MULTIBODY SYSTEMS WITH FLEXIBLE BEAMShosting.umons.ac.be/html/mecara/grasmech/ucl-flex2.pdf4 CHAPTER...

MULTIBODY SYSTEMS WITH

FLEXIBLE BEAMS

Jean-Claude Samin and Paul Fisette

March 20, 2007

Contents

1 Multibody systems with flexible beams 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The finite segment approach . . . . . . . . . . . . . . . . . . . . . 41.3 The assumed mode approach . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Description of the flexible beam . . . . . . . . . . . . . . . 51.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Joint equations . . . . . . . . . . . . . . . . . . . . . . . 151.3.4 Deformation equations . . . . . . . . . . . . . . . . . . . . 211.3.5 Symbolic computation of the equations of motion . . . . . 33

1.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . 36

iii

Chapter 1

Multibody systems withflexible beams

1.1 Introduction

Flexibility effects in multibody dynamics certainly represent one of the mostarduous aspects of the modeling of mechanical systems. There are several rea-sons for this. First of all, contrary to the rigid case, flexible bodies require bothphysical and geometrical hypotheses (e.g.: elastic linear behavior, small flexi-ble motion, ...) whose validity must be questioned a posteriori as regards theaccuracy or even the meaningfulness of the simulation results.

Next, since deformable bodies theoretically have an infinite number of de-grees of freedom, their spatial discretization is generally indispensable (exceptfor some analytically solvable cases) which naturally induces another source ofapproximation. Unfortunately, the latter is not easy to control, in particular dueto the fact that the type of spatial discretization and its refinement are oftenproblem-dependent whatever the method implemented (Rayleigh-Ritz, FiniteSegment or Finite Element methods, ...) (see [11] for instance).

Finally, as a consequence of the discretization, the increase in the number ofdegrees of freedom of flexible multibody systems has a severe impact on the costof the numerical analysis. Despite the power of modern computers and the useof parallel computation, one still needs to improve the mathematical model offlexible multibody systems in terms of size, matrix conditioning and numericaltreatment (especially the time integration of stiff systems). These difficultieshave stimulated a vast variety of approaches (see [19], [17], [5], [9], [21], [18],[1], [13] for instance) which in fact stem either from classical rigid multibodydynamics or structural mechanics : indeed, the inclusion of flexibility effectsinvolves a blend of these two disciplines.

One way to deal with flexible multibody systems emerges from researches inthe field of structural mechanics. This represents a marked departure from the

1

2 CHAPTER 1. MULTIBODY SYSTEMS WITH FLEXIBLE BEAMS

classical multibody approach presented in this book. As a matter of fact, FiniteElement Methods (FEM) may be adopted in a general sense to represent anytype of components appearing in a multibody system: rigid and elastic bodies,mechanical joints, interactions between bodies or with the external world. Insuch formulations, the global motion (including rigid motion and elastic defor-mation) of a given body is generally referred directly to the inertial frame, usingabsolute coordinates. As regards flexible bodies, an appropriate finite elementdiscretization is developed and introduced in the kinematic quantities of the dy-namical equations of motion. A joint between two bodies is generally modeledvia kinematic constraints which, in a finite element context, express relations(often trivial identification) between nodal degrees of freedom.With respect to the classical multibody approach which incorporates flexibilityeffects into its dynamical formalism, one could say that the FEM approach fo-cuses mainly on the flexible bodies and uses the global multibody motion as asource of external and dynamical loading. Since all the geometrical nonlinear-ities are generally taken into account in a FEM approach, the space integralswhich appear in the equations of motion must be evaluated at each step of thenumerical treatment [16]: this is unavoidably costly in terms of computationaltime.

