Tampereen teknillinen yliopisto. Teknillisen mekaniikan ja optimoinnin laitos.Tutkimusraportti 2004:3Tampere University of Technology. Institute of Applied Mechanics and Optimization.Research Report 2004:3

Jari Mäkinen

A Formulation for Flexible Multibody Mechanics

Lagrangian Geometrically Exact Beam Elements using Constraint Manifold Parametrization

Thesis for the degree of Doctor of Technology to be presented with due permission for publicexamination and criticism in Auditorium K1703, at Tampere University of Technology,on the 10th of December 2004, at 12 o’clock noon.

Tampereen teknillinen yliopisto. Teknillisen mekaniikan ja optimoinnin laitosTampere 2004

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ISBN 952-15-1288-1ISSN 1459-5532

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ABSTRACT

In this thesis, a general formulation for flexible multibody mechanics is given. Since multibody systems arehighly constrained, the examination of differential geometry is necessary to understand the internal geome-try and kinematics of general multibody systems. Therefore, the formulation is given in the language ofdifferential geometry: manifolds, tangent spaces and tangent tensors on manifolds, push-forward and pull-back operators, metric tensors, etc.

The rotation manifold, whose elements are rotation operators, is thoroughly investigated. This study proba-bly provides the most important contribution of the thesis: material incremental rotation vectors, materialangular velocity vectors, and material angular acceleration vectors belong to the different tangent spaces ofthe rotation manifold. Hence, the direct application of the material incremental rotation vector with standardtime integration methods yields serious problems: adding quantities which belong to the different tangentspaces.

The formulation is applied to Reissner’s geometrically exact beam theory, giving a new geometrically exactbeam element that is based on the total Lagrangian updating procedure. The element has the total rotationvector as the unknown variable and the singularity problems at the rotation angle 2π and its multiples are

handled by the change of parametrization on the rotation manifold. The consistent stiffness, gyroscopic,centrifugal, and loading tensors of the total Lagrangian formulation are given explicitly. The total Lagran-gian formulation has several benefits such as all unknown variable vectors belong to the same tangent space,no need for secondary storage variables, the path-independence property (in the static case), any standardtime integration algorithm may be used, the symmetric stiffness tensor, a simple form of the kinetic energyand all nonlinear effects are included.

In addition, the formulation is applied to parametrize the constraint manifold that arises from point-wiseholonomic constraint equations. The constraint manifold parametrization, using the total Lagrangian for-mulation, has several benefits: the minimal number of variables, objective formulation, no need for secon-dary storage variables, constraint equations are satisfied automatically, the resulting equations of motion areordinary differential equations (not differential-algebraic), and easy to apply time-dependent boundary con-ditions. The constraint manifold parametrization is particularly competitive in the flexible multibody systemwhere the number of degrees of freedom is extensive comparing with the number of constraint equations.Moreover, special beam elements, which involve holonomic constraints, are also derived as the examples ofthe formulation. These elements can be exploited as customary elements in the finite element method.

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ACKNOWLEDGEMENTS

This thesis is a result of my studies in the Institute of Applied Mechanics and Optimization at the TampereUniversity of Technology as a graduate school student during 1997-2001 and as a researcher funded by EmilAaltonen’s Foundation during 2001-2004.

I am grateful to the head of the Institute of Applied Mechanics and Optimization, Professor Juhani Koski,for the possibility of working on this fascinating subject and for offering excellent research conditions.Many thanks also to the staff of the Institute for the pleasant atmosphere.

Special thanks belong to the other half of our two-person research group, to colleague Heikki Marjamäki,for his propositions in modeling telescopic boom systems.

In addition, I would like to thank preliminary examiners for their comments to improve the manuscript.

The financial support from the National Graduate School in Engineering Mechanics and Emil Aaltonen’sFoundation is also gratefully acknowledged.

I dedicate this thesis to my dear wife, Katri, who has shown a great patience and understanding during myresearch.

Tampere, November 2004

Jari Mäkinen

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Table of Contents

ABSTRACT ................................................................................................................................3

ACKNOWLEDGEMENTS ..........................................................................................................4

LIST OF SYMBOLS ....................................................................................................................6

1 INTRODUCTION ....................................................................................................................81.1 Review of Flexible Multibody Mechanics ........................................................................9

1.1.1 Classical Formulations for Flexible Multibody Systems ............................................101.1.2 Nonlinear Finite Element Formulations for Flexible Multibody Systems ...................111.1.3 Modeling Holonomic Constraint Equations...............................................................15

1.2 Restrictions and Assumptions ........................................................................................161.3 Scope and Contribution..................................................................................................16

2 INTRODUCTION TO DIFFERENTIAL GEOMETRY ...........................................................172.1 Manifolds and Tensors on Manifolds .............................................................................172.2 Rotation Vector .............................................................................................................232.3 Lie Group and Lie Algebra ............................................................................................27

2.3.1 Compound Rotation..................................................................................................292.3.2 Isomorphisms and Tangential Transformations .........................................................31

2.4 Angular Velocities, Accelerations and Curvatures..........................................................332.5 Constraint Point-Manifolds ............................................................................................342.6 Derivatives and Constraint Field-Manifolds ...................................................................372.7 Variation, Lie Derivative and Lie Variation ...................................................................382.8 Useful Formulas ............................................................................................................41

3 GEOMETRICALLY EXACT BEAM THEORY......................................................................463.1 Virtual Work Forms.......................................................................................................46

3.1.1 Weak Balance Equations for Continuum...................................................................473.1.2 Beam Kinematics .....................................................................................................503.1.3 Virtual Work for Reissner’s Beam ............................................................................51

3.2 Constitutive Relations....................................................................................................553.3 Total and Updated Lagrangian Formulations ..................................................................57

3.3.1 On Objectivity for Strain Vector and Curvature Tensor.............................................593.4 On Symmetry of Second Variation .................................................................................633.5 Consistent Tangent Tensors for Reissner’s Beam ...........................................................65

4 PARAMETRIZATION OF CONSTRAINT MANIFOLD ........................................................694.1 Special Beam Elements..................................................................................................72

4.1.1 Switching Beam Element..........................................................................................724.1.2 Offset Beam Element................................................................................................744.1.3 Slide Beam Element .................................................................................................754.1.4 Revolute Joint Beam Elements .................................................................................77

4.2 Numerical Results .........................................................................................................794.2.1 Cantilever 45-degree bend ........................................................................................794.2.2 Helical beam ............................................................................................................804.2.3 Fast Symmetrical Top...............................................................................................814.2.4 Hooke’s Joint ...........................................................................................................824.2.5 Right-Angle Cantilever Beam...................................................................................834.2.6 Two-Component Robot Arm.....................................................................................844.2.7 Two-Component Robot Arm with Revolute Joint ......................................................85

5 CONCLUSIONS ....................................................................................................................86

REFERENCES ...........................................................................................................................87

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List of Symbols

Symbol DescriptionAdR adjoint transformation with re-

spect to RB kinematic operatorB material bodyB current placementB0 initial reference placement

b body force vectorC elasticity tensorc offset vectorC constraint field-manifoldC( )V field of V

Ci tensor, see Section 2.8

ci coefficient, see Section 2.8

d translational displacement vectorDq Fréchet derivative with respect to

qd ,A ad material and spatial differential

areasdV material differential volume

E vector, defined in Eqn (82) p. 51Ei material basis vector

ei global basis vector

e unit rotation axis vectorej unit rotation axis vector for revo-

lute jointE

n n-dimensional Euclidean vectorspace

exp exponential operator

EA axial stiffness of beamEI EI2 3, principal bending stiffnesses

F deformation gradientf force vectorG gi i, material and spatial basis vectors

G gi i∗ ∗, material and spatial dual basis

vectorsG g, material and spatial metric tensors

GA GA2 3, principal shear stiffnesses

GJ torsion stiffness of beamH Hilbert spaceh q( , )t constraint equation

I i, material and spatial identity ele-ments

I ij component of the moment of iner-

tia tensorI I2 3, diagonal the moment of inertia

componentsJ inertial tensorJ torsional moment of inertiaK stiffness tensorK σ geometric stiffness tensor

Symbol DescriptionL beam lengthL V W( , ) a linear operator from V into WLin linearizationLiso V W( , ) a linear isomorphism from V into WM mass tensorM mR R, material and spatial moment vectors

M mR R, material and spatial external moment

vectorsM manifold, finite dimensionalN0 unit normal covector

N n, material and spatial internal forcevectors

N n, material and spatial external forcevectors

P first Piola-Kirchhoff stress tensorp stress vector

O originQ orthogonal operator in an observer

transformationq displacement vector

R rotation operatorR the set of real numbersR+ the set of non-negative real numbers

s length parameterS3 three-sphereSO( )3 rotation manifold, a special Lie groupso( )3 the Lie algebra of SO( )3T tangential transformationTi stress vector

Tσ traction vector on boundary

T tensor spacet i spatial basis vector

T TX xB B0 , material and spatial tangent placements

T TX x∗ ∗B B0 , material and spatial cotangent place-

mentsT tC 0

tangent field-bundle

TM tangent bundle, finite dimensionalTxM tangent space of manifold M at x

TxC velocity field-space

Tx0C tangent field-space

T tM 0tangent point-space

matT SOR ( )3 material tangent space of rotation

matTR material vector space of rotation

matTR∗ material covector space of rotation

spatT SOR ( )3 spatial tangent space of rotation

spatTR spatial vector space of rotation

U domain( , )Ui iϕϕϕϕ a parametrization chart

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Symbol Descriptionu released degree of freedomv velocity vectorW work functionalV W, vector spaces

V W∗ ∗, covector spaces

X x, material and spatial place vectors

x0 place vector at t t= 0

X Y Z, , material points of a body

X xi i, material and spatial coordinates

t time

ΑΑΑΑR material acceleration vector

αααα R spatial acceleration vector

ΓΓΓΓ γγγγ, material and spatial strain vectors

Γ i i,γ material and spatial strain vector

components∆q change of displacement vector

δ ij Kronecker’s delta symbol

δ variation operatorδR Lie-variation with respect to R

δx virtual displacement

δW virtual work

η parameter

ΘΘΘΘR material incremental rotation

vectorθθθθR spatial incremental rotation vector~ΘΘΘΘR material incremental rotation ten-

sor~θθθθR spatial incremental rotation tensor

ΚΚΚΚ κκκκR R, material and spatial curvature

vectorsρ0 material density0 ∇,∇ material and spatial gradient op-

erators0 ∇ , ∇⋅ ⋅ material and spatial divergence

operatorsϕϕϕϕ i parametrization mapping

χχχχ() placement mapping

ψ rotation angle

ΨΨΨΨ ψψψψ, material and spatial total rotation

vectorsΩΩΩΩ ωωωωR R, material and spatial angular ve-

locity vectors

Abbreviations

Abbr. Descriptionacc accelerationaccA, accB parts of acceleration, depend on accel-

eration and on velocityc central linecent, gyro centrifugal and gyroscopiccon, appl constraint and appliedDef. definitiondof degree of freedomext,inert,int external, inertial and internalerr errorj jointload loadingm, mr, s master, master-released, and slavemat materialp. pageref referencespat spatialstr strain (energy)

Other notations

Notation Description⋅ ⋅,

ginner product with the metric tensor g

( )⋅ V dot product in the vector space V

⊗ tensor product⊗S

symmetric tensor product

V W× Cartesian product of V W and a b× the vector cross product of a b and ~x skew-symmetric tensor of axis vector

xf vector f is kept constant under differ-

entationdet,diag determinant and diagonal matrix

tr trace operator&x time derivative of x , velocity vector&&x second time derivative of x , accelera-

tion vector′x spatial derivative of x

ΨΨΨΨ C complement rotation vector of ΨΨΨΨ[ , ] Lie brackets

Jij component matrix of component Jij

B∗ adjoint of operator BR R<

>, pull-back and push-forward operators

by RR + observer transformed R⋅ Ù dualization operator, a GaÙ=

⋅ # primarization operator, f G f#= −1

◊ the end of definition

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1 INTRODUCTION

Multibody mechanics is the one of the most active research areas in applied mechanics. There are severaltextbooks and hundreds of articles about multibody mechanics. Here we utilize the name mechanics insteadof dynamics. Mechanics is the name for the branch of science whose an important field of research is dy-namics. In the earliest approach, the multibody systems were modeled as rigid bodies. Rigid multibody me-chanics is conventionally applied in robotics and in mechanisms. In the circumstances, where rigidity is notaccepted, this leads to flexible multibody mechanics. Flexibility effects can be modeled with two ways, byflexible bodies and by flexible joints. In the following chapters, we will focus on flexibility in bodies.

It is informative to itemize the various classification of mechanics as given in Fig. 1, where we considerapplied mechanics as nonrelativistic classical mechanics that includes Newtonian and Lagrangian mechanicsand a certain portion of Hamiltonian mechanics. As we can see in Fig. 1 that rigid body mechanics is on theboundary of continuum mechanics expressing that we can acquire rigid body mechanics as a limit process ofcontinuum mechanics. In addition, we observe that flexible multibody mechanics contains a part of rigidbody mechanics, pointed to flexible joints with rigid bodies. In addition, flexible multibody mechanics con-tains kinematically non-exact beam theories, such as so called corotational beams. Here we consider me-chanics as a part of mathematical science although this is matter of taste.

In this thesis, the following sentence is attempted to keep in mind:“Thus mechanics is a mathematical model, or, better, an infinite class of models, for certain aspects of na-ture”C.A. Truesdell III in A First Course in Rational Continuum Mechanics, [Truesdell 1977; p. 5]

and we agree with“Don't tell me that quantum mechanics is right and classical mechanics is wrong – after all, quantum me-chanics is a special case of classical mechanics.”J.E. Marsden & T.S. Ratiu, Introduction to Mechanics and Symmetry, [Marsden & Ratiu 1999; p. 117]

Continuum Mechanics

Mathematics• Geometry• Algebra• Analysis

• Variational Methods• Differential Geometry• Functional Analysis• Manifolds• Tensors• Group Theory• Lie Algebra• PDE and FEM• ODE and DAE

Classical Mechanics

Multibody Mechanics

Rigid Body Mechanics

Applied Mechanics

Flexible MultibodyMechanics

Fig. 1 Various classification of mechanics and their relations.

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1.1 Review of Flexible Multibody Mechanics

Flexible multibody mechanics can be formulated various ways. In flexible multibody systems, a motion canbe represented by superimposing a rigid body motion and a relative flexible motion. If additionally the rela-tive flexible motion is given in a body fixed frame (non-inertial frame), this yields the classical flexiblemultibody formulation, see [Shabana 1989] and [Yoo & Haug 1986]. In the classical formulation, there existthe rigid body variables for each flexible body as unknown variables. Thus, the classical formulation can becharacterized by the superimposed motion with the rigid body variables, and the relative displacement vec-tor given in a noninertial, body fixed frame (i.e. superimposed motion with rigid body variables and nonin-ertial frame in Fig. 2). Historically, the classical formulation comes from rigid multibody mechanics byadding flexibility in bodies.

The representation of superimposed motion can be produced, moreover, by a corotational technique, espe-cially, when using beam or shell elements. Here we regard a corotational element that has a single continu-ously rotating noninertial frame along the element. This corotational frame represents a rigid body motion,but it is given in terms of nodal variables in an inertial frame. Thus in corotational elements, there are norigid body variables explicitly. Local transverse displacements in a corotational frame are interpolated byconventional interpolation functions, as in the linear finite element analysis. Finally, the local nodal dis-placement variables are transformed into an inertial frame. Hence, a corotational element can be character-ized by a motion superimposed with a relative displacement and a corotational frame motion that are givenin terms of nodal variables in an inertial frame, i.e. superimposed motion with inertial frame and no rigidbody variables in Fig. 2.

The representation of absolute motion signifies that displacement and velocity vector fields are representeddirectly with conventional shape functions, without superimposing. In the representation of absolute motion,the variables are usually given in an inertial frame. In finite element literature, so called geometrically exactbeam and shell theory use the representation of absolute motion with an inertial frame. Geometrically exacttheory means that no other kinematic simplifications during derivation are applied than the basic kinematicassumptions. In followings, we will focus on elastic deformations and internally one-dimensional elements(beams, bars) since multibody structures are commonly characterized by its length. In addition, flexiblemultibody systems modeled by one-dimensional element: bars that take into account only axial loading, andbeams that also pay regard to bending load, is not comprehensively solved.

Rigid Multibody Mechanics Flexible Multibody MechanicsFlexibility:

Superimposed MotionMotion representionin flexible systems:

Noninertial Frame Inertial FrameFrame type ofelastic variables:

Absolute Motion

Multibody Mechanics

Corotationalformulations

Kinematically exactformulations

Classical formulations

rigid bodyvariables

no rigid bodyvariables

Fig. 2 Different multibody formulations and their relations.

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1.1.1 Classical Formulations for Flexible Multibody Systems

In classical multibody formulation for flexible systems, a rigid body motion and a relative small (infinitesi-mal) elastic deformation are superimposed producing a total motion of body, [Shabana 1989] and [Yoo &Haug 1986]. If a relative displacement is expressed in a body-fixed noninertial frame, and the linear elastictheory is assumed, the consequent stiffness tensor is constant. However, the inertial tensor in this case ishighly coupled and depends on a relative elastic deformation and a rigid body rotation. In order to define aunique displacement field, certain conditions between the body-fixed frame and the relative elastic motionhave to be imposed. These conditions are called reference conditions. With a different choice of the refer-ence conditions may reduce coupling in kinetic energy between the rigid body and elastic variables,[Agrawal & Shabana 1986]. In addition, equations can be simplified and number of variables can be re-duced by modal superposition technique usually called a component mode synthesis method, [Hurty 1965]and [Craig & Bampton 1968], or a substructuring technique [Guyan 1965]. The substructuring techniquethat originally appears from the finite element analysis is also called the superelement technique or Guyanreduction.

We note that if elastic acceleration vectors are given in a noninertial frame it is not possible derive invariantinertial tensors without additional simplifications. In addition, the transformation of vector components doesnot change the frame the vector corresponds, it only effects to vector components. In other words, the vectoris the same vector independently in which frame its components are calculated. This issue is not fully under-stood in some papers.

A way to simplify a classical multibody formulation is introduced in the paper [Nikravesh & Ambrosio1991], where a lumped mass matrix approximation is applied, and accelerations in elastic deformations aregiven with respect to an inertial frame. In the special case and choice of the body-fixed frame, this leads toan invariant diagonal mass matrix. However, elastic displacement and velocity fields are given in the body-fixed noninertial frame, hence additional equations have to be applied during the time-integration procedure.Of course, this kind of inertial tensor can be directly derived by taking a lumped mass matrix in an inertialframe and choosing a single node as reference and attaching a body-fixed frame to it. Then the referencenode represents the rigid body motion and other nodes the elastic deformation. In general, a lumped massmatrix is unacceptable since displacement and velocity fields are inconsistent although lumped approxima-tion may lead to numerically decent results.

In the paper [Simo & Vu-Quoc 1987], the authors show that the linear beam theory in rotational structuresleads to a spurious loss of bending stiffness, since in the linear elastic theory an equilibrium is calculated inan undeformed state. This problem can be overcome by introducing geometric nonlinearities. In the classicalmultibody formulation, geometric nonlinearities can be included by a nonlinear beam theory, [Shabana1989, Ch. 6.10]. This beam theory is the same as in the linear stability analysis, and utilizes linear kinemat-ics as in the linear beam theory, but an equilibrium is calculated in a deformed state. The stiffness tensor isnonlinear in general, but can be decomposed into constant tensors of which one is multiplied by an axialforce, and depending on the beam theory, by a shear force. When elastic deformations, especially transversedisplacements, become large enough, results will be unacceptable because of the linear kinematic assump-tions. For example when applying bending load, it will stretch the length of the beam for reasons of thelinear kinematics.

Higher order geometrical nonlinearities have been presented in [Mayo et al. 1995] where the authors givethree different type of geometrically nonlinear beam formulations. When axial displacements are used asunknown variables, this will yield a constant stiffness tensor, but nonlinearities are transferred into the iner-tial, constraint, and forcing terms. The formulation has an advantage that axial shape modes are not neces-sary needed in geometrically nonlinear problems. This will considerably reduce the numerical stiffness ofdifferential equations and is the main reason why this formulation is so efficient comparing to other meth-ods, see comparison in [Mayo et al. 1995] and [Mayo & Dominguez 1997]. A differential equation is calledstiff if it has time scales differing from orders of magnitude. The numerical integration was utilized by apredictor-corrector multistep method with a coordinate partitioning, [Wehage & Haug 1982]. This method isefficient for nonstiff problems while other geometrically nonlinear formulations in [Mayo et al. 1995] werehighly stiff.

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We observe that, after consistent linearization, a stiffness tensor is always nonlinear in geometrically non-linear problems. Hence in classical formulation with geometric nonlinearities, inertial and stiffness tensorsare highly nonlinear and have to computed frequently in a time integration procedure. In addition, constraintequations are extremely coupled with rigid body and relative elastic motions. Classical formulation might beappropriate for small (infinitesimal) relative elastic displacements especially in rigid-flexible body prob-lems, but appears more inefficient when nonlinearities in relative deformation become more significant.

1.1.2 Nonlinear Finite Element Formulations for Flexible Multibody Systems

We consider a finite element method as a method that projects an infinite dimensional vector space onto afinite dimensional vector space where the numerical solution is accomplished. Sometimes this procedure iscalled a ‘discretization’ that leads to misunderstandings, since the original vector space is not discretizedinto the finite dimensional discrete space; rather it is projected onto the finite dimensional vector space.Thus, we will use phrase the finite element projection instead of the finite element ‘discretization’.

Before further study, a classification of consistency might be useful. In Fig. 3, consistency-in-motion meansthat the time derivative of the displacement vector field at any spatial point is equal to the velocity vectorfield at the same point. Applying a finite element method on space, this consistency leads to the same shapefunctions for the displacement and velocity fields. If a lumped mass matrix approximation is used instead ofthe consistent mass tensor, it will lead to the inconsistency in motion although a motion is consistent at par-ticular points that are called nodes.

An inertial force vector is always linearly dependent on an acceleration vector yielding a mass tensor whichmay not depend on acceleration. Then a lumped mass matrix approximation will lead to erroneous resultswith respect to the mechanical model. Hence, we demand that the equations of motion have to be consistent,i.e. consistency in the mass tensor and consistency in the velocity dependent force vector.

The next kind of consistency in Fig. 3, consistency in tangent tensors, does not affect the accuracy of theresults with respect to the mechanical model, but has an influence on the iteration speed and the time stepsize. In general, the tangential consistency has an effect on the efficiency of the formulation, since consis-tent tangent tensors are the best linear approximations. On the other hand, computing the tangential matricesis a time-consuming procedure, so efficiency can be sped up by computing only those tangential matricesthat have a dominant status. Consistent tangent tensors are nonsymmetrical in general, but in particularcases, the tangent tensors of internal forces are symmetric tensors.

Displacement vector q

Force vector f q vt, ,b g

Velocity vector v• consistent mass tensor• consistent velocity-

dependent force vector

Tangent tensors:consistent internal force tangent tensors:stiffness and damping tensors• consistent inertial force tangent tensors:

centrifugal and gyroscopic tensors• consistent external force tangent tensors

Consistency in motion v q= &

Consistency in tangent tensorsD , ,f q v q v K q D vb g b g⋅ = ⋅ + ⋅∆ ∆ ∆ ∆

Mechanical model• path independent

Computational model• path independent

Consistency in model type

Fig. 3 Three various consistencies.

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The third kind of consistency in Fig. 3, consistency in model type, is quite an unusual type of classification.Let us consider a nonlinear static mechanical model in elasticity. This model may have multiple solutions,but they do not depend on the loading path, so a different loading path does not have an effect on the finalsolutions. However, the computational model may depend on the loading path. For example, in an updatedLagrangian procedure at an arbitrary solution step, the current solution depends on the incremental solutionsbut depends also on the current reference placement that depends on the previous reference placement. Thecurrent reference placement in the updated Lagrangian procedure contains the information about the dis-placements and strains, which are updated an incremental way. In the final placement, there is no guaranteethat the displacement and strain fields are consistent (compatible) although this inconsistency might besmall. In a total Lagrangian updating procedure, where the reference placement is permanently the initialplacement, final solutions do not depend on a loading path. We prefer the total Lagrangian updating proce-dure.

Planar corotational finite element formulations have been presented for transient problems since the late60’s and the early 70’s. In the early paper [Belytschko & Hsieh 1973], the authors introduce a corotationalbeam formulation where the internal force is computed by a corotational technique. Local transverse dis-placement interpolation is given with respect to a corotational frame that is defined as a string of two endnodes. The velocity field is interpolated differently: by absolute motion representation with global shapefunctions in an inertial frame. This yields the inconsistency-in-motion: different shape functions for thedisplacement and velocity fields. In the numerical example, the authors additionally apply a lumped massapproximation and the transient problem is computed by an explicit time-integration procedure whereby notangent matrices are needed. A very similar approach has been given more recently in [Iura & Atluri 1995]where the authors use an inertial frame for kinetic energy and a corotational frame for potential energy withdifferent shape functions for the displacement and velocity fields. This approach leads to a conventionalconstant mass matrix, but the inconsistency-in-motion exists. A consistent corotational beam element in theplane case has been given in [Bahdinan et al. 1998] where the linearization of the inertial force leads to theadditional terms for the damping and stiffness matrices.

A spatial corotational procedure for the dynamic case has been given in [Belytschko et al. 1977] where anexplicit time-integration scheme is used without the need for tangent matrices. A spatial corotational beamelement, with the consistent stiffness matrix in the static case, has been introduced in [Oran 1973] where thederivative of transformation matrices has been correctly accounted, but the element is limited to small in-crements and small local rotations. A fully consistent corotational beam with large increments and rotationsin static case has been presented in [Crisfield 1990] of which the consistent mass matrix is given [Crisfield1997; Ch. 24.19]. Corotational technique has an advantage, namely, a relatively simple form of the consis-tent stiffness matrix, giving an effective formulation especially for static case. As disadvantages, we canmention the highly coupled kinetic energy with the relative displacement and the corotational frame motion.

Another corotational planar beam element has been introduced in [Shabana et al. 1998] where the authorsapply global Hermitean shape functions for the translational displacements and for the slopes, which are thespatial derivatives of displacements at a node. The motion is then presented by the representation of abso-lute motion and the rotation is described by two independent variables, two slopes per node. The number ofdisplacement variables is then four per node in the plane case. Each element is attached to a corotationalframe that is defined as a string of two end nodes. These kinds of shape functions are unusual and may leadto spurious results in consequence of the higher order interpolation for the longitudinal displacement field.The stiffness matrix is generally nonlinear and is not consistently derived. One advantage of this formula-tion is that the kinetic energy is a quadratic function of velocities only, giving a constant mass matrix withno centrifugal and Coriolis forces (in the plane case).

An approach extended for the spatial beams has been given in [Avello et al. 1991] where a nine-parameterrepresentation of rotation is given. These parameters are orthonormal basis vectors of a moving basis hencesix orthonormal constraint conditions per node, in addition to an absolute motion representation is applied.In this beam formulation, the mass matrix is constant but singular that is equivalent to hiding constraintequations. The number of displacement variables is twelve per node with six constraint equations whichconsiderably weakens the efficiency of the formulation.

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A rotation description for a static spatial beam is derived in [Rhim & Lee 1998] where two basis vectors in across-section are used for the parametrization of rotation. These basis vectors are allowed to change theirlengths and their relative angle, yielding the additional elastic relations. The number of displacement vari-ables is nine per node if no warping parameters are applied. An advantage of this formulation is a simpletreatment of vectorial rotation parameters, but as disadvantages it can be mentioned that the classicalHookean law may yield erroneous results because of a high sensitivity with the extension of basis vectorscomparing with the other deformations, a large number of variables per node, and in addition, the momentload is not easy to apply.

A finite strain beam element has been introduced in various contents. The assumptions of beam kinematicscan be divided into two types: the Timoshenko-Reissner-hypothesis and the Euler-Bernoulli hypothesis. Inan Euler-Bernoulli beam theory, the normals of a center line remain the normals in a deformed state with noin-plane or out-of-plane warping deformations of a cross-section, i.e. the cross-section remains in-plane andits shape, and its normal and the tangent of the center line are parallel in the deformed state. This means thatthe translational displacement and rotational fields are kinematically coupled. Thus, Euler-Bernoulli beamelements are very rare in a geometrically exact beam theory, but are commonly used in a corotational tech-nique.

In the Timoshenko-Reissner beam hypothesis, transversal shears are allowed, such that a cross-section planeremains a plane in a deformed state, but the normal of a cross-section is not necessary parallel with the tan-gent of the center line. In addition, all warping effects are excluded in the Timoshenko-Reissner beam hy-pothesis. A finite-strain beam theory was introduced by Reissner for planar beams in [Reissner 1972] andfor spatial beams in [Reissner 1973] where the author derived the beam equations from the classical curvetheory.

Here we consider a beam theory as a geometrically exact beam theory if no other kinematic simplificationshave been applied during derivation than the basic kinematic hypothesis, and a continuum-consistent beamtheory which is geometrically exact, and in addition, all the in-plane and out-of plane warping effects areincluded. Hence, a beam theory gives equivalent solutions with a corresponding continuum theory, [Petrov& Géradin 1998]. In general, a warping-in-plane is due to an axial extension and an out-of-plane warping isdue to a bending, torsion, and transversal shear. In the paper [Simo & Vu-Quoc 1991], the authors extendgeometrically exact beam element with a torsional warping effect, which is a significant effect especially inbeams with thin-walled open cross-section.

In modern contents, the geometrically exact spatial beam theory with finite element implementations hasbeen mainly developed by Simo & Vu-Quoc, Cardona & Géradin and Ibrahimbegović et. al.. In the paper

[Simo 1985], the author gives a dynamic formulation for Reissner’s beam and its finite element implemen-tation in the static case is given in [Simo & Vu-Quoc 1986]. In that paper, a spin rotation vector is used as aunknown variable, and a placement is updated with the aid of a rotation tensor and an exponential mapping,where memory requirements are reduced using quaternion parameters. Here we use the phrase spin rotationvector that is a vector on a tangent space of a manifold, and two successive spin rotation vectors belong todifferent tangent spaces, thus they must not be added by the parallelogram law. The main drawbacks of thisformulation are that the consistent stiffness tensor is an unsymmetrical tensor away from an equilibrium, theneed for secondary storage variables (quaternions) and their manipulations, and the spin rotation vector fieldhas to be interpolated in an inconsistent way; moreover, a solution has a path-dependent property even whena conservative loading is applied. A finite element implementation in the dynamic case by the authors isgiven in [Simo & Vu-Quoc 1988] where spatial and material quantities are exploited leading to a large num-ber of secondary storage variables and calculations between them. This dynamic finite element implementa-tion is quite different from the static implementation; moreover, the dynamic formulation has a simplifica-tion in the Newmark time-stepping method, see [Mäkinen 2001]. An updating procedure where a spin rota-tional vector is used as a unknown variable is also called an Eulerian formulation.

In the important paper [Cardona & Géradin 1988], the authors give another finite element implementationfor Reissner-beam element with a different updating procedure. They named formulations as Eulerian, totalLagrangian and updated Lagrangian and gave a finite element implementation for an updated Lagrangianformulation with the rotation vector as a unknown variable. The updated Lagrangian formulation can bypassthe singularity problem of the total Lagrangian formulation which is singular at the rotation angle 2π andits multiples. The updated Lagrangian formulation has additional benefits such as a fully symmetrical stiff-

14

ness tensor when applying a conservative loading, and any single-step time integration algorithm can beused since, in this formulation, the changes of the rotation vector belong to the same tangent space of amanifold. The updated Lagrangian formulation requires some secondary storage variables for the curvatureand rotation vector, at every spatial integration point. The authors made some simplifications in the tangentoperator of the inertial force vector by neglecting centrifugal and gyroscopic tensors, in addition the tangentstiffness tensor was simplified. The authors have also written the text book [Géradin & Cardona 2001]where a finite element approach for flexible multibody dynamics is given.

A total Lagrangian formulation in static cases with the consistent stiffness tensor is given in [Ibrahimbego-vić et. al. 1995]. The consistent stiffness tensor, which is a symmetrical tensor, has the same form in thetotal and updated Lagrangian formulations and is considerably more complicated than the consistent stif f-ness tensor in an Eulerian formulation, which leads to an unsymmetrical stiffness tensor away from an equi-librium. Generally, a total Lagrangian formulation in a static case with a conservative loading has an im-portant property that is path-independence, whereas an updated Lagrangian formulation is path-dependent.Lagrangian formulations have a consistent interpolation, while in Eulerian formulations, the interpolationhas to apply within an approximate, inconsistent, way. As we have noted earlier, total Lagrangian formula-tions have singularity at the rotation angle 2π and its multiples that are a remarkable restriction, especiallyin dynamic cases.

Different updated Lagrangian and Eulerian formulations in dynamic cases are introduced in [Ibrahimbegović

& Al Mikdad 1998], wherein only spatial quantities are present. These formulations have been developedfrom [Simo & Vu-Quoc 1988] and they have the same simplification in the Newmark time stepping schemethat yields a reduced form of the force vector and tangent tensors, as pointed out in [Mäkinen 2001]. Wesuppose these simplifications having only an insignificant effect on numerical solutions. Given formulationrequires some secondary storage variables, such as updated Lagrangian and Eulerian formulations generallyneed.

An alternative finite element implementation has been given in [Jelenić & Crisfield 1999], which has some

good numerical properties, but falls into a category of corotational formulations, i.e. geometrically non-exact beam formulation. That is because the interpolation of rotational variable has been accomplished withrespect to an element attached (corotional) frame. Thus, a different choice of the corotational frame willyield a different solution although this selection of corotational frames can be performed such that it is in-dependent on the node numbering, as mentioned in see [Crisfield & Jelenić 1999]. Moreover, using the Pet-rov-Galerkin variational method, it will give rather simple tangent tensors but yields a situation where aresidual vector does not have the same meaning as out-of-equilibrium that appears from the principle ofvirtual work, or equivalently, from the Bubnov-Galerkin variational method.

Some elementary knowledge of differential geometry is necessary to understand the rotation vector that is avector of a tangent space of a manifold, where the manifold is a Lie-group of special orthogonal tensors.The so called engineering-approach frequently leads to coarse misunderstanding and erroneous formula-tions. Especially we note that interpolation can be consistently accomplished between nodes if the rotationalvectors of each node belong to the same tangent space, also strain measures in geometrically exact beamformulations are objective quantities even in finite element implementation. Objectivity of strain measuresis proven in Chapter 3.3.1.

