Spectral Collocation Methods for the Periodic Solutionof Flexible Multibody Dynamics∗

Shilei Han and Olivier A. BauchauDepartment of Aerospace Engineering, University of Maryland

College Park, Maryland 20742

Abstract

Many flexible multibody systems of practical interest exhibit a periodic response. Thispaper focuses on the implementation of the collocation version of the Fourier spectral methodto determine periodic solutions of flexible multibody systems modeled via the finite elementmethod. To facilitate the analysis and obtain governing equations presenting low-order non-linearities, the motion formalism is adopted. Application of Fourier spectral methods requiresglobal interpolation schemes that approximate the unknown fields over the entire period ofresponse with exponential convergence characteristics. The classical spectral interpolationschemes were developed for linear fields and hence, do not apply to the nonlinear configura-tion manifolds, such as SO(3) or SE(3), that are used to describe the kinematics of multibodysystems. Furthermore, the configuration and velocity fields are related through nonlinearkinematic compatibility equations. Clearly, special procedures must be developed to adaptthe Fourier spectral approach to flexible multibody systems. The spectral interpolation ofmotion is investigated; interpolation schemes based on the polar decomposition are proposed.Assembly of the linearized governing equations at all the grid points leads to the govern-ing equations of the spectral method. Numerical examples illustrate the performance of theproposed approach.

1 Introduction

The dynamic response of many flexible multibody systems of practical interest is periodic. Theflexible components of rotating machinery fall into that category. Consider, for instance, helicopter,wind turbine, or jet engine rotor blades, among many other systems. While the response of thesecomponents is not exactly periodic due to the presence of turbulence or manufacturing imperfections,the solution is often assumed to be periodic and the solution process focuses on the determinationof this periodic response.

The transient response of such systems is also of interest: consider a helicopter performing amaneuver or a wind turbine yawing under shifting wind directions. In such cases, system response isoften treated as approximately periodic, i.e., the period of rotation of the rotor blades is assumed tobe far smaller than the time constant of the maneuver: the maneuver is a “steady-state maneuver”and the response is taken to be periodic.

From a physical standpoint, two types of periodic problems can be identified: (1) systemssubjected to periodic loading and responding periodically and (2) self or parametrically excitedsystems exhibiting limit-cycle oscillations. This paper deals with the first type of problems where

∗Nonlinear Dynamics, 92(4), pp 1599–1618 2018.

1

the period of oscillation is known from the onset. The approaches to the determination of periodicsolutions fall into three broad categories: the time-marching method, the shooting method [1], andthe Fourier spectral method [2, 3, 4].

The periodic solution of a system can be found with time-marching algorithms. Starting fromarbitrary initial conditions, the response of the system is evaluated; as the transient response diesout due to damping, the stable periodic response remains. For systems presenting low damping, timemarching over several periods of response might be required for the transients to die out completely.Consequently, the computational effort associated with this approach is high and furthermore, timemarching algorithms typically require small time step sizes to guarantee converge. When viewedin the frequency domain, the predictions of time-marching algorithms involve all frequencies fromzero to Nyquist’s frequency; only the low frequencies, however, are of practical interest. In fact,numerical dissipation is often introduced in the integrator to filter out the higher-frequency content.

In the shooting method, the periodic boundary value problem is transformed into a sequenceof initial value problems through Newton iteration. Starting from arbitrary initial conditions, thesolution is improved iteratively to satisfy the periodic boundary conditions. In this approach, theaccurate evaluation of the Jacobian requires considerable computational effort because the methodmight fail to converge if crude approximations of the Jacobian are used.

In the Fourier spectral method, the first step is to expand the periodic response of the systemin Fourier series. In the Galerkin version of the method [2, 3, 4], the residuals are required to beorthogonal to the space spanned by Fourier series. This approach, also referred to as the “harmonicbalance method,” was first introduced by Krylov and Bogoliubov [5] and has been used widelyto determine analytical solutions of nonlinear, single-degree-of-freedom oscillator problems [6, 7].With the emergence of computers, problems involving multiple degrees of freedom can be treated:examples can be found in nonlinear circuit analysis [8], nonlinear vibration and structural dynamicproblems [9, 10, 11, 12], or rotors and bladed disks [13, 14]. When applied to systems with numerousdegrees of freedoms, the cost of the harmonic balance analysis grows rapidly with the number ofharmonics, as noted by Hall et. al [15, 16, 17].

In the collocation version of the Fourier spectral method [2, 3, 4], residuals are required to vanishat a set of grid points. Because it proceeds in the time domain, this approach avoids back and forthtransformations between the time and frequency domains, leading to a more efficient formulation.Furthermore, the solution converges exponentially as the number of grid points increases becausethe spectral collocation approach is a global, spectrally accurate method. On the other hand, allgrid points are coupled, leading to a nonlinear set of equations of a size far larger than that usedby time marching methods. The state-of-the-art approach is to use preconditioned iterative solversto obtain the solution.

In the Fourier spectral collocation approach, the displacement, velocity, and acceleration fieldsare expanded in Fourier series: simple formulæ express all fields in terms of the discrete displacementvalues at the collocation points. These formulæ perform a spectral expansion of the unknownfields over the entire period of response; the exponential convergence of this interpolation schemeguarantees the exponential convergence of the Fourier spectral method. Clearly, the ability toperform a spectral expansion of the unknown fields is a prerequisite to the application of spectralmethods.

In this paper, the Fourier spectral collocation approach will be applied to multibody systemsconsisting of flexible components undergoing arbitrarily large motion but small deformation; theflexible components are connected together by kinematic joints. In classical kinematics, rigid-bodymotions are represented by independent displacement and rotation fields. The classical spectralinterpolation schemes [4] can be applied to the displacement, velocity, and acceleration fields becausethey form linear fields. The same interpolation formulæ do not apply to the rotation field becauseit forms the Special Orthogonal group SO(3). Furthermore, the angular velocity field lies in thespace tangent to the rotation manifold. While the angular velocity field forms a linear field, it is

2

related to the rotation field through the nonlinear kinematic compatibility equations.In this paper, the kinematics of multibody systems will be described using the motion formalism

developed by Borri and Bottasso [18], Merlini and Morandini [19], and Sonneville et al. [20, 21]. Thisformulation leads to simple governing equations presenting low-order algebraic nonlinearities forthe flexible components. In contrast with classical kinematics that represent rigid-body motions byindependent displacement and rotation fields, the motion formalism treats motion as a unit; motionforms the Special Euclidean group SE(3). The challenges mentioned in the previous paragraphremain: the classical spectral interpolation schemes [4] do not apply to nonlinear configurationmanifolds such as SO(3) or SE(3) and the configuration and velocity fields are related throughnonlinear kinematic compatibility equations. Clearly, special procedures must be developed toadapt the Fourier spectral approach to the present problem and overcome these hurdles.

This paper is organized as follows. A brief introduction to the Fourier spectral method is givenin section 2. The analysis of kinematics of rigid-body motion via dual entities is introduced insection 3. The motion interpolation algorithm is developed in section 4; followed by a numericalexample for the algorithm in section 4.4. The transformation from dual entities to matrix/vectorform is discussed in section 5. The assembly of governing equations in spectral method is discussedin section 6. Numerical examples are presented in section 7. Summary and conclusions are listedat the end of the paper.

2 The Fourier spectral method, preliminaries

Consider a set of grid points equally spaced in time over a period T , tk = kT/N , k = 0, 1, . . . , N−1,where N is an odd number. Periodic function of time t, f(t), is of period T and expanded in Fourier

series as f(t) =∑(N−1)/2

j=−(N−1)/2 fj exp(Ijτ), where I =√−1 and the non-dimensional time variable is

τ = 2πt/T . The coefficients of the expansion, fj, can be found as fj =∫ 2π

0f(τ) exp(−Ijτ) dτ/(2π).

Evaluation of the integral via the trapezoidal rule yields the following approximation, fj ≈∑N−1k=0 f(τk) exp(−Ijτk)/N , where τk = 2πtk/T = 2πk/N . The trapezoidal rule is exact [4] if

f(t) ∈ Span[1, exp(±Iπt/T ), . . . exp(±I (N−1)πt/T )] and a good approximation otherwise [22, 23].Introducing the discrete expansion coefficients back into the Fourier series yields an interpolationscheme for periodic functions

f(τ) =N−1∑k=0

f(τk)

1

N

(N−1)/2∑j=−(N−1)/2

eIj(τ−τk)

=N−1∑k=0

IkN(τ)f(τk). (1)

The periodic sinc functions, IkN(τ), are defined as

IkN(τ) =1

N

(N−1)/2∑j=−(N−1)/2

eIj(τ−τk) = e−I(N−1)x1− exp(I2y)

N [1− exp(I2x)]

= e−I(N−1)xexp(Iy)

exp(Ix)

exp(−Iy)− exp(Iy)

N [exp(−Ix)− exp(Ix)]=

sin y

N sinx,

(2)

where x = (τ − τk)/2 and y = N(τ − τk)/2. The second equality in eq. (2) results from thesummation formula for geometric series and the last equality comes from the Euler formula forcomplex numbers.

