Modal Reduction Proceduresfor Flexible Multibody Dynamics∗

Valentin Sonneville†, Olivier Bruls‡ and Olivier A. Bauchau††Department of Aerospace Engineering, University of Maryland

College Park, Maryland 20742‡Department of Aerospace and Mechanical Engineering,

University of Liege, Liege, Belgium

Abstract

The comprehensive simulation of flexible multibody systems calls for the ability to modelvarious types of structural components such as rigid bodies, beams, plates, and kinematicjoints. Modal components offer additional modeling versatility by enabling the treatment ofcomplex, three-dimensional structures via modal reduction procedures based on the small de-formation assumption. The first part of this paper describes the mode-acceleration method,which appears to be the method of choice because it imposes no restriction on the selectionof the modal basis. Herting’s and Craig-Bampton’s methods are particular cases of the mode-acceleration method and although more complex, Rubin’s method is equivalent to Herting’smethod. The second part of the paper deals with modal components undergoing large motion.The problem is formulated within the framework of the motion formalism, which leads todeformation measures that are objective and tensorial; they differ from those obtained froma classical description of the kinematics of the problem based on independent displacementand rotation fields. Derivatives are expressed in the material frame, leading to a remarkableproperty: the tangent stiffness and mass matrices depend on the configuration of the modalcomponent through the deformation measures only and hence, geometrically non-linear prob-lems can be solved with constant iteration matrices. In contrast, when using the classicalformalism to represent the kinematics of the problem, the same matrices are configurationdependent, i.e, the iteration matrix must be recomputed frequently during a simulation.

1 Introduction

Multibody systems often involve fatigue sensitive components that present complex three-dimensional geometries and undergo small deformations. For those components, two questionsmust be answered: (1) what is the impact of the flexibility of the component on the dynamicresponse of the overall system and (2) what is the three-dimensional stress field in such compo-nent? The prediction of the three-dimensional stress fields in these complex components and theassessment of their fatigue life require the use of three-dimensional finite element models of the com-ponents. In applications of the finite element, fine mesh sizes are often required to capture complexgeometries, intricate material property distributions, and the resulting three-dimensional strainfields that determine structural flexibility. Fine mesh sizes result in a set of algebraic equationsthat is prohibitively large for direct dynamic simulation and leads to the appearance of spurious,

∗Computer Methods in Applied Mechanics and Engineering, to appear.

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high-frequency modes that are of little physical relevance. Finally, geometric complexity does notimply a complex dynamic response of the structure that can often be captured accurately by a smallnumber of low frequency modes.

To overcome the prohibitive size of finite element models, order-reduction techniques have beendeveloped. By projecting the original equations of motion onto a subspace of the configurationspace, they produce a reduced system of equations that represents the dynamic behavior of theoriginal system accurately within a desired frequency range. Detailed information about the originalcomponent can be obtained by expanding the solution of the reduced system in a recovery step.

Component mode synthesis techniques provide efficient tools to reduce the order of three-dimensional systems through a substructuring approach that exploits the second-order, lightly-damped nature of mechanical systems. The order-reduction process, called “modal reduction,”is based on the selection of mode shapes of the component. Several approaches have been pro-posed to analyze linear structural dynamics problems, such as Hurty’s [1], Craig-Bampton’s [2],MacNeal’s [3], Rubin’s [4], and Herting’s [5] methods. The selection of modes is application de-pendent [6–10]. Modal bases could include eigenmodes of the fully-clamped, partially-clamped,or free-floating structure; experimentally measured modes can be used as well. Other recent ap-proaches consist in projecting the equations onto modes that are not eigenmodes [11–14] or thatinclude information about the non-linear behaviour of the structure [15–20].

Flexible multibody systems involve two types of structural components: those that will betreated via multibody dynamics techniques, called “multibody components,” and those whose dy-namic behavior can be approximated using modal reduction techniques, called “modal components.”In automotive applications, for example, the suspension of the vehicle is analyzed through multi-body dynamics techniques but the behavior of the car body can be approximated using a few modesonly. Indeed, ride quality is not affected by the high-frequency modes of the body. Similar examplesabound in mechanical and aerospace applications.

The connections between the multibody and modal components are established through “in-terface” or “boundary nodes” whose treatment is intertwined with the reduction process. Twoapproaches are possible. In the first approach, the displacements at the interface nodes are in-cluded in the reduced set of variables of the modal component through an appropriate selection ofthe modes: connection to the other elements of the model is then established easily through themodel assembly procedure. In the second approach, the reduced set of variables includes the modalamplitudes of the selected modes only and the relationships between the displacements at the in-terface nodes and the modal amplitudes are treated as kinematic constraints, typically enforced viaLagrange’s multiplier technique.

The first approach involves a smaller number of variables because it eliminates the need forLagrange multipliers but leads to densely populated reduced matrices. With the second approach,it is possible to obtain diagonal reduced matrices with appropriate, yet numerically challengingoperations. This advantage, however, is offset by a larger number of unknowns (the Lagrangemultipliers used to enforce the constraints) and densely populated constraint gradients. The latterstrategy has been widely used in the literature; Geradin and Rixen [21] give a succinct derivation ofthis approach. Clearly the choice between the two approaches is a compromise between simplicityand computational efficiency. In this work, the first approach is selected.

The nature of the degrees of freedom (dofs) at the interface nodes could also be an issue whenconnecting modal components to the multibody system. Typically, multibody components, suchas nonlinear beam elements or kinematic joints, are formulated using rotation and displacementvariables at the nodes whereas standard three-dimensional finite elements involve displacementvariables only. This mismatch between dofs is often treated via multi-point constraints but physicalinsight is required to impose meaningful constraints.

Craig-Bampton’s method is the most widely used component mode synthesis technique. Ituses fully-clamped modes, i.e., the natural vibration mode shapes of the structure clamped at all

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interface nodes, a strategy that is appropriate for linear structural dynamics applications. Be-cause modal amplitudes vanish at the interface nodes, connecting modal to multibody componentsis straightforward. The popularity of Craig-Bampton’s method stems from the simplicity of itsimplementation.

Unfortunately, the use of fully-clamped modes is not necessarily appropriate for flexible multi-body systems. Consider, for example, the coupled rotor-fuselage-landing gear dynamic analysis of ahelicopter: it makes sense to treat the fuselage as a modal component whereas the rotor and landinggears are multibody components; the interface nodes are the points at which the fuselage connectsto the rotor and landing gears. Craig-Bampton’s method calls for the use of fully-clamped modes,i.e., the modes of the fuselage clamped at all interface nodes. The natural frequencies and modeshapes of these fully-clamped modes bear little resemblance to those experienced by the aircraftin flight: while the fuselage remains connected to the rotor and landing gears at all times, thefuselage is never clamped at those points. Clearly, the use of fully-clamped modes over-constrainsthe deformation modes of the fuselage. In general flexible multibody systems, modal componentswill be connected to other multibody and modal components all undergoing complex motions andhence, the fully-clamped modes used by Craig-Bampton’s method rarely form a suitable basis formodal reduction.

Rubin’s method is able to overcome these limitations by using free-floating modes, i.e., thenatural vibration mode shapes of the free-floating structure. These modes appear to be more suitablefor flexible multibody system applications but also leads to higher complexity in the description ofthe interface because modal amplitudes affect the displacements at the interface node. Consideringthe coupled rotor-fuselage-landing gear problem discussed earlier, it is clear that the inertia of therotor affects the modes of vibration of the fuselage in flight. When evaluating the free-floatingmodes, the inertia of the rotor can be taken into account. In contrast, Craig-Bampton’s methodignores this effect by clamping the fuselage at the interface points.

Although it has received little attention in the literature, Herting’s method is a versatile compo-nent mode synthesis technique that combines the simplicity of Craig-Bampton’s method with theuse of free-floating modes. It is shown that Herting’s and Rubin’s method are equivalent, i.e, theyspan the same subspace of modes and the reduced sets of variables used by the two methods arerelated by a one-to-one mapping.

In this paper, the derivation of the modal reduction process starts from the mode-accelerationmethod, which has been discussed by numerous authors [22–25]. The mode-acceleration methodis a general approach to modal reduction that does not impose restriction on the nature of thevibration modes used in the reduction process and yet, retains the simplicity of Craig-Bampton’smethod. Craig-Bampton’s and Herting’s methods are presented as particular cases of the mode-acceleration method: the former uses the modes of the fully-clamped structure whereas the latteruses the rigid-body and elastic modes of the free-floating structure.

Modal reduction techniques have been used to model flexible multibody systems for decades [26–33]. The large amplitude motions encountered in this context complicate the formulation. Theprevailing approach is to decompose the overall motion of modal components into (1) a rigid-body motion represented by the motion of a frame, called the “floating frame of reference,” and(2) a superimposed elastic motion. Typically, the elastic motion and the resulting deformationmeasures are assumed to remain small and the modal reduction procedure discussed in the previousparagraphs is then applied to the deformation measures to reduce the size of the system. In themultibody literature, this approach is referred to as the “floating frame of reference” approach [28]and numerous authors have presented variations of this technique [30,34–41].

While the decomposition of the overall motion into rigid-body and elastic motions is ratherintuitive, it is not defined uniquely, even under the small deformation assumption. Furthermore,if the modal reduction process includes the rigid-body motions of the elastic structure, as is thecase when free-floating modes are used, those rigid-body motions could be double counted, adding

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to the ambiguity of the decomposition. Many authors selected rather arbitrarily a floating framethat supports the rigid-body motions. For instance, Cardona and Geradin [37] or Bauchau andRodriguez [30, 31] attached the floating frame at an interface node of the modal component whileAmbrosio [36, 38] connected the floating frame to the center of mass of the modal component.Cardona [39] and Ellenbroek and Schilder [42] selected a floating frame that is not attached ata material point of the modal component: its position and orientation are defined as a weightedaverage of the corresponding quantities at all points of the component. In the present work, thelocation of the floating frame is determined by a linearized version of Tisserand’s criterion [43]: thelocation of the floating frame minimizes the elastic kinetic energy linearized with respect to thedeformations. This approach was adopted by several authors, e.g., [9, 40,44,45].

To handle the large amplitude motion of modal components, the local frame motion formalismproposed by Sonneville et al. [46, 47] will be used. In the motion formalism, position and rotationvariables are coupled and treated as a unit referred to as a “frame” or a “motion.” This frameworkcomes with powerful tools to describe the kinematics of the structure: the treatment of rigid-bodymotions is streamlined, which results naturally in deformation measures that are both objectiveand tensorial and in a consistent evaluation of the inertial forces. Within this framework, materialderivatives are used systematically, filtering out the geometric nonlinearities from the equilibriumequations. The formulation of geometrically nonlinear modal components becomes a natural exten-sion of that of their geometrically linear counterparts. The reduced level of nonlinearity leads toimproved computational efficiency: if the deformations do not change significantly, the governingequations for modal components can be solved with constant tangent matrices, i.e., the computa-tional burden associated with the simulation of nonlinear modal components becomes comparableto that associated with the simulation of their linear counterparts. Although the proposed methodis presented within the framework of the motion formalism, it can accommodate structural matri-ces and mode shapes obtained from finite element formulations based on the classical kinematicdescription.

The paper is organized as follows. The setting for the modal reduction of linear equationsof motion is described in section 2. The kinematics of modal components is presented and thedecomposition of the motion into rigid-body and elastic motions is introduced. The modal reductionvia the mode-acceleration method is also introduced in this section. Section 3 introduces some basicelements of the local-frame motion formalism. The reduced equations of motion for large amplitudemotions are developed in section 4. The paper concludes with numerical examples in section 5.

2 Modal reduction procedure for linear systems

This section presents a general procedure for the modal reduction of systems described by linearequations of motion. The concepts of the floating frame of reference and of the associated deforma-tion measures are introduced in this section. Although the concept of floating frame of reference issuperfluous in the linear case, it becomes an indispensable ingredient for the nonlinear formulation.Because the kinematic description is linear, the algebraic developments presented in the sectionare rather straightforward and lay the foundations for the more complex developments presentedin section 4 for multibody systems. The mode-acceleration method is discussed first and specificmethods are then derived as special cases.

2.1 Kinematics of a structural component

Consider the linear equations of motion of a spatially-discretized, free-floating structural components

M u+K u = f, (1)

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where u ∈ Rn is the vector of nodal displacements, M and K are the n × n symmetric mass andstiffness matrices, respectively, and f ∈ Rn is the vector of external loads. The term “nodal dis-placement” is used in a generic sense because array u often contains a combination of displacementand rotation components. The mass and stiffness matrices are evaluated in the unstressed, refer-ence configuration of the component, usually via the finite element method. Energy dissipationmechanisms in the structure are not addressed in this paper.

The natural vibration mode shapes of the component include six rigid-body modes, denoted U i,i = 1, 2, . . . , 6, stored as the columns of matrix U , of size n × 6. Each rigid-body mode satisfiesequation K U i = 0, i.e., they are associated with a vanishing natural frequency and based on simplekinematic considerations, can be expressed as

U =

...

...I −x0j0 I...

...