Finally, we can summarize that a total Lagrangian geometrically exact finite element formulation is com-petitive if its major drawback, singularity, could be bypassed. The total Lagrangian formulation has severalbenefits such as all unknown variable vectors belong to the same tangent vector space, no need for secon-dary storage variables, the path-independence property (in the static case), any standard time integrationalgorithm may be used, the symmetric stiffness tensor, a simple form of the kinetic energy and all nonlineareffects are included. Hence, we will develop a singularity-free, geometrically exact, finite element formula-tion with a total Lagrangian updating procedure and we will explicitly give the consistent tangent tensors.

15

1.1.3 Modeling Holonomic Constraint Equations

In this Section, we examine constraint multibody systems that are rather closely related with solving differ-ential algebraic or stiff differential equations. Here, we consider only holonomic constraint equations thatare constraint equations depending on displacement and on time, but not on velocity. In the multibody sys-tem, bodies are interconnected by joints, e.g. spherical, revolute, cylindrical, Hooke’s, helical, prismatic,and sliding joints, see [Haug 1989] or [Anonymous 1997]. These joints can be presented by holonomic con-straint equations that do not depend on time. Holonomic constraint equations that depend on time naturallyarise from the boundary conditions of the multibody system.

Constraint equations with together the equations of motion can be solved by three ways: by penalizationand/or by dualization, or by parametrization. The penalty method is the simplest approach to apply, but ithas drawbacks: the convergence rate highly depends on user given penalty factors, the converged solutiondoes not satisfy the constraint equations exactly, and the resulting system of differential equation is highlystiff. A differential equation system is called stiff if it has widely varying time scales, see [Hairer & Wanner1991].

Dualization, or more precisely, the method of Lagrange multipliers is the most extensively used method inrigid and flexible multibody mechanics, see e.g. [Haug 1989] and [Shabana 1989]. In the Lagrange multi-plier method, converged solution satisfies exactly the constraint equations, but the resulting system is dif-ferential-algebraic with the differential index-3, [Hairer & Wanner 1991]. Differential-algebraic equationswith a higher differential index are, in general, more complex to solve.

In flexible multibody systems, rigid joints, as spherical and revolute joints, can be modeled similarly as inrigid multibody systems. The modeling of flexible joints, like prismatic joints, is quite different because ofthe flexible effects of links. A prismatic joint modeling with the Lagrange multiplier method has been pre-sented in [Riemer & Wauer 1988] and more recently in detail in [Bauchau 2000]. In general, the Lagrangemultiplier method also suffers extra variables leading to double of the number of Lagrange multipliers com-paring with minimal coordinate set. The convergence rate in the Lagrange method can be sped by introduc-ing extra penalty terms, yielding the augmented Lagrange method, i.e. the combination of dualization andpenalization. An example of modeling joints with the augmented Lagrange multiplier method is given in[Cardona et al. 1991]. In this method as a drawback, coefficient matrices are less sparse than in the La-grange multiplier method.

Holonomic constraint equations generate a manifold into time-displacement vector space. If the constraintequations are continuously differentiable, the constraint manifold also has a differentiable structure, i.e. themanifold is smooth. Parametrization of constraint manifold is the oldest solution method in constrainedmechanical systems. This method is also called generalized/Lagrangian/minimal coordinate approach inanalytical dynamics, or relative coordinates in robotics. Additionally, the parametrization of constraintmanifold is equivalent with master-slave technique in the finite element method, and embedding constraintsor coordinate portioning in multibody dynamics. Usually, the constraint manifold can be parametrized onlylocally but changing parametrization charts, we could express the whole operation domain of the multibodysystem. Especially, the rotation group, also a manifold, can be represented minimally by two parametriza-tion charts.

The parametrization of constraint manifold has several benefits: the minimal number of variables, constraintequations are satisfied automatically, the resulting equations of motion are ordinary differential equations(not differential-algebraic), and easy to apply time-dependent boundary conditions. We could mentiondrawbacks as more complicated coefficient matrices, whose sparsity is similar to the matrices of the aug-mented Lagrange multiplier method. The parametrization of constraint manifold is particularly competitivein the flexible multibody system; the number of degrees of freedom is extensive comparing with the numberof constraint equations.

16

Master-slave technique for different rigid joints with large rotation has been presented in the static case in[Jelenić & Crisfield 1996] where tangent tensors are consistent. Using an equivalent approach, called coor-dinate partitioning, a formulation for rigid joints with large rotations is given in the dynamic case in[Mitsugi 1997] with non-consistent tangent tensors. Modeling rigid and flexible joints using the parametri-zation of constraint manifold with consistent tangent tensors has not been completely presented, thus furtherstudy seems to be necessary.

1.2 Restrictions and Assumptions

Although we study a general formulation of flexible multibody mechanics, we make some restrictions andassumptions during this thesis. These restrictions and assumptions are: – manifolds are Riemannian manifolds that are equipped with a metric tensor – constraint equations are holonomic and time-independent – only conservative loads are considered – a geometrically exact beam theory with Timoshenko-Reissner-hypothesis is exploited – only a material version of Lagrangian updating formulation is entirely examined – a finite element method is utilized for a numerical solution approach – linearly interpolated elements and Newmark time integration scheme are used in the numerical examples.

1.3 Scope and Contribution

In this thesis, the aim is to give a general formulation for flexible multibody mechanics. Since multibodysystems are highly constrained, the examination of differential geometry is necessary to understand the in-ternal geometry and kinematics of general multibody systems. Therefore, the formulation is given in thelanguage of differential geometry: manifolds, tangent spaces and tangent tensors on manifolds, etc.

In addition, the rotation manifold SO( )3 , whose elements are rotation operators, will be investigated thor-oughly. This study probably provides the most important contribution of the thesis: spin material rotationvectors, material angular velocity vectors, and material angular acceleration vectors belong to the differenttangent spaces of the rotation manifold. Hence, the direct application of the material incremental rotationvector with standard time integration methods yields serious problems: adding quantities which belong tothe different tangent spaces.

The formulation is applied to Reissner’s geometrically exact beam theory, giving a new geometrically exactbeam element that is based on the total Lagrangian updating procedure. The element has the total rotationvector as the unknown variable and the singularity problems at rotation angle 2π and its multiples are han-dled by the change of parametrization on the rotation manifold SO( )3 . The consistent stiffness, gyroscopic,centrifugal, and loading tensors of the total Lagrangian formulation are given explicitly.

In addition, we derive a general formulation how to parametrize the constraint manifold, which arises frompoint-wise holonomic constraint equations. The parametrization of constraint manifold using the total La-grangian formulation has several benefits: the minimal number of variables, objective formulation, no needfor secondary storage variables, constraint equations are satisfied automatically, the resulting equations ofmotion are ordinary differential equations (not differential-algebraic), and easy to apply time-dependentboundary conditions. The constraint manifold parametrization is particularly competitive in the flexiblemultibody system where the number of degrees of freedom is extensive comparing with the number of con-straint equations. Moreover, special beam elements, which involve holonomic constraints, are derived as theexamples of the formulation. These elements can be exploited as customary elements in the finite elementmethod.

17

2 INTRODUCTION TO DIFFERENTIAL GEOMETRY

In this Section, we study differential geometry very elementarily, but hopefully in a practical way. Someknowledge on differential geometry is essential comprehending the quantity of finite rotation. In addition,dividing vector spaces into material and spatial spaces is necessary since these spaces behave differently inobserver transformation and in objective derivatives (Lie-derivatives).

All the vector spaces, which we consider, have metric tensors thus they are metric vector spaces, and all thefinite dimensional manifolds are Riemannian manifolds that are embedded in an Euclidean space. Hence, wecan always choose an orthonormal set of basis vectors and we will get rid of those informative (read: terri-ble) subscript and superscript tensor notation and Christoffel symbols. Additionally, we may identify a dualvector space by its primary vector space. In classical tensor analysis, this identification is applied, but herewe make distinction between primary and dual spaces in the formulation, and the identification is accom-plished later in the finite element implementation. If the identification of dual and primary vector spaces isdone a priori, then push-forward and pull-back operations are not uniquely defined. We also itemize terms avector space and a linear space where the linear space is considered as a trivial manifold (or a linear mani-fold, or a flat manifold). Vector spaces usually appear from the tangent spaces of the manifold which aredistinct at different points of a nontrivial manifold

2.1 Manifolds and Tensors on Manifolds

In this section, we give the definitions for vector and tensor algebra on topological vector spaces1, defini-tions for manifolds, and tensor algebra on manifolds. We recommend consulting, especially, the paper[Stumpf & Hoppe 1997], and the textbooks [Wang & Truesdell 1973] or [Marsden & Hughes 1983] for ten-sors on manifolds, and textbooks [Arnold 1978] or [Abraham et al. 1983] for differentiable manifolds. Areader is assumed to be familiar with classical tensor algebra on Euclidean spaces2, text books like [Ogden1984] or [Truesdell 1977] or [Bonet & Wood 1997].

1 Definitions for covector space, dot product, and adjoint operator: The covector space V ∗ of the vectorspace V is defined by the space of linear maps V → R , i.e. V =L V∗: ( , )R . These linear maps are repre-sented by the dot product (duality pairing) defined as

⋅ ⋅∗ × → ∈: , ,V V R Rf a f ab ga ,

which have two properties: bilinearity, i.e. it is linear with respect to each of its two members, and definite,i.e. if f ∈ ∗V is fixed and f a a⋅ = ∀ ∈0 V , then a 0= . Conversely, if a ∈V is fixed and f a f⋅ = ∀ ∈ ∗0, V ,then f 0= . If f a⋅ = 0 , the vector a is said to be orthogonal to the covector f , and vice versa. Note that acovector space is also a vector space satisfying the vector space properties. Because the vector space and itsco-covector space are canonically isomorphic3, i.e. V V= ∗∗ , we have the symmetry property of the dotproduct: f a a f⋅ ⋅= .

Let F ∈L V W( , ) be a linear operator from V W→ . The adjoint operator F∗ ∗ ∗∈L W V( , ) is defined withthe aid of the dot product as

F w a w Fa a w∗ ∗⋅ ⋅= ∈ ∀ ∈ ∈R V W, ,

where the first dot product is on the vector space V , and the latter on the vector space W , see Fig. 4. ‡

On notation: we omit ⋅ -symbol when there is no source of confusion. Then the terms Fa and F a⋅ areidentical. The brackets are used for purpose of dependency, e.g. F x a( )⋅ denotes the linear operator F(x)acts (linearly) on a where the operator depends on x. In this case, the dot symbol may not be omitted. Inaddition, in the composite mapping of operators, like FG , the dot symbol is omitted.

1 We consider the topological vector space as a general vector space without explicit knowledge of a metric.2 The Euclidean space is a real, finite-dimensional, linear, inner-product space with an Euclidean metric.3Topological vector spaces are isomorphic, denoted by ≅ , if there exists a (continuous) linear bijection, called isomor-phism, between these spaces. Two vector spaces are isomorphic iff they have the same dimensions. Vector spaces arecanonical isomorphic, denoted by = , if there exists a natural (‘almost trivial’) isomorphism.

18

t w⋅ ∈b gW

Rf v⋅ ∈b gV

R

W∗

V∗

WV

F−∗

F∗

F−1

F

Fig. 4 The diagram of domains and ranges for the operator F ∈Liso V W( , ) and its derivatives.

A covector space is commonly called a dual vector space, and the element of the covector space are calledcovectors, or dual vectors, or linear forms (which never make any sense). Additionally, the dot product isalso called duality pairing, and an adjoint operator is called a dual operator. We do not make any notationaldifference between the elements in the vector and covector spaces since we desire to use the notation similarto classical tensor algebra. For example, force quantities like moment and force vectors, and Lagrange mul-tiplies are the elements of covector spaces. We will use the byte ‘co-’ instead of the word ‘dual’ because ofits simplicity and compactness.

2 Definition for inverse operator, and inverse adjoint operator: If the operator F is a linear bijection (iso-morphism), denoted F ∈Liso V W( , ) , the inverse operator F− ∈1 Liso W V( , ) exists and it is unique. The in-verse operator is defined by formulas

I F F i FF= − −1 1 and = ,

where I ∈Liso V V( , ) is the identity on V , and i ∈Liso W W( , ) is the identity on W . The inverse of theadjoint operator F∗ ∗ ∗∈Liso W V( , ) is defined similarly by formulas

i F F I F F∗ −∗ ∗ ∗ ∗ −∗= and = ,

where i ∗ ∗ ∗∈Liso W W( , ) is the identity on W ∗ , and I ∗ ∗ ∗∈Liso V V( , ) is the identity on V ∗ . Note that aninverse adjoint operator is an operator F−∗ ∗ ∗∈Liso V W( , ) , see Fig. 4. ‡

3 Definition for tensor product and tensor space: The tensor product between the vector a ∈V 1 and thecovector f ∈ ∗W is defined via the dot product by the formula

( ) ( )a f w f w a w⊗ ⋅ ⋅= ∈ ∀ ∈V W, ,

where the tensor a f⊗ belongs to the tensor space produced by V W and ∗ , i.e. a f⊗ ⊗∗∈ =V W L W V( , ) .

The tensor product is a linear mapping for each member separately, i.e. a bilinear operator, because of thebilinearity of the dot product. The tensor is called a two-point tensor if it is defined on two different vectorspaces. The general two-point tensor space T can be denoted by

T := ⊗ ⊗ ⊗ ∗ ⊗ ⊗ ∗ ⊗ ⊗ ⊗ ⊗ ∗ ⊗ ⊗ ∗V V V V W W W WL L1 24 34 1 24 34 1 24 34 1 24 34

L L

r s t u

that is the space of r-fold on the vector space V , s-fold on the covector space V ∗ , t-fold on the vector spaceW , and u-fold on the covector space W ∗ . This can be shortly denoted by the tensor space T ( , ; , )r s t u withthe order of r s t u+ + + . ‡

Note that any other permutation of vector spaces is possible, thus e.g. the notation ( , ;0, )1 0 1 could mean thetensor spaces V W⊗ ∗ or W V∗ ⊗ . In the case of one-point tensor spaces, defined on the same vector orcovector space, we use a simplified notation: e.g. the tensor space T (1,1) for the tensor spaces V V⊗ ∗ orV V∗ ⊗ defined on V , or correspondingly for the tensor spaces W W⊗ ∗ or W W∗ ⊗ defined on W .

1 This vector space V could be a covector space, or more generally, a tensor space

19

There are two possible points of view to comprehend a tensor: operational or quantitative. The operationalaspect informs ‘how it works’, and quantitative responds to ‘how much is it’. Mathematicians represent theoperational point of view and engineers the quantitative point of view. Although we will define the tensorby quantitative, we shall keep in mind its operational aspect: a tensor is a multilinear operator.

4 Definition for tensors: A tensor is defined an element of a tensor space. Thus after the property of tensorproduct, the two-point tensor T of the tensor space T ( , ; , )r s t u , given in Def. 3, is a multilinear mapping

T:V V V V W W W W∗ ∗ ∗ ∗× × × × × × × × × × × →L L

1 24 34 1 24 34 1 244 344 1 24 34L L

r s t u

R .

The two-point tensor T is an element of two-point tensor space such that it assigns a tensor for its two-pointdomain. ‡

The tensor space is a vector space itself by satisfying all vector space properties. Then we may state that thetensors are vectors and the vectors are tensors. However, we consider the first-order tensors as vectors, andthe higher-order tensors as tensors. Sometimes the tensors are characterized by their component transforma-tion laws under the change of the basis: the object is a tensor if its components change like tensor compo-nents under a coordinate transformation. For example, Christoffel symbols are not tensors. Conversely, thevectors are characterized by direction, magnitude, and, especially, by the parallelogram law: the vector canbe added to another vector by the parallelogram law. For example, it is often incorrectly claimed that thefinite rotation does not satisfy the parallelogram law, whereupon the finite rotation vector is not a vectorquantity. We keep these characterizations rather old-fashioned and they can lead to serious misunderstand-ings. The vectors and tensors may be characterized by studying if they are elements of corresponding vectorand tensor spaces, respectively.

The trace of the second order tensor is usually defined by the contraction of its components. This is a con-tradiction with the component independence of the tensor, although, the trace is component-independent.We follow the definition of the trace given in [Truesdell 1977; App. II].

5 Definition for trace and double-dot product: The trace tr ( , )∈ ×∗L V V R of the one-point tensorf a⊗ ∗ ⊗∈V V is a scalar-valued linear operator defined via the dot product as

tr :f a f a⊗ = ∈⋅b g R .

Also the trace operation for the tensor on V V⊗ ∗ can be applied by noting V V= ∗∗ , but it is not defined fortwo-point tensors. The double-dot product for the tensors f t⊗ ∗ ⊗ ∗∈V W and v w⊗ ⊗∈V W is defined viathe ordinary dot product

f t v w f v t w⊗ ⊗ = ⋅ ∈⋅ ⋅b g b g b g b g: :V W

R ,

where the subscripts indicate the vector space of the corresponding dot product. Therefore, the double-dotproduct is a mapping L ( , )V W V W∗ ∗× × × R that is a four-linear operator. ‡

All tensors, which we have considered, have been presented by the tensor product of the vectors, e.g. thetensor f a⊗ . However, a general tensor can not be expressed directly in that way. We may present a com-mon tensor with basis vectors of tensor space. Let G i , with the index i = 1 2 3, , , be an ordered basis for thevector space V and let gi ( i = 1 2 3, , ) be an ordered basis for the vector space W, then we may present ageneral second-order two-point tensor T ∈ ⊗V W by the linear combination of the basis vectors, namely(with the conventional summation)

T G g= ⊗Tij i j , (1)

where G gi j⊗ ⊗∈V W corresponds the basis vector of the tensor with the coefficient Tij ∈R . The coeffi-

cient matrix [ ]Tij ∈ ×R3 3 is called the component matrix of the tensor T with respect to the bases G i and

gi1. Higher order tensors are represented a similar way. In order to represent tensors on covector spaces,

we have to define the bases for the covector spaces.

1 The component matrix is an isomorphism between the tensor space V W⊗ and the Cartesian space R3 3× .

20

6 Definition for bases of covector spaces: Let G i and gi be ordered bases of the vector spaces V and

W, respectively. The bases (dual bases) G i∗ and gi

∗ on the covector spaces V W∗ ∗ and are defined by

formulas

G G g gi j ij i j ij∗ ∗⋅ ⋅= =δ δ, ,

where δ ij is the Kronecker’s delta symbol. Then, for example, the tensor T ∈ ⊗ ∗V W may be represented byT G g= ⊗ ∗Tij i j . ‡

We have defined a tensor algebra on a topological vector space. These vector spaces are often induced by amanifold, yielding a tensor algebra on the manifold that we define next.

7 Definition for manifold: A set M ⊂ En is a manifold with dimension d, if there exists a bijection1

ϕϕϕϕ i in:U → E from an open domain U i

d⊂ E in a d-dimensional Euclidean parameter space onto some openset in the manifold, ϕϕϕϕ ϕϕϕϕi i i i: ( )U U M→ ⊂ , such that every point of the manifold is an image under a map-ping, see Fig. 5. A pair ( , )U i iϕϕϕϕ is called a chart or a parametrization chart, and the mapping ϕϕϕϕ i is called achart mapping or a parametrization mapping ‡

8 Definition for differentiable manifold: A manifold M is a differentiable manifold if for every pointa ∈M there exist images ϕϕϕϕ1 1(U ) and ϕϕϕϕ2 2(U ) where the point a ∈M belongs to, such that the compositemapping ϕϕϕϕ ϕϕϕϕ2

11

− o is a diffeomorphism2 from ϕϕϕϕ ϕϕϕϕϕϕϕϕ11

1 1 2 2− ∩( )( ) ( )U U onto ϕϕϕϕ ϕϕϕϕϕϕϕϕ2

11 1 2 2

− ∩( )( ) ( )U U . The com-posite mapping is called the change of parametrization, see Fig. 5. ‡

We note that usually a chart mapping is defined by an inverse mapping from an open set of a manifold into aparameter space. We have defined a chart mapping differently since we could use this terminology whenconstraint equations are parametrized; also, we note the connection between the finite element method. Avector space, where a manifold is embedded, is called a embedding space; the Euclidean space E

n in Fig. 5.

ϕϕϕϕ1

ϕϕϕϕ2

U1 U2

manifold M ⊂ En

parameter space Ed

change ofparametrizationϕϕϕϕ ϕϕϕϕ2

11

−o

ϕϕϕϕ ϕϕϕϕ1 1 2 2U Ud i d i∩embedding space En

Fig. 5 A geometric interpretation for a parametrization of a manifold, when n d= 3 2 and = .

1 a mapping is a bijection if it is injective and surjective, i.e. one-to-one and onto mapping2 a diffeomorphism is a bijection with continuously differentiable mapping and its inverse mapping

21

ϕϕϕϕ( )t

M

TxM

&x x

Fig. 6 The tangent vector &x and its tangent space TxM on the manifold M at the point x .

9 Definition for tangent vector and tangent space on manifold: Let ϕϕϕϕ( )t be a parametrized vector-valuedcurve in the manifold M through the point x ∈M such that ϕϕϕϕ( )t = =0 x . The tangent of curve (or equiva-lent class of curves) ϕϕϕϕ( )t at t = 0 on the manifold M is defined as

& lim( ) ( )

, ( ) , ( )x x= − = ∈→t

t

tt

0

00

ϕϕϕϕ ϕϕϕϕ ϕϕϕϕ ϕϕϕϕ where M .

The tangent vector &x belongs to a tangent space of the manifold, namely &x x∈T M , see Fig. 6. The tangent(vector) space TxM is a set of tangent vectors at x ∈M . ‡

10 Definition for tangent bundle on manifold: A tangent bundle TM is defined a union of the tangent spaceson the manifold M at its every point

T TM = MM

: ,x x

x

b g∈U .

The dimension of the tangent bundle is twice the dimension of the manifold M . Especially, the pair of statevectors, the placement x( )t and velocity vectors v( )t , belongs to the tangent bundle, ( , )( )x v t T∈ M . ‡

For a two-point tensor, its domain of points is divided into two separate but not independent regions whichare defined in the vector spaces V W and . It is convenient to choose a material body B, containing all mate-rial points of body, for one region of the domain and another region which is obtained via a mapping of thematerial body B. The material body B is a set of points and its elements are denoted X Y Z B, , ,K∈ .

11 Definition for current and initial reference placements: Let χχχχ t B: → E3 be a smooth time-dependent em-bedding of the material body B into the Euclidean space E3 . For each fixed time t, the mapping χχχχ( , )t ⋅ isdefined as a current placement of the body B along with the current place vector x of a body-point, namely

B: ( , ), : ( , ),= = ∈χχχχ χχχχt B t X X Bx .

The initial reference placement B0 is defined as the special case of the current placement B by settingt = 0 , giving

B0 0 0: ( , ), : ( , )= = = = ∈χχχχ χχχχt B t X X BX ,

where X is an initial reference place vector. ‡

Since the initial reference placement B0 is uneffected in the observation transformation (see e.g. [Ogden1984; Ch. 2]), we call vectors and tensors defined on the initial reference placement B0 as material quanti-ties. For example, a reference place vector X is called a material place vector, and B0 the material place-ment. Sometimes the material description is named as referential or Lagrangian description, and occasion-ally, some distinction has been accomplished between these phrases.

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spatial placement B

spatial place vector x

at time t = 0

point X

body B

material place vector X

material placement B0

at moment t

spatial vector a

material vector A

global frame in E2

Fig. 7 The material body B with the body-point X , the material placement B0 with the material placevector X and the material vector A , and the spatial placement B with the spatial place vector x andthe spatial vectora .

Contrary to the material placement B0 , the current placement B and vectors and tensors defined on it areconcerned in the observation transformation. Vector and tensors defined on the current placement B arecalled spatial quantities, e.g. a current place vector x is also named as a spatial place vector, and B as aspatial placement. A spatial description is sometimes called an Eulerian description.

We will apply the phrases ‘material’ and ‘spatial’ for placements, vectors, tensors, fields, spaces and de-scriptions. A geometric interpretation of the material body B, the material placement B0 , and the spatialplacement B is given in Fig. 7. Note that placements, likewise place vectors, should be regarded as map-pings, not the images of these maps, according to Def. 11.

In Fig. 7, it is demonstrated that a body-point X B∈ , which is represented by a vector-valued mappingX: ( )= χχχχ0 X , assigns a material vector A on the material placement B0 . The material vector belongs to thetangent space of the material placement B0 , namely TXB0 , where X corresponds a base point1 of manifold.Correspondingly, the body-point X B∈ , which is represented by the mapping x: ( )= χχχχ X , assigns the spatialvector a on the spatial placement B . The spatial vector belongs to the tangent space of the spatial place-ment B , i.e. a x∈T B , where x represents a base point of manifold. Note that the placements B B0 and aremanifolds.

Now we could set V = B W = BT TX x0, giving, for example, the type of (1,1;1,1) two-point tensor at a body-

point X B∈ with mappings X = χχχχ0( )X and x = χχχχ( )X T X X x x:T T T T∗ ∗× × × →B B B B0 0 R , where TX∗B0 and

Tx∗B are the covector spaces for the vector spaces TXB0 and TxB , respectively. The two-point tensor T is

an element of multilinear operators, denoted as T X X x x∈ × × ×∗ ∗L B B B B( , )T T T T0 0 R . For the sake of sim-

plicity, we omit body points and mappings when expressing tensors and vectors, and we call the place vec-

tors X and x as the material base point and the spatial base point, respectively.

So far we have studied vectors and tensors in vector spaces without knowledge about its metric. A metric ofthe vector space is a symmetric positive-definite bilinear operator2 , called a metric tensor. Let pairs ( , )V Gand ( , )W g indicate metric vector spaces in the material and spatial representation, with the (material) met-

1 A base point is a point of the manifold where a tangent space is induced.2 a metric is a scalar valued function that induces a linear bijection (isomorphism), called a metric tensor

23

ric tensor G ∈ ∗L V V( , ) and the (spatial) metric tensor g ∈ ∗L W W( , ) . Metric tensors are used for measuringdistances and deformation, which is impossible without introducing metric. Since manifolds are embeddedin the Euclidean space E3 , we could choose metric tensors as the identity elements. This can be achieved byidentifying the metric vector spaces ( , )V G and ( , )W g with the Euclidean vector space E3 . However, thisidentifying is not accomplished at this moment since it is informative to comprehend the existence of themetric tensor in different operators like deformation and strain tensors.

12 Definition for inner product and transpose operator: The inner product for a metric vector space ( , )V Gis defined by

⋅ ⋅ × → = =⋅ ⋅, : , ( , ) , : ( )V V R a b a b Ga b a bG

aÙ ,

where the dot product is defined in Def. 1 p. 17. For simplicity, the covector Ga is often denoted by aÙ . The

tensor F G g∈L V W( , ),( , )b g , its transpose operator FT is defined via the inner product

F w v w Fv w vG g

T , , ,= ∀ ∈ ∈W V .

Hence, the transpose operator is a mapping FT ∈L W V( , ) . After the definition of the inner product, wefound a relation between the transpose FT and the adjoint operator F∗ , yielding F G F gT = − ∗1 . Note that thetranspose operator depends on metric tensors on contrary to the adjoint operator. ‡

2.2 Rotation Vector

A rotation vector is one of the most misunderstood quantities in applied mechanics. In this section, we willdemonstrate how a rotation vector and a differentiable manifold are connected. We derive a rotation opera-tor in terms of the rotation vector, see e.g. [Argyris 1982]. At this point, we assume that all vectors live inthe Euclidean space E3 . This assumption is not contradictory since vectors in any three-dimensional topo-logical vector space can be identified by an isomorphism with vectors in the Euclidean space E3

We are trying to find an expression for the rotated vector p1 in the terms of the original vector p0 , the unitrotation axis e , and the non-negative rotation angle y about the rotation axis. The original projectorvector r 0 and the rotated projector vector r 1 in the rotation plane are, see Fig. 8

r e p e

r r e p p e r r

0 0

1 0 0 0 0 13

= × ×

= + × ∈ ∈ +

b g,cos sin , , , , ,ψ ψ ψE R

(2)

where × denotes the cross product on E3 . Now the rotated vector p1 can be expressed with the aid of (2)

p p r r

p e e p e p p p e r r1 0 0 1

0 0 0 1 0 0 131

= − +

= + − × × + × ∈ ∈ +cos sin , , , , , ,ψ ψ ψb g b g E R(3)

p1

r1

p0

ψ

ψ

r0

e p× 0 sinψr1

r0

eE

3

Fig. 8 A rotational motion about e-axis where p0 is the original vector and p1 is the rotated vector.(Note that e r e p× = ×0 0 ).

24

13 Definition for rotation vector in Euclidean spaceE3 : A rotational motion can be represented by a rotationvector defined as

ΨΨΨΨ ====: , ,ψ ψe e e= ∈ ∈ +1 3E R , (4)

where the unit rotation axis vector e and the non-negative rotation angle y are oriented such that they forma right-handed screw, see Fig. 8. ‡

Note that the length of the rotation vector is equal to the rotation angle, i.e. || ||ΨΨΨΨ = ψ . Here we do not restrictthe angle of rotation; it may have any non-negative values. The rotation vector ΨΨΨΨ lives in a three-dimensional vector space that is isomorphic to the Euclidean space E3 . This issue will be realized later.

Now Eqn (3) can be written in terms of the rotation vector, which yields the expression of the rotation op-erator

p p p p

I p

1 0 0 2 0

2 0

1

1

= + × + − × ×

= + + −FHG

IKJ

sin cos

sin ~ cos ~,

ψψ

ψψ

ψψ

ψψ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ

b g

2

(5)

where the skew-symmetric tensor ~ΨΨΨΨ called the rotation tensor, is defined by formula

~,ΨΨΨΨ ΨΨΨΨa a a= × ∀ ∈E3 ,

or more formally ~

:ΨΨΨΨ ΨΨΨΨ= × .

14 Definition for rotation operator in Euclidean space E3 : A rotation operator transforms linearly and iso-metrically a vector into another vector in a rotational motion that is represented by a rotation vector. Therotation operator R ∈Liso ( , )E E3 3 is defined with the aid of the rotation vector ΨΨΨΨ ∈E3 by equation:

R I:sin ~ cos ~

,= + + − =ψψ

ψψ

ψΨΨΨΨ ΨΨΨΨ1

22 ΨΨΨΨ ,

Then in Fig. 8, the rotation operator R transforms the vector p0 into the vector p1 , i.e. p Rp1 0= . ‡

Now a rotational motion, represented by a rotation vector, is directly involved in a rotation operator. In ad-dition, the definition gives explicitly a canonical1 parametrization of the rotation operator that is a point of amanifold itself. This rotation manifold, namely SO(3), is a three-dimensional smooth manifold embedded inthe Euclidean space E3 3× that is isomorphic to the nine-dimensional Cartesian space R9 . We note that arotation operator is an element of the rotation manifold, i.e. R ∈SO( )3 .

Expanding the trigonometric terms in Def. 14 and using the realities that

~ ~,

~ ~ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ2 1 1 2 1 2 1 2 1 21 1n n n n n n− − − − −= − = −b g b gb g b gψ ψ (6)

gives the exponential representation of the rotation operator

R I= + + + + =~

!

~

!

~:exp

~ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

1

2

1

32 3

K e j . (7)

This is a significant property of the rotation operator and offers the shortest relationship between the rota-tion vector and the rotation operator. We also note that the transpose of the rotation operator is equal to thereverse rotational motion

R I RT = + + − = −sin ~ cos ~ψψ

ψψ

−−−−ΨΨΨΨ −−−−ΨΨΨΨ ΨΨΨΨe j e j b g12

2, (8)

due to the skew-symmetry of the rotation tensor, ~ ~ΨΨΨΨ ΨΨΨΨT = − .

1 There exists no effective criterion for the term canonical.

25

This yields the proper orthogonal features of the rotation operator

R R RR I

R

T T= == +

,

det ,b g 1(9)

where I is the identity element. It is evident since the inverse of the rotation operator is the reverse rotationoperator. If an operator satisfies Equation (9a) solely, there exist two possible values for its determinant,namely det( ) , R = + −1 1 , where the first value (+1) produces the preservation of the orientation.

A rotation operator can be written also with the aid of a rotation axis e, yielding

R I e e= + + − =sin ~ ( cos )~ ,ψ ψ ψ1 2 ΨΨΨΨ , (10)

This relation makes it comprehensible that the rotation operator does not depend on the multiples of therotation revolution counts, i.e. R R e( ) ( ),ΨΨΨΨ ΨΨΨΨ= + ∀ ∈2i iπ N .

Def. 14 gives a canonical parametrization of the rotation manifold SO(3). The parametrization can representa rotation operator only locally, and there exists no parametrization that is global as well as non-singular.Note that a parametrization is a mapping from an open set of Euclidean space into some open set of themanifold. The rotation vector parametrization is singular at the rotation angle equal to 2π and its multiples.It is clear that the singularity naturally appears in dynamical analysis with large rotations, and cannot beomitted. Singularity should be considered as a non-differentiable hole that must not be omitted by skipping.The singularity is due to fact that the rotation manifold is compact, and there does not exist a single con-tinuous parametrization from an open set of the Euclidean space E3 onto this compact manifold, see detailsin [Stuelpnagel 1964].

A rotation operator can be presented by higher dimensional, singularity-free representations where a unitquaternion is a four-parametric example. The coordinates of a quaternion are not independent, in fact, aquaternion produces a three-dimensional manifold into a four-dimensional Euclidean space, that is a unitthree-sphere S3 (surface) embedded in E4 . Hence, we do not speak about parametrization when consideringa mapping between different manifolds, for example in the case of a unit quaternion this mapping isS SO3 3→ ( ) .

The description of a rotation motion has been studied for a long time, so there exists a large number of dif-ferent representations of a rotation motion. Three-dimensional representations are rotation vectors, Eulerangles, Bryant angles, Rodrigues parameters (Gibbs vector), and four-parametric representations are unitquaternions (Euler-Rodrigues parameters), linear parameters, Euler rotation, and Cayley-Klein parameters,see [Spring 1986] and a historical aspect for Euler-Rodrigues parameters in [Cheng & Gupta 1989]. In addi-tion, there exists a higher dimensional representation of a rotation operator, like a rotation matrix, that has adimension equal to nine.

