesimal.

Such restrictions occur when using the FIELD element of MSC/ADAMS, a linear element thatimplements an orthotropic torsional spring based on a constant, orthotropic constitutive matrix.Similarly, the JOINTC element implemented in Abaqus/Standard describes the interaction betweentwo nodes when the second node can “displace and rotate slightly with respect to the first node,”because its formulation is based on an approximate relative rotation measure.

The formulations and implementations of flexible joints available in research and commercialcodes do not appear to allow arbitrarily large relative displacements and rotations. Moreover, inmany cases, the ordering sequence of the nodes connected to the joint matters, because the behaviorof the flexible joint is biased towards one of the nodes. This problem is known to experiencedanalysts using these codes. To the authors’ knowledge, these facts are rarely acknowledged in theliterature. It appears that little effort has been devoted to the elimination of these shortcomingsfrom the formulations found in research and commercially available codes, although the predictionsof these codes might be unexpected.

This paper presents families of finite deformation measures that can be used to characterizethe deformation of flexible joints. These deformation measures are closely related to the tensorialparameterization motion developed by [7, 8]. Because they are of a tensorial nature, these defor-mation measures are intrinsic and invariant. Numerical examples demonstrate the invariance of theformulation and the ability to tailor the joint’s constitutive behavior in the nonlinear range. Sec-tion 2 describes the configuration of the flexible joint. Section 3 presents the proposed deformationmeasures, which are derived from invariance considerations.

2 Flexible joint configuration

2.1 Kinematics of the flexible joint

Figure 1 shows inertial frame F I = [O, I = (ı1, ı2, ı3)] and a flexible joint in its reference and de-formed configurations. It consists of a three-dimensional elastic body of finite dimension and of tworigid bodies, called handle k and handle ℓ, that are rigidly connected to the elastic body. In the refer-ence configuration, the configuration of the handles is defined by frame F0 =

[B,B0 = (b01, b02, b03)

],

where B0 forms an orthonormal basis. The geometric location of points K and L, which are mate-rial points of handles k and ℓ, respectively, coincides with that of point B. The motion tensor thatbrings frame F I to frame F0 is denoted C

0(u

0, R

0), where u

0is the position vector of point B with

2

respect to point O and R0is the rotation tensor that brings basis I to basis B0.

Reference configuration

Deformed configuration

Handle l

Handle k

b01

_

b03

_

b02

_

Elastic

bodyB0

b2

_ kb1

_ k

b3

_ k

K _Fk

_Fl

_Mk

_Ml

L

b1

_ l

b3

_ l

b2

_ l

Bk

Bl

C=Ck

=

Oi1

_

i3

_

i2

_

C0=

B

C=

l

Figure 1: Configuration of the flexible joint.

In the deformed configuration, the twohandles move to new positions and the elas-tic body deforms. Materials points K andL are now at distinct locations. The config-urations of the two handles are now distinctand are represented by two distinct frames,Fk =

[K,Bk = (bk

1, bk

2, bk

3)]

and F ℓ =[L,Bℓ = (bℓ

1, bℓ

2, bℓ

3)], respectively. Motion

tensors Ck and Cℓ bring frame F0 to frames

Fk and F ℓ, respectively. The displacementvectors from point B to points K and L,are denoted uk and uℓ, respectively, and ro-tation tensors Rk and Rℓ bring basis B0 to

bases Bk and Bℓ, respectively.Relative motion tensor C brings frame

Fk to frame F ℓ and provides an intrinsicrepresentation of the motion of handle ℓ with respect to handle k. The relative displacement vectorof point L with respect to point K is denoted u = uℓ − uk and R = (RℓR

0)(RkR

0)T is the relative

rotation tensor of basis Bℓ with respect to basis Bk.The motion tensors that bring frame F I to frames Fk and F ℓ, denoted C

k and Cℓ, respectively

can be expressed as

Ck =

[(RkR

0) (u0 + uk)(RkR

0)

0 (RkR0)

], (1a)

Cℓ =

[(RℓR

0) (u0 + uℓ)(RℓR

0)

0 (RℓR0)

], (1b)

respectively. The relative motion tensor then becomes

C = CℓC

k−1. (2)

The components of relative motion tensor resolved in frames Fk and F ℓ are identical,

C∗ = Ck−1C C

k = Cℓ−1C C

ℓ = Ck−1

Cℓ =

[R∗ u∗R∗

0 R∗

], (3)

where R∗ = (RkR0)TR(RkR

0) = (RℓR

0)TR(RℓR

0) = (RkR

0)T (RℓR

0) are the components of the

relative rotation tensor resolved in bases Bk or Bℓ and u∗ = (RkR0)T (uℓ − uk).

2.2 Applied loading

The deformation of the flexible joint stems from the applied forces and moments depicted in fig. 1.At point K, the applied force and moment vectors are denoted F k and Mk, respectively; thecorresponding quantities applied at point L are denoted F ℓ and M ℓ, respectively. The loadingapplied to the flexible joint is characterized by arrays Ak and Aℓ that correspond to a translationof these loads to point O,

Ak = T k−T

{F k

Mk

}=

{F k

Mk + (u0 + uk)F k

}, (4a)

Aℓ = T ℓ−T

{F ℓ

M ℓ

}=

{F ℓ

M ℓ + (u0 + uℓ)F ℓ

}, (4b)

3

where T k and T ℓ are the translation tensors from point O to points K and L, respectively, definedas

T k =

[I (u0 + uk)0 I

], (5a)

T ℓ =

[I (u0 + uℓ)0 I

], (5b)

respectively. Because these loads are both applied at point O, the equilibrium condition resultingfrom Newton’s third law simply states

Ak +Aℓ = 0. (6)

Note the parallel between vectors Ak and Aℓ and the second Piola-Kirchhoff stress tensor [9].Indeed, they represent the true loads applied to handle k and ℓ, respectively, in their deformedconfigurations, but translated to reference point O. Although expressing equilibrium of the systemin its deformed configuration, eq. (6) is a linear function of the loads. The joint is assumed to bemassless, i.e., inertial forces associated with its motion are neglected.