In a purely multibody context, one generally assumes that the global motionof a flexible body is decomposed into a rigid body motion and a superimposedsmall deformation. It has been shown [9], [21] that this restriction, if one doesnot take care, can lead to completely erroneous results for flexible bodies sub-mitted to external or inertial loads. Although this so-called geometric stiffeningproblem can be solved by keeping second order terms in the expression of thedeformation of the material, the amplitude of the flexible motions (e.g. thetip angle of a rotating beam) must remain small in those formalisms which aregenerally based on kinematic restrictions as regards flexibility. Moreover, thespace discretization is generally based on an assumed mode technique. In thiscontext, it is apparent that the selection of an appropriate set of shape functionsfor a given multibody application is quite a delicate task and could penalize thisapproach probably more than the above-mentioned kinematic assumptions. In-deed, there is no systematic method or rule to set up an a priori suitable setof shape functions since this choice is strongly problem-dependent ([11], [20]).Moreover, it is shown in [11] that even admissible1 shape functions can be pooror inadequate to satisfy the natural boundary conditions of a system, whateverthe envisaged refinement.

Assumed modes are nevertheless selected in this chapter to discretize thebeam deformations. As regards the shape functions, we propose (as in [8]) touse monomials as an alternative to more dedicated functions. Monomials offertwo important advantages. First they are problem-independent and since theircombination approximates a Taylor series, they should be able to approximateany plausible deformation. Secondly, their “trivial” differentiation and integra-tion is most suitable to symbolic computation as detailed in section 1.3.5.

1in a Rayleigh-Ritz sense

1.1. INTRODUCTION 3

From a practical point of view, this approach offers the following two advan-tages:

• The use of an invariable set of shape functions (monomials in our case)considerably alleviates the user’s task since there is no need for a priormodal analysis to determine a set of appropriate shape functions.

• Thanks to the trivial derivation and integration of monomials, the fullysymbolic2 generation of the dynamical equations is performed with a highlevel of simplification thanks to the ROBOTRAN symbolic capabilities [1],[2]. As we shall see later, the resulting benefits are clearly revealed bythe CPU time performances of the numerical simulations, especially incomparison with a fully nonlinear finite element method [20].

The chapter is organized as follows. First the finite segment method isbriefly described [7]. While simple in concept, such an approach gives satis-factory and efficient results, particularly for planar applications. Then, basedon the Timoshenko theory, a more elaborate formalism is developed, in whichbeam flexibility is modeled via a 3D approach including shear deformation andcross-section rotary inertia. The deformation equations are obtained from thevirtual power principle, using approaches proposed in [19], [5] and taking intoaccount the previously mentioned stiffening effects. The Newton-Euler recur-sive formalism in relative coordinates of section ?? is then adapted in order totake into account the flexibility of the beam. The whole model is implementedin ROBOTRAN, including the fully symbolic differentiation of the dynamicalequations to provide the tangent matrices with respect to the generalized coor-dinates q, velocities q and accelerations q. Finally, with the help of ROBOTRAN,several experiments and numerical simulations were performed on various multi-body systems with flexible beams. For each of these, the discretization of thebeam deformations was systematically achieved by using one of the followingtechniques:

• a finite segment approach in which the beam is replaced by a sequence ofrigid segments interconnected by equivalent springs [7],

• an assumed modes method using the “power series” monomials mentionedabove [8], [1],

• an assumed modes method using shape functions resulting from a prelim-inary modal analysis [15],

• a finite element model, using an appropriate FEM code [16].

2contrary to a semi-symbolic approach ([12] for instance), in which symbolic terms cannotbe traced back to input variables


1.2 The finite segment approach

A detailed description of the so-called finite segment modeling of slender bodiescan be found in [7], in particular for the case of rectilinear beams. The methodconsists in replacing each flexible beam of the physical system by a chain ofN rigid bodies — or finite segments — connected by springs as illustrated infigure 1.1. Basic principles of structural mechanics are then used to establish the

Y(L)

X(L)

L

x

y

Figure 1.1: Finite Segment Method

relations between the force and moment components and the displacements androtations between two adjacent segments, in order to identify equivalent stiffnesscoefficients. Various configurations are treated in [7] (extension, torsion, bendingof straight or tapered segments, ...) leading to systematic rules for computingthe equivalent stiffnesses of a given chain of finite segments. The use of thismethod is illustrated in a specific problem of the tutorial (Part III, section ??).