Evaluating the first and second derivatives of interpolation scheme (1) with respect to time t

3

yields

f(τ) =N−1∑k=0

DkN(τ)f(τk), (3a)

f(τ) =N−1∑k=0

T kN (τ)f(τk), (3b)

The first and second derivative functions, denoted DkN(τ) and T kN (τ), respectively, are defined as

DkN(τ) = Ω

[cos y

2 sinx− sin y cosx

2N sin2 x

], (4a)

T kN (τ) = Ω2

[(cos2 x+ 1) sin y

4N sin3 x− cos y cosx

2 sin2 x− N sin y

4 sinx

], (4b)

where Ω = 2π/T = dt/dτ results from the transformation from time t to non-dimensional variableτ .

Figure 1 shows the interpolation functions for N = 7, k = 4, and Ω = 1.0. Evaluation of theperiodic sinc functions and their time derivatives at the grid points yields

IkN(τ`) =

1, for k = `,

0, for k 6= `.(5a)

DkN(τ`) = Ω

0, for k = `,

(−1)`−k1

2 sin[(τ` − τk)/2], for k 6= `.

(5b)

T kN (τ`) = Ω2

(1−N2)/12, for k = `,

(−1)`−k+1 cos[(τ` − τk)/2]

2 sin2[(τ` − τk)/2], for k 6= `.

(5c)

0 2π/7 4π/7 6π/7 8π/7 10π/7 12π/7 2π-4

-3

-2

-1

0

1

2

3

4

Grid points τk

Inte

rpol

atio

n f

un

ctio

ns

Figure 1: Fourier spectral interpolation func-tions: I47 (τ), solid line; D4

7(τ), dashed-dottedline; T 4

7 (τ), dashed line.

Furthermore, the periodic sinc functions satisfythe property of partition of unity,

N−1∑k=0

IkN(τ) = 1. (6)

The first- and second-order differentiation oper-ators can be represented by matrices D and T , re-spectively, both of size N ×N . The entries of D and

T located at the `th row and kth column are denoted

as D`,k = DkN(τ`) and T`,k = T kN (τ`), respectively. Itis verified easily that D and T are skew-symmetricand symmetric Toeplitz matrices, respectively.

The difference between the Galerkin and colloca-tion versions of the Fourier spectral method can nowbe explained more precisely. Consider the problemcharacterized by residual r(x, x, t) = 0, for which theperiodic solution, x(0) = x(2π), is to be determined.In the Galerkin version of the method, also called theharmonic balance method, both the unknown field, x(t), and residual, r(x, x, t), are expanded in

4

Fourier series and the coefficients of these expansions are denoted xj and rj, j = 0, 1, . . . , (N−1)/2,respectively. Orthogonality now requires the vanishing of coefficients rj, leading to a set of nonlinearalgebraic equations for coefficients xj. The collocation method is more direct: the unknown andresidual fields are interpolated by eq. (1) and the residual is required to vanish at the grid points,r(x(tk), x(tk), tk) = 0.

Both approaches, however, are equivalent. Using the trapezoidal rule leads to rj =∫ 2π

0r(τ) exp(−Ijτ) dτ/(2π) ≈

∑N−1k=0 r[x(tk), x(tk), tk] exp(−Ijτk)/N . Recasting this result in ma-

trix form yields

r0r1,cr1,s· · ·

r(N−1)/2,cr(N−1)/2,s

=

2

N

1/2 1/2 · · · 1/2cos τ0 cos τ1 · · · cos τN−1sin τ0 sin τ1 · · · sin τN−1

......

. . ....

cos[(N − 1)/2τ0] cos[(N − 1)/2τ1] · · · cos[(N − 1)/2τN−1]sin[(N − 1)/2τ0] sin[(N − 1)/2τ1] · · · sin[(N − 1)/2τN−1]

︸ ︷︷ ︸

denoted as F

r0r1r2...

rN−2rN−1

, (7)

where rj,c = r−j+rj and rj,s = (rj−r−j)/I, j = 1, 2, . . . , (N−1)/2 are the cosine and sine componentsof the Fourier expansion of the residual, respectively, and rk = r[x(tk), x(tk), tk], k = 0, 1, . . . , N−1,are the values of the residuals at the grid points. Although resulting from the trapezoidal rule, eq. (7)represents the discrete Fourier transform that relates the time and frequency domain representationsof the residuals. Indeed, transformation matrix F , of size N ×N , is invertible and its inverse is

F−1 =

1 cos τ0 sin τ0 · · · cos[(N − 1)/2τ0] sin[(N − 1)/2τ0]1 cos τ1 sin τ1 · · · cos[(N − 1)/2τ1] sin[(N − 1)/2τ1]...

......

......

...1 cos τN−1 sin τN−1 · · · cos[(N − 1)/2τN−1] sin[(N − 1)/2τN−1]

. (8)

Clearly, imposing the vanishing of the residuals at the grid points, rk = 0, or the vanishing of thecoefficients of the discrete Fourier transform of the residual, rj = 0, as done in the collation andGalerkin versions of the Fourier spectral approaches, respectively, is identical.

3 Kinematics of rigid-body motion

This section presents a short summary of the kinematics of rigid-body motion that will be used inthe rest of the paper. Additional details can be found in Borri et al. [24], Sonneville et al. [20, 21],Bauchau [25, 26], and Han and Bauchau [27]. To facilitate the analysis, rigid-body kinematicsis presented in the framework of dual algebra. A brief introduction to dual algebra is given inappendix A and more detailed presentations are found in refs. [28, 29, 30, 31, 32, 33, 34].

3.1 Motion tensor

A motion is defined as the transformation that brings inertial frame F = [O, I = (ı1, ı2, ı3)] tomaterial frame Fb = [B,B = (b1, b2, b3)]. Motion forms the Lie group SE(3) and can be representedas

c =

[R r0T 1

], (9)

where position vector r is the relative position vector of reference point B with respect to the origin,O, and rotation tensor R brings inertial basis I to material basis B. The adjoint mapping associated

5

with motion c is denoted R and its dual matrix representation is

R = R + ε rR, (10)

where parameter ε is such that εn = 0 for n ≥ 2, see details in appendix A; notation

(·) =

0 −(·)3 (·)2(·)3 0 −(·)1−(·)2 (·)1 0

(11)

indicates the skew-symmetric matrix of size 3×3 associated with a vector of size 3×1. The set of dualmatrices R forms the special dual orthogonal group because R TR = RTR+ ε(RT rTR+RT rR) = Iand det(R ) = 1. Matrix R is referred to as the motion tensor because it is a natural generalization

of rotation tensor R. The set of rotation tensors, R ∈ R3×3|RTR = I, det(R) = 1, forms the

special orthogonal group in R3×3, denoted SO(3). The set of motion tensors, R ∈ D3×3|R TR =

I , det(R ) = 1, forms the Special Orthogonal group in D3×3, denoted SO(3), which is isomorphic to

the special Euclidean group SE(3). The representation of motion defined by eq. (10) will be usedthroughout this work.

The generalized velocity resolved in the material frame is a twist defined as

v = R T R = −RT

R = ω + εv, (12)

where notation ˙(·) indicates a derivative with respect to time and taking a derivative of R TR = Iyields the second equality. Equation (12) is known as the kinematic compatibility equation: itrelates the velocity and motion fields. The velocity field is an element of so(3), the Lie algebra ofSO(3). The angular and linear velocities expressed in the material basis are denoted ω = axial(RT R)

and v = RT r, respectively. Therein, notation

axial(·) =1

2

(·)32 − (·)23(·)13 − (·)31(·)21 − (·)12

(13)

indicates the extraction of a vector of size 3× 1 from a matrix of size 3× 3.The definition of the incremental motion resolved in the material frame is similar to that of the

velocity vector in eq. (12),

∆u = R T∆R = −∆R TR = ˜∆ψ + εRT∆r, (14)

where RT∆r and ∆ψ = axial(RT∆R) represent the incremental displacement and rotation vectors,respectively, both resolved in the material basis.