3 Deformation measures

The differential work done by the applied load will be evaluated in section 3.1, and the natureof deformation measures is further discussed in section 3.2. Section 3.3 then provides invariantdeformation measures based on the tensorial parameterization of motion.

3.1 Differential work

The differential work, dW , done by the forces applied to the joint is dW = F kTduk +MkTdψk +

F ℓTduℓ +M ℓTdψℓ, where duk and duℓ are the differential displacements of point K and L, respec-

tively, and dψk = axial(dRkRkT ) and dψℓ = axial(dRℓRℓT ) the differential rotations of handles kand ℓ, respectively. Recasting this expression in a matrix form leads to

dW ={dukT , dψkT

}T kTT k−T

{F k

Mk

}

+{duℓT , dψℓT

}T ℓTT ℓ−T

{F ℓ

M ℓ

}

= dUkTAk + dU ℓTAℓ,

where the last equality follows from eq. (4). The differential motions of handles k and ℓ are definedas dUk = Axial(dCk

Ck−1) and dU ℓ = Axial(dCℓ

Cℓ−1), respectively, leading to

dUk = T k

{duk

dψk

}, (7a)

dU ℓ = T ℓ

{duℓ

dψℓ

}, (7b)

respectively, where operator Axial(·) is defined by eq. (39). Finally, introducing eq. (6), the differ-ential work becomes

dW = AℓT(dU ℓ − dUk

).

4

The term in parenthesis represents the differential relative motion, dU = dU ℓ − dUk, where dU =Axial(dC C−1). As expected, the differential work depends on the differential relative motion of the

two handles only. The differential work can now be written as dW = AℓTdU = AkT (−dU), wherethe second equality follows from the equilibrium equation, eq. (6). This statement simply impliesthat the differential relative motion of handle ℓ with respect to handle k, denoted dU is of oppositesign of that of handle k with respect to handle ℓ, as expected. In summary, the differential work iswritten

dW = ATdU , (8)

where by convention, A = Aℓ and dU = dU ℓ − dUk the differential relative motion of handle ℓ withrespect to handle k.

Let E be a set of six generalized coordinates that define the relative motion tensor uniquely, i.e.,a one-to-one mapping is assumed to exist between these generalized coordinates and the relativemotion tensor. It follows that a one-to-one mapping must exist between the relative motion tensorand increments of the generalized coordinates

dU = H(E)dE , (9)

where tensor H(E) is the Jacobian or tangent tensor of the coordinate transformation. The differen-

tial work done by the forces applied to the joint, eq. (8), now becomes dW = ATH(E)dE = LTdE ,where the generalized forces associated with the generalized coordinates are defined as

L = HT (E)A. (10)

It is assumed that the flexible joint is made of an elastic material [10], which implies that thegeneralized forces can be derived from a potential, the strain energy of the joint, denoted A,

L =∂A(E)

∂E. (11)

The differential work now becomes

dW = dET ∂A(E)

∂E= d(A), (12)

and can be expressed as the differential of a scalar function, the strain energy.

3.2 The deformation measures

In the previous section, quantities E were defined as “a set of generalized coordinates that uniquelydefine the relative motion tensor,” but were otherwise left undefined. This implies that these gen-eralized coordinates form a parameterization of the relative motion tensor, i.e., C = C(E). Becausethe strain energy of the flexible joint can be expressed in terms of these generalized coordinatesthey are, in fact, deformation measures for the flexible joint. Consequently, any parameterizationof the relative motion tensor provides deformation measures for the flexible joint.

The following notation is introduced

E =

{εκ

}. (13)

The first three components of this array form the stretch vector, denoted ε, and the last three thewryness vector, denoted κ. Although any parameterization of the relative motion tensor providesadequate deformation measures for the flexible joint, these deformation measures should be invariant

5

with respect to the choice of coordinate system. This implies that the stretch and wryness vectorsshould be first-order tensors, and hence, the deformation measure should itself be a first-ordertensor.

While the parameterization of rotation has received wide attention [11, 12, 13, 14, 15], muchless emphasis has been placed on that of motion [16, 17, 18, 19, 20]. Bauchau et al. [21, 7] haveproposed the vectorial parameterization of motion, which consists of a motion parameter vector, E ,that parameterizes the motion tensor, C = C(E). [7] and [8] have studied the parameterization ofmotion, with special emphasis on its tensorial nature. They presented a formal proof that motionparameters vectors are first-order tensors if and only if they are parallel to the eigenvectors ofthe motion tensor associated with its unit eigenvalues. Furthermore, the tangent tensor is also asecond-order tensor for motion parameters vectors.

Deformation measures should be “objective” or “frame-indifferent,” and [9] gives a precise def-inition of this concept, which implies that deformation measures should be expressed in terms oftensorial quantities and be invariant under the superposition of a rigid body motion. The relativemotion tensor remains invariant under the superposition of a rigid body motion and hence, defor-mation measures that are functions of the relative motion tensor only will share this property. Theprevious paragraphs have underlined the fact that the deformation measures should be of a tensorialnature. If this latter property is achieved, the resulting deformation measures will be objective.

