Computational Schemes for Flexible,Nonlinear Multi-Body Systems ∗

Olivier A. BauchauGeorgia Institute of Technology, School of Aerospace Engineering

Atlanta, GA, USA

Abstract

This paper deals with the development of computational schemes for the dynamicanalysis of flexible, nonlinear multi-body systems. The focus of the investigation ison the derivation of unconditionally stable time integration schemes for these typesof problem. At first, schemes based on Galerkin and time discontinuous Galerkin ap-proximations applied to the equations of motion written in the symmetric hyperbolicform are proposed. Though useful, these schemes require casting the equations of mo-tion in the symmetric hyperbolic form, which is not always possible for multi-bodyapplications. Next, unconditionally stable schemes are proposed that do not rely onthe symmetric hyperbolic form. Both energy preserving and energy decaying schemesare derived that both provide unconditionally stable schemes for nonlinear multi-bodysystems. The formulation of beam and flexible joint elements, as well as of the kine-matic constraints associated with universal and revolute joints. An automated timestep selection procedure is also developed based on an energy related error measurethat provides both local and global error levels. Several examples of simulation of re-alistic multi-body systems are presented which illustrate the efficiency and accuracy ofthe proposed schemes, and demonstrate the need for unconditional stability and highfrequency numerical dissipation.

1 Introduction

This paper is concerned with the dynamic analysis of flexible, nonlinear multi-body systems,i.e. a collection of bodies in arbitrary motion with respect to each other while each bodyis undergoing large displacements and rotations with respect to a body attached frame ofreference. The focus is on problems where the strains within each elastic body remain small.

The elastic bodies are modeled using the finite element method. The use of beam andflexible joint elements will be demonstrated for multi-body systems. The location of eachnode is represented by its Cartesian coordinates in an inertial frame, and the rotation of thecross-section at each node is represented by a finite rotation tensor expressed in the same

∗Multibody System Dynamics, 2, 1998, pp 169-225.

1

inertial frame. The kinematic constraints among the various bodies are enforced via theLagrange multiplier technique. Although this approach does not involve the minimum setof coordinates, it allows a modular development of finite elements for the enforcement of thekinematic constraints. The formulation of universal and revolute joints will be presentedhere.

The equations of motion resulting from the modeling of multi-body systems with theabove methodology present distinguishing features: they are stiff, nonlinear, differential-algebraic equations. The stiffness of the system stems from the presence of high frequenciesin the elastic members, but also from the infinite frequencies associated with the kinematicconstraints. Indeed, no mass is associated with the Lagrange multipliers giving rise toalgebraic equations coupled to the other equations of the system which are differential innature.

This paper is concerned with the time integration of the equations of motion describingthe nonlinear dynamic response of multi-body systems. The main focus is on the derivationof an algorithm presenting high frequency numerical dissipation, and for which unconditionalstability can be proven in the nonlinear case. An energy decay argument will be used toestablish stability [1].

The Newmark algorithm [2] is widely used in structural dynamics. In particular, theaverage acceleration method, also known as the trapezoidal rule, is an unconditionally sta-ble, second order accurate scheme when applied to linear problems. The classical stabilityanalysis of this scheme can be readily found in text books [3] and shows that the spectralradius remains exactly equal to unity at all frequencies. An alternate way of proving stabil-ity is based on an energy argument. Indeed, it is easily shown that the average accelerationscheme exactly preserves the total energy of the system [1].

For large finite element discretizations very high frequencies are present in the model andhigh frequency numerical dissipation is desirable, if not indispensable. Numerical dissipationcannot be introduced in the Newmark method without degrading the accuracy. Hilber,Hughes, and Taylor [4] introduced the α-method to remedy this situation. More recently,the generalized-α method [5] was introduced that achieves high frequency dissipation whileminimizing unwanted low frequency dissipation. Both methods have been successfully usedfor both linear and nonlinear problems, though unconditional stability is proved for linearsystems only.

Simo and his co-workers presented energy preserving schemes for rigid body dynamics [6],elasto-dynamics [7], and beams [8]. The unconditional stability of these schemes stems froma proof of preservation of the total energy of the system. An energy preserving schemefor nonlinear elastic multi-body systems was proposed by Bauchau [9]. In this scheme,the equations of motion are discretized so that they imply conservation of the total energyfor the elastic components of the system, whereas the forces of constraint associated withthe kinematic constraints are discretized so that the work they perform vanishes exactly.The combination of these two features of the discretization guarantees the stability of thenumerical integration process for nonlinear elastic multi-body systems.

Though energy preserving schemes perform well, their lack of high frequency numericaldissipation can be a problem [9]. First, the time histories of internal forces and velocitiescan present a very significant high frequency content. Second, it seems that the presenceof high frequency oscillations can hinder the convergence process for the solution of the

2

nonlinear equations of motion. This was observed in several examples where the dynamicresponse of the system does involve significant high frequency content. The selection of asmaller time step does not necessarily help this convergence process, as a smaller time stepallows even higher frequency oscillations to be present in the response. Finally, it seems thatthe presence of high frequency oscillations also renders strict energy preservation difficult toobtain. This could prove to be a real limitation of energy preserving schemes when applied tomore and more complex models. For such models, the use of integration schemes presentinghigh frequency numerical dissipation become increasingly desirable.

It appears that the development of “energy decaying” schemes, i.e. schemes eliminatingthe energy associated with vibratory motions at high frequency, is desirable. This is partic-ularly important when dealing with problems presenting a complex dynamic response suchas constrained flexible multi-body problems.

The key to the development of an energy decaying scheme is the derivation of an energydecay inequality [1] rather that the discrete energy conservation law which is central toenergy preserving schemes. A methodology that can systematically lead to an energy decayinequality is the time discontinuous Galerkin method [10, 11, 12] which was initially developedfor hyperbolic equations. Hughes and Hulbert [13, 14] have investigated the use of the timediscontinuous Galerkin methodology for linear elasto-dynamics. They point out that

classical elasto-dynamics can be converted to first-order symmetric hyperbolicform, which has proved useful in theoretical studies. Finite element methods forfirst order symmetric hyperbolic system are thus immediately applicable. How-ever, there seems to be several disadvantages: in symmetric hyperbolic form thestate vector consists of displacements, velocities, and stresses which is computa-tionally uneconomical; and the generalization to nonlinear elasto-dynamics seemspossible only in special circumstances.

Indeed, writing the nonlinear equations of motion of beams in this symmetric hyperbolicform does not appear to be possible.

In this paper an alternate route is taken. Practical time integration schemes that donot rely on the symmetric hyperbolic form of the equations of motion are developed. Theseschemes are of a finite difference nature, and imply an energy balance condition that isobtained by a direct computation of the work done by the discretized inertial and elasticforces over a time step. The mean value theorem guarantees the existence of discretizationsleading to these energy preservation of energy decay statements providing a rigorous proofof unconditional stability for the scheme.

Energy decaying schemes were presented by Bauchau and his co-workers for beams [15],elasto-dynamics [16], and multi-body systems [17]. Bottasso and Borri proposed both energypreserving and decaying schemes for beams [18, 19] and multi-body systems [20, 21].

The first part of this paper describes the theoretical background of the proposed method-ology and is organized in the following manner. In section 2, the symmetric hyperbolic formof the equations of motions will be discussed, and the properties of both Galerkin and timediscontinuous Galerkin approximations of these equations written in the symmetric hyper-bolic form will be presented. Practical time integration schemes that bypass the need forrecasting the equations of motion in the symmetric hyperbolic form will then be introducedin section 3. The treatment of constraint equations will be discussed in section 4.

3

In a second part of the paper, the formulation of a number of elements used in multi-bodydynamic analysis will be presented. The discretization of the inertial and elastic forces lead-ing to discrete energy preservation statements will be presented for various elements. Theformulation of geometrically exact, shear deformable beam elements, and flexible joint ele-ments will be presented in sections 6.1 and 6.2, respectively. Multi-body systems also involvenonlinear constraints. The formulation of universal and revolute joints will be presented insections 7.1 and 7.2, respectively.

An important feature of multi-body systems is the rapidly varying nature of their dynamicresponse. In numerical simulations, it is desirable, and often indispensable to use a variabletime step size which value is automatically selected by the analysis software so as to achieve anearly constant time discretization error as the simulation proceeds. A simple, yet effectivetime step selection procedure will be presented in section 8. Finally, various numericalsimulations of realistic flexible multi-body systems will be presented in section 9.

2 The symmetric hyperbolic form

2.1 Classical forms of the equations of motion

Consider a dynamical system described by a kinetic energy K = K(up, up), and a strain

energy V = V(up), where up are the degrees of freedom of the system, and ˙(.) denotes aderivative with respect to time. The Lagrangian of the system is defined as L(up, up) = K−V ,and the equations of motion of the system in Lagrangian form are then

d

dt

(L,up

)− L,up = 0. (1)

The notation (.),u is used here to indicate a derivative with respect to u, and a summationis implied by repeated indices. Hamilton’s formulation is obtained with the help of a Leg-endre transformation [22]. First, the momenta are defined as pp(ur, ur) = L,up , and theserelationships can be inverted to yield up = up(ur, pr). The Hamiltonian of the system is nowdefined

H(ur, pr) = pp up(ur, pr)− L(ur, pr). (2)

The equations of motion of the system in Hamiltonian form are then

up = H,pp ; pp +H,up = 0. (3)

2.2 The symmetric hyperbolic form

The symmetric hyperbolic form stems from a second Legendre transformation. The followingvariables are first defined

fp(ur, pr) = H,up ; vp(ur, pr) = H,pp . (4)

These relations can be inverted to yield up = up(fr, vr) and pp = pp(fr, vr). A new functionis now defined

G(fr, vr) = fp up(fr, vr) + vp pp(fr, vr)−H(fr, vr), (5)

4

timet

ft i

tm

Figure 1: The Galerkin approximation.

implying up = G,fp and pp = G,vp . It can be readily shown that the Hessians of H and Gare the inverse of each other. Hence, if H is a definite positive function, so is G. Hamilton’sequations (3) can be expressed in terms of the new variables fp and vp to find the symmetrichyperbolic form of the equations of motion

G,fpfq fq + G,fpvq vq − vp = 0; G,vpfq fq + G,vpvq vq + fp = 0. (6)

To simplify the notation, an implicit form of the equations is preferred

up(fr, vr)− vp = 0; pp(fr, vr) + fp = 0. (7)

2.3 The Galerkin approximation

In the Galerkin approximation, the equations of motion are enforced in a weak, integralmanner. Fig. 1 shows a time interval from ti to tf , and an approximate solution over thatinterval. Superscripts (.)i and (.)f will be used to indicate the value of a quantity at timesti and tf , respectively. The Galerkin approximation of the equations of motion in implicitsymmetric hyperbolic form (7) writes

∫ tf

ti

w1p[up(fr, vr)− vp] + w2p[pp(fr, vr) + fp] dt = 0, (8)

where w1p and w2p are arbitrary test functions. Integration by parts yields

∫ tf

ti

[−w1pup − w2ppp − w1pvp + w2pfp] dt + wf1pu

fp + wf

2ppfp − wi

1puip − wi

2ppip = 0. (9)

This approximation of the equations of motion enjoys remarkable properties. Indeed, select-ing the test functions as w1p = fp and w2p = vp yields

∫ tf

ti

[−fpG,fp − vpG,vp − fp vp + vp fp] dt + f fp uf

p + vfp pf

p − f ip ui

p − vip pi

p = 0. (10)

5

Timet i t j t f

t gt h

Figure 2: The time discontinuous Galerkin approximation.