Four-dimensional descriptions are topologically connected with a unit three-sphere S3 and to the properunitary group SU(2) that is a group of complex 2-by-2 matrices, and their joined algebra is an even Cliffordsubalgebra (quaternion algebra) and an algebra of 2-by-2 skew-Hermitian traceless matrices (Lie-algebrasu(2)), respectively, see [Choquet-Bruhat et al. 1989].

Correspondingly, three-dimensional descriptions are the parametrizations of the rotation manifold and theiralgebras are the cross product in the Euclidean space E3 (Lie-algebra in E3 ) and an algebra of skew-symmetric tensors (Lie-algebra so(3)). We consider three-dimensional descriptions and especially the rota-tion vector a simple, geometrical significance representation. The major drawback of the rotation vectorparametrization, singularity, can be passed by introducing another parametrization chart such that theparametrization mappings cover the rotation manifold globally.

26

15 Definition for complement rotation vector: Let a rotation vector ΨΨΨΨ with a rotation angle larger than zeroand less than perigon (full angle), i.e. 0 2< <ψ π , then its complement rotation vector ΨΨΨΨC is defined as

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨC : ,= − =2πψ ψ .

Then the rotation angle of the complement rotation vector is ψ ψC = 2π − and the rotation axis is e eC = − .‡

After substituting the complement rotation vector into Def. 14 p. 24, we notice that the rotation vector andits complement represent the same rotation operator, i.e. R R( ) ( )ΨΨΨΨ ΨΨΨΨC = . Def. 15 is a change of parametri-zation in the parameter space E3 , see Fig. 5 and Fig. 9. This change of parametrization is a continuouslydifferentiable mapping on the open domain 0 2< <ψ π , giving a smooth construction of the rotation mani-fold SO(3) at this domain. Note that the complement of a complement rotation vector is a rotation vectoritself, i.e. ( )ΨΨΨΨ ΨΨΨΨC C = , hence there is no priority over these parametrization charts.

We could represent the rotation manifold globally with these two parametrization charts. When a rotationangle exceeds straight angle (ψ > π ), we accomplish the change of parametrization according to Def. 15,giving a new rotation angle smaller than straight angle. Thus, we never get into trouble with singularity atψ = 2π . As it is illustrated in Fig. 9, the change of parametrization maps rotation angle outside of straightangle into inside of straight angle. Note that there exists no other canonical parametrization with rotationless than perigon such as those parametrizations given in Def. 15.

The zero rotation vector is an isolated point, the centre of the domain, for the parametrization change. Usinga limit process, we find out that the rotation operator approaches to the identity element when the rotationangle is decreased. Hence, we could modify the domain of the parametrization where the rotation angle isless than perigon, i.e. y < 2p including the zero rotation angle. This domain is still an open domain in theEuclidean space E3 , indeed, it is an open ball in E3 with 2p-radius.

2π

parametrization chart complement parametrization chart

π

2π

π

change ofparametrization

parametrizationmapping exp(

~)ΨΨΨΨ

parametrizationmapping exp(

~)ΨΨΨΨC

rotation manifold SO(3)

Fig. 9 The change of parametrization in the parameter space E3 for the canonical representation of therotation manifold.

27

2.3 Lie Group and Lie Algebra

The concept of Lie group and Lie algebra gives an algebraic structure for the rotation manifold. Since gen-eral groups and their special cases, Lie groups with corresponding Lie algebra, are quite unfamiliar to engi-neering literature, we give definition for these objects. We recommend the textbooks like [Choquet-Bruhatet al. 1989], [Marsden & Ratiu 1999] and [Selig 1996] for more details. As it was shown that rotation op-erators form the smooth manifold, called the rotation manifold, we show that the rotation manifold is a Liegroup, too. This issue is significant especially in composite rotations.

16 Definition for group: A group G is a set with an internal operation G G G× → by ( , )A B ABa , suchthat

1) the internal operation is associative A BC AB C A B Cb g b g= ∀ ∈, , , G ,

2) there is a unique element I ∈G called identity such that AI IA A A= = ∀ ∈, G ,

3) for each A ∈G there exists a unique element of G called the inverse of A such that A A AA I− −= =1 1 .‡

The group is called an Abelian group or a commutative group if AB BA A B= ∀ ∈, , G . For example, a set ofvectors with an internal operation, the vector addition, is an Abelian group where the identity element is thezero vector. In addition, a set of matrices equipped the matrix addition is an Abelian group with a zero ma-trix as the identity element.

If an internal operation is not commutative, then a group is called a non-Abelian group, or a non-commutative group. E.g., a set of invertable n-by-n matrices with an internal operation the matrix multipl i-cation is a non-Abelian group where the identity element is the identity matrix.

17 Definition for special orthogonal group SO(3): A special orthogonal (non-commutative) group in theEuclidean space E3 is defined by

SO( ): : , det( )3 13 3= → = = = +R R R RR I RE E linear T To t .Since SO(3) is a group it has to fulfill all group properties given in Def. 16. ‡

A rotation operator defined in Def. 14 p. 24 is also an element of the special orthogonal group as it wasshown in Equation (9) p. 25. The rotation operators form a non-commutative group with the internal opera-tion, called composite mapping, and an identity element as the identity operator I . Hence, we may denoteR ∈SO( )3 .

18 Definition for Lie group: A Lie group L is a group that is also a differentiable manifold such that an in-ternal operation L L L× → by ( , ) ,A B AB A Ba ∀ ∈L is a continuously differentiable mapping. ‡

This is short but not so an easily manageable definition. We have shown that the set of rotation operatorsform the differentiable manifold, called the rotation manifold; moreover, the set of rotation operators is anon-commutative group, special orthogonal group SO(3). So, only the continuity of internal operation hasnot been shown. To prove this, it has to be shown that a change of parametrization mappings under compo-sition is continuously differentiable on its domain. A procedure is similar to one for showing a manifold isdifferentiable. The prove that the group SO(3) is a Lie group is omitted here, but can be found in [Choquet-Bruhat et el 1989; pp. 181-182]. It is based on reality that any two composite rotations can be represented byEulerian angles, giving a differentiable change of parametrization mappings between different sets of Eule-rian angles.

19 Definition for Lie algebra: A Lie algebra l of the Lie group L is a tangent vector space at the identity,T LI , together with a bilinear, skew-symmetric brackets [ , ]a b satisfying Jacobi’s identity

a b c b c a c a b 0 a b c, , , , , , , , ,+ + = ∀ ∈l .

The skew-symmetry means that [ , ] [ , ], ,a b b a a b= − ∀ ∈l . ‡

28

How to obtain Lie brackets is still an open question and we have to define a Lie algebra adjoint transforma-tion.

20 Definition for Lie algebra adjoint transformation: An adjoint transformation AdR of the Lie algebra l isdefined by

Ad : , Ad : ,R Rb RbR Rl l L→ = ∀ ∈−1 .

Note that b ∈l is an element of the Lie algebra l and R is an element of the Lie group L. The Lie algebraadjoint transformation maps an element of the Lie algebra into another Lie algebra element. ‡

21 Determination for Lie brackets: Lie brackets can be obtained by differentiating the adjoint representationAdRb with respect to R( )η ∈L at the identity in the direction a ∈l such that R I( )η = =0 anddR a( )/dη η= =0 where h is a parameter, giving

a bR b R

,d ( ) ( )

d=

=

−η ηη η

b ge j10

.

Lie brackets is a bilinear skew-symmetric form and satisfies Jacobi’s identity, given in Def. 19. ‡

Especially, let R( ) ( )η ∈SO3 be a η -parametrized rotation operator, an element of the special orthogonal

group, given by formula R( ) exp(~

)η η= ΨΨΨΨ . Differentiating the expression exp(~

)ηΨΨΨΨ with respect to the pa-

rameter η at η = 0 gives the tangent vector space at the identity I ∈SO( )3 , yielding

dexp(~

)d

~ηη η

ΨΨΨΨ ΨΨΨΨ=

=0

. (11)

Thus, the skew-symmetric tensor ~ΨΨΨΨ belongs to the tangent space of the rotation manifold SO(3), denoted

by ~( )ΨΨΨΨ ∈T SOI 3 , where the identity I ∈SO( )3 represents a base point of the rotation manifold. The skew-

symmetric tensor ~ΨΨΨΨ is also an element of Lie algebra so( )3 for corresponding Lie group SO(3). We could

also mark so T SO( ) ( )3 3= I , i.e. Lie algebra is canonical isomorphic to the tangent space of the rotation mani-

fold at the identity. Moreover, we may denote the Lie algebra so(3) as a set of skew-symmetric operators

(tensors)

so( )~

:~ ~

3 3 3= → = −ΨΨΨΨ ΨΨΨΨ ΨΨΨΨE E linear T . (12)

We obtain the Lie brackets of the Lie algebra so( )3 by differentiating the Lie algebra adjoint representationAd

~( )R η ΘΘΘΘ with respect to η at η = 0

d(Ad~

)d

d(exp(~

)~

exp(~

))d

~ ~ ~ ~( )R η

ηη η

η η= = −

== −0

0

ΘΘΘΘ ΨΨΨΨ ΘΘΘΘ ΨΨΨΨ ΨΨΨΨΘΘΘΘ ΘΘΘΘΨΨΨΨ . (13)

Hence, the Lie brackets for the Lie algebra so(3) is [~

,~

]~ ~ ~ ~

,~

,~

( )ΨΨΨΨ ΘΘΘΘ ΨΨΨΨΘΘΘΘ ΘΘΘΘΨΨΨΨ ΨΨΨΨ ΘΘΘΘ= − ∈so 3 .

The vector cross product ( ):⋅ × ⋅ × →E E E3 3 3 in the Euclidean space E3 is a Lie algebra with Lie bracketsdefined by

[ , ]: , ,x y x y x y= × ∈E3 . (14)

The vector cross product ( )⋅ × ⋅ is a bilinear, with respect to vector addition and scalar multiplication, and askew-symmetric operator over E3 and satisfies Jacobi’s identity in Def. 19. The Lie algebra so(3) can beidentified with the cross product on E3 by formula

~,ΨΨΨΨ ΨΨΨΨa a a= × ∀ ∈E3 , (15)

where the vector ΨΨΨΨ ∈E3 is the axial vector for the skew-symmetric tensor~( )ΨΨΨΨ ∈so 3 .

29

22 Definition for Lie algebra homomorphism and isomorphism: Let l g and be two Lie algebras. A map-ping ϕ :g l→ is homomorphism, that is

ϕ ϕ ϕ[ , ] , , ,ΨΨΨΨ ΘΘΘΘ ΨΨΨΨ ΘΘΘΘ ΨΨΨΨ ΘΘΘΘgl

gb g b g b g= ∀ ∈ ,

A homomorphism is an isomorphism if the mapping ϕ : g l→ is a vector space isomorphism, i.e. a linearbijection. The Lie algebras as isomorphic, denoted l g≅ , if an isomorphism exists between these algebras.‡

The tilde mapping ~: ( )E3 3→ so is an homomorphism between the cross product on E3 and the Lie algebraso(3), i.e.

ΨΨΨΨ ΘΘΘΘ ΨΨΨΨΘΘΘΘ ΘΘΘΘΨΨΨΨ× = −b g~ ~ ~ ~ ~, (16)

see proof e.g. in [Marsden & Ratiu 1999; p. 290]. The tilde mapping is also a linear bijection giving isomor-phic correspondence between the elements of the Lie algebras, denoted by E3 3≅ so( ) . For computationalpurposes, the Lie algebra in the Euclidean space E3 is simpler than the Lie algebra so(3) and, hence, it willbe utilized in following. The Lie algebra in the Euclidean space E3 equipped with the cross product as theLie bracket is a rather unusual Lie algebra since, by our knowledge, there does not exist a Lie group whosethe Lie algebra it is.

2.3.1 Compound Rotation

A rotation operator is an element of Lie group that is a differentiable manifold as well as a non-commutativegroup. A compound of rotations is also a rotation itself and induces a Lie group structure. The compoundrotation can be defined by two different, nevertheless equivalent ways: the material description, and thespatial description.

23 Definition for material description of compound rotation: We define the material description of a com-pound rotation by the left translation mapping LeftR: ( ) ( )SO SO3 3→ as

Left R RR RR R R Rincmat

incmat

incmat: exp(

~) , , ( )= = ∈ΘΘΘΘ SO 3 ,

where R incmat is a material incremental rotation operator, and ΘΘΘΘR is a material incremental rotation vector

with respect to the base point R ∈SO( )3 . This description is called material since the incremental rotationoperator acts on a material vector space. ‡

24 Definition for spatial description of compound rotation: We define the spatial description of a compoundrotation by the right translation mapping RightR: ( ) ( )SO SO3 3→ as

RightR RR R R R R Rincspat

incspat

incspat: exp

~, , ( )= = ∈θθθθe j SO 3 ,

where R incspat is a spatial incremental rotation operator, and θθθθR is a spatial incremental rotation vector with

respect to the base point R ∈SO( )3 . This description is called spatial since the incremental rotation operatoracts on a spatial vector space. ‡

We use majuscules for material vectors and minuscules for spatial vectors. The material and spatial incre-mental rotation tensors and their rotation vectors are related by

R RR R R R RR R R Rincspat

incmat T T, , and= = =~ ~θθθθ ΘΘΘΘ θθθθ ΘΘΘΘ , (17)

where the first relation is called an inner automorphism that is an isomorphism onto itself, the second rela-

tion is a Lie algebra adjoint transformation Ad~ ~

R R RR RΘΘΘΘ ΘΘΘΘ= T , see Def. 20 p. 28, and the last relation is

another Lie algebra adjoint transformation on the Euclidean space with the vector cross product as the Lie

algebra ( , )E3 ⋅ × ⋅ .

30

exp ~ψψψψb g

exp~ΨΨΨΨd i

R

R

matT SOI ( )3

~ΨΨΨΨ

~ψψψψ

matT SOR ( )3

spatT SOR ( )3

I

I

SO( )3

spatT SOI ( )3

SO( )3

~ΘΘΘΘR

~θθθθR

Fig. 10 A geometric representation of the material (on the left) and spatial tangent spaces (on the right)on the rotation manifold.

25 Definition for material tangent space1 of rotation: Differentiating the material expression of the com-

pound rotation R exp(~

)ηΘΘΘΘ with respect to the parameter η and setting η = 0 , yields the material tangent

space at the base point R ∈SO( )3 . This material tangent space on the rotation manifold SO( )3 at any base

point R is defined as

mat with T SO SO soR R R R R( ):~

: ( ,~

)~

; ( ),~

( )3 3 3= = ∈ ∈ΘΘΘΘ ΘΘΘΘ ΘΘΘΘ ΘΘΘΘ ,

where an element of the material tangent space ~

( )ΘΘΘΘR R∈ matT SO 3 and is a skew-symmetric tensor, i.e.~

( )ΘΘΘΘR ∈so 3 . The notation ( ,~

)R ΘΘΘΘ , the pair of the rotation operator R and the skew-symmetric tensor ~ΘΘΘΘ ,

represents the material skew-symmetric tensor at the base point R ∈SO( )3 , see Fig. 10. Hence, we may

express that ~ΘΘΘΘR is a skew-symmetric tensor, or a tangent tensor, at the point R in the manifold SO( )3 . For

simplicity, we could omit the base point R by denoting ~

( )ΘΘΘΘ ∈ matT SOR 3 if there is no danger of confu-

sion. ‡

This definition is rather different than the definition found in [Simo et al. 1988] or in [Simo & Vu-Quoc1988] that reads in the form

mat for any T SO soR R R( ):~

:~ ~

( )3 3= = ∈ΘΘΘΘ ΘΘΘΘ ΘΘΘΘo t . (18)

Basically, Def 25 and (18) are equal since in (18) the rotation tensor R can be regarded as a base point of

the material skew-symmetric tensor ~

( )ΘΘΘΘ ∈so 3 . However,we note that ~ΘΘΘΘR , in Def 25, is a (material) skew-

symmetric tensor while the product R~ΘΘΘΘ is not. This can be comprehended by noticing that the left-

translation in Lie groups is defined by a product and not by an addition as in linear spaces. In linear spaces,

the left-translation to the vector Y reads Left X Y X Y:= + where the vector X is a base point. We consider

(18) rather confusing and prefer Def. 25.

26 Definition for spatial tangent space2 of rotation: A spatial tangent space on the rotation manifold SO(3)at any base point R is defined

spat with T SO SO soR R R R R( ):~

: ( ,~

)~

; ( ),~

( )3 3 3= = ∈ ∈θθθθ θθθθ θθθθ θθθθ ,

where an element of the spatial tangent space ~

( )θθθθR R∈ spatT SO 3 and is a skew-symmetric tensor, i.e.~

( )θθθθR ∈so 3 . The notation ( ,~

)R θθθθ , the pair of the rotation operator R and the skew-symmetric tensor ~ΘΘΘΘ ,

represents a spatial skew-symmetric tensor at the base point R , see Fig. 10. Again, we could omit the base

point R , i.e. ~

( )θθθθ ∈ spatT SOR 3 if there is no danger of confusion. ‡

1 Usually this space is called left-invariant vector field.2 Usually is called as right-invariant vector field

31

Rotation operators, the elements of the Lie group SO(3), are defined as linear operators R ∈L ( , )E E3 3 . Eqns(17b,c) give another interpretation to a rotation operator, it is an adjoint transformation between materialand spatial tangent spaces. Additionally, a rotational motion induces the rotation operator, since the rotationoperator maps the material place vector X ∈B0 into the spatial place vector x ∈B by the equationx R X( ) ( )t t= , i.e. R ∈L B B( , )0 . More generally, a rotation operator transforms material vectors into spatialvectors, that is R X x∈L ( , )T TB B0 .

2.3.2 Isomorphisms and Tangential Transformations

The Lie algebra so( )3 , which consists of the skew-symmetric tensors, and the Lie algebra (E3, )⋅ ×⋅ in the

Euclidean space are isomorphic with the Lie algebra isomorphism ~: ( )E3 3→ so that is the tilde mapping.

The spatial and material tangent spaces are isomorphic where the isomorphism is an adjoint transformation

Ad : ( ) ( )R R Rmat spatT SO T SO3 3→ , given in (17b). Additionally, the Lie algebra so( )3 is isomorphic in the mate-

rial tensor space with an isomorphism so T SO( ) ( )3 3→mat R by ~ ~ΘΘΘΘ ΘΘΘΘa R . Then we may express

E3 3 3 3≅ ≅ ≅so T SO T SO( ) ( ) ( )mat spatR R . (19)

Isomorphism states that for any element from a vector space we can take an element from an isomorphicvector space with a linear one-to-one correspondence. Therefore, the isomorphic spaces have the samestructure and we may associate the elements of the isomorphic spaces.

27 Definition for virtual rotation tensor and virtual rotation vector: A virtual rotation tensor δ ~ΘΘΘΘR is an

element of the corresponding tangent space T SOR ( )3 for any base point R ∈SO( )3 such that it satisfies all

linearized constraint equations, which naturally arise from joints and boundary conditions. A virtual rotation

vector δΘΘΘΘR at the base point R is the associated axial vector of the virtual rotation tensor δ ~ΘΘΘΘR . ‡

Let us consider the material form of compound rotation, given in Def. 23 p. 29, with the aid of η -parametrized exponential mappings

exp~ ~

exp~

exp~

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΘΘΘΘ+ =ηδ ηδe j e j e jR , (20)

where we are finding an incremental rotation tensor, the virtual rotation tensor δ ~ΨΨΨΨ , such that it belongs to

the same tangent space as the rotation tensor ~ΨΨΨΨ , i.e. such that δ ~

,~

( )ΨΨΨΨ ΨΨΨΨ ∈matT SOI 3 with the identity as a base

point omitted for simplicity. Note that R = exp(~

)ΨΨΨΨ , and δ ~( )ΘΘΘΘR R∈ matT SO 3 . We point out that the skew-

symmetric tensors ~ΨΨΨΨ and δ ~ΘΘΘΘR do not belong to the same tangent space of rotation as it can be verified that

exp(~

)exp(~

) exp(~ ~

)ΨΨΨΨ ΘΘΘΘ ΨΨΨΨ ΘΘΘΘ≠ + , generally. The associated rotation vector YYYY for the skew-symmetric tensor ~ΨΨΨΨ

is called the total material rotation vector whose base point is the identity. Taking the derivatives of (20)

with respect to the parameter h at η = 0 gives after the aid of isomorphism (19), see e.g. [Ibrahimbegović

et. al. 1995]

δ δψ

ψψ

ψψ ψ

ψψ

ΘΘΘΘ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨΨΨΨΨ

R

0

T

T I

R T I

=

= − − + −

= = =

⋅

⊗

→

,

sin cos ~ sin,

, exp( ), lim ( ) ,

12 3

Ù (21)

where the material tangential transformation T T= ( )ΨΨΨΨ is a linear mapping between the virtual materialtangent spaces mat matT SO T SOI R( ) ( )3 3→ . Now, we could make another verification that the virtual rotationvector δΘΘΘΘR and the virtual total rotation vector δΨΨΨΨ belong to different vector spaces on the manifold. Thisis because the tangential transformation T is equal to the identity only at ΨΨΨΨ = 0 . Note that the transforma-tion T has an effect on the base points, changing the base point I into R.

By examining the tangential transformation T in Eqn (21), we found that the transformation is non-singularwhen the rotation angle is less than perigon, i.e. ψ < 2π . It is worth noting that the tangential transformationT( )ΨΨΨΨ , the corresponding rotation operator R( )ΨΨΨΨ and the skew-symmetric rotation tensor

~ΨΨΨΨ have the sameeigenvectors. Hence, T( )ΨΨΨΨ , R( )ΨΨΨΨ , and

~ΨΨΨΨ are commutative, see [Ibrahimbegović et. al. 1995].

32

28 Definition for material vector space of rotation: For convenience, we define a material vector space onthe rotation manifold at any point R as

matT SOR R R: : , exp~

( ),= = = ∈ ∈ΘΘΘΘ ΨΨΨΨ ΘΘΘΘ ΨΨΨΨ ΘΘΘΘb g e j 3 3E ,

where an element of the material vector space is ΘΘΘΘR R∈ matT , which is an affine space with the rotation vec-tor ΨΨΨΨ as a base point and the incremental rotation vector ΘΘΘΘ as a tangent vector. ‡

Hence, the tangential transformation T is a mapping T I R: mat matT T→ . Note that the elements of this mate-rial vector space can be added by the parallelogram law only if they occupy the same affine space, i.e. iftheir associated skew-symmetric tensors belong to the same tangent space of the rotation manifold. Def. 28for a material vector space matTR should be considered as a useful and simple notation with an equivalencerelation with a material tangent space matT SOR ( )3 , defined in Def. 26 p. 30.

Respectively, we could determine the spatial tangential transformation, yielding

δ δ ψθθθθ ψψψψ ψψψψ ψψψψ ψψψψ ΨΨΨΨR T T T R= ⋅ = = = =T , ( ), exp(~), ( ) , (22)

where T is the same linear operator as in the material form (21).

29 Definition for spatial vector space of rotation: We define a spatial vector space on the rotation manifoldat any point R as

spatT SOR R R: : , exp ~ ( ),= = = ∈ ∈θθθθ ψψψψ θθθθ ψψψψ θθθθb g b go t3 3E . (23)

An element of the spatial vector space is θθθθR R∈ spatT . ‡

Hence, the spatial tangential transformation TT is a mapping between vector spaces on the rotation manifold

spat spatT TI R→ . The spatial and the material vector spaces are related by the rotation operator as given in theEqn (17c), p. 29. From Eqn (17c), it fol lows with the base point I ∈SO( )3 (note that ΨΨΨΨ ∈ matTI ).

ψψψψ ΨΨΨΨ ψψψψ ΨΨΨΨI I I II= ⇒ = , (24)

where ‘= ’ denotes the canonical isomorphism between the spatial and material vector spaces. The identity Imaps between the vector fields mat spatT TI I→ . Now, the relation between the spatial and material vectors can

be given as ( , ) ( , )ψψψψ θθθθ ΨΨΨΨ ΘΘΘΘ= I R where yyyy and YYYY represent the base points in the spatial and material vector

spaces, respectively. This relation can be written more compactly as θθθθ ΘΘΘΘR RR= , called a push-forward,

where the rotation operator should be considered as a mapping between the material and the spatial vector

spaces of rotation, R R R: mat spatT T→ , see Fig. 11. A push-forward operator maps a material vector space into a

spatial vector space (one-to-one and onto). It makes sense since the rotation operator is a two-point tensor.

We note that the push-forward operator R has no influence on the base point of the rotation manifold as it

is assumed in [Simo & Vu-Quoc 1988]. We believe that this misunderstanding arises from the definition of

the material tangent space (18). Another push-forward operator for rotation tensors is given in (17b) where~ ~θθθθ ΘΘΘΘR RR R= T is a mapping between the material and spatial tangent spaces of rotation

R R R R( ) : ( ) ( )⋅ →Tmat spatT SO T SO3 3 .

δθθθθRδΘΘΘΘR

δψψψψ IδΨΨΨΨI

TT

R

T

RT

I

spatTRmatTR

spatTImatTI

TT

R

T

RT

I

Fig. 11 A commutative diagram of virtual material and spatial rotation vectors on the rotation manifold(on the left), and their corresponding vector spaces (on the right).

33

2.4 Angular Velocities, Accelerations and Curvatures

In this Section, we give definitions in the material and spatial representations for angular velocities, angularaccelerations, and curvatures.

30 Definition for material angular velocity: A material angular velocity (skew-symmetric) tensor is definedwith the aid of rotation operator R ∈SO( )3 and its time derivative by

~: &ΩΩΩΩR R R= T

where the dot represents to the time derivative. See justification in [Marsden & Ratiu 1999; Ch. 8.6 & 15.2].‡

The rotation tensor can be viewed as a mapping, a push-forward of a material vector, R R R:mat spatT T→ be-

tween the material and spatial vector spaces. Then the material angular velocity tensor is a mapping~

:ΩΩΩΩR R Rmat matT T→ . Thus, the material angular velocity tensor is indeed a true material tensor. The skew-

symmetry can be observed by taking derivative for the equation R R IT = .

If the rotation operator is expressed with the aid of exponential mapping by R R I R Rnew = + +(~

(~

))ΘΘΘΘ ΘΘΘΘO 2 ,

where the fixed rotation R is superimposed by an infinitesimal rotation (~

)I R+ ΘΘΘΘ plus higher order terms

and substituting this into Def. 30 yields after the limit process ~ ~ΘΘΘΘR 0→

~ ~& &ΩΩΩΩ ΘΘΘΘ ΩΩΩΩ ΘΘΘΘR R R R= ⇔ = . (25)

This states that the angular velocity vector is the time derivative of the incremental rotation vector ΘΘΘΘR ;

moreover, (if the base point is omitted) ΘΘΘΘ ΘΘΘΘ ΩΩΩΩ, & , ∈matTR , which is the material rotation vector space on the

rotation manifold. The result in Eqn (25) is often given as definition for the angular velocity vector in the

elementary text books.

Similar expression and derivation can be accomplished for the spatial angular velocity tensor and vector,yielding

~ : & ,

~ ~& & ,

ωωωω ====

ωωωω θθθθ ωωωω θθθθ

R

R R R R

RR T

= ⇔ =(26)

where the spatial incremental rotation vector θθθθR , its time derivative vector &θθθθR and the spatial angular vec-

tor ωωωω R belong to the same spatial vector space on the manifold θθθθ θθθθ ωωωω, & , ∈spatTR , the base point R omitted.

31 Definition for angular accelerations: A material and spatial angular acceleration tensor and vector aredefined as the time derivative of corresponding angular velocity term, giving

~:

~&,

~( ) ,

: & , ,~ : ~& , ~ ( ) ,

: & , ,

ΑΑΑΑ ΩΩΩΩ ΑΑΑΑ

ΑΑΑΑ ΩΩΩΩ ΑΑΑΑ

αααα ωωωω αααααααα ωωωω αααα

R R R R

R R R R

R R R R

R R R R

= ∈

= ∈

= ∈= ∈

mat

mat

spat

spat

T SO

T

T SO

T

3

3

where A R and αααα R are the material and spatial angular acceleration vectors at the base R . ‡

Note that the material incremental rotation vector ΘΘΘΘR , the material angular velocity vector ΩΩΩΩR and the

material angular acceleration vector ΑΑΑΑR (majuscule of alpha-letter) belong to the same material vector

space on the rotation manifold, i.e. ΘΘΘΘ ΩΩΩΩ ΑΑΑΑR R R R, , ∈matT with the base point R I= exp(~

)ΨΨΨΨ . At separate mo-

ments, these vectors, however, occupy different vector spaces because the rotation operator depends on

time, namely R R= ( )t . The base point is moving in process of time. Vector quantities of this kind may be

called spin vectors. Spin vectors are rather tricky in numerical sense as they always occupy a distinct vector

space on a manifold. Correspondingly, the spatial spin vectors are θθθθ ωωωω ααααR R R R, , ∈spatT .

34

Angular velocity vectors and the time derivative of total rotation vectors are related by, see (21-22)

ΩΩΩΩ ΨΨΨΨ ΨΨΨΨ ΩΩΩΩ ΨΨΨΨ ΨΨΨΨ

ωωωω ψψψψ ψψψψ ωωωω ψψψψ ψψψψR I I R R I I I

R I I R R I I I

T

T

= ⋅ ∈ ∈

= ⋅ ∈ ∈

( ) & , & , ,

( ) & , & , ,

where for material description,

where for spatial description,

mat mat

Tspat spat

T T

T T(27)

where the tangential transformation depends on the total rotation vector, and the rotation operator is

R I I= =exp(~

) exp(~ )ΨΨΨΨ ψψψψ . Similar expression for the angular acceleration vector can be obtained by differenti-

ating the above formulas, giving

ΑΑΑΑ ΨΨΨΨ ΨΨΨΨ ΑΑΑΑ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

αααα ψψψψ ψψψψ αααα ψψψψ ψψψψ ψψψψR I I R R I I I I

R I I R R I I I I

T T

T T

= ⋅ ⋅ ∈ ∈

= ⋅ ⋅ ∈ ∈

&& & & , , & , &&

&& & & , , & , &&

+ where for material description,

+ where for spatial description.

mat mat

T Tspat spat

T T

T T(28)

Note that the tangential transformations T T T TI R I R, & ( , ) , & ( , )∈ ∈L Lmat matT T

spat spat and T T T T operate with the

different base points.

32 Definition for curvatures: A material curvature tensor ~ΚΚΚΚ R of s-parametrized curve is defined as

~:ΚΚΚΚ R R

RR R= = ′T Td

ds,

whose axial vector is called the material curvature vector ΚΚΚΚ R (majuscule of kappa-letter). A spatial curva-ture tensor ~κκκκ is defined, respectively

~ :κκκκ R R R= ′ T ,

where prime denotes the derivative with respect to the length parameter s. ‡

Material and spatial curvature tensors and corresponding curvature vectors are related by

~ Ad~

, adκκκκ ΚΚΚΚ ΚΚΚΚ κκκκ ΚΚΚΚ ΚΚΚΚR R R R R R R RR R R= = = =T , (29)

where the rotation operator R is a push-forward operator R R R∈L ( , )mat spatT T between material and spatialvector spaces and keeps the base point unaltered. A relation between curvature vectors and total rotationvectors becomes from an analogy of Eqns (27)

ΚΚΚΚ ΨΨΨΨ ΨΨΨΨ ΚΚΚΚ ΨΨΨΨ

κκκκ ψψψψ ψψψψ κκκκ ψψψψR I I R R I I

R I I R R I I

T

T

= ⋅ ′ ∈ ′ ∈

= ⋅ ′ ∈ ′ ∈

( ) , ,

( ) , ,

where for material description,

where for spatial description,

mat mat

Tspat spat

T T

T T(30)

where the prime denotes the derivative with respect to the length parameter s , and R is the base point.

2.5 Constraint Point-Manifolds

So far we have studied only finite-dimensional manifolds. The placement field of continuum medium takesvalues in a Hilbert space1, where chart parametrization maps vector-valued functions into vector-valuedfunctions. The placement field needs an infinite number of basis functions in order to present an arbitraryplacement field on continuum, yielding infinite-dimensional manifolds.

In multibody mechanics, constraint equations naturally arise from kinematic relations between bodies,boundary conditions and kinematic assumptions. Different joints like revolute and prismatic joints are ex-amples of point-wise kinematic relations that can be described geometrically, i.e. these joints can be pre-sented by holonomic finite-dimensional constraints. A kinematic assumption like Timoshenko-Reissnerhypothesis is correspondingly a continuous kinematic relation (infinite dimensional) and it reduces an inter-nal dimensionality by mapping a three-dimensional solid into a one-dimensional solid, called a beam.Beams are internally one-dimensional but infinite manifolds, whose generalized placement fields are pre-sented by functions. Hence, beam models have only one spatial parameter, called a length parameter.

1 Hilbert space is a complete inner-product space, and here especially a complete infinite-dimensional inner-productvector-valued function space, see Hilbert spaces e.g. in [Debnath & Mikusinski 1990]

35

A beam can be considered as a vector bundle on a manifold, where for each position of the length parameterthere is a two-dimensional vector plane where a cross-section belongs. This vector bundle occupies a set, thevolume of the beam, in the Euclidean space. We study beam kinematic assumptions in Section 3.1.2 andnow we focus on point-wise, finite-dimensional, kinematic relations. However, it should be noticed thatthese kinematic relations are points of infinite-dimensional field-manifold.

33 Definition for (holonomic) point-constraints: A holonomic point-wise constraint equation in a vectorialform is defined as a smooth mapping h: R E E× → −n n d ( d n< ) by

h x 0( , )t = ,

where the arguments are time t and the generalized place vector x( )t n∈E . Constraint equations are as-sumed to be an independent set of equations. ‡

A constraint equation, which is impossible to present in a holonomic form, Def. 33, is nonholonomic, i.e. itis not and cannot be integrated into a holonomic form. It is clear that we cannot describe nonholonomicconstraint equations, they are just kinematic relations, which are not holonomic. Different kinematic rela-tions are shown in Fig. 12, where geometric constraints include all holonomic and the so called unilateralconstraints, which are given by inequality equations with the function of time and a generalized place vectoronly. Unilateral constraints arise when modeling a kinematic relation between bodies in a contact. Espe-cially in multibody systems, a joint clearance (play) and collision problems may be modeled by a contactformulation.