In summary, the desired invariance and objectivity of the deformation measure are achieved ifand only if this deformation measure is selected to be the vectorial parameterization of motion,which further implies that the deformation measure is parallel to the eigenvector of the motiontensor associated with its unit eigenvalue.

3.3 Explicit expression of the deformation measures

The proposed deformation measures are parallel to the eigenvector of the relative motion tensorassociated with its unit eigenvalue. Because this eigenvalue has a multiplicity of two [8], two linearlyindependent eigenvectors exist, which will be selected as

N†

1=

{p0

}, and N

†

2=

{H−1up

}, (14)

where p is the Wiener-Milenkovic rotation parameter vector [22, 23, 8] and H the tangent tensorfor this parameterization. The second eigenvector is easily recognized as the Wiener-Milenkovicmotion parameter vector [8], i.e.,

N†

2= P =

{qp

}. (15)

The following notation is introduced: p = ‖p‖ = 4 tanφ/4, = pT q = pp′d, where d = nTu is theintrinsic displacement, and (·)′ denoted a derivative with respect to angle φ. Note the following limitbehaviors: (1) limφ→0 p = φ, (2) limφ→0H

−1 = I and hence, limφ→0 q = u, and (3) limd→0 = 0;

The most general measure of deformation is a linear combination of vectors N†

1and N

†

2defined

by eq. (14), which will be written as

E =

[µI λI0 µI

]P = Z(λ, µ)P, (16)

where matrix Z is defined in eq. (37), and λ and µ are arbitrary scalars. The stretch and wrynessvectors now become ε = µq + λp and κ = µp, respectively. Mozzi-Chasles’ theorem implies thata general motion can be defined in terms of the direction of axis of the motion and in termsof two scalar parameters, φ, the magnitude of the relative rotation, and d, the intrinsic relative

6

displacement. Equivalently, scalar functions λ and µ can be expressed in terms of = pT q and

p =√pTp, i.e.,

λ = λ(, p), µ = µ(, p). (17)

Although functions λ and µ are arbitrary, their limit behavior for and p→ 0 can be obtainedbased on physical arguments. The wryness vector is written as κ = µp n; since µp must be an oddfunction of φ (or p), it implies that µ is an even function of the same variable. For small values of φ(or p), the wryness vector should be equal to the infinitesimal rotation vector, i.e., limp→0 κ = φn.Introducing the expression for the wryness vector yields limφ→0 4µ tan(φ/4) n = limφ→0 µφ n = φn,which implies

limφ→0

µ = 1. (18)

Similarly, for small rotation angles, the stretch vector should equal the displacement vector, i.e.,limφ→0 ε = u. Introducing the expression for the stretch vector yields limφ→0(µH

−1u+4λ tan(φ/4) n) =

u. Given the limit behaviors of µ and H−1, the stretch vector does indeed converge to the displace-ment vector. Finally, if the relative motion is planar, the intrinsic displacement vanishes and thestretch vector should be in the plane normal to unit vector n, i.e., limd→0 n

T ε = 0. Using theexpression for the stretch vector then leads to limd→0(µp

′d+ 4λ tanφ/4) = 0, which implies

limd→0

λ = 0. (19)

In summary, eq. (16) defines the proposed deformation measures for flexible joints. Theseequations are, in fact, the nonlinear deformation-displacement relationships of the problem. Theyare not fully determined because two scalar functions, λ and µ, appear in their definition. Thesetwo functions must satisfy the limit behavior discussed earlier but are otherwise arbitrary. Specificchoices of these two scalar functions result in different families of deformation measures.

4 Formulation of flexible joints

The strain energy of the flexible joint is assumed to be a quadratic function of the deformationmeasures E∗,

A(E∗) =1

2E∗TD∗E∗, (20)

where D∗ are the components of the stiffness matrix of the flexible joint resolved in the materialframe, which are assumed to be given constants in this frame.

4.1 Elastic forces in the flexible joint

The components of relative motion tensor resolved in the material frame are given by eq. (3) andits variation becomes

δU∗= δC∗C∗−1 =

(δCk−1

Cℓ + C

k−1δCℓ)C

ℓ−1C

k

= Ck−1

(δU

ℓ− δU

k)C

k,

where the virtual motion vectors, δUk and δU ℓ, are defined in eqs. (7a) and (7b), respectively. Itnow follows that

δU∗ = Ck−1

(δU ℓ − δUk

). (21)

The virtual motion vector is related to the virtual changes in the motion parameter vector [8]by means of the tangent tensor, H, as δU∗ = H(P∗)δP∗. It then follows that δP∗ = H−1(P∗)δU∗ =

7

H−1(Pk)Ck−1(δU ℓ − δUk), where the second equality was obtained with the help of eq. (21). The

motion tensor affords the following multiplicative decomposition [8], C∗(P∗) = H(P∗)H∗−1(P∗).