The time integral clearly has a closed form solution, leading to

Gi − Gf + f fp uf

p + vfp pf

p − f ip ui

p − vip pi

p = 0. (11)

Finally, we express G in terms of the Hamiltonian H with the help of (5) to find

Hf = Hi. (12)

In summary, the Galerkin approximation (8) of the equations of motion written in sym-metric hyperbolic form implies a discrete Hamiltonian preservation statement (12). If theHamiltonian is a definite positive function, this statement implies the unconditional stabilityof integration schemes based on (8).

2.4 The time discontinuous Galerkin approximation

In the time discontinuous Galerkin approximation, the solution is allowed to present dis-continuities in the displacement and velocity fields at discrete times. Fig. 2 shows a timeinterval from ti to tf and the approximate solution over that interval. At the initial instant,the solution presents a jump. Superscripts (.)i will be used to denote the value of a discon-tinuous quantity on the left side of the jump, whereas a superscript (.)j indicates the value ofthat quantity on the right side of the jump. The equations of motion and initial conditionsare enforced in a weak, integral manner. The time discontinuous Galerkin approximation ofthe equations of motion in implicit symmetric hyperbolic form (7) writes

∫ tf

tj

w1p[up(fr, vr)− vp] + w2p[pp(fr, vr) + fp] dt + wj1p 〈up〉+ wj

2p 〈pp〉 = 0. (13)

where the notation 〈.〉 is used to denote the jump in a quantity at the initial time, i.e.〈up〉 = uj

p − uip and 〈pp〉 = pj

p − pip. This approximation of the equations of motion also

6

enjoys remarkable properties. Indeed, integrating by parts and selecting the test functionsas w1p = fp and w2p = vp yields

∫ tf

tj

[−fpG,fp − vpG,vp − fp vp + vp fp] dt + f fp uf

p + vfp pf

p − f jp ui

p − vjp pi

p = 0. (14)

The time integral clearly has a closed form solution, leading to

Gj − Gf + f fp uf

p + vfp pf

p − f jp ui

p − vjp pi

p = 0. (15)

Finally, we express G in terms of the Hamiltonian H with the help of (5) to find

Hf −Hj + f jp 〈up〉+ vj

p 〈pp〉 = 0. (16)

Since the Hamiltonian is a continuous function of ur and pr, the mean value theoremimplies

Hj = Hi + f jp 〈up〉+ vj

p 〈pp〉 − 1

2

[Hh

,upuq〈up〉〈uq〉+Hh

,uppq〈up〉〈pq〉+

Hh,ppuq

〈pp〉〈uq〉+Hh,pppq

〈pp〉〈pq〉]

= Hi + f jp 〈up〉+ vj

p 〈pp〉 − c2, (17)

where the last equality holds if the Hamiltonian is a definite positive function. Combining(16) and (17) then yields

Hf = Hi − c2,⇒ Hf ≤ Hi. (18)

In summary, the time discontinuous Galerkin approximation (13) of the equations ofmotion written in symmetric hyperbolic form implies a Hamiltonian decay inequality (18),if the Hamiltonian is a definite positive quantity. This inequality implies the unconditionalstability of time integration schemes based on (13).

2.5 Example: Linear spring-mass system

To illustrate the procedures described in the previous sections, a very simple example willbe treated here. Consider a linear spring-mass system with a kinetic energy K = 1/2 m u2,a strain energy V = 1/2 k u2, and subjected to an external force Fa(t). In this simplecase, f = k u and v = p/m, and the symmetric hyperbolic form of the equations of motionbecomes: p + ku = Fa; u − p/m = 0. The Galerkin approximation (8) for this problemwrites ∫ tf

ti

w1[u− p

m] + w2[p + ku−Fa]

dt = 0. (19)

Using a linear in time approximation for the displacement and momentum, and a constantin time approximation for the test functions, the following discrete equations are obtained

F im + F em = Fam; (20)

where the superscript (.)m denote a quantity at the mid point tm (see fig. 1). The discretizedinertial forces are

F im =muf −mui

∆t; (21)

7

and the following velocity-displacement and force-displacement relationships are used

uf − ui

∆t=

uf + ui

2; F em = k

uf + ui

2. (22)

Finally, the discretized applied forces are

Fam =1

2

∫ 1

−1

Fa(τ) dτ. (23)

∆t indicates the time step size, and τ is a nondimensional time variable such that τ = −1.0or +1.0 at times ti and tf , respectively. The properties of this integration scheme can beinvestigated using the classical techniques for the analysis of linear schemes. The spectralradius of the amplification matrix is always equal to unity, implying unconditional stability.This scheme is identical to the Newmark scheme [2] with γ = 1/2 and β = 1/4. It can bereadily shown that the discrete equations of motion (20) imply a discrete energy preservationstatement Ef = E i, where E = K + V is the total mechanical energy.

The same problem can be treated with the time discontinuous Galerkin approximation(13) which writes

∫ tf

tj

w1[u− p

m] + w2[p + ku−Fa]

dt + wj

1〈u〉+ wj2〈p〉 = 0. (24)

Using a linear in time approximation for the displacement, momentum, and test functions,the following discrete equations are obtained

F im + F eg = Fag; F ih − 1

3[F eg − f j] = Fah. (25)

The superscript (.)g denotes a quantity at the midpoint between tj and tf , whereas (.)h

denotes a quantity at the midpoint between ti and tj, see fig. 2. The discretized inertialforces are

F im =muf −mui

∆t; F ih =

muj −mui

∆t; (26)

and the following velocity-displacement and force-displacement relationships are used

uf − ui

∆t=

uf + uj

2; 3

uj − ui

∆t= − uf − uj

2; F eg = k

uf + uj

2. (27)

Finally, the discretized applied forces are

Fag =1

2

∫ 1

−1

Fa dτ ; Fah = −1

2

∫ 1

−1

Fa τdτ. (28)

It can be readily shown that the discrete equations of motion (25) imply a discrete energydecay inequality Ef ≤ E i. This is a direct consequence of (18), since the Hamiltonian is equalto the total energy of the system for this simple problem.

This can be confirmed by a conventional analysis of the scheme based on the characteris-tics of the amplification matrix. The period elongation is ∆T/T = ω4∆t4/270 + O(ω6∆t6),

8

∆t/T

Spectr

al R

adiu

s ρ

Energy Decaying

ρ∞

Generalized-α, = 0.0

ρ∞

Generalized-α, = 0.5

ρ∞

Generalized-α, = 0.9

o

+

x

*

10-2

10-1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

Figure 3: Comparison of spectral radii of various time integration schemes.

while the algorithmic damping is ζ = ω3∆t3/72 + O(ω5∆t5), where ω2 = k/m. Hence,the scheme is third-order accurate. The spectral radius, period elongation, and algorithmicdamping are shown in figs. 3, 4, and 5, respectively, as functions of ∆t/T = ω∆t/(2π). Theresults are compared with generalized-α method [5] with three different values of spectral ra-dius at infinity, ρ∞ = 0.9, 0.5, and 0.0. Fig. 4 indicates that the time discontinuous Galerkinscheme has better period elongation characteristics than generalized-α method, while fig. 5shows that its low frequency numerical dissipation characteristics are similar those of thegeneralized-α method with ρ∞ = 0.5. Asymptotic annihilation is obtained with the time dis-continuous Galerkin scheme. The scheme is unconditionally stable since the spectral radiusis always smaller than unity.

2.6 Discussion

Both Galerkin (8) and time discontinuous Galerkin (13) approximations applied to the equa-tions of motion written in the symmetric hyperbolic form (7) have been shown to providea systematic way of deriving unconditionally stable time integration schemes, provided theHamiltonian is a positive definite function. The energy decay inequality associated with thetime discontinuous Galerkin approximation implies the presence of numerical dissipation inthe resulting time integration schemes, whereas such dissipation is ruled out by the strictenergy preservation associated with the Galerkin approximation. Since the presence of nu-merical dissipation is highly desirable, the time discontinuous Galerkin approach appears tobe the most promising method.

However, both of these approaches present a major drawback: it is not always pos-sible to recast the equations of motion of general systems into the symmetric hyperbolic

9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

∆t/T

Energy Decaying

ρ∞

Generalized-α, = 0.0

ρ∞

Generalized-α, = 0.5

ρ∞

Generalized-α, = 0.9

o

+

x

*P

erio

d E

long

atio

n

Figure 4: Comparison of period elongation of various time integration schemes.

∆t/T

Alg

orith

mic

Dam

pin

g

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Energy Decaying

ρ∞

Generalized-α, = 0.0

ρ∞

Generalized-α, = 0.5

ρ∞

Generalized-α, = 0.9

o

+

x

*

Figure 5: Comparison of algorithmic damping of various time integration schemes.