34 Definition for (holonomic) constraint point-manifold: A holonomic point-wise constraint equation (Def.33) induces a d-manifold that is defined by

M =: ,t tn n d× ∈ × = ∈ −x h x 0R E Eb go t .The constraint point-manifold M is a d-dimensional smooth manifold with time as 1-parameter family. Theconstraint manifold at fixed time t t= 0 is denoted by M t0 . ‡

Although a constraint point-manifold is smooth, its master, an infinite-dimensional constraint field-manifoldin a multibody system, is usually smooth inside a body but has nonsmooth points at various joints, elbows,the sudden change of cross-section, etc, depending on different types of models. For example, a multibodysystem modeled by Reissner’s beams with a cylindrical joint has a finite jump in a rotation field, and atranslational placement field is a C0 -continuous at the cylindrical joint. It is also informative to find out thata holonomic constraint manifold does not have a boundary at all, on the contrary geometric constraints withat least one unilateral constraint do, see Fig. 13.

Geometrich x 0( , )t ≥

Holonomich x 0( , )t =

Kinematic

Nonholonomic

Fig. 12 Different types of kinematic constraints and their occupying areas.

36

holonomic constraint manifoldMt0 (without boundary)

tangent space Tx0M

x0

nonholonomicconstraint at x0

virtual displacement δ x with

nonholonomic constraint

geometric constraintmanifold (with boundary)

boundary of constraint

virtual displacement δ x without

nonholonomic constraint

Fig. 13 A geometric interpretation of kinematic constraints.

A particular interesting nonholonomic constraint is presented by an equality equation (bilateral) with thefunction of time, a placement vector, and linearly on a velocity, see e.g. nonholonomic discrete systems in[Rabier & Rheinboldt 2000] and [Rosenberg 1980]. A rolling coin on the surface without sliding is a classicexample, see Fig. 13 where the nonholonomic constraint describes the rolling coin problem.

In following, we study holonomic constraint equations, which restrict to rather simple kinematic relationlike geometric joints without clearance (play) and beam kinematic assumption (hypotheses). Hence, ourconstraint manifold can be presented by a placement vector and a possible time parameter. In addition, con-straint manifold does not have a boundary. Although holonomic constraint equations are simple, we maymodel rather complicated multibody systems with them. A virtual displacement is closely associated withconstraint manifold, see Fig. 13, and needs a proper definition. We note that virtual quantities are finite, notnecessary infinitesimal.

35 Definition for tangent point-bundle: A virtual displacement δ x at any generalized place vector x ∈En

and a fixed time t t= 0

T ttn n

tM0 0 0: ( , ) , D , , D= Mx x x h x x 0 hx xδ δ∈ × ∈ ⋅ =E E b g is surjection ,

where Dxh is a Fréchet partial derivative of holonomic constraints with respect to x at t t= 0 . The defini-tion limits the singular (nonregular) points of the constraint manifold out by demanding the derivative of theconstraints is a surjective (onto) mapping. ‡

This surjectivity request yields that dimensionality do not vary in the manifold that has importance whenaccomplishing the finite element method, the constraint manifold has a fixed dimensional independent onconstraints. It can be proven, see e.g. [Rheinboldt 1986; p. 44-45], that the null-space of Dx h , denotedkerDx h (kernel), is equal to the tangent space. Thus, T tM 0 is indeed a tangent bundle.

A vector bundle establishes a tangent vector space for each regular point of the manifold M at the fixedtime t0 . For practical reasons, we need a tangent space that is an element of the tangent bundle.

36 Definition for tangent point-space: For the fixed time t t= 0 , we could set x x0 0= ( )t , giving a tangent

point-space at the point x0 0∈Mt

T Tntx x x x

0 00M =: ( , )δ δ∈ ∈E M .Then we may denote for any virtual displacement vector δ x x∈T 0M , where the base point x0 is included inthe notation as a subscript. ‡

37

A geometric interpretation of tangent space and its element virtual displacement have been illustrated withthe holonomic and nonholonomic cases in Fig. 13.

2.6 Derivatives and Constraint Field-Manifolds

In this Section, we give definitions for Gâteaux and Fréchet differentials, and for a constraint manifoldmodeled in infinite-dimensional Hilbert spaces, called field-manifolds1 here. We assume that any operatorwe consider is Fréchet differentiable, which is stronger than Gâteaux differentiable. Hence, we use Gâteauxdifferential for more useful and simple way to calculate a Fréchet differential or derivative. We note that ifan operator is Fréchet differentiable then its Fréchet and Gâteaux differential are equal, see more details in[Oden & Reddy 1976; Ch 2].

In following definition, we consider an operator f :X H H⊂ →1 2 , later called a vector, from a set X of theHilbert space H1 into the Hilbert space H2 . The vector f is a general vector-valued nonlinear mappingbetween function spaces. We also assume that the vector f(x) is Fréchet differentiable, i.e. it has a uniqueFréchet derivative.

37 Definition for Fréchet derivative and differential: The Fréchet derivative of the vector f :X H H⊂ →1 2 atfixed x ∈X is defined as a continuous linear operator D ( ):f x H H1 2→ such that

f x u f x f x u r x u( ) ( ) D ( ) ( , )+ − = +⋅ ,

where the remainder obeys the condition lim( , )

u 0

r x u

u→=H

H

2

1

0. D ( )f x u⋅ is called Fréchet differential. ‡

A vector is called Fréchet differentiable if its Fréchet derivative exists. This derivative is also a linearizedform, or better its affine form with together f(x), for a nonlinear vector f x u( )+ at x. Def. 37 is rather sim-ple but not so practical way to calculate Fréchet derivative, hence we define Gâteaux differential for a morepractical formula to calculate Fréchet differential and derivative for Fréchet differentiable vector.

38 Definition for Gâteaux differential: The Gâteaux differential of the vector f :X H H⊂ →1 2 at fixedx ∈X is defined as a limit

D ( ) : lim( ) ( ) d

df x u

f x u f x f x u⋅ = + − =+

=→η

ηη

ηη η0 0

b g,

where the limit is to be interpreted in the norm of H 2 . The later formula is a practical and simple way tocompute the directional derivative that is the term D ( )f x u⋅ where u ∈H 1 indicates direction. ‡

We have assumed that X is a set, but next we give a more structure. We denote this structured set as C .

39 Definition for constraint field-manifold: A set C of a Hilbert space H 1 is defined as infinite-dimensionalconstraint manifold embedded in Hilbert space such that it satisfies all the kinematic constraints of theholonomic type by

C H H: ,= × ∈ × = ∈t tx h x 0R 1 2b go t ,where h x( , )t indicates an independent set of the holonomic constraint equations. A constraint manifold at afixed time t t= 0 is denoted by Ct0 . ‡

1 We use the name (infinite-dimensional) field-manifold contrast to a (finite-dimensional) point-manifold.

38

Compare the constraint field-manifold with the constraint point-manifold defined in Def. 34 p. 35. A similarway as in the point-wise case, we could define a tangent field-space which is a space of vector-valued func-tions

40 Definition for tangent field-bundle: A virtual displacement field1 δ x at any place field x ∈X 0 and thefixed time t t= 0 is defined as

T tt tC H H C0 01 1 0: ( , ) , D , , D= x x x h x x 0 hx xδ δ∈ × ∈ ⋅ =b g is surjection ,

where Dxh is the Fréchet partial derivative of the holonomic constraints with respect to x at t t= 0 . Thedefinition limits the nonsmooth isolated points of the constraint manifold out by demanding the existence ofthe Fréchet derivative. ‡

41 Definition for tangent field-space: For the fixed time t t= 0 , a tangent field-space at the base point

x0 0∈Ct is defined

T T tx x x x0 01 0C H C: ( , )= δ δ∈ ∈ ,

where the tangent field-bundle is defined in Def. 40. We may denote any virtual displacement fieldδ x x∈T 0C , where the base x0 is included in the notation as a subscript. ‡

Note that the place field x is a vector-valued function satisfying all the constraints equations (holonomic),called the constraint field-manifold C .

42 Definition for velocity field-space: A velocity field-space is closely related with the tangent field-spaceand is defined by formula

T Tx x x xC H C: & ( ,& )= ∈ ∈1n s ,where now time is free, not fixed, like in the virtual displacement. Compare with Def. 9 p. 21, a tangentvector in a finite-dimensional case. The velocity field that is an element of the velocity field-space is alsodenoted by v x x: &= ∈TC ‡

2.7 Variation, Lie Derivative and Lie Variation

In this Section, we give definition for the variation, Lie derivative and Lie variation. The concept of push-forward and pull-back operators is essential for understanding Lie derivatives and variations.

43 Definition for variation operator: The variation operator δ is defined as the special case of Fréchet dif-ferential at the fixed time t t= 0 by

δh x v h x v x h x v vx vt t t0 0 0, , : D , , D , ,b g b g b g= ⋅ + ⋅δ δ ,

where x ∈Ct0 is a place field, δ x x∈T 0C is a virtual displacement field, v x∈TC is a velocity field, andδ δv x x: &= ∈T 0C is a virtual velocity field. Moreover, D ,Dx v are Fréchet partial derivatives with respect toplace and velocity, correspondingly. ‡

The variation operator δ depends linearly on the virtual displacement and the virtual velocity. Note a minornotational difference between the virtual and variation operators, δ and δ . Calculating the place and veloc-ity variation, after Def. 43, yields

δ δx x v v= =δ δ and . (31)

This should be interpreted: the variation of place vector δx is equal, not the same thing, as the virtual dis-placement δ x . The variation has an operational meaning whereas the virtual displacement is a geometricalquantity. In generally, the variation of ‘something’ and the virtual ‘something’ are not equal, e.g. a virtualwork may exists although there does not exist a work function at all and neither the work variation.

1 a vector-valued function, more precisely

39

44 Theorem: Generally, the variation operator and the time derivative operator do not commutate.

Proof: We will prove this theorem by a counter example. Let consider the constraint equation & &x y 0+ =t ,where t represents time. Its variation is δ δ& &x y 0+ =t . On the other hand, the virtual displacement of theconstraint equation is δ δx y 0 x y 0+ = ⇔ + =t tδ δ , whose time derivative is respectively

dd

dd

δ δ δx yy 0

tt

t+ + = .

This is clearly different from δ δ& &x y 0+ =t , which is the variation of the original constraint equation. ‡

Usually in the textbooks, it is proven that reverse relation is true, but they implicitly or explicitly assumethat constraint equations are holonomic when the time derivative and variation operators are commutative.In the proof, the example is a nonholonomic one. Property that the time derivative and the variation do notgenerally commutate has been noticed at least in [Burke 1996; p. 314]. If it is assumed that the time deriva-tive and variation operators commutate, then nonholonomic constraints must not be substituted into a kineticenergy function, see e.g. [Rosenberg 1980; p. 172-173].

When deriving Hamilton’s principle from the principle of virtual work1 (d’Alembert’s principle), there isexploited the commutative assumption between the time derivative and variation. Hence, in the case of non-holonomic constraints and monogenic forces2 Hamilton’s principle is not a true variational principle, i.e. anextremum principle. This problem may be solved by considering only holonomic constraints and monogenicforces, then Hamilton’s principle is a true variational principle, see the textbook [Lanczos 1966] for moredetails.

Here we follow the paper [Stumpf & Hoppe 1997] for deriving push-forward and pull-back operators andLie derivatives. The push-forward and pull-back notation has been comprehensively utilized in the textbook[Marsden & Hughes 1983] which gives a differential geometric foundation of elasticity. In this textbook, itis assumed that there exists a smooth diffeomorphism between manifolds. However, the existence of suchmapping is rather rarely possible. Hence, we define push-forward and pull-back operators according to thepaper [Stumpf & Hoppe 1997], where no diffeomorphism between manifolds are assumed to be present.

Let the operator R X x:T TB0 → B be an invertible linear mapping between the tangent spaces of the materialand spatial manifolds. The material manifold is denoted by B0 and the spatial manifold by B . Moreover,let gi and G i be the bases for the spatial and material tangent spaces T TX xB0 and B , respectively, andlet gi

∗ and G i∗ be the corresponding dual bases for the spatial and material cotangent spaces

T TX x∗ ∗B0 and B , see Section 2.1.

45 Definition for pull-back operator: The pull-back operator by the isomorphism R X x∈Liso B( , )T T0 B forthe spatial vector a g x= ∈a Ti i B is defined by

R a R g X< := ∈−a Ti i

10d i B ,

where R x X− ∈1

0Liso B( , )T TB is the inverse operator of R. The pull-back operator by R X x∈L B( , )T T0 B forthe spatial covector f g x= ∈∗ ∗f Ti i B is defined by

R f R g X< := ∈∗ ∗ ∗f Ti id i B0 ,

where R x X∗ ∗ ∗∈L ( , )T TB B0 is the adjoint operator of R. ‡

The definition of the pull-back operator makes clear why we distinguish between vectors and covectors,their pull-back operators are quite different. A pull-back operator maps spatial vectors into material vectors,and spatial covectors into material covectors, hence we could also name the pull-back operator as the mate-rializer operator. Especially when considering objectivity and/or two-point tensors the later name, the mate-rializer operator, makes more sense. Note that the associated operator R does not need to be an isomor-phism in the pull-back operation for covectors,

1 We also include inertial forces in the form of virtual work2 a force is monogetic if there exists single scalar work potential function that depends on place, velocity and time

40

46 Definition for push-forward operator: The push-forward operator by R X x∈L B( , )T T0 B for the materialvector A G X= ∈A Ti i B0 is defined by

R A RG x> := ∈A Ti ib g B .

The push-forward operator by the isomorphism R X x∈Liso B( , )T T0 B for the material covectorF G X= ∈∗ ∗F Ti i B0 is defined by

R F R G x> := ∈−∗ ∗ ∗F Ti id i B ,

where R X x−∗ ∗ ∗∈Liso B( , )T T0 B is the inverse of R∗ . ‡

If the operator R is invertible between the material and spatial tangent spaces R X x∈Liso B( , )T T0 B , thenits adjoint, its inverse and the inverse of adjoint operators are R Rx X x X

∗ ∗ ∗ −∈ ∈Liso Liso B( , ), ( , )T T T TB B B01

0

and R X x−∗ ∗ ∗∈Liso B( , )T T0 B , respectively.

As for the pull-back operator, the definition of the push-forward operator makes clear why we distinguishbetween vectors and covectors. The push-forward operator maps the material (co)vectors into the spatial(co)vectors. The push-forward (or pull-back) operator is defined for a higher order tensor such as the push-forward (or pull-back) operator for each basis vector separately.

For example, the push-forward of the second order tensor G X X∈ ∗Liso ( , )T TB B0 0 , a material metric tensor, inthe tensor space T ( , ;0, )0 2 0 by an isomorphism F X x∈Liso B( , )T T0 B , a deformation gradient, is

F G F G G F G F G F GF x x> >= = = ∈∗⊗

∗ −∗ ∗⊗

−∗ ∗ −∗ − ∗G G T Tij i j ij i jd i ( ) ( ) ( , )1 Liso B B , (32)

where we have used the relation a Fb a b F b X⊗ ⊗ ∗= ∀ ∈( ) , T B0 . The resulting tensor F GF−∗ −1 is a spatial

deformation tensor (Cauchy deformation tensor), often denoted by c . This corresponds to classical tensoranalysis when identifying the material metric tensor by the identity G I→ and the adjoint operator by thetranspose operator F F−∗ −→ T . Hence, we get the formula for the classical spatial deformation tensorc F F= − −T 1

The deformation gradient is itself a two-point tensor, i.e. F g G x X= ∈⊗ ∗ ⊗ ∗F T Tij i j B B0 is a type of T ( , ; , )011 0 .The pull-back operator of a two-point tensor is defined as the pull-back of spatial basis vectors and covec-tors. In the case of the deformation gradient, this pull-back operator by R X x∈Liso B( , )T T0 B reads

R F R g G R g G R g G X X< < <= = = ∈⊗

∗⊗

∗ −⊗

∗⊗

∗F F F T Tij i j ij i j ij i jd i ( ) ( )10 0B B . (33)

Thus the resulting tensor R F< is a type of T ( , ; , )11 0 0 tensor, which is purely a material tensor. We note thatpush-forward and pull-back operators do not effect on tensor components, it just changes the bases.

47 Definition for Lie derivative: The Lie derivative LRc of the general tensor c( )η ∈T with respect to theisomorphic mapping R X x( ) ( , )η ∈Liso BT T0 B and the parameter η is defined by

L : ddRc R R c= FHGIKJ ∈>

<

η T .

Note that c( )η and R( )η depend on the parameter η . ‡

In Def. 47, the pull-back operator R< materializes a spatial or two-point tensor. It is known that the deriva-tive of an objective material tensor is an objective tensor, see e.g. [Ogden 1984; Sec. 2.4]. The push-forwardoperator R> is considered as the inverse of the pull-back operation where the resulting Lie derivative tensorLRc belongs to the same tensor space T as the original tensor c .

48 Definition for Lie variation: The Lie variation δRc of the general tensor c ∈T with respect to the iso-morphic mapping R X x∈Liso B( , )T T0 B is defined by

δ δ(Rc R R c: )= ∈>

< T ,

where the variation operator δ is given in Def. 43 p. 38, which is accomplished at the fixed time t t= 0 . ‡

41

As the Lie derivative, the Lie variation is an objective quantity if the original tensor is objective. The defi-nition of Lie variation is connected with a virtual displacement that can be seen by writing the Lie variationwith the aid of Gâteaux differential at the point ( , )x v and the fixed time t t= 0

δRc R R c= FHIK =>

<ddη η 0

, (34)

where the tensor c x x v v( , , )t0 + +ηδ ηδ and the operator R x x v v( , , )t0 + +ηδ ηδ depend on the virtual dis-placement δx and the virtual velocity δv . Note that e.g. the virtual displacement belongs to the tangentpoint-space Tx0M in the finite-dimensional case and to the tangent field-space Tx0C in the infinite-dimensional case.

For example, the Lie variation of the deformation tensor F g G x X= ∈⊗ ∗ ⊗ ∗F T Tij i j B B0 , a type of T ( , ; , )011 0tensor, by the rotation operator R X x∈ =Liso B( , ) ( )T T SO0 3B reads

δ δ( δ δ δ

δ δ

RF R R F R R F R R F R F

F R R F

= = = +

= +

>

< )

,

T T T

T

d i d i(35)

where we have used the result of (33). The variation of rotation operator in the material and spatial descrip-tion is

δ

δ

RR

R

RR

R

RR

RR

==

=

==

=

d exp(~

)d

~

dexp(~

)d

~

ηδη η

δ

ηδη η

δ

ΘΘΘΘ ΘΘΘΘ

θθθθ θθθθ

0

0

for material description,

for spatial description,

(36)

hence the term R Rδ T in (35) is equal to −δ~θθθθR in both descriptions because δ δ~ ~θθθθ ΘΘΘΘR RR R= T according to(17) p. 29. Finally, we have the Lie variation of the deformation tensor F with respect to the rotation op-erator R as

δ δR R x XF F F= − ∈ ⊗∗δ~

θθθθ T TB B0 , (37)

that is also called a corotational variation operator. Although the spatial virtual rotation tensorδ~ ( )θθθθR R∈ spatT SO3 , i.e. it occupies a spatial tangent space, see Def. 26 p. 30, it is also an element of the ten-sor space T Tx xB B⊗ ∗ .

2.8 Useful Formulas

In this Section, we give some useful and practical formulas which we will use in the following sections.Especially the derivatives of the tangential transformation T Eqn (21), p. 31, have an essential rule in alinearization procedure.

Comparing Eqns (29b) and (30) p. 34 and using the identity ′ = ′ψψψψ ΨΨΨΨI II , where the identity operatorI I I: mat spatT T→ , we found a formula, see [Ibrahimbegović et al. 1995]

RT T I= T . (38)

This impresses that the partial push-forward of the material tangential transformation T by the rotationoperator R is the spatial transformation TT . After Fig. 11, we may conclude that the push-forward operatorfor spin vectors is the rotation operator R, and for total vectors, the push-forward operator is the identityoperator I .

The variation of the rotation vector ΨΨΨΨ := ψe is written as

δ δ δΨΨΨΨ ==== e eψ ψ+ , (39)

where ψ is the non-negative rotation angle and e the rotation axis with unit length. Taking variation for the

both side of the equation e eÙ ⋅ = 1, where e GeÙ:= and G is a metric tensor, gives after using the symmetry

of the dot product

δe eÙ ⋅ = 0 . (40)

42

The dot product of (39) and the rotation axis e gives

δ δ δ

δ

ΨΨΨΨ ====Ù Ù Ù⋅ ⋅ ⋅+

=

e e e e eψ ψψ .

(41)

This also yields an important relation between the variation of the rotation angle and the virtual rotation

vector, giving δ =ψ δeÙ ⋅ ΨΨΨΨ , where δΨΨΨΨ is the virtual rotation vector. We note that the same expressions can

be derived for spatial rotations. Here we apply a virtual quantity since the rotation vector is a basic unknown

vector and has its geometric structure. Using above formula for δψ and substituting it into (39), we get the

variation of the rotation axis

δn I e e= − ⊗ ⋅1ψ δÙd i ΨΨΨΨ . (42)

Similar formulas may be arisen by an analogy for the time and spatial derivatives, just replacing the varia-tion operator with these derivatives.

Next, we give often-used formulas that arise from an isomorphism between the Lie algebra so( )3 , the crossproduct and their properties (vector and scalar triple products, Jacobi’s identity)

~~( ) , ~~ ,

~~ ~~ ~ ,

~~ ~~ ( ) ,

~ , ~~ ~~ ~ ~ ~ ,~~~

( )~

,

~ ~, ~ ~ ,

~ , ( ),

ab b a a b I aa a a I

ab ba ab a b b a

ab ba a b b a a b I

aa 0 ab ba ab ab ab 0

bab a b b

a a a a

Ra RaR R

= − = −

− = = −

+ = + −

= − = =

= −

= − = −

= ∈

⊗ ⊗

⊗ ⊗

⊗ ⊗

⋅

⋅

⋅

Ù Ù

Ù Ù

Ù Ù Ù

Ù

d i d ib g

d i b g

b g

a

a a

SO

2

3 2 4 2 2

2

3

~

~

~ T

(43)

where a2 and = ∈⋅( ) ,a a a bÙ E3 .

The variation of the tangential transformation (21), p. 31, is essential for deriving tangent tensors. We de-fine the tensor C1 with the aid of directional derivative of the vector T V⋅ in the direction ∆ΨΨΨΨ

C V T V V

C V V V V V V I V

13

1 2 3 4 5

( , ) : D ,

( , )~ ~

( ) ,

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ⋅ ⋅ ⋅⋅ ⋅

= ∀ ∈

= − + − + +⊗ ⊗ ⊗ ⊗

∆ ∆b ge j d i d i

E ,

1 c c c c cÙ Ù Ù Ù Ù ÙΨΨΨΨ(44)

where the coefficients ci are given by

c c

c c c

1 3 2 4

3 5 4 2 5 3

2 2

3 2 1

:cos sin

,sin cos

sin cos,

cos,

sin.

= − = + −

= − − = − = −

ψ ψ ψψ

ψ ψ ψψ

ψ ψ ψ ψψ

ψψ

ψ ψψ

(45)

Using the limit process, we obtain

lim ( , )~

ΨΨΨΨ→=

0C V V1 ΨΨΨΨ 1

2, (46)

because of the coefficient c4 1 2→ − / as ψ → 0 . The tensor C1 can be viewed as an operatorC I R1 ∈L ( , )mat matT T like the tangential transformation T . We also note that & ( , ) &T V C V⋅ ⋅= 1 ΨΨΨΨ ΨΨΨΨ .

A very similar expression in the spatial description comes from the directional derivative of the vectorT VT ⋅ in the direction ∆ΨΨΨΨ where the tensor C2 is defined via relation

C V V T V V

C V V V V V V I V

23

1 2 3 4 5

( , ) D , ,

( , ) (~

)~

( ) ,

ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ⋅ ⋅ ⋅

⋅ ⋅

= ∀ ∈

= + + + + +⊗ ⊗ ⊗ ⊗

∆ ∆Td id i d i

E

2 c c c c cÙ Ù Ù Ù Ù Ù(47)

where the coefficients are given in (45).

43

The limit process gives

lim ( , )~

ΨΨΨΨ→= −

0C V V2 ΨΨΨΨ 1

2. (48)

We note also that the tensors C V C V1 2( ) and , ( , )ΨΨΨΨ ΨΨΨΨ depend linearly on the vector V but nonlinearly on thetotal rotation vector. The tensors C C1 and 2 are connected with the aid of (38) and it yields the formula

C V C RV T VT2 1( , ) ( , ) ~ΨΨΨΨ ΨΨΨΨ= − T , (49)

where R ∈SO( )3 is a rotation operator.

We also need the time derivative of the transformation T , giving

& ( & , ) & & ~ & ~& & &T IΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ= − + + + +⋅ ⋅ ⋅ ⊗ ⊗ ⊗c c c c c1 2 3 4 5Ù Ù Ù Ù Ù ÙΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨd i d i d i d i , (50)

where the coefficients are given in (45). The limit value of the tensor &T is

lim & ( & , )~&

ΨΨΨΨ→= −

0T ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ1

2. (51)

The directional derivative of the term C V1T( , )′ ⋅ΨΨΨΨ ΨΨΨΨ can be written as

C V C V V

C V V V V I V V

V V V V

3 13

3 1 2 3 2 3

1 2 3 3

1

( , , ) D ( , ) ,

( , , ) (~

) ( )( ) (~

) ( ) ( )

(~

) ( )( ) ( ) ( ) (

′ = ′ ∀ ∈

′ = ′ + ′ + ′ + ′ + ′ +

+ ′ ′ + ′ ′ + ′ ′ +

⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅

⊗ ⊗

⊗ ⊗

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ∆ ∆T

S S

S

d id ie j

d ie j

E

c c c c c

c c c c

Ù Ù Ù Ù Ù Ù

Ù Ù Ù Ù Ù Ù

ψ′ + ′⊗ΨΨΨΨ ΨΨΨΨÙ Ù) ( ),c5

SV

(52)

where ⊗S

denotes the symmetric tensor product, and ′ci represents the derivatives of the coefficients (45),

given by

( ) : ( ) ( ), , ,

sin sin cos,

cos sin cos,

cos sin sin.

a b a b b a a b⊗ ⊗ ⊗= + ∀ ∈

′ = − −′ = − − +

′ = + + −

SE

3

1

2

4 2

2

5

3

2

6

3 3 5 8 8

7 8 15

c c

c

ψ ψ ψ ψ ψψ

ψ ψ ψ ψ ψψ

ψ ψ ψ ψ ψ ψψ

(53)

Taking the limit process, we obtain

lim ( , , ) ( ) ( )ΨΨΨΨ→

⊗′ = − ′ + ′⋅0C V V I V3 ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ1

316 S

Ù (54)

We note that the tensor C3 is symmetric since it is the second derivative of the scalar term V T⋅ ′ΨΨΨΨ .

In addition, we need the second time derivatives of the transformation T , giving

&&( && , & , ) ( & ) ( & ) ( && ) ( & ) ( & ) ( && )~

( & ) ( & ) ( && ) ( & )~&

( & )( )

~&&

T IΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ

=′

+ +FHG

IKJ −

′+ +F

HGIKJ +

+′

+ +FHG

IKJ − + +

+ +

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅⊗ ⊗

cc c

cc c

cc c c c

c

1 21 1

2 22 2

3 21 1 2 3

4

2 2

ψ ψ

ψ

Ù Ù Ù Ù Ù Ù

Ù Ù Ù Ù Ù Ù Ù

S

c5 2( && & & ).ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ⊗ ⊗+S

Ù Ù

(55)

Taking the limit process gives

lim &&( && , & , ) ( & & )~&& & &( )

ΨΨΨΨ→⊗= − − +⋅

0T IΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ1

312

13

Ù Ù . (56)

44

Variating the material angular velocity tensor Def. 30 p. 33, ~

: &ΩΩΩΩR R R= T gives

δ~ ~& ~ ~ ~ ~ΩΩΩΩ ==== ΘΘΘΘ ++++ ΩΩΩΩ ΘΘΘΘ −−−− ΘΘΘΘ ΩΩΩΩR R R R R Rδ δ δe j , (57)

whose the vector form becomes with the aid of (43c) p. 42 (Lie brackets)

δΩΩΩΩ ==== ΘΘΘΘ ++++ ΩΩΩΩ ΘΘΘΘR R R Rδ δ& × . (58)

Note that we have used the virtual incremental rotation vector δΘΘΘΘR R∈ matT , which is related with the virtual(total) rotation vector δΨΨΨΨ ∈matTI by the tangential transformation T , given in Eqn (21) p. 31. The variationof the angular rotation vector reads in terms of the total rotation vector

δΩΩΩΩ ====R T C⋅ ⋅+δ δ& ( & , )ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ1 , (59)

where we have utilized Eqn (27a) p. 34. We will use (59) in the linearization of the angular rotation vectorwhen calculating tangent tensors from virtual work forms.

The variation of the spatial angular velocity vector is, respectively

δωωωω θθθθ ωωωω θθθθ ΘΘΘΘR R R R RR= − × =δ δ δ& ( & ) , (60)

that shows a rather strange relation between the material and spatial representations.

The variation of the material angular acceleration tensor ΑΑΑΑR Def. 31 p. 33 reads

δΑΑΑΑ ΑΑΑΑR R R R R R==== ΘΘΘΘ ++++ ΩΩΩΩ ΘΘΘΘ ++++ ΘΘΘΘδ δ δ&& &× × , (61)

and in terms of the total rotations according to Eqn (28a), p. 34, yield

δΑΑΑΑ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨR T C C C= + + +⋅ ⋅ ⋅δ δ δ&& ( & , ) & ( && , ) ( & , )4 1 5d i , (62)

where the tensors C C4 and 5 are defined by the following derivative formulas

C T C T

C I

C I

4 5

4 1 5 3 5 1 5

2 2

5 32

53

2 2

( & , ) & D & & & ( & , ) D & & ,

( & , ) ( )( & ) ( & )( ) ( & ) ( )( & )

(~ & ) ( & ) ,

( & , ) ( & ) ( & & )

&ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ⋅ ⋅ ⋅ ⋅⋅ ⋅

⋅

⋅ ⋅

= =

= + + + + + +

− −

= + +′

⊗ ⊗ ⊗

⊗ ⊗

∆ ∆ ∆ ∆d i d i

d i

and

c c c c c c

c c

c cc

Ù Ù Ù Ù Ù

Ù Ù Ù

Ù Ù

ψ ( & ) ( & & ) ( ) ( & )( & )

( & )( & ) ( & )(~ & ) (

~ & ) & ( & )~&

( )( & & ).

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

Ù Ù Ù Ù Ù

Ù Ù Ù Ù Ù Ù Ù

⋅ ⋅ ⋅

⋅ ⋅ ⋅

+FHG

IKJ + +

+′

+FHGIKJ −

′− + + +

⊗ ⊗

⊗ ⊗ ⊗ ⊗

23 3

13

22 2 1 5

2c c

cc

cc c c cψ ψ

(63)

Taking the limit process, we obtain

lim ( & , ) lim ( & , ) ( & & ) ( & & )ΨΨΨΨ ΨΨΨΨ→ →

⊗= = − + ⋅0 0C O C I4 5ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ and 1

616

Ù Ù . (64)

The tensors C4 and C5 are nonsymmetric tensors in general.

We also need the inverse of transformation T −1 Eqn (21) p. 31 [Cardona & Géradin 1988] and its time de-rivatives, yielding

T I

T I

T I

−⊗

−⊗ ⊗ ⊗

−

= − + − −−

= + + + +

=′

+ +FHG

IKJ +

′+

⋅ ⋅

⋅ ⋅ ⋅ ⋅

12

11 2 3

1 1 21 1

2 22

2 22 2

2

1

2

12

( )sincos

cos sin

cos

~

& ( & , ) ( & ) ( & ) ( & & )~&

,

&& ( && , & , ) ( & ) ( & & ) ( && ) ( & ) ( &

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ψ ψψ

ψ ψ ψψ ψ ψ

ψ ψ

2+

2Ù

Ù Ù Ù Ù Ù

Ù Ù Ù Ù Ù

ΨΨΨΨ,,,,

a a a

aa a

aa ⋅ ⋅

⋅

+FHG

IKJ +

+ + + + + +

⊗

⊗ ⊗ ⊗ ⊗ ⊗

& ) ( && )

( & )( & & ) ( && && & & )~&&

,

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

a

a a

2

2 32 2 12

Ù Ù

Ù Ù Ù Ù Ù Ù

(65)

where the coefficients a ai i and ′ are given by

45

a a a

a

a

a a

1 2

2

4 4 3 2 2

1

2

2 2

2

2 2 2 2

5 2

3 2

2 24 4

2 2

2 2

2 2

2 1 2

3 3 3 3 16 32 16

2 1 2

= − = + − +−

= − −−

′ = + − + −− +

′ = − − − + − + + −− +

′ =

sincos

,cos sin

cos,

cos sin

cos ,

cos sin sin sin cos

( cos cos ),

sin sin cos cos sin cos cos

( cos cos ),

.

ψ ψψ ψ ψ

ψ ψ ψ ψψ ψ ψ

ψ ψ ψψ ψ ψ

ψ ψ ψ ψ ψ ψ ψ ψψ ψ ψ

ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψψ ψ ψ

ψ

-

(66)

Taking the limit process gives

lim ( ) ,

lim & ( & , )~&

,

lim && ( && , & , ) ( & & ) & & ~&&.

ΨΨΨΨ

ΨΨΨΨ

ΨΨΨΨ

→

−

→

−

→

−⊗

=

=

= − + +⋅

0

0

0

T I

T

T I

1

1

1

12

16

16

12

ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

Ù Ù

(67)

Finally, we get for the directional derivative for the operatorT −T

C V T V V

C V V V V I V V

63

1 2 3

1

2

( , ) : D , ,

, ( ) ( )~

,

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ⋅ ⋅ ⋅

⋅ ⋅

= ∀ ∈

= + + + +

−

⊗ ⊗ ⊗

∆ ∆T

6

d ib g d i

E

ΨΨΨΨ a a aÙ Ù Ù Ù Ù(68)

whose the limit is

lim ( , ) = 126

ΨΨΨΨΨΨΨΨ

→0C V V~ . (69)

The above formulas will be utilized in the following sections.