Equation (3) then implies CℓH∗ = CkH, and tensor W is defined as

W = (CkH)−1 = (CℓH∗)−1. (22)

Virtual changes in the motion parameter vector now become

δP∗ = W(δU ℓ − δUk). (23)

Variation in strain energy is expressed as δA = E∗TD∗δE∗ and requires evaluation of the virtualstrains, expressed as

δE∗ = B∗δP∗, (24)

where matrix B∗ is defined by eq. (48).Using eqs. (24) and (23), variation of the strain energy in the flexible joint becomes δA =

(δU ℓT − δUkT )WTBTL∗, where the generalized force vector, L∗, was defined as

L∗ = D∗E∗. (25)

The elastic forces in the joint now become

F =

{−F e

F e

}, (26)

whereF e = WTB∗TL∗. (27)

4.2 Stiffness matrix of the flexible joint

The stiffness matrix of the flexible joint stems from the linearization of the elastic forces defined byeq. (26). Linearization of F e yields

∆Fe = ∆WTN ∗ +WT∆B∗TL∗ +WTB∗T∆L∗, (28)

where N ∗ = B∗TL∗. Using eqs. (24) and (23), the last term of this expression is obtained as

WTB∗T∆L∗ =(B∗W

)TD∗

(B∗W

)(∆U ℓ −∆Uk).

The first term of eq. (28) is evaluated next. This term is recast as ∆WTN ∗ = [∆(CkH)−TN ∗+

∆(CℓH∗)−TN ∗]/2 to respect the symmetry of the problem expressed by eq. (22). Expanding thevariations leads to

∆WTN ∗ =

[−WTT ∗W −

F e

2,WTT ∗W −

F e

2

]{∆Uk

∆U ℓ

}(29)

where notation (·) is defined by eq. (41). Matrix T ∗ = [HTX +H∗TX ∗]/2, where matrices X and X ∗

are implicitly defined by the equations ∆H−TN ∗ = X∆P∗ and ∆H∗−TN ∗ = X ∗∆P∗, respectively;explicit expressions of these matrices are given in eqs. (55a) and (55b), respectively.

Finally, the second term of eq. (28) becomes WT∆B∗TL∗ = WTQ∗(L∗,P∗)∆P∗ and eq. (23)then leads to

WT∆B∗TL∗ = WTQ∗(L∗,P∗)W(δU ℓ − δUk). (30)

where the explicit expression of matrix Q∗ is given in eq. (49).

8

The linearized elastic forces can now be written explicitly as

∆F = K

{δUk

δU ℓ

}, (31)

where the stiffness matrix of the elastic joint is

K =

[D + F e/2 −D + F e/2

−D − F e/2 D − F e/2

](32)

where D = WT (B∗TD∗B + T ∗ +Q∗)W.

5 Numerical results

5.1 Change of reference point

Handle k Handle li1

_

i3

_

i2

_

F3

F2

Figure 2: Two beams connected with a flexible joint.

To illustrate the concepts developed in theprevious sections, consider the system con-sisting of two beams, each of length L= 1.2m, connected by a flexible joint, as depictedin fig. 2. The two beams have the same sec-tional stiffness properties: axial stiffness S= 43.50 MN; bending stiffness H22 = 23.26and H33 = 298.7 kN·m2 about unit vectorı2 and ı3, respectively; torsional stiffnessH11 = 28.05 kN·m2; and shearing stiffnessK22 = 4.0 and K33 = 2.81 MN, along unit vectors ı2 and ı3, respectively.

The first beam is clamped at its root and the second is loaded at its tip by concentrated forces tiploads F1 = 500α/9, F2 = 250α/9, and F3 = 200α/9 N, along unit vectors ı1, ı2, and ı3, respectively,as shown in fig. 2. The load factor α ∈ [0, 9]. The stiffness matrix of flexible joint is given by

D∗ =

2000 10 1010 2000 1010 10 2000

10001000

1000

. (33)

Two formulations will be contrasted here. In the first, the deformation measures are selected as

E† =

{ε†

κ†

}=

{(RkR

0)T (uℓ − uk)

axial(RT

0RkTRℓR

0)

}. (34)

The stretch vector, ε†, corresponds to the components of the relative displacement vector resolvedin basis Bk. The wryness vector, κ†, is the axial part of the relative rotation tensor, resolved onthe same basis; note that the components of the wryness vector selected here are identical whenresolved in basis Bk or Bℓ.

Various types of deformation measures have been used to represent the behavior of flexiblejoints, but those given by eq. (34) are rather typical; in the following sections, they will be referredto as “typical deformation measures.” In contrast, the deformation measures used in this work aredefined by eq. (16) and will be referred as “proposed deformation measures.”

9

The main claim of this paper is that the proposed strain measures are invariant with respectto the choice of reference point, i.e., are objective, a characteristic that is not shared by typicaldeformation measures. Imagine that points K and L are interchanged in fig. 2. Clearly, this doesnot modify the physical system, and hence, its response under load should be unaffected by thisinterchange. This basic invariance is satisfied by the proposed deformation measures, but not bytheir typical counterparts. To illustrate this effect, the following parameters were selected for theproposed deformation measures: λ = 0 and µ = (1 − p2/16)/(1 + p2/16)2. This choice is notimportant as the proposed deformation measures are invariant for any choice of these parameters.

For the typical deformation measures, two simulations were performed. In the first run, pointsK and L are selected as indicated in fig. 2 and in the second simulation, these two points wereinterchanged. Of course, this interchange is a modeling detail, which has no physical meaning.Yet, these two simulations yield different results because the definition of the deformation measuresmake specific reference to basis Bk, and hence, are inherently “basis sensitive.”

0 1 2 3 4 5 6 7 8 9

LOAD FACTOR, α

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

ε i [

m]

*

Figure 3: Components of the stretch vector: ε∗1

(©), ε∗2(△), ε∗

3(▽). Typical deformation mea-

sures. Reference point K: solid line, point L:dashed line.

-0.40

-0.30

-0.20

-0.10

0

0.10

0.20

0 1 2 3 4 5 6 7 8 9

κi *

LOAD FACTOR, α

Figure 4: Components of the wryness vector:κ∗1(©), κ∗

2(△), κ∗

3(▽). Typical deformation

measures. Reference point K: solid line, pointL: dashed line.