From a multibody point of view, the method offers the advantage of dealingwith flexible bodies in a purely rigid multibody context, i.e. formally, of usinga conventional multibody code. Indeed, as shown in figure 1.1, the chain offinite segments (and intermediate joints) is part of the multibody topology,and treated in the same way as the other elements. However, and this probablyrepresents the main limitation of the method, experiments show that a minimumof ...10... finite segments per beam is generally required: this strongly increasesthe length of the kinematic chains and thus the number of d.o.f., especially whendealing with 3D flexible motions. This is the reason why, from an efficiencypoint of view, this technique is certainly attractive for planar applications butbecomes less adequate in other situations; this will be quantitatively apparentin the examples of section 1.4.

1.3. THE ASSUMED MODE APPROACH 5

1.3 The assumed mode approach

1.3.1 Description of the flexible beam

The modeling of any flexible body results from various hypotheses. These arelisted and detailed below in the present case:

. Geometry : Only prismatic beams are considered, and the centroidal axisis assumed to be rectilinear in the undeformed configuration. The elasticaxis is assumed to coincide with the centroidal axis.

. Material : The beam consists of a homogeneous isotropic material whichconforms to linear elasticity.

. Deformation model [5]: 3D-model, conservation of plane cross-sections,shear deformation and cross-section rotary inertia included.

. Kinematics: The angular deformations and the curvatures of the beam areassumed to be small. As regards the angular deformations, the presenthypothesis will lead to the linearization of the 3 × 3 rotation matrix be-tween two cross-section fixed-frames (as in [9]).Since this assumption may be restrictive for some applications, a subdi-vision of the complete beam into several sub-bodies will sometimes beapplied in the present multibody approach, at the expense of course of ahigher computational cost.

. Topology: From this point of view, the beam has the same status as arigid body but it is assumed that the joint connection points (for theparent and for the children bodies) are located on the centroidal axis3.The reference body frame Xi is thus defined in such a way that its plane

Xi2, X

i3 coincides with the beam cross-section S

∆= Si (see figure 1.2)

which is attached to parent body h through joint i.

Figure 1.2 summarizes the various kinematic quantities related to the beam4.The beam is characterized by its natural length L and by a cross-sectional areaA. Let s be the current abscissa of the beam centroidal axis, i.e. the distancefrom point OE , located at the root of the beam, to the plane of a generic cross-section S of the beam in its undeformed configuration. A frame S(s) is fixedto the plane of this cross-section S, oriented so that S1 is perpendicular to theplane, while S2 and S3 are parallel to the principal axes of the cross-section.Some particular sections need to be identified:

. SE, the root section at s = 0 (3 OE) and the beam reference frame

E ∆= S(0),3This restrictive hypothesis can be circumvented in practical situations by using interme-

diate massless bodies and fictitious locked joints.4For clarity, the superscript i has been deliberately omitted for the various vectors and

variables relating to beam i.


body i (flexible beam)

body j

body h

I1

2I

3I

ˆˆ

SE

Oi

s

sO0i u r

ziS

Oj

O0j

SE Si

Sj

X( )

Xi = iS(s )jT ˆ= S(s )j jS(s )

OE

Figure 1.2: A flexible beam i in the multibody system

. Si, the section containing the parent connection point O0i at s = si (frame

Xi ∆= S(si)),

. Sj , the section containing the connection point Oj of joint j ∈ ı at s = sj

(frame Tj ∆= S(sj)).