, (2)

where x0j is the skew-symmetric matrix built from the initial position vector x0j ∈ R3 of node j. Ineq. (2), contribution [I − x0j ] is associated with displacements at node j and contribution [O I] isassociated with rotations at node j, if they appear in the formulation of the structural component.Clearly,

K U = 0, (3)

expresses the fact that rigid-body motions generate no elastic forces.The natural vibration mode shapes of the component also include elastic modes, denoted X i, i

= 1, 2, . . . , ne = n − 6, stored as the columns of matrix X, of size n × ne. Each elastic mode isthe solution of eigenvalue problem (K − ω2

iM)X i = 0, where the non-vanishing natural vibrationfrequencies are denoted ωi, i = 1, 2, . . . , ne. The frequencies are gathered along the diagonal ofmatrix ω, of size ne × ne, and hence, the elastic modes satisfy equation

KX = M X ω2. (4)

The elastic modes are orthogonal to the rigid-body modes in the metric of the mass matrix, i.e.,

UTM X = 0. (5)

This property plays a key role in this work. The elastic modes are orthogonal to each other inthe metric of the mass and stiffness matrices, i.e., (XTM X) and (XTKX) are diagonal matrices.

With the definition of the rigid-body modes given by eq. (2), matrix UTM U is not diagonal, but it

is always possible to find a linear combination of the rigid-body modes that renders matrix UTM Udiagonal.

Typically, the reduction process is applied to free-floating components that are to be connectedto other components. In such case, the structural component features rigid-body modes. In othercases, however, the structural component to be reduced could be constrained and hence, presents norigid-body mode. The simplification of the proposed methodology for this latter case is discussedin section 2.8.

2.2 Floating frame and deformation measures

It is customary to decompose the displacement field of modal components as

u = U uF + e, (6)

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Figure 1: Configuration of the modal component. For clarity of the figure, the configurations aredepicted far from each other.

as illustrated in fig. 1 in a schematic manner. First, displacement field U uF defines a rigid-bodymotion and describes the fictitious configuration of the structure if it were to be infinitely rigid;hence, this configuration is labeled “rigid-body configuration” in fig. 1. The floating frame ofreference dofs are denoted uF ∈ R6, where array uF stores the three translation and three rotationdofs of a (possibly fictitious) material point. Because matrix U is of the form given by eq. (2),this material point is located at the origin of the inertial frame in the reference configuration.Second, array e ∈ Rn defines an elastic motion that is superimposed to the rigid-body motion.Often, it is called the “elastic displacement field” but it could involve both displacement or rotationcomponents; to avoid confusion, it will be referred to as the array of “deformation measures.” Thesuperposition of these two fields defines the actual, deformed configuration of the structure, labeled“deformed configuration” in fig. 1. The decomposition expressed by eq. (6) is not unique becauseit describes the n components of the displacement field in terms of n+ 6 quantities.

Decomposition (6) can also be seen as a definition of the deformation measures,

e = u− U uF . (7)

Because the matrix of rigid-body modes is constant, the variations and time derivatives of decom-position (7) are obtained easily, δe = δu− U δuF , e = u− U uF , and e = u− U uF .

2.2.1 Definition of the floating frame

Because decomposition (6) involves six redundant variables, six constraints must be imposed to pro-vide a unique definition of the deformation measures and of the floating frame of reference. Ratherthan prescribing the motion of the floating frame of reference arbitrarily, six linear combinations ofthe deformation measures are required to vanish,

Y T e = 0, (8)

where matrix Y ∈ Rn×6 is of full column rank. A unique decomposition of the displacement fieldis now obtained. First, the displacement of the floating frame is obtained by pre-multiplication ofeq. (7) by Y T : Y T (u− U uF ) = 0. If matrix Y TU is non-singular, this implies

uF =[(Y TU

)−1Y T]u = QY u. (9)

6

The rigid-body displacement for all nodes, U uF , and the field of deformation measures now become

U uF =[

U(Y TU

)−1Y T]u = P Y

‖ u, (10a)

e =[I − U

(Y TU

)−1Y T]u = P Y

⊥u, (10b)

where the second equation follows eq. (7). Matrices P Y

⊥ and P Y

‖ are the projectors onto the subspace

orthogonal and parallel to the rigid-body modes, respectively. Because P Y

⊥U = 0 and (P Y

⊥)n = P Y

⊥,

matrix P Y

⊥ is indeed a projector and P Y

‖ = I − P Y

⊥ is the complementary projector, which satisfies

P Y

‖ U = U . Equations (10a) and (10b) can now be interpreted as follows: the rigid-body motion

and deformation measures, denoted U uF and e, respectively, are obtained by projecting the totaldisplacement field in the subspace of and in the subspace orthogonal to the rigid-body modes,respectively.

With this definition, the deformation measures are objective, i.e., they remain invariant underthe superposition of rigid-body motions [48]. Consider a given displacement field, u∗ = U u∗F + e∗,

with u∗F = QY u∗ and e∗ = P Y

⊥u∗. Under the superposition of a rigid-body motion, U α, the

displacement field becomes u∗+U α. In view of eq. (9) and eq.(10b), the floating frame of reference

and the deformation measures become QY (u∗+U α) = u∗F +α and P Y

⊥(u∗+U α) = e∗, respectively.

Thus, the superposition of a rigid-body motion is reflected in the motion of the floating frame onlywhile the deformation measures remain unaffected.

Condition (8) can also be obtained from the following minimization problem: for a given dis-placement field, u, find uF that minimizes eTAe,

uF = arg minuF∈R6

(eTAe

). (11)

where A ∈ Rn×n is an arbitrary symmetric matrix. Using eq. (7), the stationarity condition is given

by δeTAe = δuTF (UTAe) = 0, i.e., when UTAe = 0. This condition is equivalent to eq. (8) for

Y = AU . If the Hessian of the system, UTAU , is positive-definite, the stationary point is indeeda minimum.

Thus far, the only requirement on matrix Y is that Y TU be invertible. Equation (5) suggests anatural choice for the deformation measures: if they must span the subspace normal to that of therigid-body modes, they must lie in the subspace spanned by the ne elastic modes,

e = X µ, (12)

where array µ ∈ Rne stores the modal amplitudes. Equation (5) now implies

UTM e = 0, (13)

and comparison with eq. (8) yields Y = M U . Condition (13) is the solution of minimization

problem (11) for A = M . The Hessian of the system, UTM U , is positive-definite because themass matrix is positive-definite, and condition (13) leads indeed to a minimum. The same solutionis obtained if the quantity to be minimized is eTM e, leading to the following interpretation ofthe motion of the floating frame: “The motion of the floating frame minimizes the kinetic energyassociated with the deformation measures.” With condition (13), eq. (9) for the floating frame dofsbecomes

uF =[(UTM U

)−1UTM

]u = Qu (14)

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and projections (10a) and (10b) become

U uF =[

U(UTM U

)−1UTM

]u = P ‖u, (15a)

e =[I − U

(UTM U

)−1UTM

]u = P⊥u. (15b)

Operators P⊥ and P ‖ are the projectors onto the subspace orthogonal and parallel to the rigid-body

modes, respectively, both in the metric of the mass matrix.

Remark For static problems, the use of the mass matrix is awkward. A more suitable choice isY = U , which leads to invertible matrix Y TU = UTU .

2.3 Modal reduction

Modal components must be able to represent rigid-body motions exactly. Therefore, rigid-bodymodes cannot be approximated and the modal reduction applies to the deformation measures only.A “modal reduction matrix,” denoted Ψ, is introduced and the deformation measures are expressedas

e = Ψ e, (16)

where array e stores the “reduced deformation measures.” Notation • indicates quantities associatedwith the reduced system. Typically, the modal reduction matrix is a rectangular matrix withsignificantly less columns than lines, leading to a considerable reduction of the number of dofs.

In this work, the deformation measures at the interface nodes are included in the set of reduceddeformation measures because it eases the connection of modal to multibody components. The ndofs of the structure are partitioned into the nB dofs at the interface nodes and the nI internaldofs, such that nB + nI = n. The interface and internal nodes are indicated in fig. 1. Thefollowing partitions of the arrays of deformation measures and reduced deformation measures arenow introduced

e =

{eBeI

}, e =

{eBη

}, (17)

where array eB, of size nB, stores the deformation measures at the interface nodes, array eI , of sizenI , stores the deformation measures at the internal nodes, and array η, of size nη, stores the modalamplitudes. Because the array of reduced deformation measures contains both modal amplitudesand interface node deformation measures, the modal reduction process is intertwined with theselection of interface nodes. A valid partition implies nB + nη ≤ n and typically, nB + nη � n.Modal reduction matrix Ψ now takes the more explicit form

Ψ =

[I 0

ΨS

ΨD

], (18)

where matrix ΨS, of size nI×nB, is called the “static mode matrix” and matrix Ψ

D, of size nI×nη,

is called the “dynamic mode matrix.” The selection of the modal reduction matrix is discussed inthe next section, in the context of the mode acceleration method.

The deformation measures at the interface nodes are retained and the deformation measures atthe internal dofs become

eI = ΨSeB + Ψ

Dη. (19)

The partitioning of the deformation measure field introduced in eq. (17) is reflected in the followingpartitions of the stiffness, mass, rigid-body eigenmode, and elastic eigenmode matrices

K =

[KBB

KBI

KIB

KII

], M =

[M

BBM

BI

MIB

MII

], U =

[UB

UI

], X =

[XB

XI

]. (20)

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After reduction, modal components involve three types of variables: the deformation measures atthe interface nodes, eB, the modal amplitudes, η, and the floating frame dofs, uF . More specifically,

e =

{uB − UB

uFη

}= u− U uF , u =

{uBη

}, U =

[UB

0

], (21)

where array u stores the interface node dofs and the modal amplitudes of the reduced modalcomponent. Rigid-body motions are handled by the interface nodes exclusively because the portionof matrix U related to the modal amplitudes vanishes.

Remark The present framework also applies to the case where Ψ = I, i.e., all the nodes of thecomponent are retained as interface nodes. If such approach is applied to a single finite element, itleads to the corotational finite element formulation [49,50].

2.3.1 The mode-acceleration method

The mode-acceleration method has been used by numerous authors [22–25]. It is a general ap-proach to the construction of the modal reduction matrix and will be presented first because Craig-Bampton’s and Herting’s methods are special cases of the mode-acceleration method.

In view of eq. (3), the governing equations of motion of modal components, eq. (1), can be recastas K e = f −M u. In the mode-acceleration method, accelerations are expanded as follows

u = V η, V =

[VB

VI

], (22)

where the nη columns of matrix V store the modes selected for the expansion. The mode-accelerationmethod imposes no restriction on the nature of the modes stored in matrix V : they could includeeigenmodes of the constrained or unconstrained structure, rigid-body modes, or even experimentallymeasured modes.

Using partitions (17) and (20), the deformation measures at the internal nodes become eI =K−1IIfI− K−1

IIKIBeB − K−1

II[M

IBVB

+ MIIVI]η. Comparing this result with eq. (19) yields the

static and dynamic mode matrices as

ΨS

= −K−1IIKIB, Ψ

D= −K−1

II

[M

IBVB

+MIIVI

]. (23)

The static mode matrix depends on the stiffness matrix only whereas the dynamic mode matrixdepends on the stiffness matrix, the mass matrix, and the selected modes. If external loads areapplied at the internal nodes, f

I6= 0, the reduction basis could be supplemented by including some

modes related to the loading pattern; this situation is not investigated here.A key property of the mode-acceleration method is that static mode matrix Ψ

Sprovides an

exact representation of the rigid-body modes based on their values at the interface nodes only,

Ψ U =

[UB

ΨSUB

]=

[UB

UI

]= U, (24)

where the second equality stems from K U = 0, which implies KIBUB

+ KIIUI

= 0 and hence,ΨSUB

= UI.

If matrix V stores the eigenmodes of the structure under certain boundary conditions at theinterface nodes, the expression of the dynamic mode matrix can be further simplified. Indeed, theeigenmodes satisfy identity K

IBVB

+ KIIVI

= [MIBVB

+ MIIVI]ω2, where diagonal matrix ω

stores the natural frequencies associated with the eigenmodes appearing in matrix V . Multiplying

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by K−1II

leads to VI− Ψ

SVB

= K−1II

[MIBVB

+ MIIVI]ω2. In case of non-vanishing natural fre-

quencies, the dynamic mode matrix reduces to ΨD

= VI− Ψ

SVB

. Factor −ω−2 was eliminated

by scaling the modal amplitudes as η := −ω2η. This manipulation is not valid for rigid-bodymodes because their frequencies vanish. In that case, eq. (24) yields the dynamic mode matrix as

ΨD

= −K−1II

[M

IB+M

IIΨS

]UB

.

2.3.2 Floating frame and reduced deformation measures

Using eqs. (16) and (24), condition eq. (13) can be expressed in terms of reduced quantities only

UTM e = 0, (25)

where M = ΨTM Ψ is the reduced mass matrix of the modal component. Note that the rigid-bodymodes of the complete structure are not needed; only their contributions at the interface nodesappear.

First, consider the case where the modal reduction matrix is built from a subset of the elasticmodes of the unconstrained structure. In view of eq. (5), this selection of modes implies UTM Ψ =

UTM = 0 and condition (25) is satisfied for any e. Equation (21) shows that the reduced variables

are expressed in terms of the interface node dofs, the modal amplitudes, and the floating frame dofs.On the other hand, eq. (24) indicates that with the mode acceleration method, all the rigid-body

modes can be represented by the modes that are included in the static mode matrix, i.e., UTM

does not vanish. Therefore, the six constraint (25) are not satisfied and can be used to eliminatethe six degrees of freedom of floating frame. Some authors, such as Nikravesh [40], prefer to keepthe floating frame dofs as unknown variables of the reduced system and enforce conditions (25)as constraint equations via a set of six Lagrange multipliers. As shown below, the elimination ofthose dofs is a straightforward operation that is internal to the reduced component and that canbe implemented efficiently; hence, the use of Lagrange multipliers can be avoided easily.