Figures 3 and 4 show the components of the stretch and wryness vector components, respectively,versus the load factor, α ∈ [0.9], for the typical deformation measures defined by eq. (34). For eachdeformation component, the response when reference points K and L are selected as shown in fig. 2is shown in solid lines, and the response when points K and L are interchanged in shown in dashedlines. Clearly, the typical deformation measures do not yield physically meaningful predictions,because different responses are obtained for a given physical system depending on the choice of thelabeling convention for points K and L.

In contrast, the proposed deformation measures yield the same predictions when the roles ofpoints K and L are interchanged. Figures 5 and 6 show the components of the stretch and wrynessvector components, respectively, for the proposed deformation measures defined by eq. (16).

Of course, for very small deformations, the predictions based on the typical and proposed de-formation measures are identical. This result is expected because both typical and proposed de-formation measures converge to the same infinitesimal deformation measures and because identicalstiffness matrices were used.

5.2 Choice of λ and µ

The proposed deformation measures are not unique. Rather, they form families dependent ontwo arbitrary parameters, λ(ρ, p) and µ(ρ, p): each choice of these parameters yields a different

10

0 1 2 3 4 5 6 7 8 90

0.05

0.10

0.15

0.20

0.25

0.30

0.35

LOAD FACTOR, α

ε i [

m]

*

Figure 5: Components of the stretch vector: ε∗1

(©), ε∗2(△), ε∗

3(▽). Proposed deformation

measures.

-0.40

-0.30

-0.20

-0.10

0

0.10

0.20

0 1 2 3 4 5 6 7 8 9

κi *

LOAD FACTOR, α

Figure 6: Components of the wryness vector:κ∗1(©), κ∗

2(△), κ∗

3(▽). Proposed deformation

measures

deformation measure but for all choices, the objectivity and tensorial nature of the deformationmeasure is preserved. Assuming that the simple strain energy expression defined by eq. (20) is usedwith a given stiffness matrix, the choice of parameters λ and µ will alter the response of the flexiblejoint under load in the nonlinear regime.

To study the influence of the choice of parameters λ and µ on joint behavior, a very simpleexample was treated. Handle k of the joint was clamped and forces and moments were applied tohandle ℓ; the applied force vector is F T =

{125 250 375

}α/9 and MT =

{125 250 375

}α/9,

where α is the load factor.A total of sixteen combinations of parameters λ and µ were selected for the study. First, λ = 0

is selected and cases 1a, 1b, 1c, and 1d correspond to µ = 1, µ = 1+ p2, µ = 1+ p4, and µ = 1+ p6,respectively, as listed in the first row of table 1. For case 1, parameter µ is function of p only. Next,λ = 0 is selected and cases 2a, 2b, 2c, and 2d correspond to µ = 1, µ = 1 + ρ, µ = 1 + ρ2, andµ = 1 + ρ3, respectively, as listed in the second row of table 1. For case 2, parameter µ is functionof ρ only. Cases 3 and 4 are defined similarly, as listed in the third and fourth row of table 1,respectively.

Table 1: Choices of parameters λ and µ for the sixteen cases.Case a b c d

1 λ = 0 µ = 1 1 + p2 1 + p4 1 + p6

2 λ = 0 µ = 1 1 + ρ 1 + ρ2 1 + ρ3

3 µ = 1 λ = 0 p2 p4 p6

4 µ = 1 λ = 0 ρ ρ2 ρ3

In all cases, the stiffness matrix of flexible joint was selected as

D∗ =

1000 100 100100 1000 100100 100 1000

10001000

1000

(35)

Figures 7 and 8 show the components of the stretch and wryness vectors, respectively, for cases1a, 1b, 1c, and 1d. The choice of very simple polynomial expressions for µ = µ(p) is arbitrary, but

11

00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

LOAD FACTOR, α

ε i [

m]

*

Figure 7: Components of the stretch vector: ε∗1

(solid line), ε∗2(dashed line), ε∗

3(dash-dotted

line). Case 1a (©), 1b (△), 1c (▽), 1d (∗).

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

κi *

LOAD FACTOR, α

Figure 8: Components of the wryness vector:κ∗1(solid line), κ∗

2(dashed line), κ∗

3(dash-dotted

line). Case 1a (©), 1b (△), 1c (▽), 1d (∗).

all satisfy the limit behavior expressed by eq. (18). Similarly, the constant value of λ = 0 satisfiesthe limit behavior expressed by eq. (19).

For small values of the load factor, the flexible joint deformation remains small and identicalresponse is observed for all choices of parameter µ. As larger loads are applied, the deformation isno longer infinitesimal and the joint’s nonlinear response is affected by the choice of parameter µ.This effect is particularly pronounced in the wryness response, as shown in fig. 8.

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

LOAD FACTOR, α

ε i [

m]

*

Figure 9: Components of the stretch vector: ε∗1

(solid line), ε∗2(dashed line), ε∗

3(dash-dotted

line). Case 2a (©), 2b (△), 2c (▽), 2d (∗).

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

κi *

LOAD FACTOR, α

Figure 10: Components of the wryness vector:κ∗1(solid line), κ∗

2(dashed line), κ∗

3(dash-dotted

line). Case 2a (©), 2b (△), 2c (▽), 2d (∗).