10

form. In particular, it does not seem possible to cast the governing equations of constrainedmulti-body systems in the symmetric hyperbolic form. Furthermore, the time discontinuousGalerkin approach require two level of unknowns (at tj and tf ). In elasto-dynamics, threefields are required for the symmetric hyperbolic form: displacements, stresses and momenta.Hence, the final discrete equations will involve 6N unknowns, resulting in unacceptably highcomputational cost [13].

3 Practical time integration schemes

In this section, time integration schemes applicable to nonlinear elastic multi-body systemswill be developed, without resorting to the symmetric hyperbolic form of the equationsof motion. The investigation will focus on dynamical system defined by a kinetic energyK = 1/2 Mpq vp vq and a strain energy V = 1/2 Cpq εp εq. The mass matrix Mpq andstiffness matrix Cpq are symmetric and definite positive; the velocities and strains are givenas vp = Rpq(ur)uq, and εp = εp(ur), respectively. Note that the velocities are assumed to belinear functions of the up, resulting in a kinetic energy that is a quadratic form in up. Underthese conditions the total mechanical energy of the system is preserved [22].

The equations of motion of such systems simply write F ip +F e

p = Fap (t), where Fa

p (t) arethe time dependent external forces. The inertial and elastic forces, F i

p and F ep , respectively,

are

F ip =

d

dt(Rqp pq)− urDqrp pq; Dpqr = Rpq,r, (29)

F ep = Bqp fq; Bpq = εp,q, (30)

where pq = Mqr vr and fq = Cqr εr. The notation (.),p indicates a derivative with respect toup.

The energy preservation statement can be obtained by evaluating the work done by theinertial, elastic, and applied forces. The work done by the inertial forces is computed firstW i =

∫ tfti

upF ip dt = Kf − Ki. Next, the work done by the elastic forces is evaluated We =∫ tf

tiupF e

p dt = Vf − V i. Finally, the work done by the applied forces is Wa =∫ tf

tiupQq dt.

Hence, the equations of motion imply the following work balance equation

Kf −Ki + Vf − V i = Wa; ⇒ Ef − E i = Wa, (31)

where the total mechanical energy E = K + V . In the absence of externally applied forcesWa = 0 and the total energy is preserved.

Our goal is to obtain discretized equations of motion that will imply an exact energypreservation condition (31), or an energy decay inequality. At first, discretizations of theinertial and elastic forces will proposed, then energy preserving and energy decaying schemeswill be derived.

11

3.1 Discretization of inertial and elastic forces

Consider a time interval ti tf , and an approximate solution over this interval, as shown infig. 1. The following discretizations of the inertial (29) and elastic (30) forces are proposed:

F imp =

Rfqp pf

q −Riqp pi

q

∆t− uf

r − uir

∆tDm

qrp

pfq + pi

q

2; (32)

F emp = Bm

qp fmq , (33)

where the quantities Dmqrp, Bm

qp and fmq are as yet undetermined. The work done by the

discretized inertial forces is W i = (ufp − ui

p)F imp , and regrouping the term yields:

W i =uf

p − uip

∆t

[Rf

qp −uf

r − uir

2Dm

qpr

]pf

q −[Ri

qp +uf

r − uir

2Dm

qpr

]pi

q

. (34)

The following condition is now imposed:

vmq =

[Rf

qp −uf

r − uir

2Dm

qpr

]uf

p − uip

∆t=

[Ri

qp +uf

r − uir

2Dm

qpr

]uf

p − uip

∆t. (35)

These relationships now define both Dmqrp and vm

p . Note that the existence of Dmqrp satisfying

(35) is guaranteed by the mean value theorem which states that

Rfqp = Ri

qp + Rmqp,r(u

fr − ui

r); ⇒ Dmqpr = Rm

qp,r. (36)

The work done by the discretized inertial forces now becomes W i = (vfq − vi

q) Mqp vmp .

Next, the work done by the discretized elastic forces is evaluated We = (ufp −ui

p)Bmqp fm

q .The following condition is now imposed

εfq − εi

q = Bmqp (uf

p − uip). (37)

Here again, the existence of Bmqp satisfying this condition is guaranteed by the mean value

theorem which states that

εfq = εi

q + εmq,r(u

fr − ui

r); ⇒ Bqr = εmq,r. (38)

The work done by the discretized elastic forces now becomes We = (εfp − εi

p) fmp .

3.2 Energy Preserving Scheme

The discretized equations of motion for the energy preserving scheme mimic those obtainedin section (2.5) for the Galerkin approximation of the linear spring-mass problem (20)

F imp + F em

p = Famp , (39)

where F imp and F em

p are now given by (32) and (33), respectively; and Famp = 1/2

∫ 1

−1Fa

p (τ) dτ ,as in eq. (23). The work done by these discretized forces can be evaluated, as was done in

12

section (3.1). With the help conditions of (35) and (37) equations of motion (39) imply awork balance statement

(vfq − vi

q) Mpq vmp + (εf

p − εip) fm

p = Wam. (40)

The following algorithmic velocity-displacement and force-strain relationship are now se-lected (see eq. (22))

vmp =

ufp + ui

p

2; fm

p = Cpq

εfp + εi

p

2. (41)

The work balance equation (40) then becomes

Kf −Ki + Vf − V i = Wa; ⇒ Ef − E i = Wa (42)

In summary, discretization (39) implies the energy preservation statement (42) providedthat relationships (35) and (37) are satisfied, and that the algorithmic velocity-displacementand force-strain relationships (41) are used.

3.3 Energy Decaying Scheme

The discretized equations of motion for the energy decaying scheme mimic those obtainedin section (2.5) for the time discontinuous Galerkin approximation of a linear spring-masssystem (25)

F imp + F eg

p = Fagp ; F ih

p − 1

3

[F egp −Bh

qp f jq

]= Fah

p (43)

where F imp , and F ih

p are given by (32) using superscripts (.)f , (.)i and (.)j, (.)i, respectively;

F egp is given by (33) using superscripts (.)f , (.)j; and Fag

p = 1/2∫ 1

−1Fa

p dτ and Fahp =

1/2∫ 1

−1Fa

p τdτ , as in eq. (28).

The work done by the discretized inertial forces is W i = (ufp − ui

p)F imp + 3〈up〉F ih

p . Withthe help of condition (35) this becomes

W i = vmp Mpq(v

fq − vi

q) + 3vhpMpq〈vq〉. (44)

The work done by the discretized elastic forces is We = (ufp − ui

p)F egp − 〈up〉[F eg

p − Bhqp f j

q ].With the help of condition (37) this becomes

We = (εfp − εj

p) Cpq f gq + 〈εp〉 Cpq f j

q . (45)

The following velocity-displacement and force-strain relationship are now selected (see eq.(27))

vmp =

ufp + uj

p

2; 3vh

p = − ufp − uj

p

2; f g

p = Cpq

εfp + εj

p

2. (46)

The work balance equation now writes

Ef − E j + vjp Mpq 〈vq〉+ f j

p Cpq 〈εq〉 = Wa, (47)

13

0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

TIME

DIS

PLA

CE

ME

NT

AN

D V

ELO

CIT

Y

Displacement

Velocity

Figure 6: Displacement response for the trapezoidal rule (ui = 1.0).

which mirrors (16). Since the total mechanical energy is a definite positive function of thevelocities and strains, the mean value theorem implies

E j = E i + vjp Mpq 〈vq〉+ f j

p Cpq 〈εq〉 − 1

2

[Eh

,vpvq〈vp〉 〈vq〉

+Eh,vpεq

〈vp〉 〈εq〉+ Eh,εpvq

〈εp〉 〈vq〉+ Eh,εpεq

〈εp〉 〈εq〉]

= E i + vjp Mpq 〈vq〉+ f j

p Cpq 〈εq〉 − c2, (48)

which is equivalent to (17). Combining (47) and (48) then finally yields

Ef = E i − c2 +Wa; ⇒ Ef ≤ E i +Wa. (49)

In summary, discretization (43) implies the energy decay statement (49) provided thatrelationships (35) and (37) are satisfied, and that the algorithmic velocity-displacement andforce-strain relationships (46) are used.

3.4 Example: Nonlinear spring-mass system

Consider the nonlinear spring mass oscillator defined by a kinetic energy K = 1/2 m u2, astrain energy V = 1/2 k ε2, and a strain ε = u2. For this example m = k = 1.0. It is clearthat condition (37) implies Bm = uf +ui in this case, and fm = k(εf +εi)/2. The discretizedequations of motion are

muf −mui

∆t+ 2 k

(uf + ui

2

)3

= 0; (50)

14

0 5 10 150

1

2

3

4

5

6

TIME

SY

ST

EM

EN

ER

GIE

S− − Kinetic Energy

− . Strain Energy

___ Total Energy

Figure 7: Energy response for the trapezoidal rule (ui = 1.0).

0 5 10 15−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

TIME

DIS

PLA

CE

ME

NT

AN

D V

ELO

CIT

Y

Displacement

Velocity

Figure 8: Displacement response for the trapezoidal rule (ui = 2.0).

15

0 5 10 150

1

2

3

4

5

6

7

8

TIME

SY

ST

EM

EN

ER

GIE

S

− − Kinetic Energy

− . Strain Energy

___ Total Energy

Figure 9: Energy response for the trapezoidal rule (ui = =2.0).

for the trapezoidal rule,muf −mui

∆t+ Bm fm = 0; (51)

for the energy decaying scheme, and

muf −mui

∆t+ Bg f g = 0,

muj −mui

∆t− 1

3

[Bg f g −Bh k εj

]= 0; (52)

for the energy decaying scheme.Though the trapezoidal rule scheme is unconditionally stable for linear system, there is

no guarantee of stability when applied to nonlinear systems. Fig. 6 shows the response of thesystem for initial conditions u0 = 1.0, u0 = 0.0. The total energy rapidly increases as shownby fig. 7. For initial conditions u0 = 2.0, u0 = 0.0, the corresponding results are shown infigs. 8 and 9, which now show a rapid decrease in energy. The responses predicted by theenergy preserving scheme are shown in figs. 10 and 11; as expected, the total energy of thesystem is exactly preserved. Finally, figs. 12 and 13 show the responses predicted by theenergy decaying scheme; the total energy of the system decays, as expected. The results ofa convergence study shown in fig. 14 indicate second order accuracy for the trapezoidal ruleand the energy preserving scheme, and third order accuracy for the energy decaying scheme.