On the basis of the previous kinematic assumption, a linearized rotationmatrix R(s) is used to express the orientation of a given cross-section frameS(s) with respect to E:

[S(s)] = R(s)[E] (1.1)

Defining ϑ∆= (ϑ1,ϑ2,ϑ3)

T , the three Tait-Bryan angles around the 1, 2 and 3directions successively, we can write the linear approximation of the generalexpression ?? of the rotation matrix R(s) as follows:

R(s) =

1 ϑ3(s) −ϑ2(s)−ϑ3(s) 1 ϑ1(s)ϑ2(s) −ϑ1(s) 1

= E − ϑ(s) (1.2)


In particular for sections Si and Sj∈ı ,

[Xi] = R(si)[E] with R(si) = E − ϑi

(1.3)

[Tj ] = R(sj)[E] with R(sj) = E − ϑj

(1.4)

In the undeformed state, the reference position with respect to OE of a pointP which belongs to the centroidal axis is specified by its abscissa s:

−−−→OE P ref = [E]T

s00

and when the beam deforms, the current position vector of this point becomes

u(s)∆=−−−→OE P = [E]Tu(s) (1.5)

The displacement field on the centroidal axis is thus given by the function

v(s) = u(s)− s

00

(1.6)

Similarly, the reference position ur(X) of a arbitrary point X of a cross-section S with respect to point OE (see figure 1.2) is given in the undeformedstate by

ur(X)∆=−−−→OE Xref =

−−−→OE P ref + r(X) (1.7)

= [E]T

s00

+ [S(s)]T r(X)

with [S(s)] = [E] and r(X) =¡

0 r2(X) r3(X)¢T

In the current (deformed) configuration, this position vector becomes

u(X)∆=−−−→OE X = [E]Tu(s) + [S(s)]T r(X)

= [E]Tu(s) + [E]TRT (s)r(X) (1.8)

and can be written under the form

u(X) = ur(X) + ud(X) (1.9)

Using 1.6 and 1.2, the displacement vector ud(X) then reads

ud(X) = [E]T¡v(s) +

¡RT (s)−E

¢r(X)

¢= [E]T

³v(s) + ϑ(s)r(X)

´(1.10)


The functional discretization of the beam is then achieved by means of ad-missible shape functions©

τα1 (s), τα2 (s), τα3 (s)ª

and©

ρα1 (s), ρα2 (s), ρα3 (s)ª

respectively approximating the displacements v(s) on the centroidal axis and thethree angles ϑ(s) characterizing the rotational deflections of the cross sections.In a synthetic matrix form, we write

v(s) = τ(s)qt (1.11)

where

τ(s)∆=

τ1(s)τ2(s)τ3(s)

=

τ11 · · · τnt11 0 · · · · · · · · · · · · 0

0 · · · 0 τ12 · · · τnt22 0 · · · 0

0 · · · · · · · · · · · · 0 τ13 · · · τnt33

(1.12)

and

qt∆=¡

q1t1 · · · qnt1t1 q1t2 · · · q

nt2t2 q1t3 · · · q

nt3t3

¢T(1.13)

denotes the translational deformation coordinates.For the angular displacements, the discretization is

ϑ(s) = ρ(s)qr (1.14)

where

ρ(s)∆=

ρ1(s)ρ2(s)ρ3(s)

=

ρ11 · · · ρnr11 0 · · · · · · · · · · · · 0

0 · · · 0 ρ12 · · · ρnr22 0 · · · 0

0 · · · · · · · · · · · · 0 ρ13 · · · ρnr33

(1.15)

and

qr∆=¡

q1r1 · · · qnr1r1 q1r2 · · · q

nr2r2 q1r3 · · · q

nr3r3

¢T(1.16)

denotes the rotational deformation coordinates.

The choice and the number of these shape functions are generally the user’sresponsibility, and depend on the application. If the user opts for monomialsfor τ i(s) (or ρi(s)), ROBOTRAN (see section 1.3.5) will use the following set ofshape functions

(s

L), (

s

L)2, ...(

s

L)n

of the normalized variable sL , to represent the corresponding displacement vi(s)

(or ϑi(s)).