Introducing eq. (21) into condition (25) leads to UTM(u−U uF ) = 0 and eq. (14) for the floating

frame dofs becomes

uF =

[(UTM U

)−1UTM

]u (26)

The following expressions result,

U uF =

[U(UTM U

)−1UTM

]u = P ‖u, (27a)

e =

[I − U

(UTM U

)−1UTM

]u = P⊥u, (27b)

where matrices P⊥ and P ‖ are the projectors operating in the reduced modal space. Projec-

tions (27a) and (27b) echo projections (15a) and (15b), respectively, written for the completesystem. Due to the structure of the projectors, both nodal displacements and modal amplitudescontribute to the deformation measures at the boundary nodes. These equations can be interpretedas follows: the rigid-body motion and deformation measures of the reduced system, denoted U uFand e, respectively, are obtained by projecting the reduced displacement field in the subspace or-thogonal to and in the subspace of the rigid-body modes in the metric of the reduced mass matrix,respectively.

In view of eq. (21), the deformation measures become e = Ψ e = Ψ u − Ψ U uF = Ψ u − U uF ,where the last equality follows from eq. (24). Comparing this result with eq. (7) then yields

u = Ψ u. (28)

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The same modal reduction matrix is used for the displacement field and deformation measures,adding to the simplicity of the method. Because matrix Ψ is constant, the same relationship holdsfor derivatives. For instance, a time derivative of eq. (28) yields v = Ψ v, where v = u and v = ˙uare the sets of velocities and reduced velocities, respectively.

2.3.3 Reduced equations of motion

The equations of motion of the reduced system will be obtained from work and energy argumentsdeveloped in terms of the floating frame and the deformation measures because this approachgeneralizes to nonlinear problems easily. The virtual work done by the elastic forces is δWE =δuTK u and introducing eq. (7) leads to δWE = (δe + U δuF )TK(e + U uF ) = δeTK e, where thecontributions of the floating frame are filtered out due to orthogonality relationship (3). In view ofmodal reduction (16), δWE = δeT K e, where the reduced stiffness matrix is

K = ΨTK Ψ =

[KBB

+KBI

ΨS

0

0 ΨT

DKII

ΨD

]. (29)

The reduced stiffness matrix is symmetric and block diagonal because the static mode matrixsatisfies identity K

IB+ K

IIΨS

= 0. The first block is always non-singular but the second blockenjoys this property only if the dynamic mode matrix, Ψ

D, presents full column rank. While the

basis for the modal reduction, matrix V , can be chosen arbitrarily, eq. (23) shows that the dynamicmode matrix depend on this choice; the modal basis must be selected to ensure that the dynamic

mode matrix presents full column rank. In view of eq. (27b), the reduced elastic forces fE

, defined

as δWE = δuT fE

, become

fE

= PT

⊥K P⊥u = K u. (30)

Using the time derivative of decomposition (7), kinetic energy K can be written as 2K =vTM v = (e + U vF )TM(e + U vF ) = eTM e + vTF (UTM U)vF , where the last equality follows from

eq. (13). Applying modal reduction (16) and property (24) leads to 2K = ˙eTM ˙e+vTFMFvF , where

notation MF

= UTM U is introduced and the reduced mass matrix is defined as

M = ΨTM Ψ. (31)

Integration by parts of the variation of the kinetic energy leads to δeTM ¨e+ δuF (UTM U)vF . Using

the time derivatives of the projections (27), the reduced inertial forces fI, defined as δW I = −δuT f I ,

becomefI

= (PT

⊥M P⊥ + PT

‖ M P ‖)˙v = M ˙v (32)

where the second equality stems from P⊥ = I − P ‖ and PT

‖ M P ‖ = PT

‖ M = M P ‖. Note that

the inertial forces depend on the deformation measures at the interface nodes and on the modalamplitude accelerations but not on the floating frame dofs.

Finally, the reduced equations of motion become

˙u = v (33a)

M ˙v + K u = f , (33b)

where the reduced external loads are introduced as f = ΨTf .

11

2.3.4 Exact representation of eigenpairs

Let (ωj, V j) be an eigenpair of the structure subjected to specific boundary conditions and assumethat eigenvector V j, associated with eigenvalue ωj 6= 0, was selected as one column of matrix V .This leads to the following questions: is ωj a natural frequency of the reduced model? What is theassociated mode shape? Consider the following reduced vector

V j =

{V B,j

cj

}, (34)

where array V B,j stores the entries of eigenmode V j at the interface nodes and cj is an array withvanishing entries except for a unit entry at location j. Expanding this vector yields

Ψ V j =

{V B,j

ΨSV B,j + V

Icj −Ψ

SVBcj

}=

{V B,j

V I,j

}= V j. (35)

In view of property (4), ΨT (K − ω2jM)V j = 0, which then implies (K − ω2

j M)V j = 0; clearly,

vector V j is an eigenvector of the reduced system and is associated with natural frequency ωj. Insummary, if (ωj, V j) is an eigenpair of the structure subjected to specific boundary conditions andmode shape V j is included in the modal reduction basis, (ωj, V j) is an eigenpair of the reducedmodel subjected the same boundary conditions.

While this result seems to be rather technical, it is of great practical importance. Considerthe following scenario: through modal testing, an experimentalist has determined that specificeigenmodes of the system respond with significant amplitude under a given excitation. Clearly, thesemodes should be retained in the reduced model if it is to predict the dynamic behavior of the systemaccurately, and typically, the finite element model will be fine tuned to make sure it reproducesthe experimentally measured modes as accurately as possible. When using the mode-accelerationmethod, including these modes in the modal reduction basis guarantees that these modes will remaineigenmodes of the reduced system with the identical associated natural frequencies.

2.4 Herting’s method

This section presents Herting’s method [5], an application of the mode-acceleration method thatselects the rigid-body and elastic modes of the unconstrained structure as modal basis, i.e., V =[U,X], leading to the following static and dynamic mode matrices,

ΨH

S= −K−1

IIKIB, ΨH

D=[ΨH

IRMXI−Ψ

SXB

], (36)

where the first six modes of the dynamic mode matrix, called the “inertia relief modes,”

ΨH

IRM= −K−1

II(M

IBUB

+MIIUI) = −K−1

II(M

IB+M

IIΨH

S)U

B, (37)

are necessary to represent the dynamic contribution of the rigid-body modes correctly. Systemreduction is achieved by selecting a limited number of elastic modes. Because each selected interfacenode increases the number of columns of the static mode matrix, nη = n − nB modes only can beused for the dynamic mode matrix. Therefore, it must be verified that the selected nη elastic modeslead to a dynamic mode matrix presenting full column rank. In practice, if the addition of an elasticmode to the set of selected modes does not increase the column rank of the dynamic mode matrixby one, this elastic mode can be disregarded because the behavior associated with this elastic modeis already accounted for by the modes present in the set.

12

2.5 Craig-Bampton’s method

This section presents Craig-Bampton’s method [2], a mode-acceleration method that selects themodes of the constrained structure as modal basis. This implies V

B= 0 and the columns of matrix

VI

are the eigenmodes of the structure with constrained interface nodes, i.e., KIIVI

= MIIVIω2II

.This choice leads to the following static and dynamic mode matrices,

ΨCB

S= −K−1

IIKIB, ΨCB

D= −K−1

IIM

IIVI

= −VIω−2II. (38)

Factor −ω−2II

can be removed by scaling of the modal variables: η := −ω2IIη. Because the dynamic

modes are obtained from the structure constrained at the interface nodes, the maximum numberof columns of matrix V is nη = n− nB; system reduction is achieved by selecting a limited numbernη � n− nB of eigenmodes of the structure under constrained interface nodes.

2.6 Rubin’s method

This section presents Rubin’s method within the framework and notation introduced above. Incontrast with Herting’s and Craig-Bampton’s methods, Rubin’s method [4] does not appear to bea particular case of the mode-acceleration method. In Rubin’s method, the modes are obtainedfrom the unconstrained structure, which implies a six-times singular stiffness matrix, see eq. (3). Incontrast with the setting presented in section 2.3, the modal reduction is formulated as an expansionof the deformation measures in terms of elastic modes of the free-floating structure, i.e., e = X η,where η are the modal amplitudes of the selected elastic modes. This approach differs from theone presented so far because (1) it makes no reference to the interface nodes and hence, connectingmodal components to their surrounding is no longer straightforward, and (2) condition eq. (13)is satisfied automatically and the floating frame dofs must be appended to the list of dofs of thereduced system (see discussion in section 2.3.2). Nevertheless, the method can be manipulated toappear as a particular case of the mode-acceleration method: Rubin’s and Herting’s methods areproved to be equivalent.

2.6.1 Attachment modes

To overcome the connectivity issue and guarantee an exact static behavior, an additional set ofmodes, denoted A and called “attachment modes,” is introduced. The deformation measures arenow expressed as

e = X η + Aµ, (39)

where arrays η and µ store modal amplitudes. Attachment modes correspond to the static dis-placements of the structure when unit loads are applied at the interface nodes. Mathematically,this writes K A = F , where the columns of matrices A and F store the attachment modes andthe load vectors associated with unit loads applied at the boundary dofs, respectively. Note thefollowing partitions: AT = [AT

B, AT

I] and F T = [F T

B, F T

I] = [I, 0]. Because the stiffness matrix is

six times singular, system K A = F is not solvable. To remedy this problem and enforce con-dition eq. (13), the attachment modes are required to be in the subspace of the elastic modes,UTM A = 0; this constraint is enforced via Lagrange’s multiplier technique, leading to the system

of equations: K A + M U Λ = F and UTM A = 0. Solving for the Lagrange multipliers yields

Λ = (UTM U)−1UTF , leading to

K A =[I −M U(UTM U)−1UT

]F = P T

⊥F . (40)

Because the solvability condition is satisfied, the solution of this system yields the attachment modes.Note that attachment modes have units of flexibility; to avoid ill-conditioning of the governing

13

equations of the reduced component, it is recommended to scale the attachment modes so thatelastic and attachment modes are of the same order of magnitude.

To enable the straightforward connection of modal components to their environment, the modalamplitudes of the attachment modes are eliminated in favor of the dofs at the interface nodes.Partitioning eq. (39) yields eB = X

Bη + A

Bµ, which leads to µ = A−1

B(eB −XB

η). It now follows

that eI = XIη + A

Iµ = A

IA−1BeB + (X

I− A

IA−1BXB

)η and comparison with eqs. (16) and (17)yields the following static and dynamic mode matrices,

ΨR

S= A

IA−1B, ΨR

D= X

I−ΨR

SXB. (41)

The deformation measures now become

e = ΨR

{eBη

}= ΨR

{uB − UB

uFη

}= ΨRu−ΨRU uF = ΨRT

{uuF

}, (42)

where matrix T =[I −U

]. A direct identification with eq. (7) then reveals

u =[ΨR Φ

]{ uuF

}, Φ = U −ΨRU =

[0

UI−ΨR

SUB

]. (43)

The reduced set of variables includes the dofs at the interface nodes, uB, the modal amplitudes η,and the dofs of the floating frame, uF . Since the modal reduction matrix obtained here is builtfrom elastic mode contributions only, property (28) does not hold, i.e., rigid-body motions cannotbe represented by their contributions at the interface dofs only. In particular, UTM ΨR vanishes

and, in contrast with eq. (24), UI6= ΨR

SUB

, giving rise to contributions of the floating frame viamatrix Φ.

The form of matrix Φ in eq. (43) suggests that the contributions of the floating frame couldbe treated by expanding array η to include floating frame dofs, uF , and appending the columnsof matrix Φ to those of matrix Ψ

D. This approach would be valid in the linear case but cannot

be generalized to the nonlinear case since finite motions of the floating frame cannot be treated asmodal amplitudes.

2.6.2 Reduced equations of motion

Using the developments presented in the previous section, the virtual work of the elastic forcesbecomes

δuTK e = {δuT , δuTF}T TK ′ e = {δuT , δuTF}KR

{uuF

}, (44)

where K ′ = ΨRTK ΨR and the reduced stiffness matrix, KR

, is defined as

KR

= T TK ′ T =

[K ′ −(K ′U)

−(K ′U)T UTK ′U

]. (45)

Because matrices ΨR and Φ are constant, eq. (43) yields v = ΨR ˙v + Φ vF and the virtual workdone by the inertial forces becomes

δW I = −δuTM v = −{δuT , δuTF

}M

R

{˙vvF

}, (46)

where M ′ = ΨRTM ΨR, MF

= UTM U , and the reduced mass matrix is defined as

MR

= T TM ′T =

[M ′ −(M ′U)

−(M ′U)T UTM ′U +M

F

]. (47)

14

Identity UTM ΨR = 0, was used to simplify the results.The reduced equations of motion now become {

˙uuF

}=

{vvF

}(48a)

MR

{˙vvF

}+ K

R

{uuF

}= f

R(48b)

where the reduced external loads are fR

= {ΨRT ,ΦT}f . The floating frame of reference is not aphysical node and cannot be connected to the rest of the system as is the case for interface nodes.Nevertheless, the projection of the external loads on the reduction basis may result in reducedexternal loads acting on the floating frame dofs.