46

3 GEOMETRICALLY EXACT BEAM THEORY

3.1 Virtual Work Forms

In this Section, we apply the principle of virtual work for deriving the equations of Reissner’s beam theory.The same equations of motion may be derived by very different principles and approaches. One of the oldestapproaches is Newton’s second law of motion that is a fundamental principle. In Newtonian mechanics,forces are of two kinds, internal forces and external forces with respect to a corresponding mechanical sys-tem. Moreover, Newtonian mechanics is rather involved with vectorial representation and is sometimesnamed as vectorial mechanics. As we have shown in the previous section, vectors, tensor, and correspondingspaces are fundamental elements described differential geometry. However, virtual displacements and rota-tions are not included in Newtonian mechanics, but rather in Lagrangian mechanics.

In Lagrangian mechanics, forces are divided differently into constraint forces and applied forces. In addi-tion, rich mathematical methods like variational calculus and other mechanical principles are included. Interms of differential geometry, Lagrangian mechanics describes a motion on an event manifold1 with a La-grangian functional on the tangent bundle of the event manifold2. Lagrangian mechanics, and more generallyanalytical mechanics, is well suited for closed mechanical systems, but eligibility for open systems is not soclear. Moreover, the Lagrangian equations of motion as well as Hamilton’s principle have a strong algebraicand analytic nature, but their geometric meaning is not so obvious. Also, a Lagrangian functional blacks outthe interdependency of variables and their computational structure, which is needed in the finite elementmethod.

We think that there is a need for another type of mechanics between Newtonian and Lagrangian mechanics,as it is sometimes done. This mechanics could be named d’Alembertian mechanics3 where the principle ofvirtual work is its cornerstone. Here we include also inertial forces in the virtual work form. The virtualwork may be viewed as a linear form on the tangent field-bundle T tC 0 , see Def. 40 p. 38. This field-bundleis also a tangent bundle of the placement manifold at fixed time. We give definitions for the virtual work inthe finite-dimensional and infinite-dimensional cases..

49 Definition for virtual work on constraint point-manifold: Virtual work on the tangent point-bundle T tM 0

at the fixed time t t= 0 and the place vector x x( ) :t t0 0 0= ∈M is defined as a linear form by

δ δW := ⋅f x ,

where the virtual displacement δx x∈T 0M , and the force vector f f x x: ( , )= ∈ ∗t T0 0 0M which belongs to the

cotangent point-space. The tangent point-space Tx0 M is defined in Def. 36 p. 36. ‡

Note that we have restricted to holonomic constraints. Moreover, the virtual displacement δx x∈T 0M trulyoccupies the subspace of En which is a tangential space at the base x0 with dimension d . The dimension dis also the dimension of the constraint point-manifold, see Def. 34 p. 35.

Forces may be classified into external, internal (like in Newtonian mechanics) and additionally into inertialforces. We name the terms like −m&&x as inertial forces while the terms like m&&x are called accelerationforces. This may clarify the notational mess in the literature of applied mechanics that only is a controversyon words. We note that an inertial force may be regarded as an effective force. Indeed, if an external force isacting on a particle, which is otherwise free, then the inertial force may be regarded as the reaction force,hence the force equilibrium in the dynamical sense is achieved in this mechanical system.

1An event manifold is a time-placement manifold which is also a constraint manifold, i.e. the time and placement vari-ables satisfy all constraints.2This tangent bundle is called time-state manifold.3Due to Jean Le Rond d’Alembert often a misunderstood scientist who regarded mechanics as much a part of mathe-matics as geometry or algebra.

47

constraint manifold Mt0

tangent space Tx0Mplace vector x0

virtual displacement δ x

force vector f

virtual work δ δW = ⋅f x

Fig. 14 A geometric representation of the virtual work.

Another way to classify forces is used in Lagrangian mechanics where forces are separated into constraintand applied forces. Constraint forces can be verified with the aid of the virtual work since they are workless.Then we may denote that constraint forces occupy f x

con ∈ ∗ ⊥T0M that is orthogonal with Tx0M via duality

pairing. Hence, we may neglect the constraint forces in the virtual work forms.

50 Definition for virtual work on field-manifold: The virtual work on the tangent field-bundle T tC 0 at thefixed time t t= 0 and the place field x x0 0 0: ( )= ∈t tC is defined as an integral over the domain of the body B

δ δW VB

: d= ⋅z f x ,

where the virtual displacement field δx x∈T 0C , and the force field f f x x= ∈ ∗( , )t T0 0 0C which belongs to the

cotangent field-space. The tangent field-space Tx0C is defined in Def. 41 p. 38. ‡

Similarly as in the finite-dimensional case, the same classification may be realized. Especially, constraintforces occupy f x

con ∈ ∗ ⊥T0C that is orthogonal with the tangent field-space Tx0C via duality pairing.

51 Principle of virtual work: The principle of virtual work states that at a dynamical equilibrium, the virtualwork with respect to any virtual displacement, at the fixed time t t= 0 and the place vector x0 , vanishes, i.e.

δ δ δW T Tt t= ∀ ∈ ∀ ∈00 0

, or x xM C ,

where the alternatives correspond the virtual displacements on the tangent space of the constraint point-manifold, Tx0 M , and on the tangent space of the constraint field-manifold, Tx0C , see Fig. 14. ‡

3.1.1 Weak Balance Equations for Continuum

In this section, we derive the dynamic balance equation of the linear and rotational momentum in a weakform for Reissner’s beam. We will rather closely follow the paper [Cardona & Géradin 1988] for showingthat Reissner’s beam formulation is also a stress resultant formulation, consistent with continuum mechanicsat a resultant level.

We could choose the virtual work of internal forces in many ways by taking different stress tensors and theircorresponding virtual work conjugates. Although these work pairs are equivalent, the resulting computa-tional beam models may give slightly different results. The reason for this arises from material constitutiveequations. If we choose the same constant material stiffness tensors for the virtual works (which means thelinear constitutive relations between the strain and stress quantities), we get the virtual works that are notequal. This is because the connections between different stress tensors include deformation gradients andtheir Jacobians which will complicate the equivalent constitutive relations.

48

Strain measures usually depend on logarithmically, linearly, or quadratically on the principal stretches. Ifstrains are infinitesimal, all these strain measurements may be adjusted as equal. For the virtual work pair asa force quantity, we select the first Piola-Kirchhoff stress tensor. This selection is very popular for the geo-metrically exact beam theory, since the work pair for the first Piola-Kirchhoff stress tensor is the virtualdeformation gradient yielding rather a simple formulation. Additionally, this strain measure depends linearlyon the principal stretches and is therefore more closely connected with logarithmic (natural) strains thanLagrangian strains where the dependency is quadratic.

52 Definition for the first Piola-Kirchhoff stress tensor: The first Piola-Kirchhoff stress tensorP x X∈ ⊗T TB B0 is usually defined by relation

p P Nd da A= ⋅ 0 ,

where N X0 0∈ ∗T B is the unit normal covector in the cotangent of the material placement, dA and da are

differential areas in the material and spatial placements, and p x∈T B is a stress vector, see Fig. 15. ‡

The stress vector p x∈T B lies in the tangent space of the spatial placement B . We note that the materialbasis vectors , , t t t1 2 3 span the tangent space TxB . The first Piola-Kirchhoff stress tensor P is an exampleof the two-point tensor such that its stress vector belongs to the spatial vector space, its normal vector to thematerial vector space, and its differential area to the material placement.

We could write the first Piola-Kirchhoff (PK-I) stress tensor by a linear combination of stress vectors andbasis vectors, as

P T E T E T E x X= + + ∈⊗ ⊗ ⊗ ⊗1 1 2 2 3 3 0T TB B , (70)

where T T xi i= ( ) is the spatial stress vector acting on the tangent space of spatial placement TxB and Ei isthe material basis vector on the material placement, compare Fig. 15. The material and spatial place vectorsare related by a regular deformation mapping

x X X= =−χχχχ χχχχ χχχχt t01( ) : ( )d i , (71)

where we have used the same symbol for convenience, see Def. 11 p. 21. In the placement of Reissner’sbeam, the material basis vector Ei t s( , ) and the spatial stress vector Ti t s( , ) depend on the length parameters and the time t that are independent variables. The vector Ti t s( , ) adheres to unknown variables: transla-tional and rotational displacement fields, through the deformation mapping χχχχ t ( ):X B B0 → that is a pointmapping.

material placement B0

spatial placement B

B0

B

N0

P N⋅ 0 dA

n0pda

dA

da

Fig. 15 An interpretation of the first Piola-Kirchhoff stress tensor, note that P N x⋅ ∈0 dA TB although itis drawn on the material placement.

49

Before giving the balance equations of continuum, we define gradient and divergence operators. This isnecessary since in the literature of applied mechanics, the gradient and divergence operators are defineddifferently.

53 Definitions for gradient and divergence: We define the gradient of the tensor T ∈T at the place pointx ∈V as a directional derivative in the direction u ∈V by formula

∇T x uT x u

T x u( )d ( )

dD ( )⋅ ⋅= +

==η

η η 0,

where the gradient ∇T is an element of the tensor space T V⊗∗ . Moreover, we may define ∇T T: D= . A

gradient may be taken with respect to the spatial or material place vector where the material gradient is de-noted by 0∇ . The divergence operator of the tensor T ∈ ⊗T W at the point x ∈V is defined as a contractionby the double-dot product of the gradient:

∇ ∇⋅ = ∈T T I: : T ,

where the identity I ∈ ∗⊗W V . ‡

For example, the deformation gradient F can be defined as the material gradient of the deformationx X: ( )= χχχχ t by the formula F x X: ( )= 0∇ . However, the deformation χχχχ t :B B0 → is more like a point mappingthan a vector. Hence, the deformation gradient is usually defined by the tangent of the deformation:

F XX x X: ( ) *= ∈ ⊗T T Tχχχχ B B0 . (72)

This is not a contradiction because in Def. 53 we have assumed that T is a tensor (or a vector in this case).

The equations of motion of continuum with boundary conditions, in the terms of the first Piola-Kirchhoffstress tensor, can be written as

0∇

∂

∂

⋅

⋅

+ =

=

UV|W|

=

=

∗ ∗

P b x

PF FP

P N T

x x

ρ

σ σ

0

0 0

0

&&,

,

,

in

on

on

0B

B

B u

(73)

where b N T x, , , ,ρ σ0 0 are the body force vector, the density of the material body, the normal vector of the

traction boundary, the given traction vector and the given placement vector, respectively. The base points

are given in the material placement B0 , but they occupy in the tangent spaces of the spatial placement

TxB . E.g. b b X x: ( ( ))= ∈χχχχ t T B and PF X x x∗ ⊗∈χχχχ t T T( )b g B B .

The virtual work can be decomposed into three terms: external, internal and inertial virtual works with theequation

δ δ δ δW W W W= − +ext int inert , (74)

where the subscripts correspond to ‘external’, ‘internal’ and ‘inertial’. In the virtual work of internal forces,the minus sign indicates that internal forces work against the virtual displacements. Additionally, the inertialvirtual work δWinert includes the minus sign inside its form. Sometimes it is convenient to avoid additionalminus signs by introducing the virtual work of acceleration forces by the formula

δ δW Wacc inert:= − . (75)

When we apply the principle of virtual work, the governing equations are derived from the equation

δ δ δ δW W W WaccA ext int accB= − +b g (76)

where δWaccA is the virtual work of acceleration forces, which depends on the acceleration vector, andδWaccB corresponds to the terms of the acceleration virtual work like the virtual work of centrifugal forces.

50

Next, we give the form of virtual work and we show that the principle of virtual work satisfies the equationsof motion (73). Let the virtual work be stated as

δ δ δ δ ρσ σW V A V V= + − − =zz z zx b x T F gP x xg g R g

, d , d d , && de j b g∂

δBB B B00 0 0

0 0: , (77)

where the Lie derivative of the deformation gradient has been calculated in (37) p. 41 . In Eqn (77), the firsttwo terms correspond to the external virtual work δWext , the third term to the internal virtual work δWint , andthe last term to the acceleration virtual work δWacc . Note that the term of the internal virtual work δRF gP:includes also the spatial metric tensor g x x∈ ∗ ⊗ ∗T TB B and can be simplified into

( ) ( ) ( ) ( ) (~

) ( )

( ) ( ) (~

) (~

)

) ( ) (~

) ,(

δ

∇

∇ ∇

RF gP F gP F gP

x gP g F P gF P

P g x g x P g PF FP

: : :

: : :

:

= −

= − −

= − − −

∗

∗ ∗ ∗⋅ ⋅ ⋅

δ δ

δ δ δ

δ δ δ

θθθθ

θθθθ θθθθ

θθθθ

0 12

0 0 12

e jd i

(78)

where we have used the skew-symmetry of δ δ δ~ ~ ~θθθθ θθθθ θθθθT = − = − ∗g g1 and the relations to the gradient and diver-

gence operators, Def. 53.

By substituting the result (78) into the virtual work (77), we get after the use of the divergence theorem

δ δ δ ρ δ σW V A= − + + −FH IK + − =∗ ∗ ⋅ ⋅z z12

00

0 0

0PF FP g x P b x x T P Ng g

d i:( ~) , && d , dθθθθ ∇

∂B B

. (79)

Now comparing the above equation with the equations of motion (73), we found that the principle of virtual

work (77) satisfies all these equations of motion with a kinematically admissible virtual displacement. The

kinematically admissible virtual displacement field δx fulfills the essential boundary conditions (73d). We

note that the term δRF gP: also induces the balance equation of moment of momentum (73b) but the term

δF gP: does not. Also, the virtual rotation tensor δ~θθθθ depends on the virtual displacement δx in continuum,

but if the beam kinematic assumption is utilized the virtual rotation tensor δ~θθθθ is an unknown variable itself.

3.1.2 Beam Kinematics

Let , , , Oe e e e1 2 3 be a spatially fixed Cartesian frame and let , , , OE E E E1 2 3 be a material Cartesian frame,where the basis vector E1 coincides with the tangent of the center line of beam. We will assume that theplane perpendicular to the beam center line in an undeformed placement remains the undeformed plane atthe deformed body, i.e. Timoshenko-Reissner beam hypothesis is applied, see Fig. 16. We note that trans-versal shears are allowed, but no out-of-plane warping effects are included. This Timoshenko-Reissnerbeam assumption is also a holonomic field-constraint.

Let , , x x x1 2 3 be co-ordinates of a spatially fixed frame and let , , X X X1 2 3 be co-ordinates of a materialframe then the material point X in the deformed placement can be given with the aid of the spatial (float-ing) frame , , , Ot t t t1 2 3 as

x x t tt s t s X t s X t s s X X L A, , , , , ( , , ) [ , ]b g b g b g b g= + + ∈ ×c 2 2 3 3 2 3 0 , (80)

where s is the length parameter of the beam and xc determines the beam center line. The spatial frame , , , Ot t t t1 2 3 coincides with the material frame at an initial placement, i.e. t Ei i i= =, , ,12 3. Due to sheareffect, the tangent of center line at current placement d dxc s not necessarily coincide with the cross-section normal vector t1 , see Fig. 16. Eqn (80) realizes the Timoshenko-Reissner beam hypothesis, hencethe above equation is the parametrization of the constraint field-manifold that arises from the beam hypothe-sis. The beam spatial position x( , )t s can be viewed as a one-dimensional solid, i.e. an internally one-dimensional vector bundle, where this vector bundle is the cross-section plane of the beam, and time t isconsidered as an independent parameter.

The beam placement can be viewed as a Cartesian product E3 3× SO( ) where E3 refers to the translationaldisplacement and SO( )3 to the rotational displacement. The elements of SO( )3 are rotation operators Rthat are parametrized by the rotation vector ΨΨΨΨ ∈ matTR or ψψψψ ∈ spatTR .

51

e1

e2

ψinitial geometry

current geometry

E t2, 2

E t1, 1

ψ init

ψ ψ+ init

t1

t 2

Fig. 16 Beam kinematics in the plane case.

54 Definition for material placement of beam: The material placement of Timoshenko-Reissner beam at thetime t = 0 can be written with the aid of the initial rotation operator R0( )s as

B03

2 0 2 3 0 3 2 30 0: , ; ( , , ) [ , ]= ∈ = + + ∈ ×x x R e R es t s X s X s s X X L Ab g b g b g b go tE c ,

where L A, is the beam length and the cross-section area, respectively. The initial placement B0 is a con-straint field-manifold with a vector bundle. The global basis vectors and the material basis vectors are re-lated by E R ei is s i( ) ( ) , , ,= =0 12 3. The tangent space is

T X X X X X XX X E E EB03

1 1 2 2 3 3 1 2 3: ,= ∈ + + ∈E R, ,n s . ‡

55 Definition for spatial placement of beam: Correspondingly, the spatial placement of the beam can bedefined as

B : , , , , ; ( , , ) [ , ]= ∈ + + ∈ ×x x R E R Et s t s X t s s X t s s s X X L Ab g b g b g b g b g b go tE32 2 3 3 2 3 0c ,

where the rotation operator R( , )t s depends on the rotation vector ΨΨΨΨ( , )t s T∈ mat R . The spatial basis vectorsand the material basis vector are related by t REi i= . Hence, we may interpret the spatial placement B as acombination of the translation and rotation displacements, E3 3× SO( ) . The tangent space is given by

T X Xx x x R E E xB B: ( ), ,= ∈ + + ∈δ δ δE3

2 2 3 3c kinematically admissible variationn s . ‡

3.1.3 Virtual Work for Reissner’s Beam

In this section, we derive the virtual work for Reissner’s beam when the kinematic assumptions are applied.

The virtual displacement for the spatial placement of Reissner’s beam, Def. 55, at the fixed time t t= 0 withthe place vector x x0 0: ( )= t and at some point of the parameter s can be written with the material rotationrepresentation

δ δ δx x R ER x= + ∈c

~ΘΘΘΘ T

0B , (81)

where we have denoted

E E E X:= + ∈X X T2 2 3 3 0B (82)

for simplicity. The variation of the rotation operator is given in Eqn (36a) p. 41. Note that the virtual rota-

tion tensor occupies δ ~( )ΘΘΘΘR R X X∈ = ⊗ ∗

matT SO T T3 0 0B B .

52

We could also represent the virtual displacement in the terms of the spatial rotation by formula

δ δ δx x RER x= + ∈c

~θθθθ T

0B , (83)

where the spatial rotation tensor now belongs to the tensor space δ~( )θθθθR R x x∈ = ⊗ ∗

spatT SO T T3 B B . However,

we will choose the material representation of rotation because we avoid Lie derivatives in the linearization

of the virtual work forms which would arise in the spatial representation. That would complicate the lineari-

zation procedure considerable.

We also need the time derivatives of the spatial place vector, yielding

& &~

,

&& &&~ ~ ~

,

x x R E

x x R E R E

R x

R R R x

= + ∈

= + + ∈c

c

ΩΩΩΩ

ΩΩΩΩ ΩΩΩΩ ΑΑΑΑ

T

T

B

B(84)

where ~

, ~ ( )ΩΩΩΩ ΑΑΑΑR R R X X∈ = ⊗ ∗matT SO T T3 0 0B B are the angular velocity and angular acceleration tensors.

Now we could substitute the virtual displacement vector δx , Eqn (81), and the acceleration vector &&x , Eqn(84b), into the virtual work of acceleration forces δWacc which is the last term in Eqn (77) p. 50, giving

δ δ δ ρ

δ ρ δ

W V

A s sL L

acc c c

c c

= + + +

= + +

zz z⋅ ⋅

x R E x R E R E

x gx GJ G J

R R R Rg

R R R R

~, (&&

~ ~ ~) d

( && ) d (~

)d

ΘΘΘΘ ΩΩΩΩ ΩΩΩΩ ΑΑΑΑ

ΘΘΘΘ ΑΑΑΑ ΩΩΩΩ ΩΩΩΩ

0

0

0B(85)

where we have been used the center line condition:

(~

d ) (~ ~

)d~

E E E 0A X X AA Az z= + =2 2 3 3 , (86)

and the identity (43f) p. 42 that is ~~~ ~~~ , ,abab baab a b= ∀ ∈E3 . Note that GJ G J R XΑΑΑΑ ΩΩΩΩ ΩΩΩΩ,~

( )∈ =∗ ∗matT T B0 that are

elements of the material covector space of rotation, see Def. 28 p. 32. Moveover, we have denoted the iner-

tial tensor J X X∈ ⊗ ∗T TB B0 0 as

J E E=

=+

−−

L

N

MMM

O

Q

PPP

⊗∗

zJ

J

X X

X X X

X X X

A

ij i j

ij

A

,

d .22

32

32

2 3

2 3 22

0

0 0

0

0

ρ(87)

If the principal axes of the inertial tensor J are parallel to the basis vectors E E2 3 and , then the matrix ofthe inertial tensor [ ]Jij is diagonal. In following, we will use this assumption for simplicity, having for thematrix [ ]Jij

J J J J J J Jij = = +diag( , , ),1 2 3 1 2 3, (88)

where subscripts correspond to the basis vectors , , E E E1 2 3 . We note the correspondence between the vir-tual work δWacc , Eqn (85), and the equations of motion for a rigid body, i.e. the Newton-Euler equations.The formula (85) can be viewed as the lengthwise parametrized equations for the rigid body without exter-nal forces.

As for the virtual work of acceleration forces, we could write the virtual work of external forces δWext sub-stituting the virtual displacement δx , Eqn (81), into the first two terms of Eqn (77) p. 50, giving

δ δ δW s sL L

ext c= +⋅ ⋅z zx n MR Rd dΘΘΘΘ , (89)

where the external force vector n and the external material moment vector M R are denoted by

n g b X g T X

M G ER b X G ER T XR

: ( ) d ( )d ,

:~

( ) d~

( )d ,

= +

= +

z zz z

A r

A r

A A

A A

σ

σ

∂

∂

T T(90)

53

where b is the body force vector and Tσ is the traction vector on the boundary of the material placement.

The external moment vector M R can be viewed as an element of the material covector space of rotation

matTR∗ , see Def. 28 p. 32. This is because the external moment vector M R is the work conjugate of the virtual

incremental rotation vector δΘΘΘΘR .

Next, we derive the virtual work of internal forces δWint , the third term in Eqn (77) p. 50, using the beamhypothesis. The deformation gradient F , Eqn 72 p. 49, states, after substituting the beam kinematics as-sumptions (80) p. 50,

F x R E E t E t E

x R E t E R

R

R

= ′ + + +

′ + −

⊗∗

⊗∗

⊗∗

⊗∗

c

c

= +

( )~

( )~

,

s

s

ΚΚΚΚ

ΚΚΚΚ

d id i

1 2 2 3 3

1 1

(91)

where the material curvature tensor ~ :ΚΚΚΚ R R R= ′T that is defined in Def. 32 p. 34. The Lie variation of the

deformation tensor is correspondingly

δ δ δ

δ

R R

R R x X

F R R F R R x E E E

x R R x R E E

= = ′ + −

′ − ′ + ∈

⊗∗

⊗∗

⊗∗

T Tc

cT

c

=

d i d ie j

~

~ ~.

ΚΚΚΚ

ΘΘΘΘ ΚΚΚΚ

1 1

1 0δ δ T TB B(92)

Now the term δRF gP: (the virtual work of internal forces) in Eqn (77) p. 50 can be written with the aid ofEqns (70) and (92), yielding

δ δ δ

δ δ

δ δ

W V

V V

s sL L

int cT

c

Tc

Tc

T T

Tc

Tc

= ′ − ′ +

= ′ − ′ +

= ′ − ′ +

zz zz z⋅ ⋅

x R R x R E T

R x R x R T E R T

R x R x N M

R R g

R G R G

R R R

~ ~, d

~, d

~, d

(~

) d d ,

ΘΘΘΘ ΚΚΚΚ

ΘΘΘΘ ΚΚΚΚ

ΘΘΘΘ ΚΚΚΚ

δ

δ

δ

1

1 1

0

0 0

B

B B

(93)

where L corresponds to the initial length of the beam.

The material force vector N and the material moment vector M R are denoted by formulas

N G R T

M G ER T

X

R X

: d ,

:~

d ,

= ∈

= ∈

zz

∗

∗

T

T

1 0

1 0

A T

A T

A

A

B

B(94)

where G X X∈ ∗⊗

∗T TB B0 0 is the material metric tensor. The material moment vector of internal forces can beviewed as an element of matTR

∗ -space that is the material covector space of rotation, see Def. 28 p. 32.

The work conjugate of the material force vector N in Eqn (93) is the variation of the material strain vectorΓΓΓΓ , defined by the formula, and its variation reads

ΓΓΓΓ

ΓΓΓΓ ΘΘΘΘ

: ,~

,

= ′ − ∈

= ′ − ′ ∈

R x E

R x R x

X

R X

Tc

Tc

Tc

1 0

0

T

T

B

Bδ δ δ(95)

where we have used the skew-symmetry: δ δ~ ~ΘΘΘΘ ΘΘΘΘR R X XT = − ∈ ⊗ ∗T TB B0 0 which is also an element of the mate-

rial tensor space of rotation matT SOR ( )3 . The components of the material strain vector ΓΓΓΓ = Γ i iE are shown in

Fig. 17, in the plane case.

Now, we can give the internal virtual work in its material representation in addition to its spatial form

δW s sL L

int = + = +⋅ ⋅ ⋅ ⋅z zδ δ δ δΓΓΓΓ ΚΚΚΚ γγγγ κκκκN M n mR R R R R Rb g b gd d , (96)

where δR corresponds Lie variation with respect to the rotation operator R.

54

E e1 1,

Γ2

Γ1

ds = 1

t1

t 2

E e2 2,

1

γ 1

γ 2

Fig. 17 A geometric interpretation for the material strain components and for the spatial strain compo-nents, in the plane case.

The material and spatial vectors are related by the push-forward operator with the rotation operator

R X x∈Liso ( , )T TB B0 , defined by

γγγγ ΓΓΓΓ ΓΓΓΓ

κκκκ ΚΚΚΚ ΚΚΚΚ

: ,

,

: ,

: .

= = ∈

= = ∈

= = ∈

= = ∈

−∗ ∗

−∗ ∗ ∗

R R

R R

n R N R N

m R M R M

x

R R R x R

x

R R R x R

>

>

>

>

T

T T

T

T T

B

B=

B

B=

spat

spat

(97)

The components of the spatial strain vector γγγγ = γ i ie are shown in Fig. 17. The material and spatial straincomponents are connected by the relation Γ i i i it e= γ .

The inverse adjoint of rotation operator R X x−∗ ∗ ∗∈Liso ( , )T TB B0 can be written in terms of the metric tensor,

giving a simpler formula for the spatial internal force vector n , for example

R gRG

n g T x

−∗ −

∗

=

= ∈z1

1

,

d .A TA

B(98)

It is interesting to find out that the Lie variation of the spatial strain vector γγγγ and the material strain vectorΓΓΓΓ are related by formula

δ δ δ δR xR R R R Rγγγγ γγγγ γγγγ ΓΓΓΓ= = = ∈>

<d i d iT TB . (99)

So we get the virtual work term (δ (δR n Nx Xγγγγ ΓΓΓΓ⋅ ⋅=) )T TB B0 . The Lie variation appears in the spatial represen-

tation, and in the material representation the Lie variation is identical to the ordinary variation.

At this point, we should realize that we are searching for the solution of the beam displacement field xc ( )sand its rotation field R( )s that corresponds to the resultant form of the internal virtual work (96). Thisplacement manifold of Reissner’s beam may be identified by ( , ) ( )x Rc ∈ ×E3 3SO at any point of the beam.A stress tensor which gives equal internal force and moment vectors is a valid stress tensor. Hence, theequations of motion for continuum (73) p. 49 realizes only by its resultant form. This also yields the consti-tutive relation in terms of resultants quantities.

55

3.2 Constitutive Relations

We introduce the constitutive relations of Reissner’s beam and show how these relations are connected intocontinuum mechanics and a strain energy function.

Let Wstr be a strain energy function per unit volume of a hyperelastic material. The first Piola-Kirchhoffstress tensor P can also be defined from the energy function Wstr ( )F [Stumpf & Hoppe 1997] by

P gF

F x X:( )

= Œ-

ƒ1

0

∂∂

WT Tstr B B . (100)

We also assume that the strain energy function is frame-indifferent under the orthogonal transformationF QF+ = by obeying the identity

W W Wstr str str( ) ( ) ( )F QF F+ = = . (101)

This means that the strain energy function Wstr is invariant under rigid body rotation. If we set Q R= T , thenwe have via a pull-back operator the strain energy function purely in the material domain, which provides adifferent stress quantity.

The Lie variation of the energy function Wstr( )F with respect to the rotation operator R can be written withusing the above relation as

δ∂

∂δ δR R R

FF

F gP FWW

strstr= = ∈

( ): : R (102)

that is equal to the virtual work of internal forces δWint , see Eqn (77) p. 50. We get the same result forδWstr

T( )R F . The first Piola-Kirchhoff stress tensor as well as tensor gP is a two-point tensor defined onmaterial and spatial placement manifolds. We use Lie variation, push-forward and pull-back relations toderive one-point stress tensor that corresponds Eqn (102), yielding

δ δ( δ(

δ( δ(

R gP R R F R gP R F

GR P R F GR P R F I

WstrT T

T T T T

= =

= = −

∗: :

: :

) ( ) )

( ) ) ( ) )

d i(103)

where we have used R g GR∗ = T according to the definition of the transpose operator, see Def. 12 p. 23.

The Lie variation of strain energy function (103) introduces new material stress and strain tensor, defined by

ΣΣΣΣ : ,

: .

= ∈

= − ∈

∗⊗

⊗∗

GR P

H R F I

X X

X X

T

T

T T

T T

B B

B B

0 0

0 0

(104)

The material stress tensor ΣΣΣΣ = ∗ ⊗Σ ij i jE E , as well as, its work conjugate strain tensor H E E= ⊗ ∗Hij i j are bothunsymmetrical tensors and are not named in continuum mechanics. The material strain tensor H for Reiss-ner’s beam can be written by substituting the relation of the deformation gradient (91) p. 53, giving

H R x E E E

E E

R

R

= ′ + −

= +

⊗∗

⊗∗

Tc ( )

~

~,

s ΚΚΚΚ

ΓΓΓΓ ΚΚΚΚ

1 1

1

d id i

(105)

where ΓΓΓΓ is the material strain vector, Eqn (95a), and ~ΚΚΚΚ R is the material curvature tensor, Def. 32 p. 34.

We note that the rotation operator R is not, in general, equal to the rotation tensor of the polar decomposi-tion. However, if the shear components Γ Γ2 and 3 , as well as, the torsion curvature Κ 1 vanish, then R FT isequal to the right (material) stretch tensor U , which is a symmetric tensor. Thus in this particular case( Κ Γ Γ1 2 0= = =3 ), the material strain tensor H can be identified with the right stretch strain tensor, and thesymmetric part of the stress tensor ΣΣΣΣ can be identified with the Biot stress tensor, see [Ogden 1984].

Let us consider a following linear constitutive relation between the stress components of the tensor ΣΣΣΣ andthe strain components of the tensor H given by

ΣΣΣΣ = C H: (106)

where the elasticity tensor C X X X X∈ ∗ ⊗ ⊗∗

⊗T T T TB B B B0 0 0 0 is a fourth order material tensor.

56

At a cross-section of Reissner’s beam, we introduce simple constitutive relations in the component form

Σ Σ Σ11 11 21 21 31 31= = =EH GH GH, , , (107)

where E denotes Young’s modulus and G the shear modulus.

The constitutive relations (107) correspond to the engineering strains and stresses. We note that the vector

Hi i1E ER= +ΓΓΓΓ ΚΚΚΚ~ according to Eqn (105), and thus we could express the material strain vector Σ i i1E∗ as

Σ i i E G G1 1 1 2 2 3 3E E E E E E E ER∗ ∗

⊗∗ ∗

⊗∗ ∗

⊗∗= + + +⋅d i d iΓΓΓΓ ΚΚΚΚ~ . (108)

If we compare Eqn (70), p. 48, with the second Piola-Kirchhoff stress tensor and the material stress tensorΣΣΣΣ (104a), we get the strain vector at the cross-section

Σ i i1 1E GR T∗ = T . (109)

Now we could substitute the above equation into the formula of the internal force vector (94a) p. 53 thatyields after integrating over a cross-section and using the constitutive relation (108)

N E E E E E E E X= = + + ∈∗ ∗⊗

∗ ∗⊗

∗ ∗⊗

∗ ∗z Σ i i

A

A EA GA GA T1 1 1 2 2 3 3 0d d iΓΓΓΓ B , (110)

where we have used the center line condition (86) p. 52.

Similarly, we may derive for the internal moment vector (94b) p. 53, yielding

M G E E E E E E E E

G E E E E E E E E

E E E E

R R

R

R

= + + +

= + +

= + ∈

⊗∗

⊗∗

⊗∗

⊗∗

⊗∗

⊗∗

∗⊗

∗ ∗⊗

∗

⋅

⋅

⋅

zz~ ~

d

(~ ~

d )

, , , ,

E G G A

E G G A

GI EI

A

A

1 1 2 2 3 3

1 1 2 2 3 3

11 1 1 2 3

d i d i

d i

d i l q

ΓΓΓΓ ΚΚΚΚ

ΚΚΚΚ

ΚΚΚΚ

T

αβ α β α β

(111)

where in the second line we have utilized the condition (86), p. 52, and the skew-symmetry of tensor ~E , i.e.~ ~E ET = − . The component matrix of the moment of inertia tensor [ ]I ij at the cross-section can be computed

by

I

X X

X X X

X X X

Aij

A

=+

−−

L

N

MMM

O

Q

PPPz22

32

32

2 3

2 3 22

0 0

0

0

d . (112)

The moment of inertia is the second moment of area. If the principal axes of the moment of inertia tensorare parallel to the basis vector , E E2 3 , then the matrix [ ]I ij is diagonal. In following, we will use this as-sumption, for simplicity.

We introduce a linear constitutive relation in the material representation between the material strain andcurvature vectors and the internal force and moment vectors by writing

N C

M CR R

==

n

m

ΓΓΓΓΚΚΚΚ,

,(113)

where the constitutive tensors in the material frame are

C E E

C E E

nn n

mm m

= =

= =

∗⊗

∗

∗⊗

∗

C C EA GA GA

C C GJ EI EI

ij i j ij

ij i j ij

, diag ,

, diag .

2 3

2 3

b gb g (114)

Here, EA is the axial stiffness, GA GA2 3 and are the shear stiffnesses with respect to the cross-section in theprincipal directions, GJ is the torsion stiffness and EI EI2 3 and are the principal bending stiffnesses. Thecoefficients, which depend on the shear modulus G , can be adjusted such that they include some secondarywarping effects. The cross-section areas A A2 3 and include also possible shear coefficient factors, and thetorsional moment of inertia J is the moment of inertia according to Saint-Venant torsion theory, instead ofthe polar moment of inertia I I IP = +2 3 .