An increment of dual vector axial(R ) is evaluated as

∆[axial(R )] = axial(R ∆u) =1

2[tr(R )I − R T ]∆u, (15)

where notation tr(A) indicates the trace of matrix A and the second equality results from iden-

tity (38b). Similarly, an increment of dual vector axial(R1,2

) = axial(R T

1R

2) is evaluated as

∆axial(R1,2

) = axial(−∆u1R1,2

+ R1,2

∆u2)

=1

2[tr(R

1,2)I − R T

1,2]∆u2 −

1

2[tr(R

1,2)I − R

1,2]∆u1,

(16)

6

where identities (38a) and (38b) were used to obtain the last equality. Finally, an increment of dualvector axial(qR

1,2), where q is an arbitrary dual vector, is evaluated as

∆axial(qR1,2

) =axial(−q∆u1R1,2

+ qR1,2

∆u2 + ∆qR1,2

)

=

[1

2R

1,2q − q axial(R T

1,2)

]∆u2 −

[1

2R T

1,2q − axial(R

1,2)qT]

∆u1

+1

2

[tr(R

1,2)I − R

1,2

]∆q ,

(17)

where identities (38c) and (38d) were used to obtain the last equality.

4 Interpolation of the motion field

Implementations of spectral methods use interpolation formulæ (1) and (3) to approximate theunknown field and its time derivative, respectively. Because spectral interpolation formulæ (1)and (3) are linear operators, they cannot be applied to Lie groups, such as motion fields, directlyalthough these operators could be applied to any motion parameter vector [25] parameterizing themotion field.

Efficient spatial interpolation schemes [35, 36, 37] have been developed for geometrically exactbeams. These schemes assume that the deformation within an element becomes small as its sizedecreases. Relative motion parameter vectors within the element become small and interpolationof these quantities then yields excellent results. For spectral methods, however, the motion fieldmust be interpolated over one entire period. Because relative motions at the grid points cannotbe assumed to be small, relative motion parameter vectors might present singularities and theseschemes cannot be used confidently.

Typically, interpolation scheme (1) is obtained using the intuitive argument outlined in section 2,more details can be found in Hesthaven et al. [4]. The same scheme can be obtained from aminimization procedure. Let xk, k = 0, . . . , N − 1 be the values of a periodic function of time τ atgrid points τk and x(τ) the interpolated function. The interpolation of function in Euclidean spaceRn is defined by the following minimization problem

minx∈Rn

J = ‖x(τ)−N−1∑k=0

IkN(τ)xk‖2 (18)

where notation ‖ · ‖ denotes the Euclidean norm. The solution of minimization problem (18) thenyields interpolation scheme (1), where the interpolation functions are the periodic sinc functionsdefined in eq. (2). Minimization problem (18) is solved by imposing the vanishing of the variationof J , δJ = δxT [x(τ) −

∑N−1k=0 IkN xk] = 0 for any δx, which yields interpolation scheme (1). The

periodicity condition is imposed via shape functions IkN(τ).

4.1 Motion interpolation as a minimization

Figure 2 depicts the motion interpolation problem in a schematic manner. The known motiontensors at the grid points are denoted R

k, k = 0, . . . , N−1. The unknown interpolated motion tensor

at time τ is denoted R . Next, the relative motion tensors are defined: Sk

= R TRk, k = 0, . . . , N−1,

7

represent the relative motion tensors from instant τ to τk. The following notation is introduced

G =N−1∑k=0

IkN(τ)Rk, (19a)

S =N−1∑k=0

IkN(τ)Sk. (19b)

Note thatS = R TG . (20)

i1_

O

FI i2_

i3_

τ0

τ1

τ2τ-1

τ

R 0

R 1R -1

s0

s-1 s2

s1 R 2

Periodic motion

R

Figure 2: Interpolation of motion.

By analogy with minimization problem (18), the fol-lowing minimization problem is proposed for the interpo-lation of motion

minR∈SO(3)

J (R ) = ‖R − G‖2F , (21)

where the meaning of minimization of a dual function isgiven by definition (1) and notation ‖A‖F = [tr(ATA)]1/2

indicates the Frobenius norm of a square dual matrix A.Because dual matrix G is linear combination of orthogo-

nal dual matrices, it is not an orthogonal dual matrix andhence, does not represent a motion. Minimization prob-lem (21) can be interpreted as follows: motion tensor Ris the motion tensor that is as close as possible to dualmatrix G . The closeness of two dual matrices is defined as the square of the Frobenius norm of

their difference. The closest motion tensor is provided by dual polar decomposition theorem (3), asproved in appendix C.

Minimization problem (21) implies the minimization of the primal part of the objective function,minR∈SO(3) J = ‖R−G‖2F , which provides an approach for the interpolation of rotation, a problemthat has received considerable attention in many areas of engineering. A fundamental tool ofcontinuum mechanics [38, 39] is the multiplicative decomposition of the deformation gradient tensorinto a rotation tensor and a stretch tensor; typically, this operation is performed via the polardecomposition theorem [40, 41] or strategies based on quaternion algebra [42, 43]. The estimationof the attitude of a spacecraft from measured data [44] uses similar tools; because spacecraft attitudeis often specified in terms of quaternions, strategies based on quaternion algebra [45, 46] have beendeveloped. Because interpolation is a basic ingredient of the finite element method, interpolation ofrotation fields play an important role in the formulation of geometrically exact beam elements [47,48, 49, 50, 36, 51]. More recently, tools performing similar operations have been developed forcomputer graphics applications [52, 53, 54]. This paper shows that these approaches can be extendedeasily from rotation to motion by using dual entities, more details are given in appendix C.

4.2 Interpolation of velocities and accelerations

Taking the first and second time derivative of eq. (19b) and introducing identity (6) yields

N−1∑k=0

IkN(τ)Sk

+N−1∑k=0

DkN(τ)Sk

= S , (22a)

N−1∑k=0

IkN(τ)Sk

+ 2N−1∑k=0

DkN(τ)Sk

+N−1∑k=0

T kN (τ)Sk

= S . (22b)

8

The components of the velocity vector resolved in the material frame are v = R T R . A time

derivative of composition Sk

= R TRk

yields Sk

= RT

Rk

= RT

R R TRk

= −v Sk. It then follows that∑N−1

k=0 IkN Sk

= −v S and introducing this result into eq. (22a) then yields the velocity interpolation

v(τ) = −S S−1 +N−1∑k=0

DkN(τ)SkS−1. (23)

Matrix equation (23) is composed of nine scalar equations for nine unknowns: the three componentsof skew-symmetric matrix v and six components of symmetric matrix S . Conceptually, this system

can be solved by extracting the symmetric part of eq. (23), leading to six equations for the six entriesof symmetric matrix S . Introducing this solution into eq. (23) and extracting its skew-symmetricpart then yield the explicit form of the velocity interpolation. The spectral method requires theevaluation of the interpolated velocity at the grid points only, i.e., at τ = τ` and S(τ`) = I . Because

matrix S is symmetric, the interpolated velocity at the grid points reduces to

v(τ`) =N−1∑k=0

DkN(τ`)axial(S`k

), (24)

where S`k

= R T

`Rk.

A time derivative of Sk

= −v Sk

yields Sk

= −v Sk− ˙v S

kand hence,

∑N−1k=0 IkN S

k=

−v∑N−1

k=0 IkN Sk− ˙v S . Introducing this result into eq. (22b) then leads to S − 2

∑N−1k=0 D

k

N(τ)S

k−∑N−1

k=0 Tk

N(τ)S

k= −v

∑N−1k=0 I

k

N(τ)S

k− ˙v S . Finally, because S

k= −v S

k, the interpolated accelera-

tion becomes

˙v(τ) = −S S−1 − 2vN−1∑k=0

DkN(τ)SkS−1 +

N−1∑k=0

T kN (τ)SkS−1 + v v . (25)

Here again it is possible to solve this system by extracting the symmetric part of eq. (25) to findthe six entries of symmetric matrix S . Introducing this solution into eq. (25) and extracting itsskew-symmetric part then yield the explicit form of the acceleration interpolation. For applicationto the spectral method, however, it is only necessary to compute the accelerations at the grid points,

v(τ`) = axial

(−2v`

N−1∑k=0

D`,kS`k

+N−1∑k=0

T`,kS`k

). (26)

4.3 Linearization of the interpolation scheme

In the collocation version of the spectral method, the governing equations are evaluated at the gridpoints only and hence, the interpolation schemes for velocities and accelerations should be linearizedat the grid points only. Linearizing velocity interpolation scheme (24) and introducing identity (16)yields

∆v ` =1

2

N−1∑k=0

D`,k(

[tr(S`k

)I − ST`k

]∆uk − [tr(S`k

)I − S`k

]∆u`). (27)

9

Similarly, linearizing acceleration interpolation scheme (26) and introducing identity (17) leads to

∆v ` =N−1∑k=0

D`,k

([ST`k

v ` − 2axial(S`k

)vT` ]∆uk − [S`k

v` − 2v `axial(S`k

)T ]∆u` − [tr(S`k

)I − S`k

]∆v `)

+1

2

N−1∑k=0

T`,k(

[tr(S`k

)I − ST`k

]∆uk − [tr(S`k

)I − S`k

]∆u`).