2.6.3 Equivalence of Herting’s and Rubin’s methods

A cursory look at eqs. (36) and (41) reveals the close connection between Herting’s and Rubin’smethods. Herting’s method includes rigid-body modes in the expansion in contrast with Rubin’smethod that relies on an independent set of unknown variables for the floating frame. This leads todifferent reduced matrices. Nevertheless, this section shows that the two approaches are equivalent:they span the same subspace of modes and the reduced set of variables of the two methods arerelated by a one-to-one map.

The partition of eq. (40) related to the internal nodes reads KIBAB

+ KIIAI

= −(MIBUB

+

MIIUI)(UTM U)−1UT

B, which implies

ΨR

S= ΨH

S+ ΨH

IRMZB, (49)

where matrix ZB

= (UTM U)−1UT

BA−1B

. This equation expresses the static mode matrix of Rubin’smethod in terms of the static mode matrix and the inertia relief modes of Herting’s method.The displacements of the internal nodes for Rubin’s method are obtained from eq. (43) as uRI =ΨR

SeB + ΨR

DηR + (U

I−ΨR

SUB

)uF and the static mode matrix of Rubin’s method is now eliminatedusing eq. (49) to find

uRI = ΨH

SeB + ΨH

IRMZB

(eB −XB

ηR − UBuF

)+(XI−ΨH

SXB

)ηR, (50a)

uHI = ΨH

SeB + ΨH

IRMηHIRM

+(XI−ΨH

SXB

)ηHD. (50b)

The final expression of eq. (50a) is obtained by regrouping terms and using the rigid-body modesproperty, U

I= ΨH

SUB

. Equation (28) was used to evaluate the displacements of the internal nodesfor Herting’s method shown in eq. (50b) and the array of modal amplitudes of Herting’s methodwas partitioned as ηHT = {ηHT

IRM, ηHT

D}, where arrays ηH

IRMand ηH

Dstore the modal amplitudes of

the inertia relief modes and remaining dynamic modes, respectively.A comparison of eqs. (50a) and (50b) reveals that the two methods span the same subspace of

modes and that the following mapping

ηH =

{ηHIRM

ηHD

}=

[−Z

BXB−Z

BUB

I 0

]{ηR

uF

}+

[ZB

0

]eB (51)

relates the modal amplitudes of Herting’s method to those of Rubin’s method and to the dofsof the floating frame. The contribution of the floating frame in Rubin’s method is contained incontributions of the inertia relief mode amplitudes in Herting’s method, altered by contributions ofthe interface node deformations and the elastic mode amplitudes.

15

2.7 Summary

The construction of the modal reduction matrix was presented in the context of the mode-acceleration method, which imposes no restriction on the nature of the modes used for the re-duction process: they could include eigenmodes of the fully constrained, partially constrained orunconstrained structure, rigid-body modes, or even experimentally measured modes. Because thestatic mode matrix provides an exact representation of the rigid-body modes based on their valuesat the interface nodes only, see eq. (24), the floating frame dofs can be eliminated. The modal re-duction procedure preserves the eigenpairs whose mode shapes are included in the modal reductionmatrix exactly. The reduced stiffness and mass matrices are obtained from simple linear algebraoperations, see eqs. (29) and (31), respectively.

Herting’s [5] and Craig-Bampton’s [2] methods were shown to be particular cases of the mode-acceleration method; these methods are obtained by selecting the rigid-body and elastic modes ofthe unconstrained structure and the modes of the constrained structure as modal basis, respectively.These two approaches inherit all the properties of the mode-acceleration method. Rubin’s [4] methodappeared to be markedly more complex than the mode-acceleration method because it requires theevaluation of the attachment modes and cannot eliminate the floating frame dofs. Rubin’s method,however, was shown to be equivalent to Herting’s method: the two methods span the same subspaceof modes and a one-to-one map relates their sets of reduced variables.

2.8 Application to constrained structures

When unconstrained components are connected to other components of a multibody system, rigid-body modes are present, leading to decomposition (6) with the introduction of a floating frame ofreference. In some cases, however, it is required to perform the modal reduction of a constrainedcomponent, i.e., a component whose rigid-body motions are inhibited by boundary conditions.

The displacement and deformation measure fields of constrained structures are expressed asu = e = X µ, where matrix X stores the elastic modes of the constrained structure, in contrastwith the case of the unconstrained structure, for which the same fields also include contributionsfrom the floating frame, see eq. (6). Because the proposed approach eliminates the floating frame ofreference from the reduced equations, see section 2.3.3, the case of constrained structures is treatedeasily: all floating frame contributions are disregarded. The reduction procedure follows that ofthe mode-acceleration method; it uses the structural matrices and elastic modes of the constrainedstructure.

It is also possible to treat constrained structures based on the modes of the fully unconstrainedstructure. In that case, the nodes where the boundary conditions are to be imposed must beretained as interface nodes. If the rigid-body modes of the unconstrained structure are includedin the reduction process, as in Herting’s method, they become spatially truncated by the imposedboundary conditions and actually account for deformation modes of the structure, in combinationwith the selected elastic modes. In practice, this strategy is not always possible because, in theanalysis of a constrained structure, the matrices of the unconstrained system may not be availableto the analyst. Nevertheless, in the linear case, this approach is not recommended because it canbecome unnecessarily cumbersome, especially if a large number of interface nodes is required toimpose the boundary conditions.

3 The motion formalism

This section introduces the basic tools used in this work to treat large amplitude motion. Typically,in finite element based approaches to multibody dynamics analysis [29,51], kinematics is formulatedin terms of nodes presenting six dofs that can be thought of as three translations and three rotations.

16

This representation is adequate for rigid bodies, kinematic joints, and flexible components suchas beams, plates, and shells. In classical kinematic descriptions of multibody systems [29, 32],displacement and rotation fields are treated independently.

In contrast, the motion formalism treats these two fields as a single entity, leading to advan-tageous numerical properties [52–54]. In the motion formalism, the kinematics of each node ishandled by motions and changes in motion or equivalently, by frames and frame transformations,respectively, which belong to the Special Euclidean group SE(3) = SO(3)nR3, where SO(3) denotesthe Special Orthogonal group SO(3) = {R ∈ R3×3|RTR = I, det(R) = 1}. Many representations ofmotion exist such as the homogeneous representation matrix, of size 4× 4, denoted H = H(R, u),or the motion tensor, of size 6× 6, denoted C = C(R, u). These matrices are defined as

H =

[R u0 1

], C =

[R uR0 R

], (52)

respectively, where R ∈ SO(3) is a rotation tensor representing orientation, u ∈ R3 is a position ordisplacement vector, and u denotes the 3×3 skew-symmetric matrix built on the three componentsof u. Bi-quaternions provide an alternative representation, see Han and Bauchau [51, 55]. Forsimplicity, the homogeneous representation matrix representation is used here.

Consider three motions represented by matrices H1(R

1, u1), H2

(R2, u2), and H

3(R

3, u3), such

that the third is the composition of the first two, i.e., H3

= H2H

1. Performing the matrix mul-

tiplication yields R3

= R2R

1and u3 = u2 + R

2u1. Therefore, the motion formalism couples the

displacement and rotation fields: rotation tensor R2

is involved in the composition of the positionvectors. This coupling is at the heart of the motion formalism and enables the consistent treatmentof rigid-body transformations.

In the current framework, the kinematics of a node is regarded as a frame composition, Gj

=

HjG0

j, where matricesG

j, G0

j, andH

jrepresent the deformed, reference, and change in configuration

at node j, respectively, see fig. 2. The superposition of an arbitrary rigid-body motion, HR

, to thecomponent configuration is obtained by multiplying all the nodal frames, H

j, from the left to

produce a new configuration, H∗j

= HRHj, ∀j.

Figure 2: Configuration of the modal component. For clarity of the figure, the configurations aredepicted far from each other. All transformations are resolved in the inertial frame, except for HE

j

that is resolved in the floating frame (see eq. (62)).

17

3.1 Local frame derivatives

In this work, a local frame representation of derivatives is adopted [46,47,56]. In terms of homoge-neous representation matrices, this translates into

dH =

[dR du0 0

]= H

[duθ duu0 0

]= HdU , (53)

where duu = RTdu and duθ = RTdR are the position and rotation derivatives, respectively, bothresolved in the local frame. This definition introduces the 6-dimensional differential motion vectorduT = {duTu , duTθ } and, with a slight abuse of notation, operator •, defined by the last equality ineq. (53), maps a 6× 1 vector into a 4× 4 matrix.

Because the derivatives introduced by eq. (53) are resolved in the local frame, they remaininvariant under rigid-body transformations; this important property justifies the name of “localframe” or “material” derivatives. If homogeneous representation matrices H and H∗ describe twomotions related by a rigid-body transformation, H∗ = H

RH, where H

Rdescribes an arbitrary

superimposed rigid-body transformation, the material derivatives of frames H and H∗ are identical:material derivatives are not affected by superimposed rigid-body transformations.

Because motion transformations do not commute, i.e H1H

26= H

2H

1, second derivatives must

be treated carefully. In particular, consider time derivation H = HV and variation δH = HδU .

Expanding expression δ(H) = (δH ), the variation of the velocity and the time derivative of thevariational motion are related via the transpositional relationships [51],

δV = ˙δU + VδU , V =

[vθ vu0 vθ

], (54)

which defines operator •. Another useful operator follows from VT1 V2 = V2V1 as

V =

[0 vuvu vθ

]. (55)

3.2 Parameterization

A stumbling block in the manipulation of motion is its parameterization [51, 54, 55]. A minimalparameterization of rotation is a nonlinear map that describes rotation in terms of three independentparameters, stored in array p, such that R = R(p); the inverse operation is denoted p = p(R). Asproved by Stuelpnagel [57], a minimal parameterization of rotation cannot be both singularity-freeand global, i.e., situations always arise where parameter values are either not defined or not defineduniquely.

A minimal parameterization of motion is a nonlinear map that describes motion in terms of sixindependent parameters, stored in array P , such that H = H(P); the inverse operation is denotedP = P(H). Minimal parameterizations of motion inherit the problems of their rotation counter-parts: they cannot be both singularity-free and global. If they are to be used in the governingequations of modal components, motion parameters must form first-order tensors. The only param-eterizations of rotation and motion that form first-order tensors are the vectorial parameterizationsof rotation [58] and motion [51, 59], respectively. For such parameterization, the parameterizationmap and its inverse take the following form

H(P) =

[R(p

θ) T T (p

θ)pu

0 1

], (56a)

P(H) =

{pu

pθ

}=

{T−T (p

θ)u

p(R)

}, (56b)

18

where matrix T (pθ) is the tangent operator associated with the rotation parameterization, i.e.,

duθ = T (pθ)dp

θ. Note that P(I) = 0 and H(0) = I.

Consider two motions, HA

= H(PA) and HB

= H(PB), and the corresponding relative motion

HAB

= H−1AHB

, whose parameterization is PAB = P(HAB

). Differentiation yields

∆PAB = T −1(PAB)∆UB − T −1(−PAB)∆UA, (57)

where T (PAB) is the tangent operator associated with the motion parameterization. Becausematerial derivatives are used, eq. (57) expresses the variation of the relative motion parametersin terms of the motion parameters themselves, without reference to motions H

Aand H

B. For all

parameterizations of motion, the following relationships hold: T −1(0) = I, T −1(P)−T −1(−P) = P ,

and T (P)T −1(−P) = C−1(P), where C−1 is the inverse of the motion tensor,

C−1(P) =

[RT (p

θ) RT (p

θ)uT (p

θ, p

u)

0 RT (pθ)

]. (58)

In this work, Euler-Rodrigues parameters are selected because of their simplicity. The explicitexpressions for this parameterization are given in Appendix A.

4 Modal reduction procedure for multibody systems

In typical multibody applications, elastic components undergo finite motion. If the deformationmeasures of the components remain small, modal reduction is applicable and leads to considerablegains in computational efficiency. This section extends the modal reduction techniques presentedin section 2 to elastic components undergoing finite motion but small deformation. The motionformalism presented in section 3 is used to treat finite motions because it provides desirable features:(1) it leads to deformation measures that both objective and tensorial, (2) it provides a naturalcoupling between the position and rotation deformation measures, and (3) it filters out geometricnonlinearities from the equilibrium equations through the use of material derivatives, resulting inconfiguration invariant element forces and tangent matrices. For comparison, modal reduction usingthe classical description of kinematics is presented section 4.5.

4.1 Description of structural components

Typically, six dofs, three displacement and thee rotation components are defined at each node ofmultibody systems; in this work, kinematics at these nodes is treated via the motion formalismpresented in section 3. On the other hand, structural components modeled using commercial 3Dfinite element packages involve a combination of nodes presenting displacement variables only (for3D solids) and nodes presenting both displacement and rotation variables (for beams and shells).It is assumed here that the interface nodes of the structural components feature six dofs, to allowconsistent assembly with multibody components.