The beam is characterized by material and cross-sectional properties :


• A, the surface of the cross-section,

• Young’s modulus EY , shear modulus Gsh and µ, the mass per unit lengthof the beam,

• IS2 =RS

(r3(X))2 dA , IS3 =RS

(r2(X))2 dA and JSp = IS2 + IS3 , respec-tively the principal second moments of area of section S around unit vec-tors S2 and S3 and the section polar inertia around the centroid.

1.3.2 Kinematics

As mentioned in section ??, an intrinsic convention5 of the ROBOTRAN programconsiders a reference configuration of the multibody structure in which all thebody frames Xi coincide with the inertial frame I, and in which each jointaxis ei is aligned with one of the axes of the inertial frame. In addition, weassume that all the flexible beams are undeformed in this reference configuration.

Besides the computations relating to the flexible beams, we assume thatthe joint vectors ϕj and ψj as well as the joint rotation matrix Ri,j(qj) arecomputed beforehand for each joint j of the system. In accordance with section??, one has for joint j:

ψj∆= ej and ϕj

∆= 0 when j is prismatic, (1.17)

ϕj∆= ej and ψj

∆= 0 when j is revolute. (1.18)

[Xj ] = Rj,i(qj)[Xi] when the parent body i is rigid (1.19)

and [Xj ] = Rj,i(qj)[Tj ] when the parent body i is flexible (1.20)

Before computing the complete multibody kinematics, we precompute quan-tities associated with each flexible beam i, for a given value of the deformationgeneralized coordinates qt, qt, qt and qr, qr, qr. First of all, the relative angu-lar velocity vector of a cross-section frame S(s) with respect to E can beexpressed in linear approximation as (see equation ??)

Ω(s) = [S(s)]TΩ(s) = [S]TB(s).

ϑ(s) with B(s) =

1 ϑ3(s) 0−ϑ3(s) 1 0ϑ2(s) 0 1

(1.21)

The relative time derivative of this vector is given by

o

Ω (s)∆= [S(s)]T Ω(s) with Ω(s) = B(s)ϑ(s) + B(s)ϑ(s)

On the basis of 1.5, 1.6 and 1.11, we can express the relative position vectoru(s) of an arbitrary point P of the centroidal axis in frame E as well as its

5which is not restrictive


E

SE

O0i

Oi

zi

zjjbody i (flexible beam)

ˆ

ˆ

X

T

i

ˆ

ˆ

S=

= S(s )j

S

Si

Sj

jS(s )

iS(s )

(s)

d

ij

u

Oj

O0jXj

OE

Figure 1.3: Flexible beam

corresponding relative time derivatives as

u(s) = [E]Tu(s) with u(s) =¡

s 0 0¢T

+ τ(s)qto

u (s) = [E]T u(s) with u(s) = τ(s)qt (1.22)oo

u (s) = [E]T u(s) with u(s) = τ(s)qt

The computation of the joint position vector dij∆=−−−→O0iOj in the current con-

figuration is then straightforward since (see figure 1.3):

dij = u(sj)− u(si)∆= [E]Tdij

[E]with dij

[E]=

sj − si

00

+¡τ(sj)− τ(si)

¢qt

(1.23)

Its relative time derivatives with respect to E are easily deduced from 1.22:

o

dij

= [E]T dij[E]

with dij[E]

=¡τ(sj)− τ(si)

¢qt

oo

dij

= [E]T dij[E]

with dij[E]

=¡τ(sj)− τ(si)

¢qt


At this stage, all the “ingredients” are available to compute the kinematics of thetree structure in a forward recursive manner, i.e. by moving along the kinematicchains from the inertial body 0 to the terminal bodies. Recursive computationis the technique (see [14] [17] [3] for instance) we developed in section ?? andwhich consists in computing a quantity relating to a given body as a functionof those relating to its parent. Auxiliary variables are then used to store thecorresponding recursive results so that the amount of arithmetical operations isminimized.