(28)Let arrays ∆uT = ∆uT0 ,∆uT1 , . . . ,∆uT−2,∆uT−1 and ∆vvvT = ∆vT0 ,∆vT1 , . . . ,∆vT−2,∆vT−1 store

the incremental motion and velocity vectors at all the grid points, respectively. The linearizedvelocity and acceleration interpolation expressions, eqs. (27) and (28), respectively, can be restatedin compact forms as

∆vvv = Γv∆u, (29a)

∆vvv = Γa∆u = (Λ Γ

v+ Γ

d)∆u, (29b)

where matrices Γv, Γ

a, Λ, and Γ

dare all composed of N ×N sub-matrices of size 3× 3. Notation

[·]`k indicates the sub-matrices of size 3× 3 at location (`, k), `, k = 1, 2, . . . , N ,

[Γv

]`k

=1

2

−∑N−1

k=0 D`,k[tr(S`k

)I − S`k

], for k = `,

D`,k[tr(S`k

)I − ST`k

], for k 6= `.[Γd

]`k

=

−∑N−1

k=0 D`,k[S `kv` − 2v `axial(S`k

)T ]−∑N−1

k=0 T`,k[tr(S`k

)I − S`k

]/2, for k = `,

D`,k

[ST`k

v ` − 2axial(S`k

)vT` ] + T`,k[tr(S`k

)I − ST`k

]/2, for k 6= `.[Λv

]`k

=

−∑N−1

k=0 D`,k[tr(S`k

)I − S`k

], for k = `,

0, for k 6= `.

4.4 Numerical example of motion interpolation

To validate the interpolation scheme, a simple example is presented. Consider a periodic ro-tation described by the following time dependent Euler angles: precession φ = Ωt, nutationθ = π/13 sin(2Ωt), and spin ψ = π/17 [1 − cos(3Ωt)], where Ω = 9 rad/s. The rotation ten-sor is found by evaluating the following composition R = R(φı3)R(θı1)R(ψı3) and the associated

angular velocity is then ω = RT R.These rotation and angular velocity fields were interpolated based on the proposed scheme

using an increasing number of grid points, N = 3, 7, 15, 31, and 63. To assess the accuracy of theinterpolation, each grid interval [τi, τi+1] was divided into 60 subintervals and τk is the commontime for two adjacent subintervals. The interpolated rotations and velocities were evaluated andcompared to their exact counterpart, denoted R

eand ωe, respectively. The following error measures

were selected

eR =60N∑k=1

‖RT (τk)Re(τk)− I‖F

180N, eω =

60N∑k=1

‖ω(τk)− ωe(τk)‖60N‖ωe(τk)‖

.

Figure 3 shows the error measures defined by eq. (4.4) versus the number of grid points on alogarithmic plot. Clearly, both of the rotation and angular velocity fields converge exponentially,as required for the exponential convergence of spectral methods.

10

100 101 10210-15

10-10

10-5

100

Number of grid points, NTIn

terp

olat

ion

err

or

Figure 3: Interpolation error, (/): eR, (.): eω.

5 Recasting dual entities to vector/matrix form

As shown in the previous sections, the use of dual entities facilitates the derivation of kinematicequations dramatically. In dynamics problems, however, the inertial and elastic forces are notanalytical functions of the dual kinematic variables. To proceed, the dual entities are expandedinto matrix/vector form composed of their primal and dual components. For instance, dual vectora ∈ D3 is recast as vector a ∈ R6 and dual matrix A ∈ D3×3 is recast as matrix A ∈ R6×6 accordingto the following template

a = a+ εao =⇒ao

a

, (30a)

A = A+ εAo =⇒[A Ao

03

A

]. (30b)

6 Governing equations of multibody dynamics

Typical flexible multibody systems are composed of rigid-bodies, flexible components such as beams,plates, and kinematic joints. At an arbitrary time, the finite element method is used to discretizethe problem based on n structural nodes; superscript (·)i indicates the node number. The resultinggoverning equations are

V −

axial(R 1T R

1)

...

axial(R nT Rn)

= 0, (31a)

M V − V TM V + fE +GTλ = fA, (31b)

g(R 1, . . . ,R n) = 0, (31c)

where array V , of size 6n, stores the velocities at the nodes, V T = v1T , · · · , vnT, and block diagonalmatrix V = diag(v i), of size 6n × 6n, stores the nodal velocities of the system. If the system isof period T , periodic boundary conditions must be imposed: R i(0) = R i(T ) and v i(0) = v i(T ),

i = 1, 2, · · · , n.Equations (31a) are the kinematic compatibility equations for each of the structural nodes.

Equations (31b) are the dynamic equilibrium equations of the system. Matrix M , of size 6n×6n, is

11

the mass matrix of the system and array fA, of size 6n, stores the externally applied forces. Array

fE stores the elastic forces of the system, obtained by taking the directional derivative of the elastic

energy V along incremental motion array ∆U , i.e., ∆UTfE = DV [∆U ], where array ∆U stores the

nodal incremental motions, ∆UT = ∆u1T , . . . ,∆unT. Notation D(·)[∆U ] indicates the directionalderivative of quantity (·) along vector ∆U . Matrix G, of size m × 6n, is the constraint Jacobian,and array λ, of size m, stores the Lagrange multipliers used to impose the kinematic constraints.Equations (31c) are the m kinematic constraints at the mechanical joints of the system. Thekinematic constraints are associated with lower pair joints such as revolute, prismatic, cylindrical,screw, planar, and spherical joints or universal joints. The nodal momenta of the system areintroduced as P = M V , i.e., P T = p1T , · · · , pnT, where pi, of size 6× 1 denotes the momenta of

the ith node.In spectral methods, governing equations (31) are enforced at the grid points. As shown in

sections 4.2 and 4.3, the velocity and acceleration at a grid point are coupled with the correspondingquantities at all the other grid points, leading to a set of equations that couples all variables at allgrid points.

6.1 Linearized equations of motion

Because governing equations (31) are nonlinear, linearization is required for the solution process.Linearizing eqs. (31b) and (31c) leads to

M∆V +H∆V + (K +Kg)∆U +GT∆λ = −r, (32a)

G∆U = −g, (32b)

where matrix H is defined as H = −V TM−P and matrix P is a block diagonal matrix P = diag(pi),of size 6n× 6n. Notation pj, j = 1, . . . , n, indicates the matrix of size 6×6 defined as

pj =

[0 poj

poj pj

],

where arrays poj and pj are the partition of vector pj, see eq. (30a). Arrays r and g are the residualsassociated with the equilibrium equations and constraint equations, respectively.

Finally, the stiffness matrix is composed of two parts: the stiffness due to elastic deformationK ∆U = DfE[∆U ] and that due to the kinematic joints K

g∆U = D(GTλ)[∆U ].

6.2 Assembling the equations at all grid points

In the collocation version of the spectral method, equations of motion (31) are enforced at eachgrid point. The nonlinear nature of these equations has two implications: the solution processrequires the linearization step described in section 6.1 and degrees of freedom at all structuralnodes and all grid points become coupled. Arrays ∆U and ∆V, both of size 6n × N , store theincremental motions and velocities of all the structural nodes at all grid points, respectively, i.e.,∆UT = ∆u1T , . . . ,∆unT and ∆VT = vvv1T , . . . ,vvvnT, respectively.

The kinematic compatibility equations at each node, v i = R iT R , are discretized using the

spectral interpolation scheme discussed in section 4, leading to N sets of equations in the form ofeq. (24) for each node. These equations are then linearized, see eq. (29a), and assembled for all thenodes of the structure to yield

∆V = diag(Γiv)∆U. (33)

12

A similar procedure is used to discretize the nodal accelerations using eq. (26). The linearized formsof these equations, see eq. (29b), are then assembled for all nodes, leading to

∆V = diag(Γia)∆U. (34)

When evaluated at grid point τk, the linearized equations of equilibrium (35a) and constraintequations (35b) become

Mk∆V k +H

k∆V k + (K

k+K

g,k)∆uk +GT

k∆λk = −rk, (35a)

Gk

∆uk = −gk, (35b)

where subscript (·)k indicates quantities evaluated at grid point τk.To facilitate the description of the assembly process, the following Boolean matrices are intro-

duced

Ak

=

Q1

k⊗ I

6

· · ·Qn

k⊗ I

6

, Bk

= qk⊗ I

m,

where symbol ⊗ indicates the Kronecker product of matrices. All the entries of matrix Qi

k, of size

N × n, vanish except for a unit entry at location (k, i). Similarly, all the entries of array qk, of size

N × 1, vanish except for a unit entry at location k. Array L, of size mN × 1, stores the Lagrangemultipliers for all constraints at all grid points, LT = λT0 , . . . , λTN−1.