In the finite element method, nodal unknowns are displacement components. Similarly, formultibody systems, nodal unknowns are selected to be frame change components, H, rather thanframe configuration components, G. The change in configuration of all the nodes of a structuralcomponent, denoted u, consists (1) of a set of kB interface nodes featuring motion variables, denotedwith subscript (·)B and (2) of a set of internal nodes featuring a mixture of displacement and motionvariables, denoted with subscript (·)I . If the set of internal nodes vanishes, no reduction is possibleand the procedure presented here defaults to the corotational element formulation. Note that u isnot a vector because the configuration space does not form a vector space. Derivatives, however,

19

belong to the tangent space and can be associated with vectors (see section 3.1). In particular, thevariation of the change in configuration, denoted δu, becomes

δu =

{δUBδuI

}, (59)

where array δUTB = {δUT1 , · · · , δUTkB}, with δU j = H−1jδH

j, stores the components of the virtual

motion vectors at the interface nodes resolved in the material frame and array δuI stores thevariations of the quantities defined at internal nodes.

Using the same partition of the configuration space, the velocity and acceleration vectors of astructural component become

v =

{VBvI

}, v =

{VBvI

}(60)

where array VTB = {VT1 , · · · ,VTkB}, with Vj = H−1jHj, stores the components of the nodal velocity

vectors at the interface nodes resolved in the material frame and array vI stores the time derivativesof the quantities defined at the internal nodes.

The elastic forces, fE(u), acting in a structural component are nonlinear functions of the con-figuration variables. A first-order Taylor series expansion of the virtual work done by these forcesabout the reference configuration, u0, yields

δuTfE(u) = δuTfE(u0) + δuTK(u0)∆u+ h.o.t, ∆u =

{∆UB∆uI

}, (61)

where K(u0) is the tangent stiffness matrix evaluated in the reference configuration and ∆u rep-resents the motion increments from u0 to u. For simplicity, the structural component is assumedto be unstressed in its reference configuration, which implies fE(u0) = 0. The array of incremental

motion vectors at the interface nodes is ∆UTB = {∆UT1 , · · · ,∆UTkB}, with ∆U j = H−1j

∆Hj, and

array ∆uI stores the increments in the quantities defined at the internal nodes.If the higher-order terms are neglected in eq. (61), the series expansion becomes an approxima-

tion whose accuracy degrades for larger values of the finite configuration change ∆u. For practicalapplications, it is necessary to introduce a parameterization of motion to deal with the kinematicnonlinearity of motion.

The kinetic energy K of the structural component is given by 2K = vTM(u)v, where M(u) isthe configuration dependent mass matrix. The virtual work done by the inertial forces is obtainedas δWI = −δuTf I , where the inertial forces take the generic form f I(u, v, v) = M(u)v+g(u, v). Thefirst term is a linear function of accelerations and the second accounts for nonlinear gyroscopic andcentrifugal forces. The expression of this second term differs for different finite element formulations.Because it is desirable to develop modal reduction tools that are independent of the finite elementformulation used to model the structural component, they should use the mass matrix of thesystem in its reference configuration but should remain independent of the specific expression usedto evaluate the inertial forces.

For linear problems, modal reduction procedures are based on a careful selection of rigid-bodyand elastic modes of the structural component in its reference configuration. For nonlinear problems,the modal reduction procedure uses the same modes, i.e., the mode of the eigenproblem defined bymatrices K(u0) and M(u0).

4.2 Floating frame and deformation measures

A floating frame of reference is used to keep track of the large motion of the modal component.The change in configuration of this floating frame is denoted H

F, which implies that H0

F= I in the

20

reference configuration. This is consistent with eq. (6), which depends on displacement componentsu and uF only, but not on the initial configuration of the floating frame of reference. As was thecase for the linear problem, the floating frame moves with the modal component and its change inconfiguration is defined by condition (13).

For linear problems, the total displacement field is decomposed into its rigid-body and defor-mation measure parts; because both parts form linear fields, the composition is expressed by asimple, commutative addition of vectors. For nonlinear problems, as depicted in fig. 2, the samecomposition operation takes the form of a non-commutative multiplication of the correspondingmatrices,

Hj

= HFHE

j, (62)

where matrices Hj

and HF

represent the changes in configuration of the frame at node j and of

the floating frame, respectively. The elastic motion, HE

j= H−1

FHj, is resolved in the floating frame

of reference and can be interpreted as the elastic motion at node j with respect to the fictitiousrigid-body configuration obtained as the application of the floating frame transformation to thereference configuration, i.e., H

FG0

j.

Next, a parameterization of the elastic motions of the interface nodes is introduced to obtaindeformation measures. Using the notation introduced in eqs. (56), the deformation measures fornode j become

E j = P(H−1FHj), (63)

which expresses the deformation measures as functions of the nodal motions and floating frameexplicitly; note the analogy with eq. (7). The deformation measures at all interface nodes aregathered into a single array ETB = {ET1 , · · · , ETkB}. These deformation measures are non-incrementaland geometrically consistent with the finite motions at the nodes. Their definition depends onchanges of configuration only: the reference configurations of the floating frame and of the interfacenodes do not appear. The vanishing of the deformation measures at all nodes, E j = 0, ∀j implies

HE

j= I and hence, H

j= H

F, ∀j: the structure is undeformed but undergoes a rigid-body motion.

Using eq. (57), increments of the deformations measures (63) are expressed as

∆E j = Jj∆U j − F j

∆UF , (64)

where Jj(E j) = T −1(E j), F j

(E j) = T −1(−E j), J−1j Fj

= C−1j

, and Cj

is the motion tensor at node j.

Note that matrices Jj

and Fj

depend on the deformation measures only and are independent of the

current position and orientation of the system with respect to the inertial frame. This property is adirect consequence of the local frame approach adopted here. For the interface nodes, the followingnotation is introduced

∆EB = JB

(E1, · · · , EnB)∆U − F

B(E1, · · · , EnB

)∆UF (65)

where JB

(EB) = diag(J1, · · · , J

nB) and F T

B(EB) = [F T

1, · · · , F T

nB].

The complete array of deformation measures combines the deformation measures at the interfaceand internal nodes, denoted EB and eI , respectively,

e =

{EBeI

}. (66)

Because the motion formalism is used for the interface nodes whereas any formalism can be usedfor the internal nodes, a different notation is used for the deformation measures at those two sets ofnodes. Such details are unimportant because the deformation measures at the internal nodes willbe eliminated by the reduction process. The deformation measures at the internal node and their

21

kinematic relationship with the floating frame should be available to reconstruct the displacementfield associated with the condensed dofs. This post-processing step is rather straightforward and isnot addressed here.

Expanding the notation introduced in eq. (65), the derivative of the complete array of deforma-tion measures becomes

∆e = J ∆u− F ∆UF (67)

where J(e) = diag(JB, J

I) and F (e) = [F

B, F

I]. Matrices J

Iand F

Ipertain to the kinematics of

the internal dofs and are not derived here. Consider an incremental motion applied to the componentsuch that the increments of the floating frame are ∆UF = α and those of the deformation measurevanish, ∆e = 0; the resulting motion increments are thus ∆u = (J−1F )α. Clearly, matrix J−1F ,

where J−1jFj

= C−1j

(E j), represents the rigid-body modes in the present formulation.

When evaluated in the reference configuration, the derivative of the array of deformation mea-sures reads

∆e∣∣u=u0

= ∆u− U ∆UF , (68)

where matrix U = [UB, U

I]. The expression for the rigid-body modes at the interface nodes,

UT

B= [I, · · · , I], differs from that in eq. (2) because of the present developments use the motion

formalism. The expression of UI

is not relevant because the reduction process presented here doesnot make use of it. Note the analogy with eq. (7), developed for the linear problem. Similarrelationships follow for variations and time derivative, evaluated in the undeformed configuration:δe = δu− U δUF , e = v − U VF .

4.2.1 Definition of the floating frame

Using the arguments presented for the linear case in section 2.2.1, the deformation measures mustsatisfy the six constraints defined by eq. (8), Y T e = 0. Although not strictly necessary, it is assumedthat matrix Y does not depend on the configuration. For the nonlinear case, this condition provides,in general, an implicit definition of the floating frame, because e is a nonlinear function of the floatingframe dofs and of the nodal motions. Hence, the treatment of the floating frame involves two steps:(1) express increments of the floating frame dofs in terms of those of the reduced variables and (2)evaluate the floating frame dofs for a given configuration. The first step allows the elimination ofthe floating frame of reference from the set of unknown variables and the second step is needed toevaluate the deformation measures during the solution process.

A variation of eq. (8) implies Y T∆e = 0 and introducing eq. (67) then yields Y T (J ∆u −F ∆UF ) = 0. If Y TF is non-singular, this implies

∆UF =[(Y TF

)−1 (Y TJ

)]∆u = QY (e)∆u, (69)

which echoes eq. (9) for the linear case. The increments of rigid-body motions of all nodes,J−1F ∆UF and those of the deformation measures now become

J−1F ∆UF =[

J−1F(Y TF

)−1 (Y TJ

)]∆u = P Y

‖ (e)∆u, (70a)

J−1∆e =[I − J−1F

(Y TF

)−1 (Y TJ

)]∆u = P Y

⊥(e)∆u, (70b)

which echoes eqs. (10) for the linear case. Matrices P Y

⊥ and P Y

‖ are the projectors onto the subspace

orthogonal and parallel to the rigid-body modes, respectively. Because P Y

⊥(J−1F ) = 0 and (P Y

⊥)n =

P Y

⊥, matrix P Y

⊥ is indeed a projector and P Y

‖ = I − P Y

⊥ is the complementary projector, which

satisfies P Y

‖ (J−1F ) = (J−1F ). The consistent use of deformation measure increments involves

22

multiplicative factor J−1. Equations (70) generalize the developments in section 2.2.1: neglectingthe effect of deformations leads to J ≈ I and F ≈ U , and the linear case of eq. (10) is recovered.

Similar relationships hold for time derivatives: VF = QY v and e = J P Y

⊥v. For accelerations,

VF = QY v + QYv and e = J P⊥v + (J P⊥) v, where

QY

= QY J−1(J − F QY ) (71a)

(J P Y

⊥ ) = (J P Y

⊥)J−1(J − F QY ) (71b)

The expression for the time derivative of the inverse of the tangent operator appearing in matricesJ and F is given in appendix A for the Euler-Rodrigues motion parameters.

With definition (8), the deformation measures are objective because they remain invariant underthe superposition of rigid-body motions. Indeed, consider a given configuration the structure, H∗

j,

∀j, and an arbitrary rigid-body motion, Hα, that brings the structure to its new configuration, H

j=

HαH∗j, ∀j. The motion increments at the nodes can be expressed in terms of increments of the rigid-

body motion dofs ∆U j = C−1j

∆α, where Cj

is the motion tensor associated with node j and ∆α =

H−1α

∆Hα. With help of eq. (62), C

jcan be written as C

j= C

FCEj

, where CF

and CEj

are the motion

tensors associated with the floating frame and relative motions, respectively. The motion incrementsof the node j now become ∆U j = C−1

j∆α = CE−1

jC−1F

∆α = (J−1jFj)C−1

F∆α = (J−1

jFj)∆αF , where

∆αF = C−1F

∆α. Expanding this results for all nodes gives ∆u = (J−1F )∆αF . Equations (69)and (70b) now yield the increments of the floating frame dofs and of the deformation measures asQY (J−1F )∆αF = ∆αF and P Y

⊥(J−1F )∆αF = 0, respectively. Clearly, the superposition of a rigid-

body motion results in increments of the floating frame dofs but leaves those of the deformationmeasures unaffected.

As opposed to the linear case, condition (8) cannot be obtained from a minimization problemas in eq. (11). Indeed, the minimization of eTAe leads to condition F TAe = 0, which is a non-linear function of the deformation measures and cannot be satisfied if matrix Y is configurationindependent.

As was the case for the linear problem, Y = M U proves to be a convenient choice and leads to

condition (13), UTM e = 0. The motion increments of the floating frame in eq. (69) become

∆UF =[(UTM F

)−1 (UTM J

)]∆u = Q(e)∆u, (72)

which echoes eq. (14) for the linear case, and the projections in eqs. (70) become

J−1F ∆UF =[

J−1F(UTM F

)−1 (UTM J

)]∆u = P ‖(e)∆u, (73a)

J−1∆e =[I − J−1F

(UTM F

)−1 (UTM J

)]∆u = P⊥(e)∆u, (73b)

which echo eq. (15) for the linear case.The second step is the computation of the floating frame of reference H

Fand the related elastic

deformation measures for the interface nodes. Because the kinematics of motion is nonlinear, aniterative (Newton-Raphson) procedure is required: (1) Start with an initial guess, H

F, typically the

change in configuration of the floating frame at the previous time step, or the change in configurationof any interface node; (2) Compute the deformation measures, using eq. (63) for the interface nodes,and evaluate the residual of condition (13), r = UTM e; (3) If ‖r‖ ≥ ε, where ε is a small positive

number, linearize condition (13) for a fixed component configuration, i.e., r − (UTM F )∆UF = 0,

and compute an incremental correction of the floating frame configuration, ∆UF = (UTM F )−1r.

23

Update the configuration of the floating frame as HF

:= HFH(∆UF ) and continue to step (2).

Terminate the procedure when ‖r‖ < ε.For small deformations, the procedure can be simplified by using the modified Newton-Raphson

method: in that case, F ≈ U and iteration matrix (UTM F ) ≈ (UTM U) remains constant.