57

3.3 Total and Updated Lagrangian Formulations

Next, we consider a total Lagrangian updating formulation for a material rotation vector and compare it withan updated Lagrangian formulation. We also discuss difficulties about an Eulerian formulation. We exploitonly the material representation, but the spatial formulation may be considered similarly.

The updating formulations of the material rotation vector can be divided into three classes [Cardona &Géradin 1988]:1) The total Lagrangian formulation where the total rotation vector ΨΨΨΨ ∈matTI is the unknown variable2) The updated Lagrangian formulation where the updated rotation vector ΨΨΨΨR Rref refmat∈ T is the unknown

variable3) The Eulerian formulation where the incremental rotation vector ΘΘΘΘR R∈matT is the unknown variable

In total Lagrangian formulation, we have an updating procedure similar to a displacement vector

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ( ) ( ) ( )mat

i i i T+ = + ∈1 ∆ I , (115)

where the rotation vector change ∆ΨΨΨΨ ( )i always occupies the vector space matTI since the base pointI ∈SO( )3 remains fixed. The superscripts in brackets correspond to iterative steps.

We may consider the total Lagrangian updating procedure as the linear space updating procedure. This for-mulation preserves the path-independent property (in the static case) and can be regarded as a consistentupdating formulation. The major drawback is the singularities at the rotation angle ψ = 2π and its multiples,but we overcome this difficulty by the complement rotation vector with the change of parametrization, seeSection 2.2. We will study this updating procedure in following.

The updated Lagrangian formulation is closely related with the total Lagrangian, but the base pointR ∈SO( )3 is occasionally updated. The rotation angle ψ ref ref=|| ||ΨΨΨΨ should never exceed 2π . Usually, weupdate the base point at every incremental step in static analysis, and at every time step in transient dynamicanalysis. The updated Lagrangian procedure reads

YYYY YYYY YYYY

YYYY YYYY

ref ref mat

refnew

refold

ref ref

ref, at every iteration,

when updating the base point,

( ) ( ) ( )

( ) ( )exp~

, ,

i i i

n

T+= + Œ

= =

1

0

∆ I

R R 0e j(116)

where the second formula corresponds to updating the base point which we occasionally execute. The up-dated Lagrangian formulation does not preserve the path-independent property (in the static case). This isbecause the new base point R ref

new depends on the previously computed solution ΨΨΨΨref( )n and then the error of

the updated rotation vector ΨΨΨΨref( )n causes a cumulative error to the base point R ref

new . The updated and totalLagragian formulations are illustrated in Fig. 18.

initial placement B

updated Lagrangianformulation

reference placements Bref

total Lagrangianformulation

initial placement B current placement B

current placement B

Fig. 18 A Geometric representation of the total and updated Lagrangian formulations.

58

The Eulerian updating formulation for material rotation is somehow a more complicated procedure althoughthe updating formula can be written with rather a simple formula

R R R( ) ( ) SO(3) at every iterationi i i+ = ∈1 exp

~ ( )ΘΘΘΘe j . (117)

The above updating formula is a nonlinear procedure and utilizes the property of Lie group and its Lie alge-bra. In static analysis, we could employ the Eulerian formulation where there are no angular velocity andangular acceleration vectors, and therefore no time integration procedure. Algebra with rotation vectors,angular velocity and acceleration vectors or corresponding skew-symmetric tensors are more involved, andcan be properly computed by the Lie group methods of differential equations, see e.g. [Zanna 1999] and[Iserles & Nørsett 1999].

The Lagrangian formulations (total and updated) are geometrical integration methods where the constraintmanifold is parametrized. The total rotation vector, likewise the updated rotation vector, forms a localparametrization of the rotation manifold SO( )3 via the exponential mapping, but the incremental rotationvector ΘΘΘΘR R

( ) ( )i iT∈mat , at each iteration step ( )i , belongs to a different vector space. In Section 2.4, we calledquantities such as incremental rotation vectors by spin vectors. The spin vectors have a vector character at apoint, but not locally on the contrary to the total and updated rotation vectors which are vectors in localsense, i.e. they are a local parametrization of the rotation manifold.

We note that the direct application of the incremental rotation vector ΘΘΘΘR with standard time integrationmethods yields serious problems: adding quantities which belong to the different tangent space. It is wellknown that the spatial form of Newmark time integration scheme is improper for time integration purposes,but the material form of (Newmark) time integration scheme has the same unwanted quality, see [Mäkinen2001]. This produces a new concept of consistency which is defined in following.

For example, Newmark integration methods given in [Simo & Vu-Quoc 1988] can be considered as a mani-

fold inconsistent time integration method. The problem arises from knowledge that the material incremental

tensor ~ΘΘΘΘR occupies the tensor space matT SOR ( )3 and not the tensor space matT SOI ( )3 as it is explicitly as-

sumed in that paper, see the indication in Section 2.3.1. However, the Newmark integration method may

give accurate results when using a small time step size.

There also exists a manifold consistent integration method given in the paper [Cardona & Géradin 1988]where the authors utilize the Newmark integration scheme although any standard time-integration schemescan be exploited. The time-integration method is based on the updated Lagrangian formulation giving amanifold-consistent vector addition to the unknown variable, as in (116a). Only a few researches have fol-lowed with this manifold consistent approach. This is mainly because the exact tangential tensor is rathercomplicated and only the approximated form of the tangential stiffness tensor is explicitly given in this pa-per.

Next, we will derive the virtual work form in terms of the total material rotation vector ΨΨΨΨ and its virtualdisplacement δΨΨΨΨ in order to achieve the total Lagrangian formulation. We choose the material descriptionsince the linearization procedure is somehow simpler than in the spatial description, no emergence of theLie-derivative. We need to express spin vectors such as the virtual incremental rotation vector δΘΘΘΘR , theangular velocity vector ΩΩΩΩR , the angular acceleration vector ΑΑΑΑR and the curvature vector ΚΚΚΚ R in terms of thetotal rotation vector ΨΨΨΨ and δΨΨΨΨ , giving

δ δΘΘΘΘ ΨΨΨΨ ΩΩΩΩ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΚΚΚΚ ΨΨΨΨR R

R R

T T

A T T T

= =

= + = ′

, & ,&& & & , ,

(118)

where the tangential transformation is given in (21), p. 31. We note that δΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ, , & , && ∈matTI whereas spin

vectors δΘΘΘΘ ΩΩΩΩ ΑΑΑΑR R R R, , ∈matT .

59

Now, we could write the virtual work of acceleration forces δWacc (85), p. 52, in terms of the total material

vectors δΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ, , & , && , yielding

δ δ ρ δW A s sL L

acc c cT T T ~= + + +⋅ ⋅z zx gx G T JT T JT T T JT( && ) d && & & ( & ) & d0 ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨd i , (119)

that can be decomposed into the acceleration depend part δWaccA and the velocity dependent part δWaccB

δ δ ρ δ

δ δ

W A s s

W s

L L

L

accA c cT

accBT T ~

= +

= +

⋅ ⋅

⋅

z zz

x gx G T JT

G T JT T T JT

( && ) d && d ,

& & ( & ) & d ,

0 ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

d i

d i(120)

where we have used the relation T GT G∗ −= T 1 for the adjoint of the tangential transformation T .

Note that the rotational part of the virtual work of acceleration forces δWacc does not depend on the centerline displacement vector xc and its derivatives, and the translational part does not depend on the rotationvector ΨΨΨΨ and its derivatives. This is due to the kinematic assumption of Reissner’s beam (80), p. 50, sincethe displacement and rotation vectors are distinct variables.

The virtual work of external forces (89), p. 52, reads in matTI -formulation

δ δ δW s sL L

ext c= +⋅ ⋅z z ∗x n T M Rd dΨΨΨΨ , (121)

where the external material moment vector M R R∈ ∗matT so it is a spin covector. According to the above for-

mula, we may define a new external material moment vector by setting

M T MI R I:= ∈∗ ∗matT , (122)

where matTI∗ is the initial covector space of rotation in the material description.

Finally, the virtual work of internal forces (96), p. 53, can be written in the total Lagrangian formulation

δW s sL L

int = +⋅ ⋅z zδ δΓΓΓΓ ΚΚΚΚN MR Rd d , (123)

where the variations of the material strain curvature vectors are according to (95b) p. 53, (118d) and (44)p. 42

δδ δ

ΓΓΓΓ ΨΨΨΨΚΚΚΚ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

= ′ + ′= ′ + ′= ′ + ′

⋅⋅

R x R x T

T T

C TR

Tc

Tc

~δ δδδ δ

( ) ,

,

( , ) .1

(124)

Substituting the above equation into the form (123) yields

δ δ δ δW s sL L

int cT

c~= ′ + ′ + ′ + ′⋅ ⋅ ⋅−∗ ∗ ∗ ∗ ∗z z( )d ( ) ( , ) dx R N T R x N C M T MR RΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ1d ie j , (125)

where R N−∗ can be viewed as a push-forward of the material force vector N X∈ ∗T B0 , that is R N R N> = −∗ .It is clear that the virtual work of internal forces is more complicated than other virtual works forms. This ismainly because of the choice of the unknown variable, the rotation vector. We may also choose the strainand the curvature vector as unknown variables, but this yields well known compatibility problems.

3.3.1 On Objectivity for Strain Vector and Curvature Tensor

In this Section, we show that the strain and curvature vectors in the material and spatial descriptions areobjective quantities. This objectivity can be called an observer frame-indifference, see [Ogden 1984; Ch. 2].It is clear that an objective spatial quantity behaves differently than an objective material quantity, since theobserver transformation can be selected only for material description such that objective quantities are in-variant with respect to an arbitrary rigid body motion.

60

The material strain vector and curvature tensor for Reissner’s beam have been defined as, Eqn (95) p. 53and Def. 32 p. 34,

ΓΓΓΓ

ΚΚΚΚ

: ,~

: ,

= ′ −

= ′

R x E

R RR

Tc

T

1 (126)

where the rotation operator R , the center line placement xc , and the material basis vector E1 depend on thelength parameter of beam s. Let the observer transformation for the rotation operator R and for the place-ment vector xc be

R QR

x Q x c

+

+

=

= +

,

,c cb g (127)

where the orthogonal operator Q ∈SO( )3 corresponds to the rigid body rotation, and the vector c ∈E3 cor-responds to the rigid body translation, respectively. Note that the rotation operator R x X( )s T T∈ ⊗B B0 is atwo-point tensor and it acts in the observer transformation such as a deformation gradient F , see Section3.2. By substituting the observer transformations (127) into the material strain vector and curvature tensor,Eqns (126), we obtain the transformed vectors

ΓΓΓΓ ΓΓΓΓ

ΚΚΚΚ ΚΚΚΚ

+

+

= + ′ − = ′ − =

= ′ = ′ =

( ) ( ) ( ),

~(

~),

QR Q x c E R x E

QR QR R RR R

Tc

Tc

T T

b gb g b g

1 1(128)

where we have used the properties of the observer transformation: ′ = ′ =Q O c 0( ) , ( )s s , and QQ IT = . By

comparing the equations (128) with the definitions (126), we found that the material strain vector and cur-

vature tensor are both frame-indifferent: the quantities are invariant under the observer transformation. In

addition, we observe that the objectivity is also achieved if any interpolation applied to the material strain

vector and curvature tensor with the aid of the total rotation vector ΨΨΨΨ( )s T∈mat I , or via of the updated rota-

tion vector ΨΨΨΨref mat ref( )s T∈ I . In the total Lagrangian formulation, the rotation operator has the interpolation

R = exp(~

( ))YYYY s , and correspondingly in the updated Lagragian formulation, the interpolation is

R R= ref ref( ) exp(~

( ))( )s siYYYY . We note that an interpolation for the incremental rotation vector ΘΘΘΘR R( )s T∈mat can

not be directly applied, since in general, the incremental rotation vector occupies a different vector space of

rotation matTR . That is because the base point R R= ( )s depends on the length parameter s.

Correspondingly, the spatial strain vector γγγγ and the spatial curvature tensor ~κκκκ R have been defined as, Eqn(97) p. 54 and Def. 32 p. 34

γγγγ

κκκκ

: ,~ : .

= ′ −

= ′

x RE

R RR

c

T

1(129)

Subsituting the observer transformations (127) for the above spatial quantities gives

γγγγ γγγγ

κκκκ κκκκ

+

+

= ′ − =

= ′ =

Q x RE Q

QR R Q Q QR R

c

T T T

1b g ,~ ~ .

(130)

Hence, the spatial strain vector γγγγ and the spatial curvature tensor κκκκ are spatial objective quantities. Note

that the spatial curvature tensor ~κκκκ R transforms such as a second order tensor in the spatial description. Its

associated vector κκκκ R varies in the observer transformation such as a spatial vector, giving

κκκκ κκκκR RQ+ =( )s . (131)

We note that the objectivity is also achieved if any interpolation is adapted to the spatial strain vector and

the spatial curvature tensor with the aid of the total rotation vector ψψψψ( )s T∈spat I , or via of the updated rota-

tion vector ψψψψ ref spat ref( )s T∈ I .

In the paper [Crisfield & Jelenić 1999], it is incorrectly shown that the strain measures (strain and curvature

vectors) in the total and updated Lagragian finite element formulations are nonobjective quantities. How-ever, the proof given in that paper is incorrect. Next, we give the details about why it is so.

61

Let ΨΨΨΨ ΨΨΨΨ1 2, mat∈ TI be nodal vectors of total material rotation for a linearly interpolated element, and let ΨΨΨΨQ

be a total rotation vector for an objective transformation operator Q Q= exp(~

)ΨΨΨΨ , i.e for rigid body rotation.

These rotation vectors have the following component values [Crisfield & Jelenić 1999; App. C] with respect

to the global frame , , , O e e e1 2 3

Ψ Ψ Ψ1 2i i i =

1

-0.5

0.25

, =

-0.4

0.7

0.1

=

0.2

1.2

-0.5

L

NMMM

O

QPPP

L

NMMM

O

QPPP

L

NMMM

O

QPPP

, Q , (132)

hence e.g. ΨΨΨΨ1 1=Ψ i ie with the conventional summation. The linear interpolation functions for the nodal ro-tation vectors ΨΨΨΨ ΨΨΨΨ1 2 and read

N ss

LN s

s

Ls L1 21 0( ) , ( ) , [ , ]= − = ∈ . (133)

The interpolation for the total rotation field is therefore ΨΨΨΨ ΨΨΨΨ( ) ( ) ( )s N s C Ti i= ∈ mat Q . This interpolation isclearly acceptable because the nodal rotation vectors belong to the same vector space of rotation. The ob-server transformation for the rotation operator R given in (127a) yields

exp(~

) exp(~

),

exp(~

) exp(~

).

ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ1 1

2 2

+

+

=

=

Q

Q(134)

The transformed nodal rotation vector ΨΨΨΨi T+ ∈mat I can be obtained by extracting it from the rotation operatorwith the aid of Spurrier’s algorithm, [Simo & Vu-Quoc 1988]. We note that the original rotation vectorsΨΨΨΨ ΨΨΨΨ1 and 2 in the observer transformation (134) belong to the vector space of rotation matTQ . Although wemay extract the transformed nodal rotation vectors ΨΨΨΨi

+ such that they satisfy the observer transformationrelations (134), the linear interpolation is not preserved, yielding

exp( ( )~

) exp( ( )~

), [ , ]N s N s s Li i i iΨΨΨΨ ΨΨΨΨ+ ≠ ∀ ∈Q 0 (135)

This arises from the fact that the rotation manifold SO( )3 has curved character. Indeed, a linear vector val-ued function in the vector space of rotation matTQ is not linear in the different vector space of rotation matTI .The results in Fig. 19 show that the linear interpolation does not preserve in the observer transformation,settings according to Eqn (132).

Therefore, we may pronounce that the extracted nodal rotation vectors from the corresponding rotation op-erations must not be interpolated. There is an infinite number of possibilities for the interpolation since thetransformation operator Q ∈SO( )3 is arbitrary. However, in the paper [Crisfield & Jelenić 1999], the ex-tracted nodal rotation vectors are interpolated giving incorrect results for the objectivity of the total andupdated Lagrangian formulations.

Fig. 19 The components of the rotation vectors ΨΨΨΨ ΨΨΨΨ( ) ( )s s and + interpolated through the length of beam(left), and the norm || ( )|| || ( )||ΨΨΨΨ ΨΨΨΨs s and + (right). The solid line indicates the initial rotation vector ΨΨΨΨ( )sand the broken line the transformed rotation vector ΨΨΨΨ + ( )s . Solid line corresponds to the proper values.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

1.5

2

Length parameter s/L

Com

pone

nts

of r

otat

ion

vect

or

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Length parameter s/L

Nor

m o

f rot

atio

n ve

ctor

62

On the contrary Eqn (135) states that the interpolation is not preserved in the observer transformation, thisdoes not mean that the beam formulation is nonobjective. Indeed, this property is never required for beingan objective formulation. It is sufficient that the beam formulation satisfies the conditions (128). Generallyspeaking, a global interpolation on a nonflat manifold is never preserved in an observer transformation. Thisis because of the nonlinear character of the manifold, a parametrization mapping is a nonlinear mapping fora nonflat manifold. In the paper [Jelenić & Crisfield 1999], the authors utilize a corotational interpolation,

where the interpolation is carried out with respect to an element-attached frame. Hence, this interpolationnaturally is preserved in the observer transformation.

We have above assumed that the observer transformed rotation interpolation ΨΨΨΨ + ∈C T( )mat I keeps the basepoint I fixed. However, it also makes sense and can be assumed that the base point transforms under theobserver transformation (I Q→ ) giving ΨΨΨΨ + ∈C T( )mat Q . Then we could denote ΨΨΨΨ ΨΨΨΨ+ =( ) ( )s s and an inter-polation is preserved under an observer transformation. This is an important issue and clarifies the frame-indifference of geometrically exact beam formulations.

In the paper [Crisfield & Jelenić 1999; App. C], it is also shown that the updated Lagrangian formulation,

named as the incremental form, has a path dependent property. The statement is correct, but the proof is,

however, incorrect, suffering from the same problems as described above. In the proof it is assumed that two

steps via rotation vector interpolation ΨΨΨΨ( ) ( )s C T∈ mat I followed with ΨΨΨΨref mat ref( ) ( )s C T∈ I belong into the

same rotation vector space, but however it is not true. We may compute nodal values for combined rotation

by formulas

exp(~

) exp(~

) exp(~

),

exp(~

) exp(~

)exp(~

),

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ1 1 1

2 2 2

+

+

=

=ref

ref

(136)

where ΨΨΨΨ ΨΨΨΨ1 2+ + and are the nodal values for the rotation vector in the combined rotation. However, the inter-

polation is not generally valid, yielding

exp( ( )~

) exp( ( )~

)exp( ( )~

) ( , )N s N s N s s Lk k i i j jΨΨΨΨ ΨΨΨΨ ΨΨΨΨ+ ≠ ∀ ∈ref 0 . (137)

That is because of the rotation vectors ΨΨΨΨ + ∈( )s Tmat I and ΨΨΨΨ1ref mat ref( )s T∈ I belong to the different vector

spaces of rotation. The above equality arises from the expectations shown in the observer transformation,

Eqn (135). We note that, in general, the rotation vector interpolation ΨΨΨΨ + ( )s can not be decomposed into the

interpolation parts ΨΨΨΨ( )s and ΨΨΨΨref ( )s , and correspondingly, the interpolation parts can not be aggregated

into a single interpolation. Hence, the rotation interpolation has rather an extraordinary property, which

arises from the curvature feature of the rotation manifold SO( )3 .

Next, we show why the updated Lagrangian formulation has a path dependent property. Let ei be the inter-polation error of the exact rotation vector ΨΨΨΨref

( )n , and let R err,i be the error of the exact rotation operator R ref

at the i th incremental step. The error of the rotation operators at the first incremental step and at the fol-lowing steps up to the incremental step r , reads

R R I e

R R R R e

R R R R e

R R R R e

ref,1 err,1 ref

ref,2 err,2 ref,1 err,1 ref

ref,3 err,3 ref,2 err,2 ref

ref,r err ref, err, - ref

+ = +

+ = + +

+ = + +

+ = + +−

exp~ ~ ,

( )exp~ ~ ,

( )exp~ ~ ,

( )exp~ ~ ,

,( )

,( )

,( )

, ,( )

ΨΨΨΨ

ΨΨΨΨ

ΨΨΨΨ

ΨΨΨΨ

1 1

2 2

3 3

1 1

n

n

n

r r r rn

r

e je je j

e jM

(138)

Thus in the updated Lagrangian formulation, the error of the rotational operator is cumulative yielding thestatement that the updated Lagrangian formulation does not preserve the path-independent property of themechanical model. Although the mechanical model can be path-independent only in some static cases, it is adesired quality also for dynamic cases when one source of error is totally absent. Computational models thatpreserve the path-independent property can be called consistent in model type. We note that the total La-grangian formulation has this kind of property since the base point remains fixed. Hence, the rotation vectorbelongs to the fixed vector space of rotation, i.e. ΨΨΨΨ ( )( )i s T i∈ ∀mat I .

63

3.4 On Symmetry of Second Variation

Next, we follow the presentation given in the papers [Makowski & Stumpf 1995] and [Simo 1992] to showwhy a tangent stiffness tensor can be nonsymmetrical. This is somehow rather involved issue and we give asimple explanation for the nonsymmetry of the second variation. Detailed derivation could be found in theabove papers. We consider a finite dimensional point-manifold M without any restriction. Let us introducea r -parametrized curve on the manifold M such that

aaaa aaaa aaaa aaaa: , ( ), ( ), ( )E1 0 0Æ = = ¢M r ra q qd , (139)

where δq is the tangent to the curve αααα at the point q of the manifold M , see Fig. 20. Let W:M → R be awork function on the manifold M . The work function depends on the displacement vector q ∈M . The firstvariation of the work function W at the point q ∈M in the direction δq q∈T M reads

δWW r

r rWq q q; :

d ( )

dD ( )δ δb g b g b g=

== ⋅αααα

αααααααα0

0 , (140)

that is very similar to the variation defined on a vector space. The first variation δW depends linearly onthe direction δq , hence we may denote δW = ⋅f qδ , where the force vector f q( ) belongs to the cotangentialspace Tq

∗M . As usual, we call the point q0 ∈M as a critical point of function, an equilibrium, if the varia-tion δW( ; )q q0 δ vanishes for arbitrary δq q∈T M . At the critical point, the corresponding force vector f q( )0

vanishes, i.e. f q 0( )0 = .

In order to give the second variation of function on the manifold, we introduce another curve with the fol-lowing properties

bbbb bbbb bbbb bbbb: , ( ), ( ), ( )E1 Æ = = ¢M s sa q q0 0D , (141)

where ∆q q∈T M is the tangent to the curve ββββ . We denoted the tangency by ∆ in order to indicate the dif-ference from δq . Note that the virtual displacement δq( ) ( )s T s∈ ββββ M depends on the curve ββββ if the base pointvaries according to the curve ββββ , see Fig. 20.

Hence, we could write the second variation of the work function W on the manifold M

δ2

0 0

0 0 0 0

WW s

s s

s s

s sq q q

f q

f q q f q q

; , :d ( )

d

d ( ) ( )

d

D ( ) ( ) ( ) D ( ) ,

δδ δ

δ δ

∆

∆ ∆

b g b g b g b gc h

b gd i b g b g b gd i

==

==

= +

⋅

⋅ ⋅ ⋅ ⋅

ββββ ββββ ββββ

ββββ ββββ ββββ ββββββββ ββββ

(142)

where the first term is often denoted by H( ; , )q q qδ ∆ that is the Hessian of the function W .

αααα( )r

M

Tq0M

δqq

ββββ( )s

RW( )q

∆q

Fig. 20 A geometric presentation of the parametrized curve αααα:E1 → M and the work functionW:M → R on the manifold M .

64

The Hessian of the function W could be denoted by

H W( ; , ): D ( ), ( )q q q q qδ δ∆ ∆= ⊗αβαβαβαβ αααα ββββ0 0b g b g: , (143)

that is always a symmetric form on the Riemannian manifold M , i.e. H H( ; , ) ( ; , )q q q q q qδ δ∆ ∆= since theFréchet partial derivatives commute D Dαβαβαβαβ βαβαβαβαW W= for the smooth function W. However, the second termin the formula (142) that reads

f q qββββ ββββββββ( ) D ( )0 0b g b gd i⋅ ⋅δ ∆ (144)

is generally nonsymmetric, unless q q0 0= ( ( ))ββββ is a critical point (an equilibrium). At the critical point, wehave the force vector f q 0( )0 = , giving the symmetric second variation of the function W . The nonsymmet-rical term (144) vanishes also if Riemannian manifold M is a flat manifold. Then we get for the derivativeD ( ( ))ββββ ββββδq O0 ≡ .

We conclude that the second variation δ2W( ; , )q q qδ ∆ is a symmetric form if q0 ∈M is a critical point ofthe function W, or if the manifold M is a flat (Euclidean) manifold. The beam placement can be identifiedby the manifold ( , ) ( )d R ∈ ×E3 3SO , where d is the translational displacement and R is the rotation opera-tor. If we present rotation by the (material) incremental rotation vector ΘΘΘΘR R∈matT , where the base point Rdepends on solution, we have a nonsymmetric stiffness tensor away from a critical point. However, at acritical point , i.e. an equilibrium, we have a symmetric stiffness tensor.

The symmetry of stiffness tensor can be achieved at arbitrary point by the parametrization of the manifold.

The parametrization mapping ϕϕϕϕ maps from an open set in an Euclidean space into an open set of the mani-

fold M , see Fig. 21. The parametrized work function W oϕϕϕϕ :U ⊂ →E R is a mapping from an Euclidean

set into the set of real numbers R . Since the set of an Euclidean space is a flat manifold, the nonsymmetric

term of the second variation (144) will always vanish. Especially, the rotation manifold SO( )3 could be

parametrized by the rotation vector ΨΨΨΨ ∈ =matTI E3 , where the parametrization mapping is the exponential

mapping, ΨΨΨΨ ΨΨΨΨa exp(~

) ( )∈SO 3 .

αααα( )r

M

δq

q

ββββ( )s

RW( )q

∆q

s

rparametrizationchart in E2

parametrizationmapping ϕϕϕϕ

decompositionmapping Woϕϕϕϕ

W( )q

Fig. 21 A geometric presentation for the parametrization of the two-manifold M .

65

3.5 Consistent Tangent Tensors for Reissner’s Beam

In this Section, we will derive consistent tangent tensors for geometrically exact Reissner’s beam. We willnot introduce a finite element approximation, giving infinite dimensional tangent tensors. However, thefinite element implementation is straightforward and we will discuss this issue in the following section.Tangent tensors arise from the linearization of the virtual work forms. We call the tangent of the internalforce vector by stiffness tensor, and the tangent of the external force vector by loading tensor. We decom-pose the tangent of the acceleration force vector into acceleration, velocity, and displacement dependentterms, giving mass, gyroscopic, and centrifugal tensors, respectively.

The total and updated Lagrangian formulations yield identical tangent tensors. However, the updated La-grangian formulation requires secondary storage variables like the curvature vector and the rotation operatorat the previously converged solution, see [Cardona & Géradin 1988]. In the total Lagrangian formulation,secondary storage variables are completely avoided since the reference placement is always the initialplacement. We note that consistent gyroscopic, centrifugal and loading tensors for the total and updatedLagrangian formulations have not been presented anywhere by author’s knowledge. The consistent stiffnesstensor of Reissner’s beam with the total Lagrangian formulation has been firstly introduced in [Ibrahimbe-gović et al. 1995]. This formulation suffers from singularity at the rotation angle ψ = 2π and its multiplesand is therefore restricted to the rotation angle ψ < 2π .

The linearization of the virtual work δ δW( , & ,&&; )q q q q around the state point ( , & )q q0 0 ∈TC at the time t t= 0

can be written as

Lin δ δ δ δ δ δ δ δW W W W W( , & ,&& ; ) ( , & ; ) ( ,&&; ) D D &&

q q q q q q q q q q q qq qb g = − + +⋅ ⋅0 0 0 0accA ∆ ∆ , (145)

where Dq denotes Fréchet partial derivative with respect to q that denotes the displacement field includingthe translational displacement and rotational quantities, i.e. q d: ( , )= ∈ΨΨΨΨ C . We have denoted the change ofthe base point by (∆ ∆q q, & ) that avoids confusion with the virtual displacement field δq q∈TC . The virtualwork δW0 depends on the state point ( , & )q q0 0 but not on the acceleration field &&q . Hence, the virtual work ofacceleration forces δWaccA could be written via a bilinear form

δ δ δW sL

accA( ,&&; ) ( ) ( &&) dq q q M q q0 0= ⊗z ΨΨΨΨ : , (146)

where we have denoted M ( )ΨΨΨΨ0 for the mass tensor that depends on the current rotation field ΨΨΨΨ0 at the timet t= 0 . According to Eqn (120a), we have the mass tensor M ( )ΨΨΨΨ for Reissner’s beam

Mg O

O G T JT( ):ΨΨΨΨ =FHG

IKJ

Aρ0Td i . (147)

Therefore, the acceleration virtual work δWaccA is excluded from the virtual work form δW0 which includesthe following virtual work forms, see Section 3.3,

δ δ δ δ δ δ δ δW W W W0 0 0 0 0 0 0( , & ; ): ( ; ) ( ; ) ( , & ; )q q q q q q q q q q= − +ext int accBb g , (148)

computed at the point ( , & )q q0 0 ∈TC at the time t t= 0 .

The translational displacement field of beam center line d is defined by

d x x( , ): ( , ) ( , ) , [ , ],t s t s t s s L= − = ∈ ∈c c 0 3 0E (149)

where xc( , )t s is the current place vector of center line. Now, the displacement field-manifold C , at anypoint of it, can be viewed as the vector space E3 × matTI that is a flat manifold because the base pointI ∈SO( )3 remains fixed. Hence, the tangent field-bundle TC is also a flat manifold. This flatness arisesfrom natural seasons: we have parametrized the rotational manifold SO( )3 and this parametrization chartbelongs to three-dimensional Euclidean space.

In cases when the tangent field-bundle TC is not a flat manifold, the tangent field-space δq q∈TC dependson the base point q ∈C . Thus, the dependency has to be taken account when linearizing the virtual workform, as shown in Section 3.4.

66

Next, we give the linearization for the virtual work forms δ δ δW W Wacc ext and , int , separately. The virtual workof acceleration forces δWaccA is already in a linear form with respect to the acceleration field &&q . The masstensor given in (147), however, depends on the displacement field q . In addition, the form δWaccB does notdepend on the translational displacement field d which simplifies the linearization procedure considerably.We denote the rotation field by C T( )mat I and the translational displacement field by C( )E3 , hence the dis-placement manifold C is equal to C T( )E3 × mat I . Its element is therefore C ∋ = ∈ ×q d I( , ) ( )ΨΨΨΨ C TE3

mat and atan arbitrary length parameter s we have correspondingly ( ( ), ( ))d Is s TΨΨΨΨ ∈ ×E3

mat .

The linearization of the acceleration virtual work δWacc , Eqn (85) p. (52), at the state point ( , & )q q0 0 ∈TCyields

Lin δ δ δ δ δ δW W W Wacc acc acc acc( , & ,&&; ) ( , & ,&& ; ) D D &&q q q q q q q qb g = + +⋅ ⋅0 0 ΨΨΨΨ ΨΨΨΨΨΨΨΨ ΨΨΨΨ∆ ∆ , (150)

where the linear virtual forms DΨΨΨΨ ΨΨΨΨδWacc⋅∆ and D &&ΨΨΨΨ ΨΨΨΨδWacc⋅∆ could be given by

D ( )

D & ( & )&

ΨΨΨΨ

ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

δ δ

δ δ

W s

W s

L

L

acc cent

acc gyro

= d ,

d ,

⋅

⋅

⊗

⊗

zz=

∆ ∆

∆ ∆

K

C

:

:(151)

and where the centrifugal and gyroscopic tensors, K Ccent gyro and , are denoted by

K G C J J T J J C J C C

C G T J J T JC

centT ~

gyroT ~

: (~

, )~

( ) ( & , ) ( & , ) ( && , ) ,

:~

( ) ( & , ) .

= + + − + +

= − +

2 0 0 0 0 1 0 0 5 0 0 1 0 0

0 0 4 0 0

ΩΩΩΩ ΩΩΩΩ ΑΑΑΑ ΨΨΨΨ ΩΩΩΩ ΩΩΩΩ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

ΩΩΩΩ ΩΩΩΩ ΨΨΨΨ ΨΨΨΨ

e j d i

e j (152)

Above we have utilized the formulas given in Section 2.8. The angular velocity field ΩΩΩΩ0 ∈C T( )mat R and theangular acceleration field ΑΑΑΑ0 ∈C T( )mat R are computed by Eqns (118), p. 58, via the total material rotationvector and its time derivatives ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ0 0 0, & , && ( )∈C Tmat I at the fixed time t t= 0 . The centrifugal and gyroscopictensor are, in general, nonsymmetric tensors. This nonsymmetry arises from the kinetic energy that dependson the displacement field q as well as the velocity field &q . The kinetic energy T that we may interpret asa velocity dependent potential reads for Reissner beam

T sL

( , & ) ( ) ( & & ) dq q M q q q= ⊗z12 : , (153)

where the symmetric mass tensor M q( ) is given in Eqn (147). Solely, a displacement dependent potentialgenerates a symmetric tangent tensor in the linearization procedure excluding the mass tensor which is pref-erable a linear operator than a tangent tensor.

Correspondingly, the linearization of the external virtual work δWext at the point q0 ∈C yields

Lin δ δ δ δ δ δW W Wext ext ext( ; ) ( ; ) D ( , )q q q q q q qqb g = + ⋅0 0 ∆ , (154)

where the term Dq qδWext⋅ ∆ is denoted with the aid of the loading tensor K load

D ( )dq K q qδ δW sL

ext load= ⊗z : ∆ . (155)

We have assumed that the external forces and moments do not depend on the velocity field &q . The externalforce or moment fields are called conservative if they are obtained via a displacement dependent potentialfunctional. On a linear space (i.e. on a flat manifold), the conservative loading and the symmetry of theloading stiffness tensor are equivalent issues.