With these definitions, the unknowns at grid point τk can be expressed as ∆V k = AT

k∆V,

∆V k = AT

k∆V, ∆Uk = AT

k∆V, and ∆λk = BT

k∆L. Introducing these relationships into eqs. (35),

left multiplying eqs. (35a) and (35b) by Ak

and Bk, respectively, and summing over all grid points

leads to

M∆V + H∆V + K∆U + GT∆L = −R, (36a)

G∆U = −C, (36b)

where the following quantities are defined,

M =N−1∑k=0

AkM AT

k, H =

N−1∑k=0

AkHkAT

k,

K =N−1∑k=0

Ak(K

k+K

g, k)AT

k, G =

N−1∑k=0

BkGkAT

k,

R =

r0...

rN−1

, C =

g0...

gN−1

.

Introducing the linearized velocity and acceleration interpolation (33) and (34) yield the finalgoverning equations of the spectral method[

S GT

G 0

]∆U∆L

= −

RC

, (37)

where S = M diag(Γia) + H diag(Γi

v) + K. For the beam components, matrices M , H

k, and K

kand

hence, matrices M, H, and K are highly sparse matrices with entries concentrated in a small bandnear the diagonals. The unknowns at different grid points, however, are coupled together because ofthe off-diagonal entries of matrices Γi

aand Γi

v. Consequently, matrix S features a large bandwidth

and hence, application of a direct factorization scheme to this matrix is computationally inefficient.

13

6.3 Configuration updating

An initial guess of the motion and velocity field R i(τk) and v i(τk), i = 1, 2, . . . , n, for all the

structural nodes is given to initiate the spectral method. The Lagrange multipliers can be set to zero.Solving linearized governing equation (37) at each Newton iteration leads to increments ∆ui(τk) andλτk . The motion field is then updated by exponential map R i(τk) = R i(τk) exp[∆ui(τk)]. When the

solution is converged, the motion tensors and velocity vectors at all grid points are obtained. Themotion tensors, velocities, and accelerations at any time can be evaluated using the interpolationschemes in sections 4.1 and 4.2 as a post-processing step.

7 Numerical examples

To validate the proposed approach, three numerical examples are presented.

7.1 Planar, rigid mechanism with damper

AO

B

i2

_

i1

_

b1

_

b2

_

bar 1

bar

2

θ1 = Ωt

θ2

Revolutejoint

Figure 4: Configuration of a pla-nar, rigid mechanism.

The first example deals with the planar, rigid mechanism de-picted in fig. 4. This very simple example is intended to demon-strate the computation advantage of spectral methods. Bar 1is connected to the ground via a revolute joint at point O andto bar 2 by means of a revolute joint at point A. Bars 1 and2 are rigid and of lengths 0.5 ft and 14.3 ft, respectively. Themass per unit length of bar 1 and 2 is 0.12 lb/ft. The angularvelocity of bar 1 is prescribed as Ω = 50 rad/s. At point A,a torsional spring-damper element connects bars 1 and 2. Thesystem is initially at rest.

The spring-damper element is modeled as a Zener solid el-ement; the moment in the element is M = K∞θ2 +Kbα, whereinternal state α is governed by the following evolution equa-tion, τbα + α = θ2. The stiffness constants of the model areselected as K∞ = 2.0 × 105 lb·ft/rad and Kb = 1.5 × 105 lb·ft/rad. Finally, relaxation time τbcontrols the amount of dissipation in the joint.

Clearly, the periodic solution of the mechanism corresponds to the configuration where bars 1and 2 are aligned and rotating with period T = 2π/Ω. Relative degree of freedom θ2 vanishes inthe steady periodic solution. First, the periodic solution of the problem was obtained using theproposed spectral method; one single Newton iteration was sufficient to solve eqs. (37) accurately.

Next, the periodic solution of the system was obtained using the time marching approach. Thegeneralized-α scheme was used to integrate the equations of motion using a constant time stepsize ∆t = 3 ms, which corresponds to about 25 time steps per revolution; initial conditions wereselected as θ1 = θ2 = α = 0 rad and θ1 = Ω and θ2 = α = 0 rad/s. Fig. 5 depicts the time historyof displacement of point B along the direction of unit vector b2 for six values of the relaxation time,τb = T/20, T/10, 3T/20, T/5, T/4 and 3T/10. The time history of relative rotation θ2 is shown infig. 6.

Due to the initial conditions, bar 2 vibrates in the rotating frame until the associated energyis dissipated in the damper. Because the system is lightly damped, bar 2 still vibrates after eightrevolutions, but the amplitude of the motion has deceased dramatically. In theory, the periodicsolution cannot be reached in a finite number of revolutions; in practice, some criterion is requiredto stop the time marching procedure and declare the response to be “almost periodic.” For lightly

14

-6

-4

-2

0

2

4

6

0 2T 4T 6T 8T7T 8T

-3

-2

-1

0

1

2

3 10-2

Time t

Dis

pla

cem

ent

u2

[ft]

Figure 5: Time history of displacement of pointB along b2 direction: τb = T/20 (), τb = T/10(), τb = 3T/20 (.), τb = T/5 (+), τb = T/4 (∗),τb = 3T/10 ().

0 2T 4T 6T 8T-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

7T 8T-3

-2

-1

0

1

2

3 10-3

Time t

An

gle θ 2

[rad

]

Figure 6: Time history of relative rotation θ2:τb = T/20 (), τb = T/10 (), τb = 3T/20 (.),τb = T/5 (+), τb = T/4 (∗), τb = 3T/10 ().

damped systems, spectral methods are far more efficient than their time marching counterparts;furthermore, they provide the true periodic solution of the problem.

7.2 Spatial four-bar mechanism

bar 3

A

C

D

i1_

B

O

k

i2_

i3_

θ

bar 2

bar 1

0.40m

0.85m

Revolute jointUniversal joint

Spherical jointTorsional spring

Figure 7: Configuration of a spatialfour-bar mechanism.

The second example deals with the flexible, spatial four-bar mechanism depicted in fig. 7. Bar 1 is connected tothe ground via a revolute joint at point A and to bar 2by means of a spherical joint at point B. In turns, bar 2is connected to bar 3 via a universal joint at point C andfinally, bar 3 is connected to the ground via a revolutejoint at point D. For the reference configuration of thesystem depicted in fig. 7, the coordinates of points A, B,C, and D are xA = (−0.4, 0, 0) m, xB = (−0, 75, 0, 0.652)m, xC = (0, 0.85, 0.2)m , xD = (0, 0.85, 0) m, respectively.

Bars 1 and 2 are of square cross-section of size 16 by16 mm; Bar 3 has a square cross-section of size 8 by 8mm. The three bars are made of steel, whose mechanicalcharacteristics are Young’s modulus E = 207 GPa andPoisson’s ratio ν = 0.3. The sectional stiffness propertiesobtained through a cross-sectional analysis [55, 56, 57]are listed in table 1. The sectional mass properties areas follows: mass per unit span m00 = 1.997 and 0.4992 kg/m, moments of inertia per unit spanm22 = m33 = 42.60 and 2.662 mg·m2/m for bars 1 and 2, and bar 3, respectively.

Table 1: Sectional stiffness properties of the barsAxial Shearing Shearing Torsional Bending BendingS [MN] K22 [MN] K33 [MN] H11 [N·m2] H22 [N·m2] H33 [N·m2]

Bar 1 & 2 52.99 16.88 16.88 733.5 1131 1131Bar 3 13.25 4.220 4.220 45.84 70.66 70.66

The revolute joint at point A includes a torsional spring of stiffness constant k = 18 N·m/rad.

15

In the reference configuration, the three bars are unstressed and the torsional spring is un-stretched.At point D, the rotation of bar 3 is prescribed as θ = 0.3t+ [1− cos(0.3t)]/3 + [sin(0.6t)]/10 + [1−cos(0.9t)]/21 rad. The problem is periodic with a period T = 20.944 s.

0 T/4 T/2 3T/4 T0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

Time

Dis

pla

cem

ent x 3

at B

[m

]

Figure 8: Coordinate x3 of point B: N = 3 (),N = 5 (), N = 7 (.), N = 11 (+), N = 21 (∗).

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 T/4 T/2 3T/4 TTime

Vel

ocit

y x 3

at

B [

m/s

]

.

Figure 9: Velocity component v3 at point B:N = 3 (), N = 5 (), N = 7 (.), N = 11 (+),N = 21 (∗).