4 Enforcement of the constraints

4.1 Work done by the forces of constraint

Consider a multi-body system subjected to holonomic constraints Cp(ur) = 0. If the Lagrangemultiplier method is used to enforce these constraints, a constraint potential Vc = λpCp is

16

0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

TIME

DIS

PLA

CE

ME

NT

AN

D V

ELO

CIT

YDisplacement

Velocity

Figure 10: Displacement response for the energy preserving scheme.

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TIME

SY

ST

EM

EN

ER

GIE

S

− − Kinetic Energy

− . Strain Energy

___ Total Energy

Figure 11: Energy response for the energy preserving scheme.

17

0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

TIME

DIS

PLA

CE

ME

NT

AN

D V

ELO

CIT

Y

Displacement

Velocity

Figure 12: Displacement response for the energy decaying scheme.

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

TIME

SY

ST

EM

EN

ER

GIE

S

− − Kinetic Energy

− . Strain Energy

___ Total Energy

Figure 13: Energy response for the energy decaying scheme.

18

101

102

103

104

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

NUMBER OF TIME STEPS (LOG)

ER

RO

R (

LOG

)

o Trapezoidal Rule

+ Energy preserving Scheme

* Energy Decaying Scheme

Figure 14: Convergence study.

19

added to the strain energy of the system, where λp are the Lagrange multipliers. Thecorresponding forces of constraint are

F cp = Bqpλq; Bpq = Cp,q. (53)

The work done by these forces is Wc =∫ tf

tiupF c

p dt =∫ tf

tiλpCp dt. Since Cp must vanish

at all times, Cp = 0, and Wc = 0, i.e. the work done by the forces of constraint vanishesexactly.

Another method of enforcing constraints is the penalty method. In this approach, aconstraint potential Vc = 1

2Ppq CpCq is added to the strain energy of the system. Ppq = p δpq

is the penalty matrix. There is a close analogy between the constraint potential and thestrain energy of an elastic medium: the constraint can be viewed as a strain quantity andthe penalty as a stiffness coefficient. As the “stiffness” p increases, the “strains” Cp becomesmaller and smaller. At the limit, we let the penalty coefficient increase to infinity so as todrive the constraints to zero, while the product PpqCq/2 goes to a finite value λp. This writes

Vc = limp →∞Cp → 0

(1

2Ppq Cq) Cp = λpCp. (54)

Hence, the Lagrange multiplier method can be viewed as the limiting case of the penaltymethod when the penalty coefficient goes to infinity.

4.2 Energy preserving scheme

When the penalty method is used, the forces of constraint can be discretized as if they wereelastic forces, see section (3.2). The discretized forces of constraint are F cm

p = Bmqp fm

q , andthe following condition is imposed (see eq. (37))

Cfq − Ci

q = Bmqp (uf

p − uip), (55)

which defines Bmqp. The work done by the forces of constraint is then Wc = Vcf − Vci. This

approach presents two major drawbacks: the constraints are not enforced exactly since afinite value of the penalty coefficient must be used, and as a result, the work done by theforces of constraint does not vanish.

To avoid these problem, the Lagrange multiplier approach will be used. The discretizedforces of constraint can be readily obtained through the limiting process mentioned earlier

F cmp = Bm

qp λmq , (56)

and their work then becomes Wc = (Cfp − Ci

p) λmp . The Lagrange multipliers λm

p are nowadditional unknowns of the problem. The additional equations required to solve the problemare obtained by enforcing the exact vanishing for the work done by the discretized forces ofconstraint. Since λm

p 6= 0, this implies Cfp − Ci

p = 0. In order to avoid the drift phenomenon,it is preferable to enforce the condition Cf

p = 0 at each time step.In summary, the discretization of the forces of constraint (56) together with relationship

(55) and the discrete constraint Cfp = 0 imply the exact vanishing of the work done by the

discretized forces of constraint.

20

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

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

TIME

BO

B D

ISP

LAC

EM

EN

TS

X−Displacement

Y−Displacement

Figure 15: Displacement response for the trapezoidal rule.

4.3 Energy decaying scheme

The Lagrange multiplier method for the energy decaying scheme is also obtained as thelimiting case of the penalty approach. The discretized forces of constraint are (see eq. (43))

F cgp = Bg

qp λgq ; F ch

p = −1

3

[F cgp −Bh

qp λjq

]. (57)

The work done by these forces become Wc = (Cfp −Cj

p) λgp +(Cj

p−Cip) λg

p, and vanishes only ifCf

p −Cjp = 0 and Cj

p −Cip = 0. Here again it is preferable to enforce Cf

p = Cjp = 0 to avoid the

drift phenomenon. In summary, the discretization of the forces of constraint (57) togetherwith the discrete constraints Cf

p = Cjp = 0 imply the exact vanishing of the work done by the

discretized forces of constraint.

4.4 Example: the pendulum problem

Consider the pendulum problem defined by a kinetic energy K = 1/2 m upup, a potentialenergy V = −m g u2, and the constraint C = 1/2(upup− l2) = 0, where l is the length of thependulum. For this example m = 1.0 kg, l = 0.5 m; v0 = 1.695 m/sec; and g = 9.81 m/sec2.It is clear that condition (55) implies Bm

p = ump , where um

p = 1/2 (ufp + ui

p). The governingequations for the trapezoidal rule and energy preserving schemes are

mufp −mui

p

∆t+ Bm

p λm = mgp. (58)

For the trapezoidal rule the constraint condition is Cm = 1/2 [ump um

p − l2] = 0, whereas forthe energy preserving it is Cf = 1/2 [uf

pufp − l2] = 0. Finally, the governing equations for the

21

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−4

−2

0

2

4

6

TIME

BO

B V

ELO

CIT

IES

X−Velocity

Y−Velocity

Figure 16: Velocity response for the trapezoidal rule.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

TIME

LAG

RA

NG

E M

ULT

IPLI

ER

Figure 17: Lagrange multiplier response for the trapezoidal rule.

22

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5

0

5

10

15

20

25

TIME

SY

ST

EM

EN

ER

GY

− − Kinetic Energy

− . Potential Energy

___ Total Energy

Figure 18: Energy response for the trapezoidal rule.

energy decaying scheme are

mufp −mui

p

∆t+ Bg

p λg = mgp; (59)

mujp −mui

p

∆t− 1

3

[Bg

p λg −Bhp λj

]= 0; (60)

subjected to two constraint condition Cf = 0, and Cj = 1/2 [ujpu

jp − l2] = 0.

Fig. 15, 16 and 17 show the time history of the pendulum displacements, velocities, andLagrange multiplier, respectively, for the trapezoidal rule. Though the displacement historyis accurately predicted, the velocities and Lagrange multipliers present violent oscillations ofa purely numerical origin. The sharp rise in total energy shown in fig. 18 clearly indicatesthe unstable nature of this scheme. Figs. 19 to 22 show the corresponding results for theenergy preserving and decaying schemes which are in very close agreement. All predictedhistories are smooth. The total energy is exactly preserved for the energy preserving scheme,and for the energy decaying scheme, the amount of dissipated energy is very small for thissimple problem.

5 Kinematic notations and conventions

The kinematic description of bodies and joints in their undeformed and deformed configu-rations will make use of three orthogonal triads. First, an inertial triad is used as a globalreference for the system; it is denoted SI with unit vectors ~ı1, ~ı2, and ~ı3. A second triad S0,with unit vectors ~e01, ~e02, and ~e03 is attached to the body and defines its orientation in the

23

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

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

TIME

BO

B D

ISP

LAC

EM

EN

TS

X−Displacement

Y−Displacement

Figure 19: Displacement response for the energy preserving and decaying schemes.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

TIME

BO

B V

ELO

CIT

IES

X−Velocity

Y−Velocity

Figure 20: Velocity response for the energy preserving and decaying schemes.

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5x 10

−4

TIME

LAG

RA

NG

E M

ULT

IPLI

ER

Figure 21: Lagrange multiplier response for the energy preserving and decaying schemes.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

TIME

SY

ST

EM

EN

ER

GY

− − Kinetic Energy

− . Potential Energy

___ Total Energy

Figure 22: Energy response for the energy preserving and decaying schemes.

25

reference configuration. Finally, a third triad S∗ with unit vectors ~e1, ~e2, and ~e3 defines theorientation of the body in its deformed configuration.

Let ~u0 and ~u be the displacement vectors from SI to S0, and S0 to S∗, respectively, andR0 and R the rotation tensors from SI to S0, and S0 to S∗, respectively. In this work, allvector and tensor components are measured in either SI or S∗. For instance, the componentsof vector ~u measured in SI and S∗ will be denoted u and u∗, respectively, and clearly

u∗ = RT0 RT u. (61)

Similarly, the components of tensor R measured SI and S∗ will be denoted R and R∗,respectively. For brevity sake, a compact notation will be used to deal with the componentsof the linear and angular vectors simultaneously. For instance,

v =

[vv

](62)

defines the six components of the velocity vector consisting of the three components linearand angular velocities denoted v and v, respectively. The following 6 × 6 operators areintroduced

R0 =

[R0 00 R0

]; R =

[R 00 R

]; U [.] =

[0 0[.] 0

]. (63)

The skew-symmetric matrix formed with the components u will be denoted u.