4.3 Modal reduction

The modal reduction procedure presented in section 2.3 for linear problems is generalized here tononlinear problems. The configuration space of the reduced component is denoted u and consists ofuB, the set of change of configuration of the interface nodes, and η, the vector of modal amplitudes.The deformation measures and their reduced counterparts are related through the modal reductionmatrix, Ψ, as in eq. (16),

e = Ψ e, (74)

and the set of reduced variables is partitioned in a manner that echoes eq. (17) as eT = {ETB, ηT}.The evaluation of the deformation measures at the interface nodes is based on eq. (63).

For convenience, the following reduced quantities are introduced ∆uT = {∆UTB,∆ηT}, vT =

{VTB, ηT}, and ˙vT = {VTB, ηT}. Due to eq. (65), increments of the reduced deformation measuresbecome

∆e =

{∆EB∆η

}=

{JB

∆UB − FB∆UF

∆η

}= J ∆u− F ∆UF , (75)

where J(EB) = diag(JB, I) and F (EB) = [F

B, 0]. The rigid-body motions are thus handled exclu-

sively by the interface nodes, since the part of F related to the modal amplitudes vanishes.

4.3.1 Floating frame and reduced deformation measures

As discussed in section 2.3.2, the floating frame dofs can be eliminated when the modes selected tobuild the modal reduction matrix Ψ are able to represent the rigid-body modes, see eq. (24); this

condition is met when using the mode acceleration method and condition (25), UTM e = 0, also

applies to the nonlinear case.Retracing the steps presented in section 4.2.1, increments of the floating frame dofs are expressed

in terms of increments in the reduced variables using eq. (75) and the derivative of eq. (25) to find

∆(UTM e) = U

TM(J ∆u− F ∆UF ) = 0, leading to

∆UF =(UTM F

)−1 (UTM J

)∆u = Q(EB)∆u, (76)

that echoes eq. (26). The following expressions result

J−1F ∆UF =

[J−1F(UTM F

)−1 (UTM J

)]∆u = P ‖(EB)∆u, (77a)

J−1

∆e =

[I − J−1F

(UTM F

)−1 (UTM J

)]∆u = P⊥(EB)∆u, (77b)

that echo eqs. (27). Projections (77) echo projections (73) written for the complete system. Due tothe structure of the projectors, both nodal displacements and modal amplitudes contribute to thedeformation measures at the boundary nodes. Similar relationships hold for velocities and eqs. (71)apply for the reduced matrices.

The procedure described in section 4.2.1 for the evaluation of the floating frame dofs can bemodified to involve reduced quantities only: the residual now becomes r = U

TM e and the resulting

corrections to the floating frame dofs are ∆UF = (UTM F )−1r.

24

Equation (75) yields the increments in the complete deformation measure field as

∆e = Ψ∆e = Ψ J ∆u−Ψ F ∆UF . (78)

In contrast with the linear case, identification with eq. (67) is not possible because, in general,Ψ F 6= F . Starting from eq. (67) written in the form ∆u = J−1(∆e+F ∆UF ) = J−1(Ψ∆e+F ∆UF ),a relationship between ∆u and ∆u can be obtained from eqs. (76) and (77) to find

∆u =[J−1(Ψ J P⊥ + F Q)

]∆u =

[J−1Ψ J + J−1(F −Ψ F )Q

]∆u (79)

which reduces to eq. (28) in the linear case. Because of the nonlinear kinematics, this relationshipdepends on matrices J and F , which are deformation-dependent contributions of the size of theoriginal system.

4.3.2 Reduced equations of motion

The linearized virtual work done by elastic forces about the reference configuration of the elasticcomponent, eq. (61), is expanded using eqs. (68) and (74) to find

δuTfE(u) ≈ δuTK(u0)∆u =(δeTΨT + δUTFUT

)K(u0)

(Ψ ∆e+ U ∆UF

)= δeT K(u0)∆e, (80)

where eq. (3) is used to filter out the contributions of the floating frame. Although this expressionholds for infinitesimal deformations about the reference configuration only, the vectorial nature ofthe deformation measures suggests that for small deformations, the energy of the modal componentis represented appropriately by a strain energy function of the form eT K(u0)e/2, where the reduced

stiffness matrix K is given by eq. (29). This manipulation implies a small deformation assump-tion, but leads to a reduced strain energy that depends on variables of the reduced system only.

Using (77b), the virtual work done by the elastic forces becomes δuT fE

, where the reduced elasticforces are

fE

(e) = (J P⊥)T K e. (81)

Although stiffness matrix K remains constant, the reduced elastic forces are nonlinear functionsof the reduced deformation measures because of the nonlinear kinematics. They do not depend,however, on the configuration of the structural component with respect to an inertial frame andtherefore, remain invariant under the superposition of a rigid-body motion. Using eq. (77b), incre-ments in the reduced elastic forces become

∆fE

= (J P⊥)T K (J P⊥) ∆u+ ∆(J P⊥)T K e

≈ (J P⊥)T K (J P⊥) ∆u = KT

∆u.(82)

For small deformations, the second term in the tangent stiffness matrix is negligible. The tangentstiffness matrix depends on the deformation measures but is independent of the configuration of thestructural component with respect to an inertial frame, leading to improved numerical efficiency: aslong as the deformation measures do not change significantly, the tangent stiffness can be evaluatedonce only and does not need to be updated even if the component undergoes large amplitude mo-tions. It is convenient to evaluate the tangent stiffness matrix in the reference configuration, whereKT

= K = ΨTK(u0)Ψ, see eq. (29). If the deformation measures change noticeably, the tangentstiffness matrix should be re-evaluated to ensure convergence of the iterative solution process.

The kinetic energy K = vTM v/2 can be expressed in terms of the time derivative of the reduceddeformation measures and of the floating frame velocity as

2K ≈(Ψ ˙e+ U VF

)TM(u0)

(Ψ ˙e+ U VF

)= ˙eTM ˙e+ VTFPF , (83)

25

where the last equality follows from the definition of the floating frame of reference in eq. (25),

M is the reduced mass matrix given by eq. (31), notation MF

= UTM(u0)U was introduced,

and PF = MFVF is the momentum associated with the floating frame. Equation (76), recast as

VF = Q v, shows that the velocity of the floating frame is a function of the deformation measures,

of the interface node velocities, and of the time derivative of the modal amplitudes. The smalldeformation assumption leads to an expression of the reduced kinetic energy that depends onvariables of the reduced system only.

To obtain the inertial forces, the variation of the floating frame velocity is expressed via transpo-sitional relationship (54) to find δVF = ˙δUF + VF δUF . The variation of the kinetic energy becomesδ ˙eTM ˙e+ ( ˙δUF + VF δUF )TPF and integration by part then yields δeTM ¨e+ δUTF (M

FVF − VTFPF ).

Equations (76) and (77b) then yield the reduced inertial forces

fI(EB, v, ˙v) = M

I ˙v +[(J P⊥)TM (J P⊥) + Q

TM

F(Q)

]v − QT VTFPF , (84)

where MI(EB) = (J P⊥)TM (J P⊥) + Q

TM

FQ. Using eqs. (71), this expression reduces to

fI(EB, v, ˙v) = M

I ˙v + MI( ˙J − ˙F Q)v − QT VTFPF . (85)

The first term of this expression is linear in the reduced accelerations. The second term can beinterpreted as Coriolis-type forces because the time derivatives of matrices J and of F dependon the rate of change of the deformation measures at the boundary nodes. Finally, the thirdterm is quadratic in the floating frame velocities and describes gyroscopic and centrifugal effects.The reduced inertial forces are nonlinear functions of the reduced deformation measures but donot depend on the configuration of the structural component with respect to an inertial frame.Under the assumption of small deformation state and small rate of change of deformation, thedifferentiation of the inertial forces can be approximated as

∆fI ≈ M

I∆ ˙v + G

I∆v, (86)

whereGI(EB, v) = −QT

[VTF MF

+ PF]Q. (87)

The mass and gyroscopic matrices are independent of the configuration of the structural componentwith respect to an inertial frame but depend on the deformation measures.

With help of eq. (79), the virtual work done by the externally applied forces becomes δuTf =

δuT f , where f = (Ψ J P⊥ +F Q)TJ−Tf are the reduced externally applied forces. Due to the non-

linear kinematics, this relationship depends on matrices J and F , which are deformation-dependentcontributions of the size of the original system. In the context of modal reduction, it is desirableto avoid such computation. For small deformations, matrices J

I≈ I and F

I≈ U

I: the reduced

forces become independent of the internal variables. Certain types of externally applied forces, suchas aerodynamic forces, depend on the configuration of the component with respect to an inertialframe. In this case, the consistent evaluation of the externally applied forces requires the knowledgeof the configuration of the complete structural component; this more complex issue is not addressedhere.

Finally, the reduced equilibrium equations of the reduced component become fI+f

E= f , where

eq. (79) yields the reduced applied loads as f = [J−1(Ψ P⊥ − P ‖)J ]Tf . The complete computation

procedure is detailed in table 1. The reduced equations of motion for the nonlinear case reduce tothose for the linear case, eqs. (33a), when the Coriolis forces, gyroscopic forces, centrifugal forces,and nonlinear contributions of the boundary node deformation measures are disregarded.

26

With the systematic use of material derivatives, the elastic and inertial forces are functions of thedeformation measures only; they do not depend on the configuration of the structural component,i.e., its position and orientation with respect to the inertial frame. The same properties are sharedby the stiffness and mass matrices. Consequently, the governing equations remain invariant underthe superposition of rigid-body modes. Finally, this invariance impacts the computational efficiencyof nonlinear modal components favorably [46,60].

4.4 Application to constrained structures

If a linear structure is fully constrained, the floating frame can be ignored altogether. For nonlinearstructures, however, the floating frame describes the rigid-body motion of the structure and itsoverall elastic motion. When boundary conditions inhibit rigid-body motion, the floating frameremains relevant because it describes overall elastic motion. Within the proposed framework, thebest strategy is to apply the boundary conditions at a set of interface nodes after the modalreduction.

4.5 Comparison with the classical kinematic description

This section contrasts the classical and motion formalism kinematics to further highlight the featuresof the latter. The formulation of modal components using classical kinematics can be found innumerous publications, e.g. [30, 37,61,62].

4.5.1 Classical deformation measures

Figure 3 shows the modal component in its reference and deformed configurations; it also showsthe rigid-body configuration, a fictitious configuration that assumes the structure to be rigid. Thefloating frame of reference defines this fictitious configuration. Unless stated otherwise, the compo-nents of all tensors are resolved in the inertial frame. In the reference configuration, the positionvector of node j is expressed as x0j = x0F + R0

Frj, where x0F and R0

Fare the position vector and

orientation tensor of the floating frame, respectively, and rj = R0T

F(x0j − x0F ) are the components

of the relative position vector of node j with respect to the origin of the floating frame resolved inthe floating frame. The rotation tensor at node j and of the floating frame are denoted as R0

jand

R0

F, respectively. For simplicity, it is assumed that the orientations of the floating frame and of the

nodal frames coincide in the reference configuration, i.e., R0

j= R0

F, ∀j.

The deformed configuration of the component is obtained by superposing a local motion ontoa fictitious rigid-body configuration described by the motion of the floating frame. The rotationtensor at node j is expressed as R

j= R

FSj, where S

jare the component of the relative orientation

of the tensor at node j with respect to the fictitious rigid-body configuration resolved in the floatingframe. Similarly, the position vector of node j is expressed as xj = xF + R

F(rj + dj), where dj

is the relative position vector of node j with respect to its position in the fictitious rigid-bodyconfiguration, resolved in the floating frame. To complete the kinematic description, the changein orientation of the rotation tensors at node j and of the floating frame, denoted Q

jand Q

F,

respectively, are introduced: Rj

= QjR0

jand R

F= Q

FR0

F.

The components of relative position vector dj and of relative orientation tensor Sj, both resolved

in the floating frame, now define the elastic displacements of the component

dj = (QFR0

j)T (xj − xF )−R0T

j(x0j − x0F ), (88a)

Sj

= R0T

j(QT

FQj)R0

j. (88b)

27

• Initialize Reduced Matrices (M , K, Ψ, U)

M = ΨTM Ψ see eq. (31)

K = ΨTK Ψ see eq. (29)

MF

= UTM U see eq. (83)

• Compute Floating Frame And Deformation Measures (u)HF

= H1

while

for all interface nodesE j = P (H−1

FHj) see eq. (63)

Fj

= T −1(−E j) see eq.(64)

r = UTM e see eq. (25)

St = (UTM F )−1

if ‖r‖ < ε break

∆UF = St r

HF

:= HFH(∆UF )

for all interface nodes

Jj

= Fj

+ E j see eq. (64)

Q = St (UTM J) see eq. (72)

J P⊥ = J − F Q see eq. (77b)

• Compute Elastic Forces (e, J P⊥)

fE

= (J P⊥)T K e see eq. (81)

KT

= (J P⊥)T K (J P⊥) see eq. (82)

• Compute Inertial Forces (v, ˙v, Q, J P⊥)

MI

= (J P⊥)TM(J P⊥)

fI

1= M

I ˙v

VF = Q v see eq. (72)

fI

2= −QT VTF MF

VF˙e = (J P⊥)v see eq. (73b)

for all interface nodes˙Jj

= (T −1)˙(E j, E j)˙Fj

= ˙Jj− ˆE j

fI

3= M

I( ˙J − ˙F Q)v

fI

= fI

1+ f

I

2+ f

I

3see eq. (85)

GI(EB, v) see eq. (87)

Table 1: Computation procedure

28

Figure 3: Configuration of the modal component. For clarity of the figure, the configurations aredepicted far from each other. All tensors are expressed with respect to the inertial frame, exceptrj, dj and S

jthat are expressed with respect to the floating frame (see eq. (88)).