Each of the three bars is meshed using eight, two-node beam elements. The periodic solution ofthe problem is obtained using the proposed spectral method with a varying number of grid points,N = 3, 5, 7, 11, and 21. Figures 8 and 9 depict the position and velocity components of point B,both along unit vector ı3, respectively. The predictions are quite inaccurate when 3 grid points areused, which is equivalent to including the constant and first-order harmonics to approximate thesolution. As the number of grid points increases, the solution converges rapidly: the predictionsusing 11 and 21 grid points are nearly identical. The same behavior is observed in fig. 10, whichshows the driving torque at point D.

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 T/4 T/2 3T/4 TTime

Act

uac

tor

torq

ue

Figure 10: The driving torque at point D: N = 3 (), N = 5 (), N = 7 (.), N = 11 (+), N = 21(∗).

7.3 Planar four-bar mechanism

Figure 11 depicts a flexible four-bar mechanism. Bar 1 is of length 0.12 m and is connected tothe ground at point A by means of a revolute joint. Bar 2 is of length 0.24 m and is connectedto bar 1 at point B with a revolute joint. Finally, bar 3 is of length 0.12 m and is connected to

16

bar 2 and the ground at points C and D, respectively, by means of two revolute joints. In thereference configuration, the bars of this planar mechanism intersect each other at 90 degree anglesand the axes of rotation of the revolute joints at points A, B, and D are normal to the plane ofthe mechanism. To simulate an initial defect of the mechanism, the axis of rotation of the revolutejoint at point C is rotated by +5 degrees about unit vector ı2 indicated in fig. 11. The angularvelocity of bar 1 at point A is prescribed as Ω = 0.6 rad/s for the duration of the simulation.

0.24 mbar 2

bar

1

bar 3

A

B C

D0.

12 m

Revolute jointsBeams

Ω = 0.6 rad/s

i2_

i1_

8 mm16 mm

16 m

m

8 m

m

b3_

b2_

b3_

b2_

Misalignedaxis of rotation

Bars 1&2sections

Bar 3section

Figure 11: Configuration of the four-bar mechanism.

If the bars were infinitely rigid, no motion would be possible because the mechanism locks.For elastic bars, motion becomes possible, but generates large, rapidly varying internal forces andmoments at time t = T/4 and 3T/4, when the three bars align in one line. In the present analysis,each of the three bars is meshed using 5 three-node beam elements. The periodic solution isobtained using the proposed spectral method with a varying number of grid points, N = 5, 7, 11,and 21. The reference solution is from Bauchau et al. [58]: time-marching algorithms were usedfor three complete revolutions; the third revolution is presented in the figures. The component ofdisplacement along unit vector ı1, denoted u1, at point C is shown in fig. 12. At point C of bar 2,the Euler angles (sequence 3-1-2) defining the orientation of bar 2 with respect to the inertial basisare computed and the first angle of the sequence, φ, is shown in fig. 13. The relative rotation, θ, atthe revolute joint at point D is depicted in fig. 14. The prediction of the spectral method agreeswell with the reference solutions for N ≥ 11 except in the region near t = T/4 and 3T/4.

0 T/4 T/2 3T/4 T-0.15

-0.10

-0.05

0

0.05

0.10

0.15

Time

Dis

pla

cem

ent u

1 [m

]

Figure 12: Displacement u1, at point C: timemarching (), N = 5 (), N = 7 (.), N = 11(/), N = 15 (+), N = 21 (∗).

0 T/4 T/2 3T/4 T-10

0

10

20

30

40

50

60

70

Time

Eu

ler

angl

e ϕ

[d

eg]

Figure 13: Rotation, φ, at tip C of Bar 2 atpoint: time marching (), N = 5 (), N = 7(.), N = 11 (/), N = 15 (+), N = 21 (∗).

The components of axial force, F1, and bending moment, M2, along unit vector b1 and b2,respectively, at the mid-span of bar 1, are depicted in figs. 15 and 16, respectively. Bending momentM2 varies smoothly over the period; good correlation with the reference solution is observed when7 or more grid points are used. The axial force exhibits sharp changes at t = T/4 and 3T/4. Gibbs

17

0 T/4 T/2 3T/4 T-2.0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

Time

Rel

ativ

e ro

tati

on θ

[ra

d]

Figure 14: The relative rotation, θ, at revolutejoint D: time marching (), N = 5 (), N = 7(.), N = 11 (/), N = 15 (+), N = 21 (∗).

0 T/4 T/2 3T/4 T-1000

-800

-600

-400

-200

0

200

Time

Axi

al f

orce

F1 [

N]

Figure 15: Axial force, F1, at the mid-span ofBar 1: time marching (), N = 5 (), N = 7(.), N = 11 (/), N = 15 (+), N = 21 (∗).

phenomenon is observed in the predictions of the spectral method: higher-order harmonics appearto try to capture these sharp variations.

0 T/4 T/2 3T/4 T-20

0

20

40

60

80

100

120

140

Ben

din

g m

omen

t M

2 [N

.m]

Time

Figure 16: Bending moment, M2, at the mid-span of Bar 1: time marching (), N = 5 (), N = 7(.), N = 11 (/), N = 15 (+), N = 21 (∗).

8 Conclusions

The collocation version of the Fourier spectral method was implemented to determine the periodicsolutions of flexible multibody systems modeled via the finite element method. To facilitate theanalysis and obtain governing equations presenting low-order nonlinearities, the motion formalismwas adopted, i.e., the configuration of the system was represented by the Lie group SO(3). Whilethe application of Fourier spectral methods to multibody systems is rather straightforward, thedevelopment of spectral interpolation schemes for nonlinear configuration manifolds such as SO(3)or SO(3) is not.

This paper has presented spectral interpolation schemes for SO(3) and SO(3). The proposedapproach is based on a minimization principle, that generalized the interpolation schemes used forlinear fields. To facilitate the process, the kinematic analysis was formulated in terms of dual entities.The interpolation of rotation fields was recast as a minimization problem which can be solved withthe help of the polar decomposition theorem. This theorem was generalized to dual matrices and

18

was shown to provide the solution of the minimization problem that leads to the interpolation ofmotion fields. These novel tools provide an approach to the interpolation of periodic motion fieldsthat are exponentially convergent, a prerequisite to the implementation of spectral methods.

Once the configuration, velocity, and acceleration fields have been interpolated accurately, thegoverning equations for all nodes and all grid points are assembled. The global nature of the inter-polation schemes yields governing equations that couple the variables at all nodes and grid pointsleading to iteration matrices with a larger number of degrees of freedom and a wider bandwidththan those encountered in time marching processes.

Numerical examples have illustrated the accuracy of the proposed approach. When the periodicresponse exhibits sharp variations, Gibbs phenomenon arises, as expected for Fourier solutions.

The interpolation schemes developed in this work provide new tools for approximating timedependent and spatially varying configurations in SO(3) and SO(3) with applications in mechanics,control, robotics, and computer graphics. Given the size and bandwidth of the iteration matrixresulting from the application of Fourier spectral collocation methods, computationally efficientsolution techniques must be developed. A parallel implementation of the solution process is par-ticularly attractive; Krylov subspace iterative solvers with circulant pre-conditioners should workefficiently for this type of problem.

A Dual numbers, vectors, matrices, and functions

Dual numbers were first introduced in the nineteenth century by Clifford [28]. Typically, they arewritten as a = a + εao, where a and ao are referred to as the primal and dual parts, respectively,and parameter ε is such that εn = 0 for n ≥ 2. The product of two dual scalars now becomesab = (a+ εao)(b+ εbo) = ab+ ε(abo + aob). The zero and identity dual numbers are 0 = 0 + ε0 and1 = 1 + ε0, respectively. The inverse of a dual scalar is 1/a = 1/a− εao/a2.

A dual vector is composed of two vectors of the same size a = a+ εao. The inner product of twodual vectors is aT b = (a + εao)T (b + εbo) = aT b + ε(aT bo + ao T b). The cross product of two dual

vectors of size 3× 1 is ab, where notation (·) is the skew-symmetric dual matrix associated with a.Dual vector a is unit if ‖a‖ =

√aT a = 1, which is equivalent to aTa = 1 and aTao = 0.

Similarly, a dual matrix is composed of two matrices of the same size A = A+εAo. The transpose

of a dual matrix is AT = (A+ εAo)T = AT + εAoT . The identity dual matrix is defined as I = I+ ε0.The product of a dual matrix by a dual vector is found easily as A a = Aa + ε(Aao + Aoa). Dual

matrix A is orthogonal if ATA = ATA+ ε(ATAo +AoTA) = I , which is equivalent to ATA = I and

symm[ATAo] = 0.Consider a dual function of a dual variable, J (a). Function J is assumed to be analytic [59, 33],

i.e., it is of the following form: J (a) = J(a)+εaoJ ′(a), where J(a) is the primal part of dual functionJ (a) and notation (·)′ indicates a derivative with respect to a. The derivative of dual function Jwith respect to dual variable a is a dual number, dJ (a)/da = J ′(a) + εaoJ ′′(a). Two importantobservations can be made: (1) the primal part of an analytic function depends on the primal partof its dual variable only and (2) the dual part of an analytic function is a linear function of the dualpart of its dual variable. Because the magnitude of a dual number is not defined, dual numberscannot be compared. The primal part of a dual function, J(a), can reach a minimum or maximumwhereas its dual part, aoJ ′(a), never reaches a minimum or maximum in a open set because it is alinear function of ao. Minimization of a dual function is not meaningful.