Of course, parameters λ and µ are arbitrary functions of both variables ρ and p. Case 1, in whichparameter µ is a function of variable p only, is a special case. In case 2, parameter µ is selected tobe a simple polynomial function of variable ρ only, as shown in the second row of table 1. Figures 9and 10 show the components of the stretch and wryness vectors, respectively, for cases 2a, 2b, 2c,and 2d. The nonlinear response of both stretch and wryness vector components is significantlyaffected by the choice of parameter µ.

Finally, two additional cases, cases 3 and 4, were treated where parameter λ is selected to be afunction of variables p and ρ, respectively, while keeping a constant value of parameter µ = 1, aslisted in the last two rows of table 1, respectively. Here again, simple polynomial expressions wereselected and the limit behaviors expressed by eqs. (18) and (19) were satisfied. Figures 11 and 12

12

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

LOAD FACTOR, α

ε i [

m]

*

Figure 11: Components of the stretch vector:ε∗1(solid line), ε∗

2(dashed line), ε∗

3(dash-dotted

line). Case 3a (©), 3b (△), 3c (▽), 3d (∗).

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

κi *

LOAD FACTOR, α

Figure 12: Components of the wryness vector:κ∗1(solid line), κ∗

2(dashed line), κ∗

3(dash-dotted

line). Case 3a (©), 3b (△), 3c (▽), 3d (∗).

show the components of the stretch and wryness vectors, respectively, for cases 3a, 3b, 3c, and 3d.Finally, figs. 13 and 14 show the components of the stretch and wryness vectors, respectively, forcases 4a, 4b, 4c, and 4d.

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

LOAD FACTOR, α

ε i [

m]

*

Figure 13: Components of the stretch vector:ε∗1(solid line), ε∗

2(dashed line), ε∗

3(dash-dotted

line). Case 4a (©), 4b (△), 4c (▽), 4d (∗).

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6 7 8 90

κi *

LOAD FACTOR, α

Figure 14: Components of the wryness vector:κ∗1(solid line), κ∗

2(dashed line), κ∗

3(dash-dotted

line). Case 4a (©), 4b (△), 4c (▽), 4d (∗).

The goal of the simple examples presented in the previous paragraphs is to show that the non-linear behavior of the flexible joint is strongly affected by the functional dependency of parametersλ and µ on variables ρ and p. Of course, in general, the two parameters can be selected to befunctions of both variables, i.e., λ = λ(ρ, p) and µ = µ(ρ, p). Softening or stiffening behavior canbe obtained by tailoring the functional dependency of parameters λ and µ on variables ρ and p.

The attention has focused thus far on the strain energy given by eq. (20), which is a quadraticexpression of the deformation measures, A = E∗TD∗E∗/2. This leads to the linear relationshipbetween the generalized forces and proposed deformation measures, L∗ = D∗E∗, see eq. (25). Thislinear relationship, however, is deceptively simple.

Indeed, eq. (27) shows that the relationship between the elastic forces in the joint and theproposed deformation measures is F e = WTB∗TL∗. Next, the relationship between the externally

13

applied loads and proposed deformation measures is obtained with the help of eqs. (4) as

{F k

Mk

}= T kTWTB∗TD∗Z(λ, µ)P∗, (36a)

{F ℓ

M ℓ

}= T ℓTWTB∗TD∗Z(λ, µ)P∗, (36b)

where eq. (16) was used to express the proposed deformation measures in terms of P∗, the relativemotion of the joint’s two handles. Although the stiffness matrix, D∗, is constant, the relationshipbetween the externally applied loads and the joint’s deformation measures is nonlinear becausematrices W and B∗ are nonlinear functions of the deformations measures, see eqs. (22) and (48),respectively, and the relative motion of the two handles, P∗, is also a nonlinear function of the han-dle’s relative motion. These observations explain the nonlinear load-deformation behavior exhibitedin figs. 4 to 14.

In practice, the joint’s constitutive laws can be obtained from a two step procedure. First, exper-imental measurements must be obtained for infinitesimal deformations of the joint and lead to theidentification of the entries of the constant stiffness matrix, D∗. Next, experimental measurementscharacterizing the joint’s behavior in the nonlinear range must be obtained. The optimal functionaldependencies of parameters λ and µ on variables ρ and p can then be determined using suitableparameter identification methods.

Of course, it is not guaranteed that suitable functions, λ = λ(ρ, p) and µ = µ(ρ, p), can befound, which will model the observed behavior of the joint accurately. In such case, more complexstrain energy expressions could be selected in an attempt to better capture the observed constitutivebehavior of the joint; in general, A = A(E∗). Of course, the existence of a strain energy functionimplies that material behavior is conservative, which will not always be true.

6 Conclusions

This paper has focused on the constitutive behavior of elastic bodies of finite dimension, typicallycalled flexible joints in structural and multibody dynamics. Physically meaningful deformationmeasures were proposed that are objective and of a tensorial nature; an explicit expression of thesemeasures was derived. Equipped with these deformation measures, constitutive laws for the flexiblejoint were derived by assuming the existence of a strain energy function that is a quadratic formof these deformation measures. Because all the quantities involved in the formulation are objectiveand tensorial, the predicted joint behavior presents the required invariance with respect to changesof basis or reference point.

Numerical examples were presented that demonstrate the invariance of the predicted behaviorwith respect to the choice of reference point, even in the nonlinear range; in contrast, typical formu-lations found in the literature up to date do not appear to present these desirable characteristics.The proposed deformation measures are not unique: their definition depends on the choice of twoparameters, which are functions of the relative rotation and the intrinsic relative displacement atthe joint. Numerical examples presented in the paper show that the choice of these two parametersaffects the response of the joint in the nonlinear regime significantly. Consequently, the proposeddeformation measures form families, and the choice of the functional dependency of the parametercan be selected to tailor the nonlinear response of the joint.