6 Formulation elastic elements

6.1 Formulation of beam elements

Beams can be defined as elastic bodies whose volume is that spanned by a cross-sectiontranslating along a smooth reference line. The cross-section lies in the plane defined byvectors ~e02, ~e03 and ~e2, ~e3 in the undeformed and deformed configurations, respectively, asdepicted in fig. 23. The kinetic and strain energies of the beam are

K =1

2

∫ L

0

v∗T M∗v∗ dx1; V =1

2

∫ L

0

e∗T C∗e∗ dx1, (64)

respectively. L is the length of the beam; x1 the curvilinear coordinate along the referenceline; M∗ and C∗ the components of the sectional inertial and stiffness tensor, respectively;and v∗, and e∗ the components of the sectional velocity and strain vectors, respectively. The6 × 6 inertial matrix M∗ is fully populated, allowing the modeling of rotary inertia effectsand the offset of the sectional mass center with respect to the reference line. Similarly, the6 × 6 stiffness matrix C∗ is fully populated, allowing the modeling of shearing deformationeffect, the offset of tension center and shear center with respect to the reference line, andall elastic couplings that might arise from the use of tailored composite materials. Thevelocity-displacement and strain-displacement relationships are

v =

[vv

]=

[uv

]; e =

[ee

]=

[(u′0 + u′)−RR0 ı1

e

], (65)

26

e03

e01

e02

i2

i3

i1

e1

e2

e3

u R,

u0

R0

,

Undeformed Configuration

DeformedConfiguration

Figure 23: Beam in the undeformed and deformed configurations.

27

where (.)· and (.)′ denote derivatives with respect to time and x1, respectively; v are thecomponents of the sectional angular velocity vector, with ˜v = R RT ; and e the components ofthe sectional elastic curvature vector, with ˜e = R′RT . These relationships are geometricallyexact, i.e. are valid for arbitrarily large displacements and rotations, although the strainsare assumed to remain small. Virtual variations in sectional velocities and strains are

δv∗T =(δd

T − δdT U [˜u])RR0; (66)

δe∗T =(δd′T − δdT U [u′0 + u′]

)RR0, (67)

where δdT = (δuT , δψT ) are the virtual displacements and rotations. The virtual rotations

are defined as δψ = δR RT .The equations of motion of the beam will be derived from Hamilton’s principle

∫ tf

ti

∫ L

0

(δv∗T M∗v∗ − δe∗T C∗e∗T + δWa

)dx1 dt = 0, (68)

where δWa is the virtual work done by the externally applied forces. The equations of motionof the beam are found by introducing eqs. (66) and (67), and using the strain and velocityexpressions, eqs. (65), to find(RR0 p∗

)·+ U [˜u] RR0 p∗ − (RR0 f ∗

)′ − U [u′0 + u′] RR0 f ∗ = q; (69)

where the sectional momenta and elastic forces are defined as p∗ = M∗ v∗ and f ∗ = C∗ e∗,respectively; and q are the external forces.

An energy preserving discretization of these equations of motion is now sought. Theinertial and elastic forces are discretized according to eqs. (32) and (33), to yield the followingdiscretized equations of motion

RfR0 p∗f−RiR0 p∗

i

∆t+ U [

uf − ui

∆t] RaR0

p∗f

+ p∗i

2−(RbR0 f ∗

m)′ − U [u′0 + u′m] RbR0 f ∗

m= q

m, (70)

where the elastic forces f ∗m

are as yet undetermined; um = (uf + ui)/2; and the rotationoperators are defined in Appendix B which also discusses the discretization of finite rotations.

The work done by the discretized inertial forces can be computed by premultiplying theseforces by the incremental displacements and rotations, then integrating over the span of thebeam to find

W i =

∫ L

0

[uf − ui

r

]T[RfR0 p∗

f−RiR0 p∗

i

∆t

+U [uf − ui

∆t] RaR0

p∗f

+ p∗i

2

]dx1, (71)

where r are the components of the Rodrigues parameters used to measure the incrementalrotations, see Appendix B. Regrouping terms, this work becomes

W i =

∫ L

0

uTf − uT

i

∆tRmR0

[(G∗ − r∗

2

G∗ + G∗T

2) p∗

f

−(G∗T +r∗

2

G∗ + G∗T

2) p∗

i

]+

rT

∆tRmR0 (G∗ p∗

f−G∗T p∗

i). (72)

28

This result mirrors eq. (34). With the help of eqs. (A4) and (A5), the mid-point velocity v∗mdefined in eq. (35) becomes

v∗m =

RT

0 RTa

uf − ui

∆tRT

0 RTm

r

∆t

, (73)

and the work done by the discretized inertial forces reduces to

W i = v∗Tm M∗ (v∗f − v∗i ). (74)

We now turn to the work done by the discretized elastic forces which can be computedby premultiplying these forces by the incremental displacements and rotations, integratingover the span of the beam, then integrating by parts to find

We =

∫ L

0

[u′f − u′i

r′

]T

+

[uf − ui

r

]T

U [u′0 + u′m]

Rb f ∗

mdx1. (75)

Regrouping terms, this work becomes

We =

∫ L

0

f∗Tm

G∗ + G∗T

2

[(I − r∗

2) RT

0 RTm (u′0 + u′f )

−(I +r∗

2) RT

0 RTm (u′0 + u′i)

]+ f

∗Tm

RT0 RT

mGHT r′

dx1. (76)

With the help of eq. (A6), this reduces to

We =

∫ L

0

[f∗Tm

(e∗f − e∗i ) + f∗Tm

(e∗f − e∗i )]

dx1. (77)

Combining eqs. (74) and (77) it is now clear that discretization (70) implies the followingenergy balance equation

∫ L

0

[v∗Tm M∗ (v∗f − v∗i ) + f ∗T

m(e∗f − e∗i )

dx1 = ∆Wa

m, (78)

which mirrors eq. (40). Finally, the following velocity-displacement and force-displacementrelationships are selected (see eq. ( 41)

v∗m =v∗f + v∗i

2; f ∗

m= C∗ e∗f + e∗i

2. (79)

The energy balance equation then implies Ef − Ei = ∆Wam, the discrete energy preservation

condition. In summary, the energy preserving formulation for beams consists of the dis-cretized equations of motion (70) together with relationships (79). This energy preservingformulation can be readily extended to an energy decaying formulation by following the stepsoutlined in section 3.3.

29

6.2 Formulation of flexible joint elements

Consider two bodies denoted with superscripts (.)k and (.)l, respectively, linked together bylinear and torsional springs at a point. In the reference configuration, the flexible joint isdefined by two coincident triads Sk

0 = S l0 = S0. In the deformed configuration, the two

triads undergo relative displacements and rotations, inducing deformations s in the flexiblejoint. These deformations are selected as the relative displacements and rotations of the twobodies

s∗ = RT0 RkT u; s∗ =

1

2

g32 − g23

g13 − g31

g21 − g12

=

1

2

R∗32 −R∗

23

R∗13 −R∗

31

R∗21 −R∗

12

= n∗ sin φ, (80)

where u = uk − ul is the relative displacement between the bodies; R∗ij the components of

the relative rotation tensor R∗ = RT0 RkT RlR0; φ the magnitude of this relative rotation; n∗

the unit vector about which it takes place; and

gαβ = ekTα el

β. (81)

The strain energy in the flexible joint now writes

V =1

2s∗T C∗ s∗, (82)

where C∗ are the components of the flexible joint stiffness tensor. This 6× 6 stiffness matrixis fully populated allowing the modeling of the various linear and torsional stiffnesses, as wellas potential elastic couplings. The elastic forces F e in the flexible joint are readily foundfrom variations of the strain energy

δV =

δuk

δψk

δul

δψl

T

·

− RkR0 f∗

S f∗ − u RkR0 f

∗

RkR0 f∗

− S f∗

=

δuk

δψk

δul

δψl

T

· F e, (83)

where f ∗ = C∗s∗. The operator S is defined as

S =1

2[h32 − h23, h13 − h31, h21 − h12] , (84)

wherehαβ = ek

α elβ. (85)

An energy preserving discretization is now sought and these elastic forces are discretizedaccording to eq. (33)

F em =

− RkaR0 f

∗m

Sm f∗m− um Rk

aR0 f∗m

RkaR0 f

∗m

− Sm f∗m

, (86)

30

where the rotation operators are defined in Appendix B which also describes the discretiza-tion of the finite rotations;

Sm =1

2[h32m − h23m, h13m − h31m, h21m − h12m] ; (87)

and

eαm =eαf + eαi

2; hαβm = ek

αm elβm. (88)

The work done by these discretized elastic forces writes

We = f∗Tm

RT0 RkT

a

[(uf − ui) + um rk

]+ f

∗Tm

STm (rk − rl). (89)

Regrouping terms and using eqs. (A5) and (A6) then yields

We = f∗Tm

(s∗f − s∗i ) + f∗Tm

(s∗f − s∗i ) = f ∗Tm

(s∗f − s∗i ). (90)

Finally, the following force-displacement relationship

f ∗m

= C∗(s∗f + s∗i )/2 (91)

is selected (see eq. (41)) and the work done by the discretized elastic forces becomeWe = Vf−Vi. In summary, the energy preserving formulation for flexible joints consists of the elasticforce discretization (86) together with the constitutive laws (91). This energy preservingformulation can be readily extended to an energy decaying formulation by following thesteps outlined in section 3.3.

7 Formulation of constraint elements

7.1 Formulation of universal joint elements

Consider two bodies denoted with superscripts (.)k and (.)l, respectively, linked togetherby a universal joint, as depicted in fig. 24. In the undeformed configuration, the universaljoint is defined by two triads Sk

0 and S l0 with a common origin, and ~e k

03 is orthogonal to ~e l03.

The kinematic constraint associated with a universal joint implies the orthogonality of thecorresponding vectors in the deformed configuration

C = ekTα el

β = gαβ = 0, (92)

where α = β = 3. Of course, in the deformed configuration, the origin of the triads isstill coincident; this constraint is readily enforced within the framework of finite elementformulations by Boolean identification of the corresponding degrees of freedom. As discussedin section 4, holonomic constraints are enforced by the addition of a constraint potential λ C,where λ is the Lagrange multiplier. The forces of constraint F c corresponding to eq. (92)are readily obtained as

δC · λ =

[δψk

δψl

]T

·[

λ hαβ

− λ hαβ

]=

[δψk

δψl

]T

· F c. (93)

31

u = u0 0

K L

R K

i3

R 0

K

i1

i2

R 0

L

R L

u = uK L

e03

K

e03

L

e3

K e3

L

Figure 24: Universal joint in the undeformed and deformed configurations.