19

B Identities of matrix and vectors

This section presents a set of useful identities that are used throughout the paper. The identitiesinvolve a dual matrix, A, of size 3× 3, and a dual vector, a, of size 3× 1.

axial(aA) =1

2[tr(A)− A]a, (38a)

axial(Aa) =1

2[tr(A)− AT ]a, (38b)

axial(b aA) =

[1

2AT b − axial(A)bT

]a, (38c)

axial(bAa) =

[1

2Ab − baxial(A)T

]a, (38d)

tr(aA) = −2aTaxial(A), (38e)

tr(a aTA) = aT symm(A)a. (38f)

These identities can be verified easily.

C The polar decomposition theorem

The polar decomposition theorem can be stated as follows.

Theorem 1 (Polar decomposition theorem). An invertible matrix, G ∈ R3×3, can be decomposedinto the product of a rotation tensor, R ∈ SO(3), by a symmetric matrix, S, as G = RS. MatricesR and S are defined uniquely if matrix [tr(S)I − S] is required to be positive-definite.

Proof. Let the spectral decomposition of positive-definite matrix GTG be UTdiag(λ1, λ2, λ3)U ,

where positive eigenvalues, λi, i = 1, 2, 3, satisfy λ1 ≤ λ2 ≤ λ3. In view of identity STS =

(RTG)T (RTG) = GTG, symmetric matrix S can be chosen as UTdiag(±√λ1,±

√λ2,±

√λ3)U .

Of these eight choices, two only, UTdiag(±√λ1,√λ2,√λ3)U , render matrix [tr(S)I − S] positive-

definite. The sign of the lowest eigenvalue is determined by sign of det(G) = det(R) det(S) = det(S):choose the positive or negative sign if det(G) is positive or negative, respectively.

Remark 1. Polar decomposition theorem (1) differs slightly from the traditional polar decompo-sition theorem used in continuum mechanics [39, 60]. The proof above shows that eight differentsymmetric matrices S satisfy multiplicative decomposition G = RS. The solution is made uniqueby imposing an additional condition: in the traditional and present versions of the theorem, matri-ces S and [tr(S)I − S] are required to be positive-definite, respectively. When det(G) > 0, the twotheorems are identical.

Given an invertible matrix G ∈ R3×3, the following question is asked: which rotation tensorR ∈ SO(3) is as close as possible to G? The problem is stated as

minR∈SO(3)

J(R) = ‖R−G‖2F , (39)

where the closeness of the two matrices is defined as the square of the Frobenius norm of theirdifference.

Theorem 2 (Closest rotation tensor). The rotation tensor that satisfies minimization problem (39)is that provided by polar decomposition theorem (1).

20

Proof. Defining S = RTG, the objective function becomes J = tr[(R−G)T (R−G)] = tr[I+GTG−S−ST ] = tr(I+GTG)−2tr(S) and its variation is δJ = −2tr(δS) = 2tr(δψ S), where δψ = RT δR.

Finally, trace identity (38e) yields δJ = −4δψTaxial(S) = 0, which implies that matrix S must besymmetric. For the objective function to reach its minimum, its Hessian must be positive-definite.Taking a second-order variation yields δ2J = −2δψTaxial(δS), leading to δ2J = 2δψT [tr(S)I−S]δψ,where matrix [tr(S)I−S] is the Hessian. Clearly, polar decomposition theorem (1) provides uniquelydefined rotation tensor R and matrix S that satisfy the two required conditions: S is symmetricand [tr(S)I − S] is positive-definite.

The polar decomposition theorem is now generalized to dual matrices.

Theorem 3 (Dual polar decomposition theorem). An invertible dual matrix, G ∈ D3×3, can be

decomposed into the product of a dual orthogonal matrix, R ∈ SO(3), by a symmetric dual matrix,

S , as G = R S . Matrices R and S are defined uniquely if it is also required that matrix [tr(S)I −S]

be positive-definite, where S is the primal part of dual matrix S .

Proof. To prove the theorem, dual matrices R and S will be constructed and the solution will be

shown to be unique. First, dual identity G = R S is expanded as (G+ εGo) = (R + εRo)(S + εSo),

which implies

G = RS, (40a)

Go = RSo +RoS. (40b)

Equation (40a) expresses theorem (1), i.e., rotation tensor R and symmetric matrix S are defined

uniquely. Equation (40b) implies RTGo = So + (RTRo)S, where matrix RTRo = z is antisymmetricbecause motion tensor R is orthogonal. Because matrix So must be symmetric, axial(So) = 0, and

extracting the axial part of this equation yields axial(RTGo) = axial[zS]. Identity (38a) now yields

[tr(S)I−S]z = 2axial(RTGo), a linear system that can be solved to find z. The dual parts of motion

tensor R and symmetric matrix S are found as Ro = Rz and So = RTGo − zS, respectively.

Given an invertible dual matrix G ∈ D3×3, the following question is asked: which motion tensor

R ∈ SO(3) is as close as possible to G? By analogy with eq. (39), the following minimization

problem is introducedmin

R∈SO(3)J (R ) = ‖R − G‖2F . (41)

As discussed earlier, the minimization of a function of dual numbers is devoid of meaning. Therefore,notation “min J ” is now defined as follows.

Definition 1 (Minimization of a dual function). The minimization of a dual function of dualvariables, J (a), implies the satisfaction of two conditions: (1) the variation of the objective functionmust vanish, δJ (a) = 0 and (2) the primal part of the objective function must achieve a minimum.

As discussed in appendix A, the primal part of a dual function can reach a minimum whereasits dual part never reaches a minimum in a open set because it is a linear function of the dual partof its variable. Definition (1) does not require the minimization of the dual part of the function,bypassing the problem. Note that the vanishing of the variation of the primal part of the objectivefunction is a prerequisite for its minimization. An alternative statement of definition (1) reads: theminimization of a function of dual variables implies the minimization of its primal part and thestationarity of its dual part.

21

Theorem 4 (Closest motion tensor). The motion tensor that satisfies dual minimization prob-lem (41) is that provided by dual polar decomposition theorem (3).

Proof. Defining S = R TG , the objective function becomes J = tr[(R −G)T (R −G)] = tr[I + GTG −S − ST ] = tr(I + GTG)− 2tr(S) and its variation is δJ = −2tr(δS) = 2tr(δu S), where δu = R T δR .

Finally, trace identity (38e) yields δJ = −4δuTaxial(S) = 0, which implies that dual matrix S must

be symmetric. The objective function is evaluated as J = ‖R − G‖2F + 2ε[tr(GTGo) − 2tr(So)].The minimization of its primal part is expressed by minimization problem (39), which is solved bythe polar decomposition theorem, see theorem (2). Clearly, dual polar decomposition theorem (3)provides uniquely defined motion tensor R and matrix S that satisfy the two required conditions:

S is symmetric and [tr(S)I − S] is positive-definite.

References

[1] U.M. Ascher, R.M.M. Mattheij, and R.D. Russell. Numerical Solution of Boundary ValueProblems for Ordinary Differential Equations. Society for Industrial and Applied Mathematics,1995.

[2] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral Methods in Fluid Dynamics.Springer-Verlag, Berlin, Heidelberg, 1988.

[3] J.P. Boyd. Chebyshev & Fourier Spectral Methods. Springer-Verlag, Berlin, Heidelberg, 1989.

[4] J.S. Hesthaven, S. Gottlieb, and D. Gottlieb. Spectral Methods for Time-Dependent Problems.Cambridge University Press, Cambridge, 2007.

[5] N.M. Krylov, N.N. Bogoliubov, and S. Lefschetz. Introduction to Nonlinear Mechanics. Prince-ton University Press, 1947.

[6] C. Hayashi. Nonlinear Oscillations in Physical Systems. Princeton University Press, 1964.

[7] A.H. Nayfeh and D.T. Mook. Nonlinear Oscillations. John Wiley & Sons, New York, 1979.

[8] Kundert. K.S., J.K. White, and A. Sangiovanni-Vincentelli. Steady-State Methods for Simulat-ing Analog and Microwave Circuits. Kluwer Academic Publishers, Boston, Dordrecht, London,1990.