This paper has focused on an expression of the strain energy that depends on the proposeddeformation measures in a quadratic manner, leading to a linear relationship between the generalizedforces and deformations measure. Despite this linearity, the relationship between externally appliedforces and deformations is nonlinear. More general joint constitutive behavior could be obtained by

14

considering more general strain energy expressions. Investigating energy dissipation in flexible jointbased on the time rate of change of the proposed deformation measure is another possible extensionof the this work.

A Notational conventions

To simplify the writing of this seemingly complicated expression, the following notation is intro-duced. First, tensor Z, a function of two scalars, α and β, is introduced

Z(α, β) =

[βI αI0 βI

]. (37)

Second, the generalized vector product tensor is defined

N =

[n m0 n

]. (38)

Notation N does not indicate a 6 × 6 skew-symmetric tensor, but rather the above 6 × 6 tensorformed by three skew-symmetric sub-tensors. By analogy to notation a = axial(a), the followingoperator is introduced

N = Axial(N ). (39)

Consider two vectors defined as

V =

{vω

}, P =

{pq

}.

The well-known property of the vector product, ab = −ba, then generalizes to

VP = −PV, (40a)

VTP = PV , (40b)

where the following notation was introduced

P =

[0 pp q

]. (41)

Finally, the following identity results

P V = P V + VT P. (42)

B Linearization of functions λ and µ

Scalar λ and µ were introduced in eq. (17) as functions of and p. Using the chain rule forderivatives, the variation of µ is δµ = µδ+µpδp, where notations (·) and (·)p indicate derivativeswith respect to variable and angle p, respectively. Because = p∗T q∗ and p2 = p∗Tp∗, it follows

that δ = p∗T δq∗ + q∗T δp∗ and pδp = p∗T δp∗, and hence,

{δpδp

}=

[p∗T q∗T

0T p∗T

]δP∗. (43)

15

The variations of parameters λ and µ now become

{δλδµ

}= Λ

1(λ, µ)

[p∗T q∗T

0T p∗T

]δP∗. (44)

where matrix Λ1is defined as

Λ1(λ, µ) =

[λ λp/pµ µp/p

]. (45)

Because parameters λ and µ are even functions of variable p, their derivatives with respect to thesame variable are odd functions of p and hence, λp/p and µp/p present no singularity when p→ 0.

Consider two arbitrary arrays, LT ={ℓT mT

}and PT =

{qT pT

}, and two arbitrary scalars,

α = α(, p) and β = β(, p). Matrices Z and Z are defined implicitly by the following two identities,

δZ(α, β)L = Z(L, α, β,P)δP and δZT (α, β)L = Z(L, α, β,P)δP, respectively. It is shown easilyshown that

δZ (α, β)L =

[m ℓ0 m

]{δαδβ

}, (46a)

δZT (α, β)L =

[0 ℓℓ m

]{δαδβ

}. (46b)

Introducing eq. (44) now yields the following explicit expressions for matrices Z and Z,

Z(L, α, β,P) =

[m ℓ0 m

]Λ

1(α, β)

[pT qT

0T pT

], (47a)

Z(L, α, β,P) =

[0 ℓℓ m

]Λ

1(α, β)

[pT qT

0T pT

]. (47b)

C Variations of the strain measures

The proposed strain measures are defined by eq. (16) and variations of these quantities are expressed

as δE∗ =[Z(λ, µ) + Z(P∗, λ, µ,P∗)

]δP∗, where matrix Z is defined by eq. (47a). This implies that

δE∗ = B∗δP∗, where matrix B∗ is defined as

B∗ = Z(λ, µ) + Z(P∗, λ, µ,P∗). (48)

The linearization of the elastic forces requires the evaluation of matrix Q∗, implicitly defined by

δB∗L∗ = Q∗δP∗. The previous results yield

Q∗ =Z(ǫ, γ) + Z(P∗, λ, µ,L∗) + Z(L∗, λ, µ,P∗)

+

[p∗ q∗

0 p∗

]S

[p∗T q∗T

0T p∗T

],

(49)

where the following scalar functions were defined

{αβ

}=

[p∗T q∗T

0T p∗T

]L∗,

{ǫγ

}=

[λ λp/pµ µp/p

]{αβ

}, (50)

and

S =

[αλ + βλp/p αλp/p+ β(λpp − λp/p)/p

2

αµ + βµp/p αµp/p+ β(µpp − µp/p)/p2

]. (51)

16

D Linearization of the tangent tensor

The linearization of the elastic forces also requires linearization of the tangent tensor. When usingthe Wiener-Milenkovic motion parameterization, the tangent tensor is expressed as

H−1(P) = Z(χ0, χ0)− P/2 + Z(χ2, χ2)PP , (52a)

H∗−1(P) = Z(χ0, χ0) + P/2 + Z(χ2, χ2)PP. (52b)

The linearization of the tangent tensor is achieved by defining matrices X and X ∗ implicitly defined

by the following expressions, ∆H−TL = X (L)∆P and H∗−TL = X ∗(L)∆P . Tedious algebra yieldsexplicit equations of these matrices as

X (L) =Z(L, χ0, χ0,P) + Z(NP, χ2, χ2,P)

− L/2 + ZT (χ2, χ2)(N + PT L), (53a)

X ∗(L) =Z(L, χ0, χ0,P) + Z(NP, χ2, χ2,P)

+ L/2 + ZT (χ2, χ2)(N + PT L), (53b)

where N = PTL = LP and matrix Z is defined by eq. (47b).The Wiener-Milenkovic rotation parameterization is defined as p = p(φ)n, where p(φ) =

4 tanφ/4. Table 2 lists the expressions for all the scalar functions appearing in the above expressionwhen the Wiener-Milenkovic motion parameterization is used.