To obtain unconditionally stable schemes for constrained systems, these forces of constraintmust be discretized so that the work they perform vanishes exactly. The following discretiza-tion is proposed

F cm =

[sλm hαβm

− sλm hαβm

], (94)

where s is a scaling factor for the Lagrange multipliers, and λm the unknown mid-point valueof this multiplier. The work done by these constraint forces is computed as follows

∆Wc = sλm(rkT − rlT ) hαβm = sλm iTα RT0 RkT

m

[Gk + GkT

2rk

T

Gl + GlT

2+

Gl + GlT

2rl

T Gk + GkT

2

]Rl

mRT0 iβ

= sλm (Cf − Ci), (95)

where the last equality was obtained with the help of eq. (A5). It is now clear that the workdone by the discretized constraint forces vanishes if Cf − Ci = 0. In order to avoid the driftphenomenon, it is preferable to enforce the condition Cf = 0 at each time step. In summary,discretization (94) together with the constraint Cf = 0 leads to the vanishing of the workdone by the forces of constraint. This energy preserving formulation can be readily extendedto an energy decaying formulation by following the steps outlined in section 4.3.

7.2 Formulation of revolute joint elements

Consider two bodies denoted with superscripts (.)k and (.)l, respectively, linked togetherby a revolute joint, as depicted in fig. 25. In the undeformed configuration, the revolute

32

i2

R = R0 0

K L

R K

u = u0 0

K L

RL

u = uK L

i3

i1

e01

e02

e03

e1

K

e3

Ke

3

L=

e2

L

e1

L

e2

K

φ

Figure 25: Revolute joint in the undeformed and deformed configurations.

joint is defined by coincident triads Sk0 = S l

0. In the deformed configuration, no relativedisplacements are allowed and the corresponding triads are allowed to rotate with respectto each other in such a way that ~e k

3 = ~e l3 . This condition implies the orthogonality of

~e k3 to both ~e l

1 and ~e l2 . These two kinematic constraints are both given by eq. (92) with

α = 3, β = 1, and α = 3, β = 2, respectively, and are enforced in the manner described inthe previous section. The relative rotation φ between the two bodies is defined by adding tothe revolute joint formulation a third constraint

C = gαα sin φ + gαβ cos φ = 0, (96)

where α = 1 and β = 2. Here again, this constraint is enforced via the Lagrange multipliertechnique. The corresponding forces of constraint F c are readily obtained

δC · λ =

δψk

δψl

δφ

T

·

λ (hαα sin φ + hαβ cos φ)− λ (hαα sin φ + hαβ cos φ)

λ (gαα cos φ− gαβ sin φ)

=

δψk

δψl

δφ

T

· F c. (97)

Here again, these forces of constraint must be discretized so that the work they performvanishes exactly. The following discretization is proposed

F cm =

sλm (hααm sinm φ + hαβm cosm φ)− sλm (hααm sinm φ + hαβm cosm φ)

sλm (gααm cos φm − gαβm sin φm)

, (98)

where sinm φ = (sin φf +sin φi)/2 and sin φm = sin(φf +φi)/2, with similar notations for thecosine function; and gαβm = (gαβf + gαβi)/2. The work done by these forces is

∆Wc = sλm (rkT − rlT )[hααm sinm φ + hαβm cosm φ

]

+sλm 2 sinφf − φi

2[gααm cos φm − gαβm sin φm] . (99)

33

Note that the increment in relative rotation was selected as 2 sin(φf − φi)/2. The first termcan be handled in the same manner as in eq. (95), and trigonometric identities are used toreduce the second term to find

∆Wc = sλm [(gααf − gααi) sinm φ + (gαβf − gαβi) cosm φ

+gααm(sin φf − sin φi) + gαβm(cos φf − cos φi)]

= sλm (Cf − Ci). (100)

In summary, discretization (98) together with the constraint Cf = 0 leads to the vanishingof the work done by the forces of constraint. Here again, this energy preserving formulationcan be readily extended to an energy decaying formulation by following the steps outlinedin section 4.3.

8 Time step size adaptation procedure

The response of constrained multi-body system often rapidly varies in time, prompting theneed for an automated time step size adaptation procedure. The first step toward thedevelopment of such procedure is the choice of a measure of the error associated with thetime discretization of the equations of motion. Within the framework of the energy decayingscheme, the following error measure e is proposed

e =Ed

Ei

, (101)

which corresponds to the amount of energy Ed dissipated by the scheme, normalized by theinitial energy. For energy decaying schemes, the numerically dissipated energy Ed = c2 is apositive quantity, as shown in eq. (48). It is an integrated quantity that reflects errors inboth velocities and strains at all points of the structure. Note that for the exact solution nonumerical dissipation should take place and e = 0.

Consider, at first, the energy decaying discretization of the single degree of freedom,linear oscillator described in section 2.5. The error measure can be explicitly computed as

e = 1− ρ2 =(ω∆t)4

(ω∆t)4 + 4(ω∆t)2 + 36, (102)

where ρ is the spectral radius of the amplification matrix. This relationship can be invertedto find

ω∆t =

√2

e +√

9e− 8e2

1− e≈√

6 4√

e, (103)

which gives the time step size required to achieve a specified error level. It will be convenientto write this relationship in the following manner

∆tnew

∆told

≈ 4

√e

eold

, (104)

34

h = 6 m

1 m

Revolute joint

l = 4.25 m

e = 0.25 m

Figure 26: The windmill problem.

where ∆tnew is the time step size that will achieve a desired error level e, if a previous timestep size ∆told corresponded to an error level eold.

We now turn to nonlinear, flexible multi-body systems. The error measure given byeq. (101) is still a rigorous error measure for such system since no numerical dissipationoccurs in the exact solution. However, the relationship between the error measure and thetime step is no longer given by eq. (102). Nevertheless, the time step size update formulaof eq. (104) was found to be a reliable estimate of the time step size required to achieve adesired level of accuracy in complex multi-body systems.

9 Numerical Examples

9.1 The windmill resonance problem

The first example deals the windmill resonance problem depicted in fig. 26 and will be usedto illustrate the different behavior of the energy preserving and decaying schemes. Considera four-bladed rotor mounted on an elastic tower. The tower has a height h = 6.0 m, bendingstiffnesses I22 = I33 = 3.87 MN.m2, a torsional stiffness GJ = 2.97 MN.m2, and a mass

35

0 0.5 1 1.5 2 2.5 3−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

TIME [sec]

TO

WE

R L

AT

ER

AL

TIP

DIS

PLA

CE

ME

NT

[m]

Figure 27: Time history of the tower tip lateral deflection for two rotor speed Ω = 15 rad/sec(solid line), and 30 rad/sec (dashed line).

0 0.5 1 1.5 2 2.5 3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

TIME [sec]

BLA

DE

LA

G A

NG

LE

Figure 28: Time history of the lag angles for the four blades for two rotor speed Ω =15 rad/sec (solid line), and 30 rad/sec (dashed line).

36

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

ROTOR CENTER OF MASS X [m]

RO

TO

R C

EN

TE

R O

F M

AS

S Y

[m]

Figure 29: Time history of the location of the rotor center of mass for two rotor speedΩ = 15 rad/sec (solid line), and 30 rad/sec (dashed line).

per unit span mT = 12.72 kg/m. A 30 kg concentrated mass is located at the top of thetower. A nacelle is connected to the tip of the tower and projects 1.0 m forward. Thenacelle properties are identical to those of the tower. The rotor hub is located at the tipof the nacelle and is represented by a 20 kg concentrated mass. Each blade is uniform andhas a length l = 4.25 m, a mass mb = 12.75 kg, in- and out-of-plane bending stiffness ofI33 = 4.71 kN.m2, and I22 = 0.547 kN.m2, respectively, and the blade’s hinge located atdistance e = 0.25 m from the hub. The tower and nacelle are discretized with two and onecubic beam elements, respectively, whereas each blade is discretized with three cubic beamelements. A revolute joint located at the tip of the nacelle allows rotation of the rotor, andfour revolute joints model the four lead-lag hinges of the blades.

At first, two cases will be studied with initial rotor speeds Ω = 15 rad/sec and 30 rad/sec,which correspond to stable, and unstable rotor speeds, respectively. Fig. 27 shows the timehistory of the tower tip lateral deflection. The lower rotor speed is in the stable operationrange, whereas the higher speed is in the unstable range. Fig. 28 shows the correspondinglag angles for the four blades of the rotor. The exponential growth of these quantities clearlydemonstrate the catastrophic effect of the unstable behavior of the system. The mechanismtriggering this instability is better understood by looking at the time history of the locationof the rotor center of mass as viewed by an inertial observer, as depicted in fig. 29. As theinstability sets in, the rotor’s center of mass spirals around as a result of the blade’s lagmotion. This motion excites the tower at its natural frequency and initiates the instability.

Next, the behavior of the system will be studied under the effect of a constant torqueT = 500 M.m applied at the rotor hub. The rotor is initially rotating at a constant angularvelocity Ω = 15 rad/sec, i.e. in the stable regime. At that instant, the constant torqueis applied and the rotor speed increases, driving the system into an unstable regime. Thesudden application of the hub torque excites high frequency vibrations in both tower and

37

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

TIME [sec]

RO

TO

R A

NG

ULA

R S

PE

ED

Figure 30: Time history of the rotor speed under the effect of an applied hub torque, usingthe energy preserving scheme.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

TIME [sec]

BLA

DE

LA

G A

NG

LE [r

ad]

Figure 31: Time history of the lag angles for the four blades under the effect of an appliedhub torque, using the energy preserving scheme.

38

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

TIME [sec]

RO

TO

R A

NG

ULA

R S

PE

ED

Figure 32: Time history of the rotor speed under the effect of an applied hub torque, usingthe energy decaying scheme.

rotor. First, this problem is treated with the energy preserving scheme. Fig. 30 shows theincreasing rotor speed due to the applied torque. However, very high frequency vibrationsare superposed to the slow rise in rotor speed. Since no physical damping is present in thesystem, and no numerical dissipation is provided by the energy preserving scheme, theseoscillations do not damp out. The same phenomenon is observed in fig. 31 for the lag anglesof the four blades. The presence of these high frequencies hinders the convergence processat each time step, and requires an increasing number of iterations to be used. At timet = 4.466 sec, convergence fails, and the simulation is terminated.