The relationship between the classical kinematics presented here and the motion formalism pre-sented in the previous sections is made explicit via the following manipulations starting from eq. (62),

HE

j= H−1

FHj

=[GFG0−1F

]−1 [GjG0−1j

]= G0

j

[(G

FG0

Fj)−1G

j

]G0−1j

(89)

where G0

Fj= G0−1

FG0

j= H(I, rj) is the relative frame transformation between the reference configu-

ration of node j and that of the floating frame. The last bracketed term, (GFG0

Fj)−1G

j= H(S

j, dj),

is the frame transformation defined by the elastic displacements appearing in eq. (88). Clearly, theclassical elastic displacements, (S

j, dj) and those used in the present work, G0

jH(S

j, dj)G

0−1j

are

identical within the contribution of the initial configuration. The derivation using the motion for-malism amounts to the chain of frame transformations expressed by eq. (89), in contrast with itscounterpart based on the classical kinematics, which is far more cumbersome.

Although the elastic displacements for both approaches are related intimately, the definitionsof the deformation measures differ markedly. Typically, classical approaches define the array ofdeformation measures at node j as

E j(dj, Sj) =

{djκj

}, (90)

where the position and the curvature parts of the deformation measures are defined as the elasticdisplacement components themselves, dj, the skew-symmetric part of the rotation tensor, κj =

axial(Sj) = (S

j−ST

j)/2, respectively. The curvature part corresponds to a parameterization of the

elastic rotation by the linear parameters [29, 51] but any other rotation parameterization could beused.

In contrast, the motion formalism uses a vectorial parameterization of the relative motion as thedeformation measures, see eq. (56b). Consequently, the position part of the deformation measuresbecomes T−T (κj)dj: position and rotation components are coupled because no rotation parame-

terization exists for which T−T (κj) = I,∀κj. The vectorial parameterization of motion is based

on invariants of the motion tensor [51, 55, 59]; in contrast, deformation measures ETj = {dTj , κTj } in

eq. (90) do not share that property. Of course, for infinitesimal deformations, T−T (κj ≈ 0) ≈ I andthe two approaches coincide.

Equation (90) can be viewed as a non-vectorial parameterization map E j(uEj , RE

j) of relative

motion HE

j= H(RE

j, uEj ) = G0

jH(S

j, dj)G

0−1j

introduced in eq. (62). Using the notation introduced

29

in eq. (64), variations of these deformation measures expressed in terms of material derivativesbecome

Ji(HE

j) =

[RE

j0

0 12

[tr(RE

j)I − (RE

j)T]] , F

i(HE

j) =

[I −uEj0 1

2

[tr(RE

j)I −RE

j

]] . (91)

If eqs. (64), derived from the motion formalism, are replaced by eqs. (91), obtained from classicalkinematics, the modal reduction procedure based on the motion formalism reduces to its classicalcounterpart. This simple substitution enables the direct comparison of the two approaches.

4.5.2 Material and inertial derivatives

The developments presented in this paper make systematic use of the material derivatives. Thisstrategy leads to intrinsic contributions of the elastic and inertial forces to the equilibrium equations,i.e., contributions that depend on the deformation measures but remain independent of the positionand orientation of the component with respect to an inertial frame. In contrast, classical modalcomponent formulations are based on inertial derivatives: dX T = {dxT , dψT}, where dψ = dQQT

are the components of the differential motion vector resolved in the inertial frame. The relationshipbetween the inertial and material differentials, denoted dX and dU , respectively, is

dX = RC0−TdU , R =

[R 00 R

]. (92)

When the differential motion vector is resolved in the inertial frame, increments in deformationmeasures become

∆E j = JjC0TjRT

j∆X j − F j

C0TFRT

F∆X F , (93)

where ∆X j and ∆X F are the increments for node j and for the floating frame, respectively. Incre-ments in deformation measures now depend explicitly (1) on the initial configuration of the floatingframe through motion tensor C0

Fand (2) on the present orientation of the structural component

with respect to an inertial frame through matrix R. In contrast, when using material derivatives,increments in deformation measures are independent of an initial configuration of the floating frameand of the present configuration of the component, see eq. (64). Clearly, when using material framederivatives, the geometric nonlinearities associated with large motions are filtered out of the reducedequilibrium equations. This important property is lost when inertial derivatives are used.

Remark The mass and stiffness matrices generated by a finite element package that is basedon the classical, uncoupled kinematics in the inertial frame can be used within the present localframe framework, provided that transformation (92) with R = I is performed on the interface nodecontributions after the modal reduction.

5 Numerical examples

Several numerical examples are presented in this section to illustrate the capabilities of the proposedapproach. For all examples, the mode-acceleration method uses the matrices of the unconstrainedcomponents; the nodes needed to interconnect components, apply external loads, and to enforcethe boundary conditions are retained as interface nodes. At each interface node, the static modematrix includes three or six static modes for two- or three-dimensional problems, respectively. Timeintegration of the reduced equations of motion was performed using a global parameterization-freegeneralized-α scheme [63,64]. For comparison, predictions are presented for the proposed approachand for the classical definition of the deformation measures described in section 4.5, see eq. (90).

30

5.1 Static problem: 45 degree bend beam

Bathe and Bolourchi [65] discussed the static problem of a circular beam cantilevered at the rootand subjected to a tip concentrated load acting in the direction normal to the plane of the beam.The beam has a radius of curvature of 100 m, spans a 45 degree circular arc, and presents thefollowing mechanical properties: axial stiffness, EA = 10 MN, shear stiffnesses, GA22 = GA33 =4.167 MN, torsional stiffness, GJ = 83.34e4 Nm2, bending stiffnesses EI22 = EI33 = 833.4 kN·m2.The tip load, P = 1 kN, is applied in ten equal increments. The displacement components at thefree end of the beam are investigated.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

(a) Motion-based deformation measures.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

(b) Classical deformation measures.

Figure 4: Out-of-plane component of tip displacement (m) versus load level for the 45 degree bendbeam. Reference solution (−), one modal element (◦−), two modal elements (?−), four modalelements (?−). Each modal element is made of two interface nodes.

The beam is modeled with modal elements, each featuring two interface nodes at their endpoints. Twelve static modes, those related to the two interface nodes, are used for the modalreduction. Three approaches are considered for this problem: the beam is modeled (1) with onesingle modal element (12 dofs), (2) with two modal elements, one for each half of the beam (18 dofs),and (3) four modal elements, one for each quarter of the beam (30 dofs). For this static problem,the floating frame is defined using Y = U in eq. (8). Figures 4 and 5 show the tip out-of-plane andin-plane deflections, respectively, versus load for the proposed and classical deformation measures;the fully nonlinear beam solution of the problem is used as reference.

When classical deformation measures are used, convergence could not be achieved with a singlemodal element but both approaches converge towards the reference solution as the number of modalelements increases. Clearly, the proposed deformation measures provide more accurate predictions,particularly for the in-plane displacement components. The number of iterations to reach a equi-librium solution at each load step was lower for the motion-based deformation measures than fortheir classical counterparts.

5.2 Rotating beam

Wu and Haug [66] described a dynamic problem consisting of a cantilevered beam of length L = 8m connected to the ground via a revolute joint. The root rotation angle at the joint, denoted φ, is

31

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-40

-35

-30

-25

-20

-15

-10

-5

0

(a) Motion-based deformation measures.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-40

-35

-30

-25

-20

-15

-10

-5

0

(b) Classical deformation measures.

Figure 5: In-plane component of tip displacement (m) versus load level for the 45 degree bend beam.Reference solution (−), one modal element (◦−), two modal elements (?−), four modal elements(?−). Each modal element is made of two interface nodes.

prescribed with the following schedule,

φ(t) = ωT

{τ 2/2 + (cos 2πτ − 1)/(2π)2, for τ < 1,

τ − 1/2, for τ ≥ 1,(94)

where T = 15 s, ω = 4 rad/s and τ = t/T . For τ < 1, the root rotation undergoes a sharp angularacceleration. The motion is characterized by geometric nonlinearities and the proper treatment ofthe centrifugal forces, the last term of the inertial forces in eq. (85), is essential. For τ ≥ 1, the rootrotation is driven at a constant angular velocity and the motion is characterized by small amplitudevibrations. The beam’s mechanical properties are as follows: axial stiffness EA = 5.03 MN, shearstiffness GA22 = GA33 = 1.94 MN, bending stiffness EI22 = EI33 = 566 N·m2, mass per unit lengthm = 0.201 kg/m, and moment of inertia m22 = m33 = 22.7 mg·m2/m. The problem is simulatedfor 20 s with a constant time step size of 2 ms. The spectral radius of the generalized-α scheme isρ∞ = 0.

The beam was modeled using a single modal element featuring two interface nodes located atthe root and tip of the beam. In the first modeling approach, the static modes associated with thetwo interface nodes are used for the modal reduction (6 dofs). In the second modeling approach,the lowest three bending modes of the beam clamped at its root were added to the modal basis (9dofs). Figures 6 and 7 show the transverse and axial components of the beam’s tip displacement,respectively, resolved in a rotating frame attached at the root of the beam. The reference solutionshown in these figures is obtained using a fully nonlinear beam model.

When using the motion-based deformation measures, the modal element captures the dynamicresponse of the beam accurately. Adding three vibration modes does not improve the correlationnoticeably in the first phase of the motion. A marginal improvement is observed in the secondphase of the motion because it is of a purely vibratory nature. Although the iteration matrix wasevaluated in the reference configuration once only, convergence was achieved at each time step inabout two iterations. When using the classical deformation measures, poor correlation is observedwith the fully nonlinear beam solution; the addition of three vibration modes does not improve thecorrelation. Clearly, the classical deformation measures fail to capture the geometric nonlinearityof the problem. Because the three bending vibration modes characterize the linear, vibratory

32

0 2 4 6 8 10 12 14 16 18 20

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

(a) Motion-based deformation measures.

0 2 4 6 8 10 12 14 16 18 20

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

(b) Classical deformation measures.

Figure 6: Time history (s) of the transverse tip displacement (m) for the rotating beam example.Reference solution (−), one modal element with two interface nodes (◦−), one modal element withtwo interface nodes + three clamped-free vibration modes (?−).

behaviour of the beam only, their addition does not improve the correlation, as expected.Several authors [30, 67] have used classical formulations to treat this example and reported

poor correlation with the exact solution. Cardona and Geradin [67] pointed out that the lack ofcorrelation stems from the inaccurate evaluation of the reduced inertial forces. Good accuracy couldbe achieved, for example, with substructring techniques [66] or when modal derivatives are includedin the modal basis [18].

5.3 Beam on a spherical joint

Figure 8: Beam on a spherical joint, sec-tion 5.3.

Cardona [39] has presented the three-dimensionalproblem of a cantilevered beam of length L = 141.42m mounted on a spherical joint shown in fig. 8.The beam presents the following mechanical prop-erties: axial stiffness EA = 18.9 MN, shear stiff-nesses GA22 = GA33 = 7.27 MN, torsional stiffnessGJ = 10.9 MN·m2, bending stiffnesses EI22 = EI33= 14.175 MN·m2, mass per unit length m = 70.2g/m, and moments of inertia (13.5,6.75,6.75) kg·m.A moment M(t) is applied for 10.2 s at the sphericaljoint. After 15 s, a transverse force F (t) is applied for 0.4 s at the tip of the beam to trigger athree-dimensional response,

M(t) = 1000

t, t < 0.2,

0.2, 0.2 ≤ t ≤ 10,

10.2− t, 10 < t < 10.2,

0, t ≥ 10.2,

F (t) = 500

0, t ≤ 15,

t− 15, 15 ≤ t ≤ 15.2,

15.4− t, 15.2 ≤ t ≤ 15.4,

0, for t ≥ 15.4.

(95)

The problem is simulated for 50 s with a constant time step size of 20 ms; the spectral radius of thegeneralized-α scheme is ρ∞ = 0.99. The angular velocity at the root about the inertial direction inwhich the moment has been applied is observed.

The beam was modeled using a single modal element featuring two interface nodes located atthe root and tip of the beam. In the first modeling approach, the static modes associated with the

33

0 2 4 6 8 10 12 14 16 18 20

-0.025

-0.02

-0.015

-0.01

-0.005

0

(a) Motion-based deformation measures.

0 2 4 6 8 10 12 14 16 18 20

-0.025

-0.02

-0.015

-0.01

-0.005

0

(b) Classical deformation measures.

Figure 7: Time history (s) of the axial tip displacement (m) for the rotating beam example. Refer-ence solution (−), one modal element with two interface nodes (◦−), one modal element with twointerface nodes + three clamped-free vibration modes (?−).

two interface nodes are used for the modal reduction (12 dofs). In the second and third modelingapproaches, the six and twelve lowest frequency clamped-free vibration modes are added to themodal basis, respectively, for a total of 18 dofs and 24 dofs, respectively.