Table 2: Scalars functions associated with the Wiener-Milenkovic motion parameterization.Quantity Value Quantity Value

ν 1/(1 + p2/16) ε 1/(1− p2/16)p′ 1 + p2/16 ε ε2/8ζ1 ν2/ε ζ2 ν2/2ζ1 ν2(1− 4ν)/8 ζ2 −ν3/8σ0 ν σ2 ν2/8σ0 −ν2/8 σ2 −ν3/32χ0 1/ν χ2 1 / 8χ0 /8 χ2 0

Matrices X and X ∗ defined in eqs. (53a) and (53b), respectively, now simplify considerably.

Indeed, eqs. (37) and (47b) imply ZT (χ2, χ2) = I/8, Z(NP, χ2, χ2,P) = 0, and

Z(L, χ0, χ0,P) =1

8

[0 ℓℓ m

] [pT qT

0T pT

], (54)

and finally,

X (L) = Z(L, χ0, χ0,P)− L/2 + (N + PT L)/8, (55a)

X ∗(L) = Z(L, χ0, χ0,P) + L/2 + (N + PT L)/8. (55b)

References

[1] L. Anand. On H. Hencky’s approximate strain-energy function for moderate deformations.Journal of Applied Mechanics, 46:78–82, March 1979.

17

[2] L. Anand. Moderate deformations in extension-torsion of incompressible isotropic elastic ma-terials. Journal of the Mechanics and Physics of Solids, 34(3):293–304, 1986.

[3] M. Degener, D.H. Hodges, and D. Petersen. Analytical and experimental study of beamtorsional stiffness with large axial elongation. Journal of Applied Mechanics, 55:171–178, March1988.

[4] R. Ledesma, Z.-D. Ma, G. Hulbert, and A. Wineman. A nonlinear viscoelastic bushing elementin multibody dynamics. Computational Mechanics, 17(5):287–296, September 1996.

[5] J. Kadlowec, A. Wineman, and G. Hulbert. Elastomer bushing response: Experiments andfinite element modeling. Acta Mechanica, 163(5):25–38, 2003.

[6] P. Masarati and M. Morandini. Intrinsic deformable joints. Multibody System Dynamics,23(4):361–386, 2010.

[7] O.A. Bauchau and L.H. Li. Tensorial parameterization of rotation and motion. Journal ofComputational and Nonlinear Dynamics, 6:031007 1–8, 2011.

[8] O.A. Bauchau. Flexible Multibody Dynamics. Springer, Dordrecht, Heidelberg, London, New-York, 2011.

[9] L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Inc.,Englewood Cliffs, New Jersey, 1969.

[10] O.A. Bauchau and J.I. Craig. Structural Analysis with Application to Aerospace Structures.Springer, Dordrecht, Heidelberg, London, New-York, 2009.

[11] T.R. Kane. Dynamics. Holt, Rinehart and Winston, Inc, New York, 1968.

[12] J. Argyris. An excursion into large rotations. Computer Methods in Applied Mechanics andEngineering, 32(1-3):85–155, 1982.

[13] M.D. Shuster. A survey of attitude representations. Journal of the Astronautical Sciences,41(4):439–517, 1993.

[14] A. Ibrahimbegovic. On the choice of finite rotation parameters. Computer Methods in AppliedMechanics and Engineering, 149:49–71, 1997.

[15] O.A. Bauchau and L. Trainelli. The vectorial parameterization of rotation. Nonlinear Dynam-ics, 32(1):71–92, 2003.

[16] J. Angeles. On twist and wrench generators and annihilators. In M.F.O. Seabra Pereira andJ.A.C. Ambrosio, editors, Computer-Aided Analysis Of Rigid And Flexible Mechanical Systems,pages 379–411, Dordrecht, 1993. NATO ASI Series, Kluwer Academic Publishers.

[17] M. Borri, L. Trainelli, and C.L. Bottasso. On representations and parameterizations of motion.Multibody Systems Dynamics, 4:129–193, 2000.

[18] T. Merlini and M. Morandini. The helicoidal modeling in computational finite elasticity. Part I:Variational formulation. International Journal of Solids and Structures, 41(18-19):5351–5381,2004.

[19] T. Merlini and M. Morandini. The helicoidal modeling in computational finite elasticity. PartII: Multiplicative interpolation. International Journal of Solids and Structures, 41(18-19):5383–5409, 2004.

18

[20] E. Pennestrı and R. Stefanelli. Linear algebra and numerical algorithms using dual numbers.Multibody System Dynamics, 18:323–344, 2007.

[21] O.A. Bauchau and J.Y. Choi. The vector parameterization of motion. Nonlinear Dynamics,33(2):165–188, 2003.

[22] T.F. Wiener. Theoretical Analysis of Gimballess Inertial Reference Equipment Using Delta-Modulated Instruments. PhD thesis, Massachusetts Institute of Technology, Cambridge, Mas-sachusetts, 1962. Department of Aeronautical and Astronautical Engineering.

[23] V. Milenkovic. Coordinates suitable for angular motion synthesis in robots. In Proceedings ofthe Robot VI Conference, Detroit MI, March 2-4, 1982, 1982. Paper MS82-217.

19