On the other hand, fig. 32 shows the results of the simulation of the same problemusing the energy decaying scheme. Though high frequency oscillations are present at thebeginning of the simulation due to the sudden application of the hub torque, the energydecaying scheme now provides high frequency numerical dissipation that rapidly dampsout the energy associated with the high frequency vibrations. Typically, 4 iterations arerequired at each time step, as compared to 7 with the use of the energy preserving scheme.Furthermore, the simulation can proceed for a longer period of time, up to t = 5.36 secas depicted in fig. 33. By that time, very large blade angular motions are observed, andthe exponential growth gives place to a limit cycle type behavior involving complex blademotions.

9.2 The four bar mechanism problem

The second problem deals with a four bar mechanism problem depicted in fig. 34 and isused to demonstrate the automated time step size selection procedure described in section 8.Bar 1 is of length L1 = 0.12 m and is connected to the ground at point A by means ofa revolute joint. Bar 2 is of length L2 = 0.24 m and is connected to bar 1 at point B

39

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

TIME [sec]

BLA

DE

LA

G A

NG

LE [r

ad]

Figure 33: Time history of the lag angles for the four blades under the effect of an appliedhub torque, using the energy decaying scheme.

A

B C

D

Revolute jointsΩ = 20 rad/sec

Misaligned axis

of rotation

0.24 m

0.12 m

Figure 34: The four bar mechanism problem.

40

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

0

2

4

6

8

10

TIME [sec]

RO

TA

TIO

NS

[rad

]

Figure 35: Time history of rotations of the system. Relative rotations at points A (4) andD (2), and absolute rotation of bar 2 (+).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−3

TIME [sec]

OU

T−

OF

−P

LAN

E D

ISP

LAC

EM

EN

TS

[m]

Figure 36: Time history of out-of-plane displacements at points B (solid line) and C (dashedline).

41

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−7

−6

−5

−4

−3

−2

−1

0

1x 10

4

TIME [sec]

BA

R1

RO

OT

FO

RC

ES

[N]

Figure 37: Time history of forces at the root of bar 1, in local axes. Axial forces (4), in-planeshear forces (2), and out-of-plane shear force (+).

with a revolute joint. Finally, bar 3 is of length L3 = 0.12 m and is connected to bar2 and the ground at points C and D, respectively, by means of two revolute joints. Inthe reference configuration the bars of this planar mechanism intersect each other at 90degree angles and the axes of rotation the revolute joints at points A, B, and D are normalto the plane of the mechanism. However, the axis of rotation the revolute joint at pointC is at a 5 degree angle with respect to this normal to simulate an initial defect in themechanism. A torque is applied on bar 1 at point A so as to enforce a constant angularvelocity Ω = 20 rad/sec. If the bars were infinitely rigid, no motion would be possible asthe mechanism locks. For elastic bars, motion becomes possible, but generates large internalforces. Bar 1 has the following physical characteristics: axial stiffness EA = 40 MN , bendingstiffnesses EI22 = EI33 = 0.24 MN.m2, torsional stiffness GJ = 0.28 MN.m2, and mass perunit span m = 3.2 kg/m. Bars 2 and 3 have the following physical characteristics: axialstiffness EA = 40 MN , bending stiffnesses EI22 = EI33 = 24 kN.m2, torsional stiffnessGJ = 28 kN.m2, and mass per unit span m = 1.6 kg/m.

If the four revolute joints had their axes of rotation orthogonal to the plane of themechanism, the response of the system would be purely planar, and bars 1 and 3 wouldrotate at constant angular velocities around points A and D, respectively. The initial defectin the mechanism causes a markedly different response. Fig. 35 shows the time history of therelative rotations at points A and D, as well the absolute rotation of the mid point of bar 2.Bar 1 rotates at a constant angular velocity under the effect of the applied torque, but bar 3now oscillates back and forth, never completing an entire turn. When the direction of rotationof bar 3 reverses, bar 2 undergoes large rotations, instead of near translation. Furthermore,the response is three dimensional as shown in fig. 36 which depicts the time history of out-of-plane displacements at points B and C. Point C undergoes a 3 mm maximum out-of-planedisplacement.

42

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2000

0

2000

4000

6000

8000

10000

TIME [sec]

BA

R1

RO

OT

MO

ME

NT

S [N

m]

Figure 38: Time history of moments at the root of bar 1, in local axes. Torque (4), in-planebending moment (2), and out-of-plane bending moment (+).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−6

−4

−2

0

2

4

6

8x 10

4

TIME [sec]

BA

R2

MID

−S

PA

N F

OR

CE

S [N

]

Figure 39: Time history of forces at mid-span of bar 2, in local axes. Axial forces (4),in-plane shear forces (2), and out-of-plane shear force (+).

43

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

TIME [sec]

BA

R2

MID

−S

PA

N M

OM

EN

TS

[Nm

]

Figure 40: Time history of moments at mid-span of bar 2, in local axes. Torque (4), in-planebending moment (2), and out-of-plane bending moment (+).

The time history of the three components of the internal force at the root of bar 1 areshown in fig. 37, whereas fig. 38 shows the time history of components of twisting and bendingmoments at the same location. The corresponding quantities at mid-span of bar 2 are shownin figs 39 and 40. These large internal forces are all caused by the initial imperfection of themechanism.

For this problem, the automated time step size selection procedure described in section 8was used. Fig. 41 shows the time history of the time step size which varied as much as by afactor of six during the simulation. The desired local error level, as defined by eq. (101), wasset to be e = 1.0× 10−06 and a new time step size was evaluated at each time step accordingto eq. (104). Fig. 42 shows the time history of the cumulative error measure obtained bysumming the error at each time step. This result shows that less than 0.013% error wasaccumulated over the entire simulation.

9.3 The tail rotor transmission problem

The last problem deals with the supercritical transmission of a helicopter tail rotor. Fig. 43shows the configuration of the problem. The aft part of the helicopter is modeled and consistsof a 6 m fuselage section which connects at a 45 degree angle to a 1.2 m projected lengthtail section. This structure supports the transmission to which it is connected at points Mand T by means of 0.25 m support brackets. The transmission is broken into two shafts,each connected to flexible joints at either end. Shaft 1 is connected to a revolute joint atpoint S, and gear box 1 at point G. Shaft 2 is connected to gear box 1 and gear box 2 whichtransmits power to the tail rotor with its axis of rotation normal to the plane of the drawing.The gear ratios in gear boxes 1 and 2 are 1:1 and 2:1, respectively.

The fuselage has the following physical characteristics: axial stiffness EA = 687 MN ,bending stiffnesses EI22 = 19.2 MN.m2, EI33 = 26.9 MN.m2, torsional stiffness GJ =

44

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2x 10

−3

TIME [sec]

TIM

E S

TE

P S

IZE

[sec

]

Figure 41: Time history of the time step size.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−4

TIME [sec]

NO

RM

ALI

ZE

D T

OT

AL

ER

RO

R

Figure 42: Time history of the cumulative error measure.

45

Fuselage

Tail

R M

T

Shaft 1

Flexible joint

Shaft 2

G

H

Revolute joint

S

i1

i2

Figure 43: Configuration of the tail rotor transmission problem.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

TIME [sec]

SH

AF

T 1

MID

−S

PA

N D

ISP

LAC

EM

EN

TS

[m]

Figure 44: Time history of shaft 1 mid-span transverse deflections. u2: dashed line; u3: solidline.

46

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

SHAFT1 MID−SPAN DISPLACEMENT X2 [m]

SH

AF

T1

MID

−S

PA

N D

ISP

LAC

EM

EN

T X

3 [m]

Figure 45: Motion of shaft 1 mid-span point projected onto the ~ı2, ~ı3 plane.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−6

−4

−2

0

2

4

6x 10

−3

TIME [sec]

TA

IL T

IP D

ISP

LAC

EM

EN

TS

[m]

Figure 46: Time history of tail tip transverse displacements. u2: dashed line; u3: solid line.

47

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−300

−200

−100

0

100

200

300

400

500

600

TIME [sec]

SH

AF

T 1

MID

−S

PA

N M

OM

EN

TS

[Nm

]

Figure 47: Time history of shaft 1 mid-span bending moments in local axes. M2: dashedline; M3: solid line.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−200

−100

0

100

200

300

400

TIME [sec]

SH

AF

T 1

RO

OT

FO

RC

ES

[N]

Figure 48: Time history of shaft 1 root forces in local axes. F2: dashed line; F3: solid line.

48

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1500

−1000

−500

0

500

1000

1500

TIME [sec]

FU

SE

LAG

E R

OO

T F

OR

CE

S [N

]

Figure 49: Time history of fuselage root forces in local axes. F2: dashed line; F3: solid line.

8.77 MN.m2, and mass per unit span m = 15.65 kg/m. The properties of the tail are onethird of those of the fuselage. Shafts 1 and 2 have the following physical characteristics: axialstiffness EA = 22.9 MN , bending stiffnesses EI22 = 26.7 kN.m2 and EI33 = 27.7 kN.m2,torsional stiffness GJ = 22.1 kN.m2, and mass per unit span m = 0.848 kg/m. The cen-ter of mass of the shaft has a 1 mm offset with respect to the shaft reference line. Thesmall difference in bending stiffnesses together with the center of mass offset are meant torepresent an initial manufacturing imperfection or an unbalance in the shaft. The stiffnessproperties of the flexible couplings are as follows: axial stiffness 5.0 kN/m, transverse stiff-nesses 1.0 MN/m, torsional stiffness 0.1 MN.m/rad, and bending stiffnesses 0.1 kN.m/rad.Finally, gear boxes 1 and 2 have a concentrated mass of 5.0 kg each, and the tail rotor a15.0 kg mass with a polar moment of inertia of 3.0 kg.m2.