0 5 10 15 20 25 30 35 40 45 50

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

45.5 45.6 45.7 45.8 45.9 46 46.1 46.2 46.3 46.4 46.5

-0.02

0

0.02

0.04

0.06

0.08

17.5 17.6 17.7 17.8 17.9 18 18.1 18.2 18.3 18.4 18.5

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

(a) Motion-based deformation measures.

0 5 10 15 20 25 30 35 40 45 50

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

45.5 45.6 45.7 45.8 45.9 46 46.1 46.2 46.3 46.4 46.5

-0.02

0

0.02

0.04

0.06

0.08

17.5 17.6 17.7 17.8 17.9 18 18.1 18.2 18.3 18.4 18.5

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

(b) Classical deformation measures.

Figure 9: Time history (s) of the root angular velocity (rad/s) for the beam on a spherical jointexample. Reference solution (−), one modal element with two interface nodes (◦−), one modalelement with two interface nodes + six clamped-free vibration modes (?−), one modal elementwith two interface nodes + twelve clamped-free vibration modes (?−).

Figure 9 shows the root angular velocity about the inertial direction in which the momenthas been applied; the reference solution was obtained using a fully nonlinear beam model. In thisexample, the use of the proposed and classical deformation measures yield the same level of accuracy.The modal element with two interface nodes does not captures the dynamic response of the beamaccurately. Adding vibration modes improves the correlation considerably. Although the iterationmatrix was evaluated in the reference configuration once only, convergence was achieved at each timestep in about three iterations. The accuracy of the solution with classical deformation measuresreported by Cardona [39] degrades significantly after the vertical impulse is applied. Ellenbroek

34

and Schilder [42] obtained an accurate solution with classical deformation measures and attributedthis success to a proper definition of the floating frame.

5.4 Flexible slider-–crank mechanism

Crank

ConnectingrodO

S

Slider

B

R

MB

N

Figure 10: Flexible slider-crank mechanism, sec-tion 5.4.

Wu and Tiso [18] have described the planar,flexible slider–crank mechanism depicted in fig-ure 10. The crank of length Lc = 10 m is at-tached to the ground at point O and to the con-necting rod at point M via revolute joints. Atits other end, the connecting rod of length Lr =20 m is attached to the slider, modeled by a rev-olute joint connected in series with a groundedprismatic joint at point S. The mechanical prop-erties of the crank and connecting rod are asfollows: axial stiffness EA = 11.2 GN, bendingstiffness EI = 149.3 MN·m2, mass per unit length m = 432 kg/m, and moment of inertia, m22 =5759 g·m. In the initial configuration, the crank and connecting rod line up along the horizontal.The rotation at the root of the crank is prescribed as φ(t) = 3πt2/50 and hence, reaches 270 degreesin 5 s. The crank’s root rotation is then locked and remains at 270 degrees, causing large oscillationsof the system. The problem is simulated for 10 s with a constant time step size of 1 ms; the spectralradius of the generalized-α scheme is ρ∞ = 0.95.

The correlation focuses on the transverse displacement components at the mid-span of the crankand connecting rod denoted M and N, respectively. These displacement components are resolvedin frame R, attached to the material point at the root of the crank, and frame B, attached to thematerial point at the root of the connecting rod, respectively. Note that Wu and Tiso [18] useddifferent frames to report their predictions.

0 1 2 3 4 5 6 7 8 9 10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(a) Motion-based deformation measures.

0 1 2 3 4 5 6 7 8 9 10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(b) Classical deformation measures.

Figure 11: Time history (s) of the transverse displacement (m) at point M. Reference solution (−),one modal element with three interface nodes (◦−), one modal element with three interface nodes+ thee free-free vibration modes (?−).

The crank and connecting rod were each modeled as modal elements with two interface nodeslocated at their end points and one interface point at mid-span. In the first modeling approach,the static modes associated with the interface nodes are used for the modal reduction (18 dofs).In the second modeling approach, the lowest three bending modes of the free-floating crank and

35

connecting rod were added to their respective modal bases (24 dofs). Figures 11 and 12 show thetransverse displacement components at points M and N, respectively, resolved in frames R and B,respectively. The reference solution shown in these figures is obtained using a fully nonlinear beammodel.

When using the motion-based deformation measures, the modal elements capture the responseof the system accurately. The addition of vibration modes does not have a noticeable effect onthe response of the crank but improves the accuracy of the response of the connecting rod slightly.The iteration matrix was kept to its constant value evaluated in the reference configuration andconvergence was achieved at each time step in less than about 4 iterations.

0 1 2 3 4 5 6 7 8 9 10

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

(a) Motion-based deformation measures.

0 1 2 3 4 5 6 7 8 9 10

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

(b) Classical deformation measures.

Figure 12: Time history (s) of the transverse displacement (m) at point N. Reference solution (−),one modal element with three interface nodes (◦−), one modal element with three interface nodes+ thee free-free vibration modes (?−).

When using the classical deformation measures, the modal elements also capture the responseof the system accurately but the addition of vibration modes degrades the correlation for theconnecting rod. This degradation was also reported by Wu and Tiso [18]. Clearly, the formulationsbased on classical deformation measures are unreliable.

5.5 Out-of-plane loading of a right angle beam

F(t)_

i1_

i2_

i3_

R

E

T

Figure 13: Configuration of theright-angle beam.

Simo and Vu-Quoc [68] described the dynamic system consist-ing of two beams, each of length L = 10 m, connected to-gether at a ninety degree angle to form the elbow depicted infig. 13. The elbow is cantilevered and an out-of-plane concen-trated load F(t) is applied, where

F (t) = 50

t, t ≤ 1,

2− t, 1 < t < 2,

0, t ≥ 2.

(96)

The mechanical properties of the beams are as follows: axial stiffness EA = 1 MN, shear stiffnessesGA22 = GA33 = 1 MN, torsional stiffness GJ = 1 kN·m2, bending stiffnesses EI22 = EI33 = 1kN·m2, mass per unit length m = 1 kg/m, and mass moments of inertia per unit span m22 = m33

= 10 kg·m2/m. Due to the configuration of the system, a large, three-dimensional motion of thestructure is observed. The problem is simulated for 30 s with a constant time step size of 10 ms;the spectral radius of the generalized-α scheme is ρ∞ = 0.90.

36

Three different modeling strategies were used: (1) the system is modeled using a single modalelement featuring three interface nodes, two located at the ends points, and one at the elbow (18dofs), (2) the system is modeled using two modal elements, one for each of the beams and eachfeaturing two interface nodes (18 dofs), and (3) the system is modeled using four modal elements,one for each half of the beams and each featuring two interface nodes (30 dofs). Three additionalmodeling strategies were considered: the first six free-free vibration modes of each sub-componentwas added to the corresponding modal basis leading to 18 + 6 = 24, 18 + 2×6 = 30, and 30 + 4×6= 54 dofs, for modeling strategies (1), (2), and (3), respectively. For each modeling strategy, the 6dofs at the root of the beam are constrained by the boundary condition.

0 5 10 15 20 25 30

-10

-8

-6

-4

-2

0

2

4

6

8

10

(a) Motion-based deformation measures.

0 5 10 15 20 25 30

-10

-5

0

5

10

(b) Classical deformation measures.

Figure 14: Time history (s) of the out-of-plane tip displacement (m) for the right-angle beamexample without vibration modes. Reference solution (−), one modal element (◦−), two modalelements (?−), four modal elements (4−).

Figures 14 and 15 show the out-of-plane displacement at the free end of the system for thethree modeling strategies and the same strategies supplemented with additional vibration modes,respectively. The reference solution shown in these figures is obtained using a fully nonlinearbeam model. When using the proposed deformation measures, the following observations are made.While the single modal element with three interface nodes captures the overall motion of the system,significantly more accurate solutions are obtained by increasing the number of modal elements andby incorporating vibration modes. Excellent correlation is obtained with the reference solutionwhen using the third modeling strategy.

For the classical deformation measures, convergence could not be achieved with the single modalelement. Improved accuracy is obtained with the other two modeling strategies. Adding vibrationmodes to the second and third modeling strategies degrades and improves solution accuracy, respec-tively. Here again, formulations based on classical deformation measures are found to be unreliable.Although the iteration matrix evaluated in the reference configuration once only, convergence wasachieved at each time step in about two iterations.

6 Conclusions

This paper describes a general approach to the modal reduction of geometrically non-linear struc-tures. For linear systems, the motion is decomposed into a rigid-body motion and superimposeddeformation measures. To ensure a unique decomposition, six conditions are imposed on the de-formation measures that are shown to be objective. Next, modal reduction is applied to the de-formation measures using the mode-acceleration method. The advantages of this method were

37

0 5 10 15 20 25 30

-10

-8

-6

-4

-2

0

2

4

6

8

10

(a) Motion-based deformation measures.

0 5 10 15 20 25 30

-10

-5

0

5

10

(b) Classical deformation measures.

Figure 15: Time history (s) of the out-of-plane tip displacement (m) for the right-angle beamexample with the first six free-free vibration modes of each modal element. Reference solution (−),one modal element (◦−), two modal elements (?−), four modal elements (4−).

highlighted: (1) any type of vibration mode can be used, (2) due to the presence of static modes,rigid-body modes are represented exactly based on their values at the interface nodes, and (3) eigen-pairs of the complete system that are included in the modal basis are represented exactly by thereduced system. Well-known methods such as Craig-Bampton’s, Herting’s, and Rubin’s methodswere shown to be particular cases of the mode-acceleration method. Because rigid-body modes arerepresented exactly, the floating frame of reference dofs can be eliminated.

Next, geometrically nonlinear problems were investigated within the framework of the motionformalism. By analogy to the linear problem, the overall motion is decomposed into a rigid-bodymotion (floating frame of reference) and superimposed elastic motions. The deformation measuresat the interface nodes are defined through a vectorial parameterization of motion of the elasticmotions. Six linear conditions are imposed on the deformation measures to provide objectivity anda unique decomposition.

The motion formalism provides the theoretical basis for a sound treatment of the kinematics ofthe problem: (1) it treats the position and rotation fields of motion as a unit, (2) it streamlinesthe treatment of rigid-body motion, (3) it leads to deformation measures that are of a tensorialnature, (4) it couples the displacement and rotation parts of the deformation measures throughthe tangent operator of the rotation part of the vectorial parameterization. In contrast, classicalkinematic approaches treat motion as separate position and rotation fields leading to deformationmeasures that are uncoupled and are not of a tensorial nature.

The evaluation of the reduced elastic and inertial forces uses constant reduced stiffness and massmatrices and nonlinear projection operators that depend on the reduced variables only. The costof this evaluation is thus independent of the size of the original system.

All derivatives used in this work are material derivatives. This approach yields numerous ad-vantages: (1) the reduced elastic and inertial forces depend on the configuration of the modalcomponent only through the deformation measures, thereby leading to low-order geometric nonlin-earities and (2) system tangent matrices such as the mass, stiffness, and gyroscopic matrices remainnearly constant throughout a simulation; geometrically non-linear problems can be solved with aconstant iteration matrix. In classical kinematics formulations based on an inertial representationof the derivatives the reduced forces and the tangent matrices depend on the orientation of thecomponent with respect to an inertial frame and the iteration matrix has to be updated frequently.

Numerical examples have been presented to demonstrate the accuracy, robustness and numerical

38

efficiency of the proposed approach. Even with a small number of modal elements, the formulationis able to capture geometrically nonlinear effects accurately; the addition of vibration modes im-proves the response systematically. For all examples, the predictions of the proposed formulationoutperform those of their classical counterparts.

Future research will investigate the efficient treatment of distributed externally applied loadswithin the proposed framework.

A Euler-Rodrigues motion parameterization

Parameterization formulas relate a transformation matrix and the motion parameters

H(P) =

[R(p

θ) T T (p

θ)pu

0 1

], (97a)

P(H) =

{pu

pθ

}=

{T−T (p

θ)u

p(R)

}, (97b)

The formulas related to Euler-Rodrigues motion parameters are given in this section. The appeal forthis parameterization is its low computation cost. In particular, it does not involve any trigonometricfunctions and the number of operations is limited as compared to other parameterizations.

The rotation parameters are obtained from a rotation matrix as

p(R) =1√

trace(R) + 1(R−RT ) (98)

The rotation matrix is obtained from the rotation parameters as

R(p) = I + p0p+1

2pp, p0 =

√1− pTp/4 (99)

The tangent operator and its inverse for the rotation parameterization read

T (p) =1

p0I − 1

2p+

1

4p0pp (100)

T−1(p) = p0I +1

2p (101)

The time derivative of the inverse of the tangent operator is given by

(T−1)˙(p, p) =1

2˜p−

pT p

4p0I (102)

The tangent operator and its inverse for the motion parameterization read

T (P) =

[T (p

θ) T

+(P)

0 T (pθ)

](103)

T −1(P) =

[T−1(p

θ) T−(P)

0 T−1(pθ)

](104)

where, with ρ = pTθpu,

T+

(P) =ρ

4p30(I +

1

4pθpθ)−

1

2pu +

1

4p0(pθpu + pupθ) (105)

T−(P) =−ρ4p0

I +1

2pu (106)

39

The time derivative of the inverse of the tangent operator is given by

(T −1)˙(P , P) =

[(T−1)˙(p

θ, p

θ) (T−)˙(P , P)

0 (T−1)˙(pθ, p

θ)

](107)

where

(T−)˙(P , P) =1

2pu −

1

4p0

(ρ

4p20(pTθpθ) + pT

θpu

+ pTθpu

)I (108)

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