[9] A. Cardona, T. Coune, A. Lerusse, and M. Geradin. A multiharmonic method for non-linearvibration analysis. International Journal for Numerical Methods in Engineering, 37(9):1593–1608, 1994.

[10] V. Jaumouille, J.-J. Sinou, and B. Petitjean. An adaptive harmonic balance method for predict-ing the nonlinear dynamic responses of mechanical systems - Application to bolted structures.Journal of Sound and Vibration, 329(19):4048–4067, 2010.

[11] A. LaBryer and P.J. Attar. A harmonic balance approach for large-scale problems in nonlinearstructural dynamics. Computers & Structures, 88(17-18):1002–1014, 2010.

[12] T. Detroux, L. Renson, L. Masset, and G. Kerschen. The harmonic balance method for bi-furcation analysis of large-scale nonlinear mechanical systems. Computer Methods in AppliedMechanics and Engineering, 296:18–38, 2015.

22

[13] G. Von Groll and D.J. Ewins. The harmonic balance method with arc-length continuation inrotor/stator contact problems. Journal of Sound and Vibration, 241(2):223–233, 2001.

[14] K.Y. Sanliturk and D.J. Ewins. Modeling two-dimensional friction contact and its applicationusing harmonic balance method. Journal of Sound and Vibration, 193:511–523, 1996.

[15] K.C. Hall, J.P. Thomas, and W.S. Clark. Computation of unsteady nonlinear flows in cascadesusing a harmonic balance technique. AIAA Journal, 40(5):879–886, 2002.

[16] J.P. Thomas, E.H. Dowell, and K.C. Hall. Modeling viscous transonic limit cycle oscillationbehavior using a harmonic balance approach. Journal of Aircraft, 41(6):1266–1274, 2004.

[17] C. H. Custer, J. P. Thomas, E. H. Dowell, and K. C. Hall. A nonlinear harmonic balance methodfor the CFD code OVERFLOW 2. In Interational Forum on Aeroelasticity and StructuralDynamics (IFASD), Paper 2009-050, Seattle, WA, June 7-10 2009.

[18] M. Borri and C.L. Bottasso. An intrinsic beam model based on a helicoidal approximation.Part I: Formulation. Part II: Linearization and finite element implementation. InternationalJournal for Numerical Methods in Engineering, 37:2267–2309, 1994.

[19] T. Merlini and M. Morandini. The helicoidal modeling in computational finite elasticity. PartI: Variational formulation. Part II: Multiplicative interpolation. International Journal of Solidsand Structures, 41(18-19):5351–5409, 2004.

[20] V. Sonneville, A. Cardona, and O. Bruls. Geometrically exact beam finite element formu-lated on the special Euclidean group SE(3). Computer Methods in Applied Mechanics andEngineering, 268(1):451–474, 2014.

[21] V. Sonneville and O. Bruls. A formulation on the special Euclidean group for dynamic analysisof multibody systems. Journal of Computational and Nonlinear Dynamics, 9(4), 2014.

[22] J.A.C. Weideman. Numerical integration of periodic functions: A few examples. The AmericanMathematical Monthly, 109(1):21–36, 2002.

[23] L.N. Trefethen and J.A.C. Weideman. The exponentially convergent trapezoidal rule. SIAMReview, 56(3):385–458, 2014.

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

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

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

[27] S.L. Han and O.A. Bauchau. Manipulation of motion via dual entities. Nonlinear Dynamics,85(1):509–524, July 2016.

[28] W.K. Clifford. Preliminary sketch of biquaternions. Proceedings of the London MathematicalSociety, s1-4(1):381–395, 1871.

[29] J.M. McCarthy. An Introduction to Theoretical Kinematics. The MIT Press, Cambridge, MA,1990.

23

[30] J.M.R. Martinez and J. Duffy. The principle of transference: History, statement and proof.Mechanism and Machine Theory, 28(1):165–177, 1993.

[31] J. Angeles. The application of dual algebra to kinematic analysis. In J. Angeles and E. Za-khariev, editors, Computational Methods in Mechanical Systems, volume 161, pages 3–31.Springer-Verlag, Heidelberg, 1998.

[32] I.S. Fischer. Dual Number Methods in Kinematics, Statics and Dynamics. CRC Press, BocaRaton, 1999.

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

[34] D. Condurache and A. Burlacu. Dual tensors based solutions for rigid body motion parame-terization. Mechanism and Machine Theory, 74:390–412, 2014.

[35] I. Romero. The interpolation of rotations and its application to finite element models ofgeometrically exact rods. Computational Mechanics, 34(2):121–133, 2004.

[36] O.A. Bauchau and S.L. Han. Interpolation of rotation and motion. Multibody System Dynamics,31(3):339–370, 2014.

[37] V. Sonneville, O. Bruls, and O.A. Bauchau. Interpolation schemes for geometrically exactbeams: a motion approach. International Journal of Numerical Methods in Engineering, 2017.To appear.

[38] G. Grioli. Una proprieta di minimo nella cinematica delle deformazioni finite. Bollettinodell’Unione Matematica Italiana, 2:252–255, 1940.

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

[40] L.C. Martins and P. Podio-Guidugli. A variational approach to the polar decomposition theo-rem. Rendiconti Accademia Nazionale dei Lincei, Serie VIII, 66:487–493, 1979.

[41] L.C. Martins and P. Podio-Guidugli. An elementary proof of the polar decomposition theorem.The American Mathematical Monthly, 87(4):288–290, 1980.

[42] J. Lu and P. Papadopoulos. On the direct determination of the rotation tensor from thedeformation gradient. Mathematics and Mechanics of Solids, 2(1):17–26, 1997.

[43] C. Bouby, D. Fortune, W. Pietraszkiewicz, and C. Vallee. Direct determination of the rotationin the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient.Journal of Applied Mathematics and Mechanics, 85(3):155–162, 2005.

[44] G. Wahba. A least squares estimate of satellite attitude. SIAM Review, 7(3):409–409, 1965.

[45] P. Davenport. A vector approach to the algebra of rotations with applications. TechnicalReport X-546-65-437, NASA, November 1965.

[46] F.L. Markley and D. Mortari. Quaternion attitude estimation using vector observations. Jour-nal of the Astronautical Sciences, 48(2/3):359–380, 2000.

[47] A. Cardona and M. Geradin. A beam finite element non-linear theory with finite rotation.International Journal for Numerical Methods in Engineering, 26:2403–2438, 1988.

24

[48] A. Ibrahimbegovic, F. Frey, and I. Kozar. Computational aspects of vector-like parameteri-zation of three-dimensional finite rotations. International Journal for Numerical Methods inEngineering, 38(21):3653–3673, 1995.

[49] M.A. Crisfield and G. Jelenic. Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proceedings of the Royal So-ciety, London: Mathematical, Physical and Engineering Sciences, 455(1983):1125–1147, 1999.

[50] P. Betsch and P. Steinmann. Frame-indifferent beam element based upon the geometricallyexact beam theory. International Journal for Numerical Methods in Engineering, 54:1775–1788,2002.

[51] E.S. Gawlik and M. Leok. Embedding-based interpolation on the special orthogonal group.SIAM Journal on Scientific Computing, 2017. to appear.

[52] B. Horn. Closed-form solution of absolute orientation using unit quaternions. Journal of theOptical Society of America A, 4(4):629–642, April 1987.

[53] M. Moakher. Means and averaging in the group of rotations. SIAM Journal on Matrix Analysisand Applications, 24(1):1–16, 2002.

[54] R. Hartley, J. Trumpf, Y.C. Dai, and H.D. Li. Rotation averaging. International Journal ofComputer Vision, 103(3):267–305, 2013.

[55] O.A. Bauchau and S.L. Han. Three-dimensional beam theory for flexible multibody dynamics.Journal of Computational and Nonlinear Dynamics, 9(4):041011 (12 pages), 2014.

[56] S.L. Han and O.A. Bauchau. Nonlinear three-dimensional beam theory for flexible multibodydynamics. Multibody System Dynamics, 34(3):211–242, July 2015.

[57] S.L. Han and O.A. Bauchau. On Saint-Venant’s problem for helicoidal beams. Journal ofApplied Mechanics, 83(2):021009 (14 pages), 2016.

[58] O.A. Bauchau, P. Betsch, A. Cardona, J. Gerstmayr, B. Jonker, P. Masarati, and V. Son-neville. Validation of flexible multibody dynamics beam formulations using benchmark prob-lems. Multibody System Dynamics, 37(1):29–48, May 2016.

[59] F.M. Dimentberg. The screw calculus and its applications. Technical Report AD 680993,Clearinghouse for Federal and Scientific Technical Information, Virginia, USA, April 1968.

[60] G.H. Golub and C.F. van Loan. Matrix Computations. The Johns Hopkins University Press,Baltimore, second edition, 1989.

25