Next, we show that under gravitation the corresponding loading stiffness tensor is symmetric. It is clear thatin this case, the external force field n is constant, therefore we will study the external moment field due togravitation, see Eqn (90a) p. 52. The virtual work of external forces δWext and the external moment covectorfield M I I∈ ∗C T( )mat under gravitation read according to Eqn (121), p. 59,

δ δ δW s AL A

extT T ( ; ) d ,

~dΨΨΨΨ ΨΨΨΨ ΨΨΨΨ= =⋅z zM M GT ER bI I , (156)

where the tangential transformation ΨΨΨΨ ΨΨΨΨa T( ) depends on the rotation field.

67

ds

spatial placement B

t 3

t 2

E3

E3

body forceb E( ( ))χχχχ t

material placement B0

E

global frame , , e e e1 2 3

Fig. 22 An external force loading under gravitation.

The linearization of the virtual work form δWext (156a) at the point ΨΨΨΨ0 ∈C T( )mat I gives the loading stiffnesstensor K load

D ( )d ,

: ( , ) ,

:~

d , :~

d ,

#

#

q

R

R

K q q

K G C M T b T

M ER b b E R b

δ δW s

A A

L

A A

ext load

loadT

cross

Tcross

T

where

~

=

= +

= =

⊗z

z z

: ∆

2 0

0 0

0 0

ΨΨΨΨ

d i

(157)

where the body force vector b b E x: ( ( ))= ∈χχχχ t T B and the cross-section vector E E E X:= + ∈X X T2 2 3 3 0B , seeFig. 22. The tensor C2 is defined in Section 2.8. The moment fields M R R

# ( )∈C Tmat and M R R∈ ∗C T( )mat arerelated by

M G MR R# = −1 , (158)

where G R R∈ ∗ ⊗ ∗mat matT T is a material metric tensor and M R is defined in Eqn (90a), p. 52. We note that usu-

ally the loading stiffness tensor and the corresponding external moment vector vanish due to the center linecondition (86), p. 52.

The loading stiffness tensor (157) is indeed a symmetric tensor. In general, the tensor

C ab T abT2(~ , ) ~~ΨΨΨΨ + T (159)

is symmetric for all a b, ∈E3 . We note that taking the linearization of the moment field M R R∈ ∗C T( )mat , Eqns

(90b) and (89) p. 52, does not produce a symmetric loading stiffness tensor, since C T( )mat R∗ is a cotangent

field-space on the nonflat manifold SO( )3 by contrast to the cotangent field-space C T( )mat I∗ which can be

identified with C( )E3 via an isomorphism. We may consider the external moment field M I I∈ ∗C T( )mat

(156b) as a conservative loading since its tangent tensor is symmetric and hence could be integrated into a

potential functional.

The symmetry of the loading tensor K load , (157), can be also noticed by another way since there exists awork functional Wext

W VV

ext : , d= z RE bg 0

0

, (160)

whose variation gives exactly the external virtual work (156a). Moreover, the second variation of the workfunctional (160) will be symmetric since the total rotation vector belongs to the linear space, i.e.ΨΨΨΨ ∈C T( )mat I . See Section 3.4 for the symmetry of second variation on a flat manifold.

68

Next, we linearize the internal virtual work form δWint given in Eqn (123), p. 59. We define a kinematic

operator between the variation of the material strain and curvature fields and the virtual displacement fields.Collecting the terms from Eqn (124), we have

δδ

ΓΓΓΓΚΚΚΚ R

B qFHGIKJ = ⋅δ $ , (161)

where the generalized virtual displacement field δ $q and the kinematic operator B are defined as

δ :=$ : ,( )

( , )q

x

BR O R x T

O T C=

′′

F

HGGI

KJJ

′′

FHG

IKJ

δδδ

c T Tc

~

ΨΨΨΨΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ1

, (162)

where the tensor C1 is given in Section 2.8. The kinematic operator B can be viewed as a linear operatorfrom the linear space C T T T( )x I IB × ×mat mat into the tangent field-space C T T( )X RB0 × mat on the manifold( ( ))B0 3× SO . We note that relation (161) is the exact form without any approximation. Now, the internalwork form δWint , Eqn (123) p. 59, can be given via the inner product, see Def. 12 p. 23

δ δ δW s sL L

intT= ⋅ =z z× × ×

B q F q B FG G g G G

$ , d $ , dint intb g b g , (163)

where ( )G G× and ( )g G G× × are product metrics with the metric tensors diag( , )G G and diag( , , )g G G .The generalized internal force field F X Rint ( )∈ ×C T TB0 mat is defined by

FG N

G M

N

MR Rint

#

#: :=FHG

IKJ =FHGIKJ

−

−

1

1. (164)

We note that the force field Fint is metric independent because of the definitions for the vectors N M and ,Eqn (94) p. 53. Moreover, the vector field Fint can be given in terms of the material strain and curvaturevectors and with the aid of the constitutive relations, Eqn (113-114) p. 56

F C CC O

O CRint , :=

FHGIKJ =

FHG

IKJNM NM

N

M

ΓΓΓΓΚΚΚΚ

, (165)

where we have defined CNM that corresponds to the generalized elasticity tensor of beam.

Now we could linearize the internal work form δWint , Eqn (163) at the point q d0 0 0= ∈( , )ΨΨΨΨ C in the direc-tion ∆ ∆ ∆ ∆$ ( , , )q x= ′ ′c ΨΨΨΨ ΨΨΨΨ , giving

Lin δ δ δ δ δ δW W Wint int int( ; $ ) ( ; $ ) D ( ; $ ) $$q q q q q q qqb g = + ⋅0 0 ∆ , (166)

where the linear form D $$q qδWint⋅ ∆ is denoted with the aid of the material stiffness tensor K mat and the geo-metric stiffness tensor K σ

D $ ( ) ( $ $ $ ) d , $ : diag , ,

: ,

:

~

( , )~

( , ) ( , , ) (~

, )~

( )

,

,

#

# #

q

R

R R

q K K G q q G g G G

K B C B

K

O O RNT

O O C M

T NR C M C M C NR x T N R x T

δ δσ

σ

W sL

int mat

matT

NM

T T T Tc

T Tc

~

⋅ = + =

=

=−

′ + ′ + ′

F

HGGG

I

KJJJ

z ⊗∆ ∆: e j

0 0

2 0

2 0 3 0 0 2

ΨΨΨΨΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

(167)

where the material stiffness tensor K mat arises from the linearization of the vector Fint with the aid of thekinematic relation (161) and the geometric stiffness tensor K σ arises from the linearization of the kinematicoperator B .

We note that the material stiffness tensor K mat is clearly symmetric because of the symmetry of the elastic-ity tensor CNM . Additionally, the geometric stiffness K σ is also a symmetric tensor since the tensor C3 ,given in Section 2.8, and the tensor having the form of (159) are symmetrical in the lower right corner ofK σ .

69

4 PARAMETRIZATION OF CONSTRAINT MANIFOLD

In this Chapter, we consider constraint point-manifolds defined in Def. 34 p. 35 that arise from point-wiseholonomic constraint equations. The usual geometric joints of a multibody system like spherical, revolute,cylindrical, universal, helical, prismatic, and sliding joints can be presented via holonomic constraint equa-tions that only depend on displacement at corresponding geometric points. All these constraints generate asmooth point-manifold that can be parametrized. Also, the principle of virtual work and its geometricstructure are naturally related with the parametrization of the constraint manifold.

Let us consider a Newtonian problem in a n-dimensional Euclidean space En

f q q M q 0( , & ) &&− ⋅ = , (168)

where f q q( , & ) is a generalized force vector, M is a mass tensor, and &&q is an acceleration vector. The prob-lem is also subjected to holonomic constraints h:E En n d→ − that generate a d -manifold embedded in En

given by

M : , ( , )= ∈ ∈ = ∈ −t tn n dR E Eq h q 0o t . (169)

We assume that the manifold M is regular at every point, i.e. the derivative D ( , )qh ∈ −L E En n d is surjectiveat every point of the manifold. This assumption is equal to the full-rank condition and is not restricting as-sumption. In differential geometry, the mapping h which satisfies the above assumption is called a submer-sion.

Now, applying the principle of virtual work into the problem (168) on the manifold (169) yields

δ δW = − ⋅ ⋅ =f q q M q q( , & ) &&b g 0 . (170)

The generalized force vector f can be split into the applied force vector f qappl ∈ ∗T

0M and into the constraint

force vector f qcon ∈ ∗ ⊥T

0M , whose virtual work vanishes. The virtual displacement δq belongs to the tangent

space Tq0M , given by

T tnq qq h q q 0

0 0 0M = δ δ∈ ⋅ =E D ( , )o t , (171)

see Section 2.5 for more details.

Suppose that we can divide the displacement vector q ∈En into the master-released displacement vectorqmr ∈Ed and into the slave displacement vector qs ∈ −En d such that the derivative D s mr sq h q q( , , )t is an iso-morphism, i.e. a linear bijection. Here, we utilize the terminology: ‘master’, ‘released’, and ‘slave’ that isconventionally used in finite element literature. After the implicit function theorem, there exists a uniquemapping φφφφ( , ):t d n dqmr R E E× → − on some neighborhood such that

$ ( ): , ( , ))h q h q q 0t t t, ( ,mr mr mr= =φφφφ . (172)

Hence, the slave displacement can be given in terms of the master-released displacement vector, i.e.q qs mr= φφφφ( , )t . Moreover, the parametrization of the manifold M , Eqn (169), can be now written as

ϕϕϕϕ ϕϕϕϕ φφφφ: ( , ) ( , ): , ( , )R E× → =d t t tM q q q qmr mr mr mra b g . (173)

The parametrization mapping ϕϕϕϕ realizes the constraints h q 0( , ( , ))t tϕϕϕϕ mr = according to Eqn (172), seeFig. 23. Sometimes, constraints are given by a time-independent equation as h q q q q 0( , ): ( )mr s s mr= − =φφφφ thatgives a natural global parametrization of the manifold M , since D s mr sq h q q I( , , )t = . We could view the timevariable as an independent parameter that usually arises from displacement boundary conditions. We notethat the derivative of the parametrization (173) with respect to the master-released displacement vector qmr ,that is D ( )q qmr mrϕϕϕϕ , is an injective (one-to-one) operator everywhere. In differential geometry, this kind ofmapping is called an immersion.

70

constraint manifold M ⊂ En

qmr

parameter space Ed

parametrization mapping ϕϕϕϕ( )qmr

q

embedding space En

Fig. 23 A geometric representation of the parametrization of the constraint manifold.

Next, we consider a time-independent parametrization mapping ϕϕϕϕ :E Ed n→ ⊂M

q q= ϕϕϕϕ( )mr , (174)

that we use for pull-backing the virtual work form (170) on the manifold M into the Euclidean space Ed .The variation of the displacement vector q ∈M in terms of the master-released displacement vectorqmr ∈Ed reads

δq B q

B qq

==

⋅δ mr

mrmr

,

: D ( ),ϕϕϕϕ(175)

where we have defined an injective kinematic operator B q q:T Td d nmrE = E E→ ⊂M between the tangent

spaces. We also need the time derivatives for q that are given via the kinematic operator

& & ,

&& && & & ,

q B q

q B q B q

q

q

= ∈

= + ∈

⋅⋅ ⋅

mr

mr mr

T

T

M

M(176)

where the time derivative of kinematic operator &B depends on the displacement qmr and linearly on thevelocity vector &qmr . This can be noticed by observing

& D &

& & D ( ) ( & & ) ,

B B q

B q q q q

q

q

=

=

⋅⋅ ⊗

mr

mr

mr

mr mr mr mr

,2 ϕϕϕϕ :

(177)

where the latter equation expresses a quadratic dependency of the velocity vector &qmr .

Now, the principle of virtual work (170) on the manifold M can be written in the parameter space Ed ac-cording to the relations (174) and (175), yielding

δ δq B f q q MB q MBq qmr mr mr mr mr mr⋅ ∗ − − = ∀ ∈( , & ) && & & ,d i 0 Ed , (178)

where the constraint equations (169) are satisfied automatically because of the parametrization (174). Theabove equation can be viewed as a pull-back operator for the covector f ∈ ∗E n , see Def. 45 p. 39.

71

Linearizing the virtual work (178) at the fixed time t t= 0 around the state point ( , & )q qmr0 mr0 ∈ ×E Ed d in thedirection ( , & )∆ ∆q qmr mr gives

δ δq f M q C q K q qmr mr0 mr mr mr mr mr mr mr⋅ ⋅ ⋅− − − = ∀ ∈&& & ,∆ ∆b g 0 Ed , (179)

where the generalized force vector f mr0 , the generalized stiffness, damping, and mass tensors are, respec-tively

f B f q q MBq

K B f q q MBq MBq

C B f q q MBq

M B MB

q

q

mr0 mr0 mr0 mr0

mr mr mr mr mr

mr mr mr mr

mr

mr

mr

: ( , & ) & & ,

: D ( ( , & ) && & & ) ,

: D ( ( , & ) & & ) ,

: .

&

= − ∈

= − + + ∈

= − + ∈

= ∈

∗ ∗

∗ ∗ ×∗

∗ ∗ ×∗

∗ ∗ ×∗

c hc hc h

E

E

E

E

d

d d

d d

d d

(180)

The above equations are the most fundamental relations to derive force vectors and their tangent tensorswhen the parametrization mapping exists: ϕϕϕϕ :Ed → M . In Fig. 24, there is shown a geometric structure ofthe virtual work and the corresponding tangential space Tq0

∗ M where the applied force f appl belongs. Usu-ally, we could give the slave displacement vector qs in terms of the master-released displacement vectorqms via mapping q qs mr= φφφφ( ) , hence the parametrization mapping ϕϕϕϕ yields from Eqn (173).

We note that the generalized force vector f includes as well as external, internal, and velocity dependentacceleration forces via the relation

f q q f q f q q f q q( , & ) ( ) ( , & ) ( , & ),= − − ∈ ∗ext int accB E n , (181)

where f f fext int accB and , denote external, internal, and velocity dependent acceleration force vectors. Notethat q q= ϕϕϕϕ( )mr by the parametrization (174). In Newtonian mechanics, it is convenient to separate forcesinto external and internal forces, in d’Alembertian mechanics into applied and constraint forces.

We could define the following relations

K f q q C f q q

K f q

K f q q q C f q q q

q q

q

q q

: D ( , & ), : D ( , & )

: D ( )

: D ( , & ,&&), : D ( , & ,&&)

&

&

= =

=

= =

int int

loadext

centacc

gyroacc

(182)

that belong to the Euclidean space E∗ ×∗n n , the embedding space.

constraint manifold M ⊂ En

qmr

parameter space Ed

ϕϕϕϕ( )qmr

q0

f mr

f appl

Tq0

∗M

embedding space En

Fig. 24 A Geometric structure of the applied force vector on the manifold and in the parameter space.

72

Substituting relations (181-182) and into (180) yields

f B f q f q q f q q MBq

K B K K K B C C B MB K f f f

C B C C B MB

M B MB

mr0ext

mrint

mr mraccB

mr mr mr0

mr load cent gyro mrext int acc

mr gyro

mr

: ( ) ( , & ) ( , & ) & & ,

: ( ) ( ) & && ( ) ,

: ( ) & ,

: ,

= − − − ∈

= − + + + + + + − + + ∈

= + + ∈

= ∈

∗ ∗

∗ ∗ ×∗

∗ ∗ ×∗

∗ ∗ ×∗

d id id i

E

E

E

E

d

d d

d d

d d

σ

2(183)

where the acceleration force f f facc accA accB= + ∈ ∗E n , and the geometric stiffness tensor K σmr is defined by

K f B fqσmr mr( ): D ( )= ∗ , (184)

such that the vector f is kept constant under the differentiation, denoted by f . The tensor K σmr is a sym-

metric tensor since according to the definition (175b) we have K f fqσmr ( ) D ( )mr

= ⋅2 ϕϕϕϕ .

Equations (183) are important relations when deriving the generalized force vector and the correspondingtangent tensors. This is because the tangent tensors in the parameter space Ed are given in terms of the tan-gent tensors in the embedding space En where the tangent tensors are known. Also, we need the kinematicoperator B and its time derivatives and its Frechét derivative. We note that the force vector− + +f f fext int acc in the tensor K σmr has an opposite direction compared with the constraint force vectorf q

con ∈ ∗ ⊥T0M . This naturally follows from the principle of virtual work (170).

4.1 Special Beam Elements

In this Section, we explicitly give some special beam elements that are derived with the aid of the parame-trization of the constraint point-manifold. These beam elements are the switching beam element, the offsetbeam element, the sliding beam element, the revolute joint beam element.

4.1.1 Switching Beam Element

In total Lagrangian representation, we could avoid singularities at the rotation angle equal to 2π and itsmultiples by switching the rotation vector ΨΨΨΨ into its complement rotation vector ΨΨΨΨ C that reads, see Def. 15p. 26

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ φφφφ ΨΨΨΨC: ( ( )),= − = =2πψ ψq , (185)

when the rotation angle ψ exceeds straight angle, i.e. ψ > π . Thus, the complement rotation angleψ ψC = −2π will be less than straight angle, i.e. ψ C < π , and the singularity at 2π is avoided, Fig. 9 p. 26.

In analysis with large rotations, there often occurs a situation where the rotation vector has been switchedinto the complement rotation vector at the one end of the beam element, and the rotation vector has re-mained the same at the other end, since the rotation angle is less than straight angle. In Fig. 25, we give anexample of this situation: at the node 1 the rotation vector ΨΨΨΨ1

C has been switched, and at the node 2 therotation vector ΨΨΨΨ2 still less that straight angle hence it is not switched. Now, the complement rotation vec-tor ΨΨΨΨ2

C presents the slave displacement qs ∈E3 , and the nodal vector ( , , , )d d1 1 2 2ΨΨΨΨ ΨΨΨΨC , where di is thetranslational displacement vector at the node i , presents the master-released displacement vector qmr ∈E12 .

In Fig. 25, the variation and time derivatives of the complement rotation vector ΨΨΨΨ2C read after Eqn (185)

δ δΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

2C

s

Cs

Cs s

=

=

= +

⋅⋅⋅ ⋅

B

B

B B

2

2 2

2 2 2

,& & ,&& && & & ,

(186)

where the slave kinematic operator B Bs s= ( )ΨΨΨΨ2 and its derivative & & ( , & )B Bs s= ΨΨΨΨ ΨΨΨΨ2 2 are

B I e e

B e I e e e e e

s

s

= − + ∈

= + + − ∈

⊗×∗

⊗ ⊗ ⊗×∗⋅ ⋅

( ) ,

& ( & ) ( & & ) ( & ) ,

1 2 2

2 3

3 3

23 3

π π

πψ ψ

ψ

Ù

Ù Ù Ù Ù Ù

E

EΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ(187)

73

Switching beam

ΨΨΨΨ3ΨΨΨΨ1C

ΨΨΨΨ2

ΨΨΨΨ1C

ΨΨΨΨ2C

Ordinary beam

Ordinary beam

Cantilever beam systemunder loadings

Fig. 25 A typical situation when a switching beam element is needed. The switching beam is constructedfrom an ordinary beam element in addition to the parametrization change at the node 2.

where e G= =ΨΨΨΨ / : ()ψ and ()Ù . Since the vector ΨΨΨΨ2C presents the slave displacement vector qs and the nodal

vector( , , , )d d1 1 2 2ΨΨΨΨ ΨΨΨΨC presents the master-released displacement vector qmr , then the displacement vector qin the embedding space E15 is equal to

qq

q

d

d:=FHGIKJ =

F

H

GGGGGG

I

K

JJJJJJ

∈mr

s

1C

C

1

2

2

2

15

ΨΨΨΨ

ΨΨΨΨΨΨΨΨ

E . (188)

Now, according to the above relation, the kinematic operator B and its time derivatives & &&B B and read

BI

O BB

O

O BB

O

O B=FHG

IKJ ∈ =

FHG

IKJ =

FHG

IKJ

×∗

×∗

×∗ ×∗

×∗

×∗

×∗

12 12

3 9

15 12 12 12

3 9

12 12

3 9s s sb g d i d iE , && , &&

&& , (189)

where the slave kinematic operator Bs and its derivative &Bs are given in Eqns (187), and &&Bs writes

&& ( && ) (& & ) && && & & & &

( && ) (& & ) ( & )(& & ) ( & ) & .

B e e I e e e e

e e e e e e e e e e B

s

s

= + + + + + +

- + + + -

◊ ◊

◊ ◊ ◊ ◊

ƒ ƒ ƒ ƒ

ƒ ƒ ƒ

2

6 2

2

2

π

πy

y y

Ù Ù Ù Ù Ù Ù

Ù Ù Ù Ù Ù Ù Ù

YYYY YYYY YYYY YYYY YYYY YYYY

YYYY YYYY YYYY YYYY

d i

d i(190)

In order to get the master-slave force vector f mr0 and the master-slave tangent tensors K Cmr mr, in additionto the mass tensor M mr in the parameter space E12 , we also need the geometric stiffness tensor K σmr whichreads, see Eqns (183-184)

KO O

O K

K f e I f e e f f e e e

σσ

σ ψ

mrs

s s s s

=FHG

IKJ ∈

= + + − ∈

∗ ×∗ ∗ ×∗

∗ ×∗

∗ ×∗

⊗ ⊗ ⊗∗ ×∗⋅ ⋅

9 9 9 3

3 9

12 12

23 32 3

s

s

E

E

,

( ) ( ) ,π Ù Ù Ù Ù

(191)

where the slave force vector f s ∈ ∗E 3 has the degrees of freedom corresponding to the slave displacement

ΨΨΨΨ2C .

74

The force vector f mr0 , the tangent tensors K Cmr mr, , and the mass tensor M mr in the parameter space E12

can be obtained by Eqn (183) when the vector f , the tangent tensors K K K Cload cent gyro, , , , and the mass ten-sor M in the embedding space E15 have been computed. We get these embedded quantities via the assem-bly procedure since we know the vectors and tensors for the ordinary beam element that corresponds to thenodal vector ( , , , )d d1 1 2 2ΨΨΨΨ ΨΨΨΨC C , which is a part of the nodal vector (188) in the embedding space. The othervectors and tensors, which correspond to the nodal rotation vector ΨΨΨΨ2 , vanish for the switching beam in theembedding space E15 .

Finally, we note that the complement of a complement rotation vector is equal to a rotation vector itself, i.e.ΨΨΨΨ ΨΨΨΨCC = , hence the switching beam element in Fig. 25 is equivalent to the case where at the node 2 therotation vector ΨΨΨΨ2

C has been switched, and at the node 1 the rotation vector ΨΨΨΨ ΨΨΨΨ1 1( )= CC is still less thanstraight angle thus it is not switched.

4.1.2 Offset Beam Element

In this section, we introduce a special beam element called an offset beam element, where the master andthe slave nodes are rigidly connected, see Fig. 26. It is often convenient to use offset elements for instancewhen modelling a beam structure where the rotational joints are not on the beam center line. Firstly we re-trench the degrees of freedom and secondly, if the rotational joint is moved to the center line, we do notneed a topological change in the finite element model.

We assume the kinematic relation between the master ‘m’ and slave node ‘s’ as follows, see Fig. 26,

d d R I c Is m m s m mat= + − ∈ = ∈( ) , ,E3 ΨΨΨΨ ΨΨΨΨ T (192)

where R Rm m= ( )ΨΨΨΨ is the current rotation operator, see Def. 14 p. 24, c is the initial offset vector from thebeam center line, dm is the master translational displacement vector at the node ‘m’, and ds is the slavetranslational displacement vector at the slave node ‘s’. The rotation vectors of master and slave nodes areequal, i.e. ΨΨΨΨ ΨΨΨΨs m= , since the master and slave node pair is assumed rigidly connected. The slave displace-ment vectorqs , the master-released displacement vector qmr , and the displacement vector q in the embed-ding space E18 read

qd

q

d

dq

q

qss

smr

m

m

mr

s

: , : , :=FHGIKJ ∈ =

F

H

GGGG

I

K

JJJJ∈ =

FHGIKJ ∈

ΨΨΨΨΨΨΨΨ

ΨΨΨΨ

E E E6

1

1 12 18. (193)

Taking the variation of the master-slave kinematic relation Eqn (192) gives

δq Bd

BI R cT

O Is sm

ms

m m=FHGIKJ =

−FHG

IKJ ∈×∗

×∗ ×∗

×∗δδΨΨΨΨ

,~

3 3

3 3 3 3

6 6E . (194)

master node ‘m’

beam element

slave node ‘s’

offset vector c

node 1

Fig. 26 An offset beam element where the slave node ‘s’ is rigidly connected to the master node ‘m’.

75

Now after Eqn (194), the kinematic operator B and its time derivatives &B and &&B read according to thenotation (193), see Eqns (175-176) p. 70,

BI

O BB

O

O BB

O

O B=FHG

IKJ ∈ =

FHG

IKJ =

FHG

IKJ

×∗

×∗

×∗ ×∗

×∗

×∗

×∗

12 12

6 6

18 12 12 12

6 6

12 12

3 6s s sb g d i d iE , && , &&

&& , (195)

where the derivatives of the slave kinematic operator are

&

~ ~ ~&, && (

~ ~ ~)~

~ ~ & ~&&B

O R cT R cT

O OB

O R cT cT cT

O Os

m m m m m m m m m m m m m=− −F

HGIKJ = − + + +FHG

IKJ

×∗

×∗ ×∗

×∗

×∗ ×∗

3 3

3 3 3 3

3 3

3 3 3 3

2ΩΩΩΩ ΑΑΑΑ ΩΩΩΩ ΩΩΩΩ ΩΩΩΩe j , (196)

where the tangential transformation T Tm m= ( )ΨΨΨΨ is given in Eqn (21), p. 31, and the angular velocity and

acceleration vectors ΩΩΩΩ ΑΑΑΑm m mat , ∈ TR at the master node ‘m’ are given in terms of YYYY YYYY YYYYm m m mat, & , && Œ TI in Eqns

(27-28) p. 34

Finally, the geometric stiffness tensor reads

KO O

O K

K GT c R g f T GC cR g f

ss

s

mrs

s mT

mT

s m 2 mT

s m

~

=FHG

IKJ Œ

= + Œ

* ¥* * ¥*

* ¥*

* ¥*

- - * ¥*

9 9 9 3

3 9

12 12

1 1 3 3

E

E

,

~ ~ , ,c h c hYYYY

(197)

where the slave force vector fs Œ*E 3 corresponds to the degrees of freedom of the slave displacement vector

ds.

Finally, the force vector f mr0 , the tangent tensors K Cmr mr, , and the mass tensor M mr in the parameter space

E12 can be obtained by Eqn (183) p. 72, when the vector f , the tangent tensors K K K Cload cent gyro, , , , and the

mass tensor M in the embedding space E18 have been computed. We get these embedded quantities via the

assembly procedure since we know the vectors and tensors for the ordinary beam element that corresponds

to the nodal vector ( , , , )d d1 1ΨΨΨΨ ΨΨΨΨs s , the part of the nodal vector q , Eqn (193c), in the embedding space E18 .

The other parts of vectors and tensors, which corresponds to the nodal vector ( , )dm mΨΨΨΨ , vanish for the offset

beam in the embedding space E18 .

4.1.3 Slide Beam Element

In this section, we introduce a special element called a slide beam element. The basic concept of the slidebeam element is shown in Fig. 27. The slide beam element is a combination of two ordinary beams: masterand slave beam elements where the slave element is kinematically connected into the master beam element.The slave node follows the flexible center line of the master beam element, such that the slave node has onetranslational degree of freedom u along the tangent of the master center line. A rotation in the slave node isassumed to be the same as the corresponding rotation in the master beam element, and the released transla-tional degree of freedom u ∈[ , ]0 1 has no effect on the rotation of the slave node. Although we exploit ageneral slide formulation without restricting to a specific finite element interpolation, we give the slidebeam element with linear shape functions.

We have the following kinematic assumptions for the slave node ‘s’

d X d Xs m m m s

s m m

= = + −

= ∈

⋅

⋅

=

=

∑

∑

ΦΦΦΦ

ΨΨΨΨ ==== ΦΦΦΦ

i i i

i i

u

u u

i

n

i

n

( ) ,

( ) , , [ , ] ,

η

η η

d i1

1

0 1ΨΨΨΨ(198)

where n is the number of the master nodes 'mi ' in the master element, ds refers to the nodal translationaldisplacement of the slave node ‘s’, X is the initial nodal position for the corresponding node, ΨΨΨΨ is thenodal rotation vector, and ΦΦΦΦmi

( )η is the interpolation tensor with the length parameter η .

76

master element

master node 'm1'

master node 'm2 '

slave node ‘s’

released dof u free node ‘f’

slave element

Fig. 27 The basic concept of a linearly interpolated slide element, where u is the released degree offreedom.

The slave displacement vector qs , the master-released displacement vector qmr in the parametrizationspace, and the displacement vector q in the embedding space read

qd

q

q

q

q

qq

qss

smr

m

m

f

mr

s

1

f f: , : , : ,=FHGIKJ ∈ =

F

H

GGGGGG

I

K

JJJJJJ

∈ =FHGIKJ ∈

+ + + +

ΨΨΨΨE E E6 6 1 6 7

M

n

u

n dof n dof (199)

where q f corresponds to the nodal displacement vector for the free nodes of the slave element. The freenodes are others than the slave node in the slave element; in addition, we denote doff for the degrees offreedom of the free displacement vector q f .

Taking the variation of the master-slave kinematic relation Eqn (198) gives

δq B q B O b

O

Ob

X d

s s mr s 1

m

m

m m m

m m

f= = ∈

=FHG

IKJ ∈ =

′ +

′

F

H

GGGG

I

K

JJJJ∈

×∗ + +

×∗ =

=

∗⋅

⋅

∑

∑

( ) , ( ): ( ) ( ) ( ) ,

( )( )

( ), ( )

( ) ( )

( ).

u u u u u

uu

uu

u

u

nn dof

ii

n

i

ni

i

i i i

i i

δ ΦΦΦΦ ΦΦΦΦ

ΦΦΦΦΦΦΦΦ

ΦΦΦΦ

ΦΦΦΦ

ΦΦΦΦ

Kb g E

E E

6 6 1

6 6 1

1

6

ΨΨΨΨ

(200)

Now, the kinematic operator B , see definition in Eqn (175) p. 70, and its time derivatives & &&B B and readaccording to the notation (199)

BI

BB

O

BB

O

B=FHGIKJ ∈ =

FHGIKJ =

FHGIKJ

+ + ×∗ + +

s s s

f fE6 7 6 1n dof n dof , &&

, &&&&

. (201)

If we exploit a linear interpolation for both master and slave beam element, we have for the interpolationtensors (n dof= 2, f = 6), see Fig. 27,

′ = − ′ = ′′=×∗ ×∗ ×∗ΦΦΦΦ ΦΦΦΦ ΦΦΦΦ1 I I O6 6 2 6 6 6 6, , i , (202)

that yields rather a simple form of the slave kinematic operator and its time derivatives

B I I O b B I I O b B I I O b

bX d X d

bd d

bd d

I I

s s s

m m m m

m m

m m

m m

m m

m m

2 1

2 1

2 1

2 1

= − ∈ = =

=+ − −

−FHG

IKJ =

−−

FHG

IKJ =

−−

FHG

IKJ =

×∗

×∗

( ) , & & & & , && && && && ,

, && &

& &, &&

&& &&

&& &&, .

1 6 19

6 62 2 1 1

2 1

u u u u u ub g d i d iE −−−− −−−−

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

(203)

Finally, the geometric tensor K σmr , see the definition in Eqn (184) p. 72, in the parameter space E19 reads

K f

O O O f

O O f

O 0σmr s

s

s

symm.

( ) =

−F

H

GGGG

I

K

JJJJ∈ ∗ ×∗

0

19 19E , (204)

where the slave force vector f s ∈ ∗E 6 corresponds to the degrees of freedom of the slave displacement vectorqs .

77

Finally, the force vector f mr0 , the tangent tensors K Cmr mr, , and the mass tensor M mr in the parameter spaceE19 can be obtained by Eqn (183) p. 72, when the vector f , the tangent tensors K K K Cload cent gyro, , , , and themass tensor M in the embedding space E25 have been computed. We get these embedded quantities via theassembly procedure since we know the vectors and tensors for the master beam element and for the slavebeam element that correspond to the nodal vector ( , , , )q q q qm m f s1 2 , the part of the nodal vector q , Eqn(199c), in the embedding space E25. The other parts of vectors and tensors, which correspond to the releasedvariable u , vanish for the slide beam in the embedding space E25.

4.1.4 Revolute Joint Beam Elements

In this Section, we give a special beam element with a rigid revolute joint at the one end, see Fig. 28. Be-cause of large rotations, we must not use the same elimination procedure as in the plane case.

A revolute joint has the following kinematic relation for the rotation operators

R R Rs m j= , (205)

where R Rs s= ( )ΨΨΨΨ is the rotation operator for the slave node ‘s’ , R Rm m= ( )ΨΨΨΨ is the rotation operator forthe master node ‘m’, and R R ej j= ( )ϕ is the rotation operator for the revolute joint, where ϕ is the rotationangle (positive or negative without restricting its size), and ej is the initial rotation axis of the revolutejoint.

Taking the variation of (205) yields the kinematic relation between the master, slave, and released dis-placements

δ δ δΘΘΘΘ ΘΘΘΘs jT

m j= +R e ϕ , (206)

where δΘΘΘΘ m mat m∈ TR and δΘΘΘΘ s mat s∈ TR are the variations of the material incremental rotation vectors for the

master and slave nodes, respectively, and ejδϕ is the variation of the relative material rotation, which be-

longs to the vector space mat j matm sT TR R R= because of the axis of the rotation ej remains constant.

Substituting the relation δΘΘΘΘ ==== ΨΨΨΨs s sT δ , where T Ts s= ( )ΨΨΨΨ the tangential transformation maps between the

tangential spaces from matTI onto mat sTR , see Eqn (21) p. 31, for Eqn (206) gives

δΨΨΨΨΨΨΨΨ

s sm

s s jT

m s j

=FHGIKJ

= − −

B

B T R T T e

δδϕ

,

,1 1c h(207)

where the slave kinematic operatorBs ∈ ×∗E3 4 and T Tm m= ( )ΨΨΨΨ .

master node ‘m’

released dof ϕ

free node ‘f’

slave element

slave node ‘s’

free node ‘f’

slave node ‘s’

master node ‘m’

Fig. 28. A revolute joint beam element.

78

The slave displacement vectorqs , the master-released displacement vector qmr and the displacement vectorq on the embedding space E16 read

q q

d

d qq

qs s mr

f

f

m

m

mr

s

= ∈ =

F

H

GGGGGG

I

K

JJJJJJ

∈ =FHGIKJ ∈ΨΨΨΨ

ΨΨΨΨ

ΨΨΨΨE E E3 13 16, , .