The lowest two natural frequencies of shaft 1 were found to be ω1 = 47.5 and ω2 =49.1 rad/sec. According to linear theory, the system is stable when the shaft angular velocityis below ω1 or above ω2, but unstable between theses two speeds. The initial condition ofthe simulation was selected as the static equilibrium configuration of the system when shafts1 and 2 are rotating at an angular velocity Ω = 45 rad/sec. A torque Q is applied to shaft1 at point S and has the following time history

Q(t) = 0 t < 1 sec

= 8 (1− cos 2πt) 1 < t < 2 sec (105)

= 0 t > 2 sec.

As a result, shaft 1 which is initially operating in a stable regime (Ω = 45 rad/sec) acceleratesand is forced to operate through the critical speed range. The simulation was run with aconstant time step ∆t = 5 × 10−04 sec for a total time of 5 sec, corresponding to about40 revolutions of shaft 1. Fig. 44 shows the time history of shaft 1 mid-span transversedeflections. Up to about 1.6 sec into the simulation, the response remains small. At that

49

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

TIME [sec]

FU

SE

LAG

E R

OO

T M

OM

EN

TS

[Nm

]

Figure 50: Time history of fuselage root bending moments in local axes. M2: dashed line;M3: solid line.

point, the shaft angular velocity increases due to the applied torque, and a linear growthin transverse displacement amplitude is observed as expected in the critical speed zone. Asthe shaft angular velocity continues to increase, the unstable zone is crossed and responseamplitudes level off. The motion of the shaft mid-span point projected onto the ~ı2, ~ı3 planeis depicted in fig. 45 which shows the whirling motion associated with the crossing of thecritical speed zone. Of course, shaft 1 motions induce vibrations in the fuselage as shown infig. 46 which depicts the time history of tail tip transverse displacements.

The large amplitude vibrations of shaft 1 during the crossing of the critical speed zonegenerate significant mid-span bending moments and root forces, both measured a bodyattached coordinate system, that are depicted in figs. 47 and 48, respectively. Of course,these vibrations induce large loads at the fuselage root section; figs. 49 and 50 show the rootforces and bending moments, respectively.

All the results described here were obtained the energy decaying scheme. Fig. 51 showsthe time history of the cumulative error measure obtained by summing the error at each timestep; less than 0.0002% error was accumulated during the entire 10,000 step simulation.

10 Conclusions

In this paper, two novel time integration schemes for flexible, nonlinear multi-body systemswere presented. First, the energy preserving scheme which combines two features: an energypreservation statement for the elastic bodies, and the vanishing of the work done by theforces of constraint. This results in exact preservation of the total energy for the nonlinearmulti-body system and unconditional stability is achieved. The scheme presents second orderaccuracy and has no numerical dissipation.

Second, the energy decaying scheme combines two features: an energy decay inequality

50

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

−6

TIME [sec]

NO

RM

ALI

ZE

D T

OT

AL

ER

RO

R

Figure 51: Time history of the cumulative error measure.

for the elastic bodies, and the vanishing work done by the forces of constraint. This results inthe decay of the total energy of the multi-body system and unconditional stability is achieved.This scheme presents third order accuracy and high frequency numerical dissipation.

Simple numerical examples have been presented to illustrate the proposed integrationschemes. The energy decaying scheme clearly provides the unconditional stability and highfrequency numerical dissipation characteristics required for the analysis of flexible, nonlinearmulti-body problems.

The formulation and discretization of specific elements were presented. The discretiza-tions of the inertial and elastic forces associated with beam and flexible joint elements weredescribed in detail, as well as the discretizations of the constraint forces associated withuniversal and revolute joints. Finally, an automated time step size selection procedure waspresented that uses an error measure based on the amount of energy dissipated in the nu-merical scheme. Both local and global error levels can be evaluated in complex simulations.

A number of simulations of realistic multi-body systems were presented that illustrate theversatility and accuracy of the proposed schemes, and demonstrate the need for unconditionalstability and high frequency numerical dissipation.

References

[1] T.J.R. Hughes. Analysis of transient algorithms with particular reference to stabilitybehavior. In T. Belytschko and T.J.R. Hughes, editors, Computational Methods forTransient Analysis, pages 67–155. North-Holland, Amsterdam, 1983.

[2] N.M. Newmark. A method of computation for structural dynamics. Journal of theEngineering Mechanics Division, 85:67–94, 1959.

51

[3] T.J.R. Hughes. The Finite Element Method. Prentice Hall, Inc., Englewood Cliffs, NewJersey, 1992.

[4] H.M. Hilber, T.J.R. Hughes, and R.L. Taylor. Improved numerical dissipation for timeintegration algorithms in structural dynamics. Earthquake Engineering and StructuralDynamics, 5:282–292, 1977.

[5] J. Chung and G.M. Hulbert. A time integration algorithm for structural dynamicswith improved numerical dissipation: The generalized-α method. Journal of AppliedMechanics, 60:371–375, 1993.

[6] J.C. Simo and K. Wong. Unconditionally stable algorithms for rigid body dynamics thatexactly preserve energy and momentum. International Journal for Numerical Methodsin Engineering, 31:19–52, 1991.

[7] J.C. Simo and N. Tarnow. The discrete energy-momentum method. Conserving algo-rithms for nonlinear dynamics. ZAMP, 43:757–792, 1992.

[8] J.C. Simo, N. Tarnow, and M. Doblare. Non-linear dynamics of three-dimensionalrods: Exact energy and momentum conserving algorithms. International Journal forNumerical Methods in Engineering, 38:1431–1473, 1995.

[9] O.A. Bauchau, G. Damilano, and N.J. Theron. Numerical integration of nonlinear elas-tic multi-body systems. International Journal for Numerical Methods in Engineering,38:2727–2751, 1995.

[10] C. Johnson. Numerical Solutions of Partial Differential Equations by the Finite ElementMethod. Cambridge University Press, Cambridge, 1987.

[11] C. Johnson, U. Navert, and V. Pitkaranta. Finite element methods for linear hyperbolicproblems. Computer Methods in Applied Mechanics and Engineering, 45:285–312, 1984.

[12] C. Johnson and V. Szepessy. Convergence of a finite element method for a nonlin-ear hyperbolic conservation law. Technical Report 1985-25, Mathematics Department,Chalmers University of Technology, Goteborg, Sweden, 1985.

[13] T.R.J. Hughes and M. Hulbert. Space-time finite element formulations for elasto-dynamics: Formulation and error estimates. Computer Methods in Applied Mechanicsand Engineering, 66:339–363, 1988.

[14] M. Hulbert. Space-Time Finite Element Methods for Second-Order Hyperbolic Equa-tions. PhD thesis, Stanford University, 1989.

[15] O.A. Bauchau and N.J. Theron. Energy decaying scheme for non-linear beam models.Computer Methods in Applied Mechanics and Engineering, 134:37–56, 1996.

[16] O.A. Bauchau and T. Joo. Computational schemes for nonlinear elasto-dynamics. In-ternational Journal for Numerical Methods in Engineering, 45:693–719, 1999.

52

[17] O.A. Bauchau and N.J. Theron. Energy decaying schemes for nonlinear elastic multi-body systems. Computers and Structures, 59:317–331, 1996.

[18] C.L. Bottasso and M. Borri. Energy preserving/decaying schemes for non-linear beamdynamics using the helicoidal approximation. Computer Methods in Applied Mechanicsand Engineering, 143:393–415, 1997.

[19] C.L. Bottasso and M. Borri. Integrating finite rotations. Computer Methods in AppliedMechanics and Engineering, 164:307–331, 1998.

[20] C.L. Bottasso, M. Borri, and L. Trainelli. Integration of elastic multibody systems byinvariant conserving/dissipating algorithms. Part I: formulation. Computer Methods inApplied Mechanics and Engineering, 190:3669–3699, 2001.

[21] C.L. Bottasso, M. Borri, and L. Trainelli. Integration of elastic multibody systems by in-variant conserving/dissipating algorithms. Part II: numerical schemes and applications.Computer Methods in Applied Mechanics and Engineering, 190:3701–3733, 2001.

[22] C. Lanczos. The Variational Principles of Mechanics. Dover Publications, Inc., NewYork, 1970.

[23] T.R. Kane and D.A. Levinson. Dynamics: Theory and Applications. McGraw-Hill, Inc.,New York, 1985.

Appendix A Rodrigues parameters

A common representation of finite rotations [23] is in terms of Rodrigues parameters r =2u tan φ/2, where φ is the magnitude of the finite rotation and u the components of the unitvector about which it takes place. The following notation is introduced

r0 = cos2 φ

2= 1 / (1 +

rT r

4). (A1)

The finite rotation tensor R, and the rotation tensor G corresponding to a finite rotation ofmagnitude φ/2 about the same axis write

R(r) = I + r0 r +r0

2rr; G(r) = I +

√r0

2r +

r0

4(1 +√

r0)rr. (A2)

The relationship between angular velocities ω and time derivatives of Rodrigues parametersis ω = H r, where

H(r) = r0(1 +1

2r). (A3)

These operators enjoy the following properties

RRT = I; Rr = r; GGT = I; Gr = r; R = GG. (A4)

rG + GT

2=

G + GT

2r = G−GT . (A5)

(I − r

2

)G =

(I +

r

2

)GT =

(G + GT

2

)−1

. (A6)

53

Appendix B Discretization of finite rotations

Let Ri Rf be the components of the rotation tensor R at times ti and tf , respectively.The incremental rotation R = RfR

Ti can be divided in two equal parts such that R =

GG. In this work, the incremental rotation is parameterized with Rodrigues parameters r.The expressions for R(r) and G(r) are given in Appendix A. A mid-point rotation is noweasily defined as Rm = GRi. The following relationships summarize the properties of thediscretization of the finite rotations

G = RmRTi G = RfR

Tm R = RfR

Ti

G∗ = RT0 RT

i RmR0 G∗ = RT0 RT

mRfR0 R∗ = RT0 RT

mRfRTi RmR0.

(B1)

The following operator are extensively used

Ra =Rf + Ri

2; Ra =

[Ra 00 Ra

]; Rb =

[Ra 00 HGT RmR0

]. (B2)

54