ϕ

(208)

Now, the kinematic operator and its time derivatives & &&B B and read according the notation (208)

BI

O BB

O

O BB

O

O B=FHG

IKJ ∈ =

FHG

IKJ =

FHG

IKJ

×∗

×∗

×∗ ×∗

×∗

×∗

×∗

16 13

3 9

19 13 16 13

3 9

16 13

3 9s s sb g d i d iE , && , &&

&& , (209)

where the time derivatives of the slave kinematic tersor are

& & & & & ,

&& && && && & & & & & & && ,

B T R T T R T T R T T e

B T R T T R T T R T T R T T R T T R T T e

s s-1

jT

m s-1

jT

m s-1

jT

m s-1

j

s s-1

jT

m s-1

jT

m s-1

jT

m s-1

jT

m s-1

jT

m s-1

jT

m s-1

j

= + +

= + + + + +

d id i2 2 2

(210)

Finally, the geometric tensor K σmr , compare with Eqn (184) p. 72, in the parameter space E13 reads

K

O O O O 0

O O O 0

O O 0

K kσ

ϕ

ϕϕ

mr

m m m

symm.

=

F

H

GGGGGG

I

K

JJJJJJ

∈ ∗ ×∗

ΨΨΨΨ ΨΨΨΨ ΨΨΨΨ

k

E 13 13 , (211)

where the subtensors are

K GC G R T f T R GC G f T R T

k T R GC G f T e T R e T f

e C G f T e

ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ

m m

m

2 j s s m m j 6 s s s1

jT

m

m j 6 s s s1

j m j j s s

j 6 s s s1

j

= + ∈

= − ∈

= ∈

− −∗ −∗ ∗ −∗ − − ∗ ×∗

∗ −∗ − − ∗ −∗ ∗ −∗ ∗ ×∗

− − ∗ ×∗⋅

1 1 3 3

1 3 1

1 1 1

, , ,

, ~ ,

, ,

ΨΨΨΨ ΨΨΨΨ

ΨΨΨΨ

ΨΨΨΨ

d i d id i

d i

E

E

E

ϕ

ϕϕk Ù

(212)

where f s ∈ ∗E 3 corresponds to the degrees of freedom of the slave rotation vector ΨΨΨΨs . The tensor C6 is de-fined in Section 2.8.

Finally, the force vector f mr0 , the tangent tensors K Cmr mr, , and the mass tensor M mr in the parameter spaceE13 can be obtained by Eqn (183) p. 72, when the vector f , the tangent tensors K K K Cload cent gyro, , , , and themass tensor M in the embedding space E16 have been computed. We acquire these embedded quantities viathe assembly procedure since we know the vectors and tensors for the master beam element and for the slavebeam element that correspond to the nodal vector ( , , , )d df f m sΨΨΨΨ ΨΨΨΨ , the part of the nodal vector q , Eqn(208c), in the embedding space E25. The other parts of vectors and tensors, which correspond to the nodalvector ( , )ΨΨΨΨm ϕ , vanish for the revolute joint beam in the embedding space E16 .

The above special beam elements, which involve holonomic constraints, can be exploited as customary ele-ments in the finite element method. Therefore, the assembly procedure of elements simplifies substantiallycompared with the method in [Jelenić & Crisfield 1996] where constraints are taken account in the assembly

procedure.

79

4.2 Numerical Results

In this Section, we present several numerical examples to illustrate the performance of the proposed totalLagrangian beam element. We choose the simplest finite element approximations based on 2-node iso-parametric beam element with linear interpolation functions. Two-point Gaussian quadrature is used in thenumerical computations of the acceleration vectors and their tangential tensors, and avoiding locking phe-nomena, a one-point quadrature is used for the internal vectors and their tangential tensors.

4.2.1 Cantilever 45-degree bend

e1

e2

e3

F

F

EA GA GA

GJ EI EI

L

=

= = = ⋅

= ⋅ = = ⋅=

600

10 5 10

705 10 833 10

25

72 3

6

52 3

5

π

,

. , .

Fig. 29 The cantilever 45-degree bend beam under the free-end force.

The first example presents a geometrically nonlinear analysis of a cantilever 45-degree bend (with the radiusR = 100, yielding the length of the cantilever beam L = 25π ) placed in the horizontal plane with a verticalstatic force applied at the free end, see Fig. 29. This example was introduced by [Bathe & Bolourchi 1979],and it is the one of the most frequently used example. The length of the rotation vector remains smaller than2π , so that a single rotation parametrization chart is used during computations. The cantilever beam is di-vided into eight 2-node beam elements with the cross-section properties given in Fig. 29.

The free-end displacements are given in Table 1 compared with the other authors’ results. The comparingresults in Table 1, we found a minor difference between each other that is due to the difference of the con-stitutive relations, interpolation functions, rotation parametrization or even cross-section properties. How-ever, our model and the model presented in [Ibrahimbegović et al. 1995] give practically the same responsebecause of the identical rotation parametrization, i.e. the total Lagrangian updating formulation. The finalsolution is obtained by applying the load in six equal load step with the full Newton-Raphson iterationmethod. We also observed the identical quadratic convergence rates as in [Ibrahimbegović et al. 1995].

Table 1. The cantilever bend free-end displacement components (di is the displacement component inthe direction ei ).

Reference Displacement d1 Displacement d2 Displacement d3

Present – 23.696 53.497 – 13.668[Bathe & Bolourchi 1979] – 23.5 53.4 – 13.4[Simo & Vu-Quoc 1986] – 23.48 53.37 – 13.5[Cardona & Géradin 1988] – 23.67 53.50 – 13.73[Crisfield 1990] – 23.87 53.71 – 13.63[Ibrahimbegović 1995] – 23.746 53.407 – 13.601[Ibrahimbegović et al. 1995] – 23.697 53.498 – 13.668

80

4.2.2 Helical beam

e1

e2

e3

m

n

n m

EA GA GA

GJ EI EI

L

= =

= = =

= = ==

50 200

10

10

10

2 34

2 32

π

Fig. 30 Straight cantilever beam under the force n and the moment mR R∈ ∗spatT load.

This example was introduced by [Ibrahimbegović 1997] where the straight cantilever beam is at the free-end

under the force vector n and the spatial moment vector mR R∈ ∗spatT load both in the direction e3 , see

Fig. 30. This type of moment load is nonconservative. The purpose of the example is to show how our

switching beam element, described in Section 4.1.1, works in the situation when the free-end rotation angle

goes over 2π . If the moment load acts only, the free-end rotation angle is equal to 20π that corresponds ten

revolutions.

During the computation, the spatial moment vector mR R∈ ∗spatT , which is a spin vector, is mapped into the

initial vector space of rotation matTI∗ via Eqns (122) p. 59 and (97d) p. 54 giving M T R mI R= ∗ ∗ . This

mapping is necessary in order to get the external nodal load vector of the total Lagrangian beam element

where the rotation vector ΨΨΨΨ( )s T∈mat I .

At the free-end, the out-of-plane displacement component (in the direction e3 ), when loading is increasedinto its final value, is shown in Fig. 31. We have very similar but not identical results compared with [Ibra-himbegović 1997], where out-of-plane displacement is almost symmetrical around zero. We observe thedecrease of finite element error when the cantilever beam is divided into 200 elements compared with themodel of 100 elements.

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

100 elements

200 elements

Displacement

Loa

ding

Fig. 31 The free-end out-of-plane displacement curve versus loading.

81

4.2.3 Fast Symmetrical Top

O e E2 2,

e E1 1,

e E3 3,

mg

mgl

J J

J

== ==

20

5

11 2

3

O e E2 2,

e E1 1,

e E3 3, EA GA GA

GJ EI EI

J J J

A L

mg

= = =

= = == = == ==

2 36

2 33

1 2 3

0

10

10

1 05

135 1

40

.

.ρ

mg

Fig. 32 The fast Symmetrical tops at the initial position, modelled by a rigid body (on the left) and aReissner’s beam element (on the right).

Next, we examine how our beam element acts as a rigid body in a dynamical case. We consider, in this ex-ample, the motion of the fast symmetrical top with one point fixed originally given in [Simo & Wong 1991],Fig. 32. The Newmark time integration scheme is used with scheme parameters β = 1 4/ and γ = 1 2/ , i.e.

the trapezoidal rule.

We obtain the acceleration force vector and its tangent tensors, and the mass tensor of the rigid body ele-ment in the total Lagrangian formulation from Eqns (120) p. 59, (147) p. 65, and (152) p. 66 by keeping theunknown variables, ( , )x ΨΨΨΨ , constant and integrating over the beam length. The top is under the gravitationload −mge3 , see Fig. 32, that cause the moment load in the rigid body element and the force load in thebeam element. The numerical constants, the initial conditions, and the external moment M I in the materialframe , , E E E1 2 3 are for the rigid body element as fol lows:

J E E E E E E

M T E R E

E E T

I I

= ⋅ + ⋅ + ⋅

= − × ∈

= ⋅ = ⋅ =

⊗∗

⊗∗

⊗∗

∗ ∗ ∗ ∗ ∗

−

⋅⋅

5 5 1

0 3 50

1 1 2 2 3 3

3 3

0 1 0 3 01

0 0

mgl T( ) ( )

. , , & ( )

ΨΨΨΨ

ΨΨΨΨ ΩΩΩΩ ΨΨΨΨ ΨΨΨΨ ΩΩΩΩ

d i mat (213)

where m g l, , are the mass of the top, the gravity constant, and the distance of the top gravity center withrespect to the origin O. The initial conditions mean that the angular velocity of the top is about eight revo-lutions per time unit; therefore, a switching beam element is required.

The numerical results of time history, the nutation and precession angles, for the top modelled by the rigidbody element and by the beam element are shown in Fig. 33. Two beam models are exploited: ‘Stiff beam’where constants are given in Fig. 32 and ‘Flexible beam’ where the stiffness constants are reduced into the100th part. Fig. 33 indicates the identical behaviour for the ‘Rigid body’ and the ‘Stiff beam model’ as ex-pected. The results are identical with the computation given in [Mäkinen 2001] where a different rotationupdating scheme, an updated Lagrangian formulation, is used.

0 0.5 1 1.50.3

0.305

0.31

0.315

0.32

0.325

0.33

0.335

Time t [s]

Ang

le o

f nut

atio

n [

rad]

Rigid bodyStiff beamFlexible beam

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

Time t [s]

Ang

le o

f pre

cess

ion

[rad

]

Rigid body

Stiff beam

Flexible beam

Fig. 33 Time history for the notation and precession angles.

82

4.2.4 Hooke’s Joint

e1

e2

α = 45oe3

EA GA GA

GJ EI EI

J J J

A L

= = == = == = == =

2 34

2 36

1 2 3

0

10

10

20 10

1 10ρ ,

L

M tA ( )

400

05. 1 tL

A

B

Fig. 34 Hooke’s joint modelled by two revolute joint beam element.

In the next dynamical example, we explore how Hooke’s joint with 45-degree angle may be modelled by ourrevolute joint beam elements, see Fig. 34. In this case, the finite element model contains two revolute jointbeam elements, Section 4.1.4, such that their master nodes of the local ends are connected together. The firstjoint beam element has the initial rotation axis of the revolute joint ej1 in the direction e2 , and the secondjoint beam has the initial rotation axis ej2 in the direction e3 . In the model, the translational degrees offreedom are frozen. In addition, the beam model is subjected to a moment load at the point A, and the mag-nitude of this load follows the pattern of a hat function, as shown in Fig. 34. Thus after the end of loading,the average angular velocity is 05. .

The computations are carried by the Newmark scheme with the constant time step h = 0 05. . The computedtime history of the difference angle || || || ||ΨΨΨΨ ΨΨΨΨB A− between the rotation vectors is shown in Fig. 35. The re-sponse curve has discontinuations because the rotation vector ΨΨΨΨ ∈matTI is mapped into its complementaryrotation vector ΨΨΨΨ C

mat C∈ TI , see Def 15 p. 26, as expected. Therefore, the response curves shown in Fig. 35for the ‘Beam’ and ‘Exact’ curves, where the latter represents the theoretical difference angle for rigidHooke’s joint, have identical results except the transient response at the beginning. Although the responsecurve of the beam model generates discontinuation, internally the model behaves smoothly, i.e. discontinua-tion indicates only the change of parametrization.

0 5 10 15 20 25 30−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time t [s]

Diff

eren

ce a

ngle

[ra

d]

Beam

Exact

Fig. 35 Time history of the difference angle for Hooke’s joint.

83

4.2.5 Right-Angle Cantilever Beam

e1

e2

e3

F t( )EA GA GA

GJ EI EI

J J J

A L

= = =

= = == = == =

2 34

2 33

1 2 3

0

10

10

20 10

1 10

;

,ρ

L

L

F t( )

5

1 2 t

Fig. 36 The right-angle cantilever beam under loading.

This test example is adopted from [Simo & Vu-Quoc 1988]. An L shape cantilever beam, with materialproperties shown in Fig. 36, is subjected to an out-of-plane concentrated load applied at the elbow. Themagnitude of this load follows the pattern of a hat function, as shown in Fig. 36. The computations are car-ried by the Newmark scheme with the constant time step h = 0 2. . Two finite element models are build: onemodel has 4 linear beam elements and another has 20 elements, totally. The cantilever undergoes a finitefree vibration with combined bending and torsion after the removal of the applied load. In this example, thelength of the rotation vector remains less than π .

The computed time history response for the elbow and tip out-of-plane displacements is shown in Fig. 37.These results correspond to the similar results given in [Ibrahimbegović & Al-Mikdad 1998] and [Simo &

Vu-Quoc 1988]

0 5 10 15 20 25 30−10

−5

0

5

10

15

Time t [s]

Elb

ow d

ispl

acem

ent

20 elements

4 elements

0 5 10 15 20 25 30−10

−5

0

5

10

15

Time t [s]

Tip

dis

plac

emen

t

20 elements

4 elements

Fig. 37 The right-angle cantilever beam. The elbow and tip out-of-plane displacements time history.

84

4.2.6 Two-Component Robot Arm

e1

e2

e3

EA GA GA

GJ EI EI

J J J

A L

= = == = == = == =

2 36

2 35

1 2 3

0

10

10

2 1

1 5ρ

L

w t t( ) ( )= 2θ

θ( )t

L

θ( )t

15.

05. tspherical

joint

Fig. 38 Two-component robot arm: problem data.

This example, introduced in [Ibrahimbegović & Al Mikdad 1998], considers a simple multibody system

composed of two flexible beams connected by a spherical joint. The system is brought in motion by impos-ing simultaneously a displacement in the out-of-plane direction e3 and a rotation about this axis, see inFig. 38. Two finite element models are build: one model has 8 equal beam elements and another has 40 ele-ments. The computations are carried by the Newmark scheme with the constant time step h = 0 01. . In thisexample, the length of the rotation vector exceeds 2π hence switching beams are required.

The computed time history response for the tip horizontal, vertical, out-of-plane displacement componentsand the length of the tip displacement vector is shown in Fig. 39. These results correspond to the similarresults given in [Ibrahimbegović & Al Mikdad 1998] and [Ibrahimbegović & Mamouri 2000].

0 0.5 1 1.5 2 2.5 3−16

−14

−12

−10

−8

−6

−4

−2

0

Time t [s]

Tip

hor

izon

tal d

ispl

acem

ent 40 elements

8 elements

0 0.5 1 1.5 2 2.5 3

0

2

4

6

8

10

12

Time t [s]

Tip

ver

tical

dis

plac

emen

t

40 elements

8 elements

0 0.5 1 1.5 2 2.5 3−2

−1

0

1

2

3

4

5

6

7

8

9

Time t [s]

Tip

out

−of

−pl

ane

disp

lace

men

t 40 elements

8 elements

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14

16

18

20

Time t [s]

Tip

dis

plac

emen

t nor

m

40 elements

8 elements

Fig. 39 The robot arm: The time history of the tip displacement components and its length for the 8 and40 element meshes.

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4.2.7 Two-Component Robot Arm with Revolute Joint

This example is modified form the previous one: a simple multibody system composed of two flexiblebeams connected by a revolute joint with the initial rotation axis ej in the direction e3 . The system isbrought in motion by imposing simultaneously a displacement in the out-of-plane direction e3 and a rota-tion about this axis as in the previous example, see in Fig. 38. Two finite element models are build: onemodel has 8 equal beam elements and another has 40 elements. The computations are carried by the New-mark scheme with the constant time step h = 0 01. .

The computed time history response for the tip horizontal, vertical, out-of-plane displacement componentsand the length of the tip displacement vector is shown in Fig. 40.

This example completes the illustrations of numerical examples for the proposed total Lagrangian beamelments.

0 0.5 1 1.5 2 2.5 3−16

−14

−12

−10

−8

−6

−4

−2

0

Time t [s]

Tip

hor

izon

tal d

ispl

acem

ent 40 elements

8 elements

0 0.5 1 1.5 2 2.5 3−1

0

1

2

3

4

5

6

7

8

Time t [s]

Tip

ver

tical

dis

plac

emen

t40 elements

8 elements

0 0.5 1 1.5 2 2.5 3−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time t [s]

Tip

out

−of

−pl

ane

disp

lace

men

t

40 elements

8 elements

0 0.5 1 1.5 2 2.5 3

0

2

4

6

8

10

12

14

16

18

Time t [s]

Tip

dis

plac

emen

t nor

m

40 elements

8 elements

Fig. 40 The robot arm: The time history of the tip displacement components and its length for the 8 and40 element meshes.

86

5 CONCLUSIONS

In this thesis, we have given a general formulation for flexible multibody mechanics. Since multibody sys-tems are highly constrained, the examination of differential geometry has been necessary to understand theinternal geometry and kinematics of general multibody systems. Therefore, the formulation has been givenin the language of differential geometry: manifolds, tangent spaces and tangent tensors on manifolds, push-forward and pull-back operators, metric tensors, etc.

The rotation manifold, whose elements are rotation operators, has been thoroughly investigated. This studyprobably provides the most important contribution of the thesis: material incremental rotation vectors, mate-rial angular velocity vectors, and material angular acceleration vectors belong to the different tangent spacesof the rotation manifold. Hence, the direct application of the material incremental rotation vector with stan-dard time integration methods yields serious problems: adding quantities which belong to the different tan-gent spaces.

The formulation has been applied to Reissner’s geometrically exact beam theory, giving a new geometri-cally exact beam element that is based on the total Lagrangian updating procedure. The element has the totalrotation vector as the unknown variable and the singularity problems at the rotation angle 2π and its multi-ples are handled by the change of parametrization on the rotation manifold. The consistent stiffness, gyro-scopic, centrifugal, and loading tensors of the total Lagrangian formulation have been given explicitly. Thetotal Lagrangian formulation has several benefits such as all unknown vectors belong to the same tangentspace, no need for secondary storage variables, the path-independence property (in the static case), anystandard time integration algorithm may be used, the symmetric stiffness tensor, a simple form of the kineticenergy and all nonlinear effects are included.

In addition, we have derived the general formulation how to parametrize the constraint manifold, whicharises from point-wise holonomic constraint equations. The parametrization of constraint manifold using thetotal Lagrangian formulation has several benefits: the minimal number of variables, no need for secondarystorage variables, constraint equations are satisfied automatically, the resulting equations of motion areordinary differential equations (not differential-algebraic), and easy to apply time-dependent boundary con-ditions. The constraint manifold parametrization is particularly competitive in the flexible multibody systemwhere the number of degrees of freedom is extensive comparing with the number of constraint equations.Moreover, special beam elements, which involve holonomic constraints, have also been derived as the ex-amples of the formulation. These elements can be exploited as customary elements in the finite elementmethod.

As further study, more special elements involving different holonomic constraint equations can be derived.Expecially, time dependent holonomic constraints require a further research. The constraint manifoldparametrization can moreover be exploited for a special type of nonholonomic constraints: geometric con-straints, where the constraint manifold has a boundary, see Section 2.5. These types of constraints arise fromcontact problems. In addition, the given total Lagrangian formulation can be utilized for deriving a geomet-rically exact shell element based on the total rotation vector as a unknown variable, giving a singularity-freeshell element without the need for secondary storage variables.

87

REFERENCES

1. Abraham, R., Marsden, J. E., Ratiu, T., (1983), Manifolds, Tensor Analysis, and Applications, Addison-Wesley, Massachusetts.

2. Agrawal, O.P., Shabana, A.A., (1986), “Application of Deformable-Body Mean Axis to Flexible Multi-body System Dynamics”, Computer Methods in Applied Mechanics and Engineering, 56, pp. 217-245.

3. Anonymous, (1997), “Abbreviations/Symbols for Terms in TTM”, Mechanism and Machine Theory,32(6), pp. 641-666.

4. Argyris, J., (1982), “An Excursion into Large Rotations”, Computer Methods in Applied Mechanics andEngineering, 32, pp. 85-155.

5. Arnold, V.I., (1978), Mathematical Methods of Classical Mechanics, Springer-Verlag, New York.

6. Avello, A., Garcia de Jalon, J., (1991), “Dynamics of Flexible Multibody Systems using Cartesian Co-ordinates and Large Displacement theory”, International Journal for Numerical Methods in Engineer-ing, 32, pp. 1543-1563.

7. Bathe, K.-J., Bolourchi, S., (1979), “Large Displacement Analysis of Three-Dimensional Beam Struc-tures”, International Journal for Numerical Methods in Engineering, 14, pp. 961-986.

8. Bauchau, O.A., (2000), “On the Modeling of Prismatic Joints in Flexible Multi-Body Systems”, Com-puter Methods in Applied Mechanics and Engineering, 181, pp. 87-105.

9. Behdinan, K., Stylianou, M.C., Tabarrok,B., (1998), “Co-rotational Dynamic Analysis of FlexibleBeams”, Computer Methods in Applied Mechanics and Engineering, 154, pp. 151-161.

10. Belytschko, T., Hsieh, B.J., (1973), “Non-Linear Transient Finite Element Analysis with Convected Co-ordinate”, International Journal for Numerical Methods in Engineering, 7, pp. 255-271.

11. Belytschko, T., Schwer, L., Klein, M.J., (1977), “Large displacement, transient analysis of spaceframes”, International Journal for Numerical Methods in Engineering, 11, pp. 65-84.

12. Bonet J., Wood, R.D., (1997), Nonlinear Continuum Mechanics for Finite Element Analysis, CambridgeUniversity Press, Cambridge.

13. Burke, W.L., (1996), Applied Differential Geometry, Cambridge University Press, Cambridge.

14. Cardona, A., Géradin, M., (1988), “A Beam Finite Element Non-Linear Theory with Finite Rotations”,International Journal for Numerical Methods in Engineering, 26, pp. 2403-2438.

15. Cardona, A., Géradin, M., Doan, D.B, (1991), “Rigid and Flexible Joint Modelling in Multibody Dy-namics using Finite Elements”, Computer Methods in Applied Mechanics and Engineering, 89, pp. 395-418.

16. Cheng, H., Gupta, K.C., (1989), “An Historical Note on Finite Rotations”, ASME Journal of AppliedMechanics, 56, pp. 139-145.

17. Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M., (1989), Analysis, Manifolds and Physics,Part I: Basics, North-Holland, Amsterdam.

18. Craig, R.R., Bampton, M.C.C, (1968), “Coupling of Substructures of Dynamic Analysis”, AIAA Jour-nal, 6(7), pp. 1313-1319.

19. Crisfield, M.A., (1990), “A Consistent Co-Rotational Formulation for Non-linear, Three-Dimensional,Beam Elements”, Computer Methods in Applied Mechanics and Engineering, 81, pp. 131-150.

20. Crisfield, M.A., (1997), Non-Linear Finite Element Analysis of Solids and Structures, Vol. 2: AdvancedTopics, John Wiley & Sons, Chichester.

21. Crisfield, M.A., Jelenić, G., (1999), “Objectivity of Strain Measures in the Geometrically Exact Three-DinemsionalBeam Theory and Its Finite-Element Implementation”, Proceedings of the Royal Society of London A, 455, pp.1125-1147.

22. Debnath, L., Mikusinski, P., (1990), Introduction to Hilbert Spaces with Applications, Academic Press,Boston.

23. Géradin, M., Cardona, A., (2001), Flexible Multibody Dynamics: A Finite Element Approach, JohnWiley & Sons, Chichester.

24. Guyan, R.J., (1965) "Reduction of Stiffness and Mass Matrices", AIAA Journal, 3(2), pp. 380.

25. Hairer, E., Wanner, G., (1991), Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin.

26. Haug, E.J., (1989), Computer Aided Kinematics and Dynamics of Mechanical Systems:Vol. 1 BasicMethods, Allyn and Bacon, Boston.

88

27. Hurty, W.C., (1965), “Dynamic Analysis of Structural Systems using Component Modes”, AIAA Jour-nal, 3(4), pp. 678-685.

28. Ibrahimbegović, A., (1995), “An Finite Element Implementation of Geometrically Nonlinear Reissner’s

Beam Theory: Three-Dimensional Curved Beam Elements”, Computer Methods Applied Mechanics andEngineering, 122, pp. 11-26.

29. Ibrahimbegović, A., (1997), “On the Choice of Finite Rotation Parameters”, Computer Methods in Ap-plied Mechanics and Engineering, 149, pp. 49-71.

30. Ibrahimbegović, A., Al Mikdad, M., (1998), “Finite Rotations in Dynamics of Beams and Implicit

Time-Stepping Schemes”, International Journal for Numerical Methods in Engineering, 41, pp. 781-814.

31. Ibrahimbegović, A., Frey, F., Kozar, I., (1995), “Computational Aspects of Vector-Like Parametrization

of Three-Dimensional Finite Rotations”, International Journal for Numerical Methods in Engineering,38, pp. 3653-3673.

32. Ibrahimbegović, A., Mamouri, S., (2000), “On the Rigid Components and Joint Constraints in Nonlinear

Dynamics of Flexible Multibody Systems Employing 3D Geometrically Exact Beam Model”, ComputerMethods Applied Mechanics and Engineering, 188, pp. 805-831.

33. Iserles, A., Nørsett, S.P., (1999), “On the Solution of Linear Differential Equations in Lie Groups”,Phil. Trans. Roy. Soc. London Ser. A, 357, pp. 983-1019.

34. Iura, M, Atluri, S.N., (1995), “Dynamic Analysis of Planar Flexible Beams with Finite Rotations byusing Inertial and Rotational Frames”, Computers & Structures, 55(3), pp. 453-462.

35. Jelenić, G., Crisfield, M.A., (1996), “Non-Linear ‘Master-Slave’ Relationship for Joints in 3-D Beams

with Large Rotations”, Computer Methods in Applied Mechanics and Engineering, 135, pp. 211-228.

36. Jelenić, G., Crisfield, M.A., (1999), “Geometrically Exact 3D Beam Theory: Implementation of a

Strain-Invariant Finite Element for Statics and Dynamics”, Computer Methods in Applied Mechanicsand Engineering, 171, pp. 141-171.

37. Lanczos, C., (1966), The Variational Principles of Mechanics, University of Toronto Press, Toronto.

38. Makowski, J., Stumpf, H., (1995), “On the ‘Symmetry’ of Tangent Operators in Nonlinear Mechanics”, ZAMM (Z.Angew. Math. Mech.) Applied Mathematics and Mechanics, 75(3), pp. 189-198.

39. Marsden J.E., Hughes, T.J.R., (1983), Mathematical Foundations of Elasticity, Prentice-Hall,Englewood Cliffs.

40. Marsden, J.E., Ratiu, T.S., (1999), Introduction to Mechanics and Symmetry : A Basic Exposition ofClassical Mechanical Systems, Springer-Verlag, New York.

41. Mayo, J., Dominguez, J., (1997), “A Finite Element Geometrically Nonlinear Dynamic Formulation ofFlexible Multibody Systems using a New Displacements Representation”, ASME Journal of Vibrationand Acoustics, 119, pp. 573-581.

42. Mayo, J., Dominguez, J., Shabana, A.A., (1995), “Geometrically Nonlinear Formulations of Beams inFlexible Multibody Dynamics”, ASME Journal of Vibration and Acoustics, 117, pp. 501-509.

43. Mitsugi, J., (1997), “Direct Strain Measure for Large Displacement Analyses on Hinge Connected BeamStructures”, Computers and Structures, 64(1-4), pp. 509-517.

44. Mäkinen, J., (2001), ”Critical Study of Newmark-Scheme on Manifold of Finite Rotations”, ComputerMethods in Applied Mechanics and Engineering, 191(8-10), pp. 817-828.

45. Nikravesh, P.E., Ambrosio, J.A.C., (1991), “Systematic Construction of Equations of Motion for Rigid-Flexible Multibody Systems Containing Open and Closed Kinematic Loops”, International Journal forNumerical Method in Engineering, 32, pp. 1749-1766.

46. Oden , J.T., Reddy, J.N., (1976), Variational Methods in Theoretical Mechanics, Springer-Verlag, Ber-lin.

47. Ogden, R.W., (1984), Non-Linear Elastic Deformations, Ellis Horwood, Chichester.

48. Oran, C., (1973), “Tangent Stiffness in Space Frames”, ASCE Journal of the Structural Division, 99,ST6, pp 987-1001.

49. Petrov, E., Géradin, M., (1998), “Finite Element Theory for Curved and Twisted Beams Based On ExactSolutions for Three Dimensional Solids, Part I: Beam Concept and Geometrically Exact Nonlinear For-mulations”, Computer Methods in Applied Mechanics and Engineering, 165, pp. 43-92.

50. Rabier, P.J., Rheinboldt, W.C., (2000), Nonholonomic Motion of Rigid Mechanical Systems from a DAEViewpoint, SIAM Society for Industrial and Applied Mathamatics, Philadelphia.

51. Reissner, E., (1972), “On One-Dimensional Finite-Strain Beam Theory: The Plane Problem”, Journal ofApplied Mathematics and Physics (ZAMP), 23, pp. 795-804.

89

52. Reissner, E., (1973), “On One-Dimensional Large-Displacement Finite-Strain Beam Theory”, Studies inApplied Mathematics, 52, pp. 87-95.

53. Rheinboldt, W.C., (1986), Numerical Analysis of Parametrized Nonlinear Equations, John Wiley &Sons, New York.

54. Rhim, J., Lee, S.W., (1998), “A Vectorial Approach to Computational Modelling of Beams UndergoingFinite Rotations”, International Journal for Numerical Method in Engineering, 41, pp. 527-540.

55. Riemer, M., Wauer, J., (1988), “Flexible Robot Models with Revolute and Prismatic Joits - Handling ofClosed Loops”, Mechanics Research Commications, 15(6), pp. 381-387.

56. Rosenberg, R.M., (1980), Analytical Dynamics of Discrete Systems, Plenum Press, New York.

57. Selig, J.M., (1996), Geometrical Methods in Robotics, Springer-Verlag, New York.

58. Shabana, A.A., (1989), Dynamics of Multibody Systems, John Wiley & Sons, New York.

59. Shabana, A.A, Hussien, H., Escalona, J., (1998), “Application of the Absolute Nodal Coordinate For-mulation to Large Rotation and Large Deformation Problems”, ASME Journal of Mechanical Design,120, pp. 188-195.

60. Simo, J.C., (1985), “A Finite Strain Beam Formulation. The Three-Dimensional Dynamic Problem. PartI”, Computer Methods in Applied Mechanics and Engineering, 49, pp. 55-70.

61. Simo, J.C., (1992), “The (Symmetric) Hessian for Geometrically Nonlinear Models in Solid Mechanics:Intrinsic Definition and Geometric Interpretation”, Computer Methods Applied Mechanics Engineering,96, pp. 189-200.

62. Simo, J.C., Marsden, J.E. Krishnaprasad, P.S., (1988), “The Hamiltonian Structure of Nonlinear Elas-ticity: The Material and Convective Representation of Solids, Rods, and Plates”, Archive for RationalMechanics and Analysis, 104, pp. 125-183.

63. Simo, J.C., Wong, K.K, (1991), “Unconditionally Stable Algotithms for Rigid Body Dynamics thatExactly Preserve Energy and Momentum”, International Journal for Numerical Method in Engineering,31, pp. 19-52.

64. Simo, J.C., Vu-Quoc, L., (1986), “A Three-Dimensional Finite Rod Model. Part II: Computational As-pects”, Computer Methods in Applied Mechanics and Engineering, 58, p. 79-116.

65. Simo, J.C., Vu-Quoc, L., (1987), “The Role of Non-Linear Theories in Transient Dynamic Analysis ofFlexible Structures”, Journal of Sound and Vibration, 119(3), pp. 487-508.

66. Simo, J.C., Vu-Quoc, L., (1988) “On the Dynamics in Space of Rods Undergoing Large Motion - AGeometrically Exact Approach”, Computer Methods in Applied Mechanics Engineering, 66, pp. 125-161.

67. Simo, J.C., Vu-Quoc, L., (1991), “A Geometrically-Exact Rod Model Incorporating Shear and Torsion-Warping Deformation”, International Journal of Solids and Structures, 27(3), pp. 371-393.

68. Spring, K., (1986), “Euler Parameters and the Use of Quaternion Algebra in the Manipulation of FiniteRotations: A Review”, Mechanism and Machine Theory, 21(5), pp. 365-373.

69. Stuelpnagel, J., (1964), “On the Parametrization of the Three-Dimensional Rotation Group”, SIAM Re-view, 6(4), pp. 422-430.

70. Stumpf, H., Hoppe, U., (1997), “The Application of Tensor Algebra on Manifolds to Nonlinear Contin-uum Mechanics – Invited Survey Article”, ZAMM (Z. Angew. Math. Mech.) Applied Mathematics andMechanics, 77(5), 327-339.

71. Truesdell, C., (1977), A First Course in Rational Continuum Mechanics, Academic Press, New York.

72. Wang, C.-C., Truesdell, C., (1973), Introduction to Rational Elasticity, Noordhoff, Leyden.

73. Wehage, R.A., Haug, E.J., (1982), “Generalized Coordinate Partitioning for Dimension Reduction inAnalysis of Constrained Dynamic Systems”, ASME Journal of Mechanical Design, 104, pp. 247-255.

74. Yoo, W.S., Haug, E.J., (1986), “Dynamics of Articulated Structures, Part I Theory”, Journal of Struc-tural Machines, 14(1), pp. 105-126.

75. Zanna, A., (1999), “Collocation and Relaxed Collocation for the Fer and the Magnus Expansions”,SIAM J. Numer. Anal, 36(4), pp. 1145-1182.