O0j

ωωi

ωω hO

j

ˆXi

Gi

Oi

dii

O0i

O0h

body ibody h

xxi

g

zi

I1

2I

3I

Oh pi

p

dhiz

Base : 0

Figure 1.4: Forward kinematics

Vector / tensor formulation

This forward kinematic recursive scheme (see figures 1.4 and 1.3) is first pre-sented in vector/tensor form for legibility.

• Initialization:ω0 =.ω0

= 0 and α0 = −gFor j = 1 : Nbody

i = index of the parent of body j (1.24)

• Absolute angular velocities ωj and ωE of frames Xj and E:


— if body i is rigid:

ωj = ωi +ϕj qj (1.25)

end.

— if body i is flexible:

ωE = ωi −ΩXi

ωTj

= ωE +ΩTj

ωj = ωTj

+ϕj qj (1.26)

(where ΩXi ∆= Ω(si) and ΩT

j ∆= Ω(sj) are computed from 1.21)

end.

• Absolute angular accelerations.ωj

and.ωE

of frames Xj and E:


.ωj

=.ωi+ ωj .ϕj qj +ϕj qj (1.27)

end.


.ωE

=.ωi−

o

ΩXi

− ωi .ΩXi

(1.28)

βE∆= ˜ω

E+ ωE . ωE (1.29)

.ωTj

=.ωE

+o

ΩTj

+ ωTj

.ΩTj

(1.30)

.ωj

=.ωTj

+ ωj .ϕj qj +ϕj qj (1.31)

end.

βj∆= ˜ω

j+ ωj . ωj (1.32)

• Absolute position vectors (see equations ?? and ??):

pj = pi + zi + dij , for attachment point Oj on body i,

xi = pi + zi + dii, for the center of mass of body i. (1.33)

• Absolute velocity vectors:

.pj

=.pi+

.zi+

.

dij

=.pi+

o

zi+ωi . zi +

.

dij

(1.34)


• Absolute accelerations:

..pj

=..pi+ (

oo

zi+2 ωi .

o

zi+ ωi . ωi . zi + ˜ω

i. zi) +

..

dij

Defining (as in ??)

αj∆=

..pj+

oo

zj

+2 ωj .o

zj −g (1.35)

we find the recursive formula:

αj = αi + βi . zi + 2 ωj .o

zj

+oo

zj

+..

dij

(1.36)

which becomes


αj = αi + βi . (zi + dij) + 2 ωj .o

zj

+oo

zj

(1.37)

end.


α∗i ∆= αi + βi . zi (1.38)

αj = α∗i + 2 ωj .o

zj

+oo

zj

+βE .dij[E]

+ 2 ωE .o

dij

[E]+

oo

dij

[E]

(1.39)

end.

• end.

Matrix formulation

Written in its final matrix form, this recursive algorithm becomes:

• Initialization:ω0 = ω0 = 0 and α0 = −g

For j = 1 : Nbody

i = index of the parent of body j (1.40)

• Absolute angular velocities of frames Xj and E:— if body i is rigid:

ωj = Rj,i(qj)ωi + ϕj qj (1.41)

end.



ωE = RT (si)¡ωi − Ω(si)

¢(1.42)

ωTj

= R(sj)ωE +Ω(sj)

ωj = Rj,i(qj)ωTj

+ ϕj qj

end.

• Absolute angular accelerations ωj and ωE of frames Xj and E:


ωj = Rj,i(qj)ωi + ωjϕj qj + ϕj qj

end.


ωE = RT (si)³ωi − Ω(si)− ωiΩ(si)

´(1.43)

βE = ˜ωE

+ ωEωE (1.44)

ωTj

= R(sj)ωE + Ω(sj) + ωTj

Ω(sj) (1.45)

ωj = Rj,i(qj)ωTj

+ ωjϕj qj + ϕj qj (1.46)

end.

βj = ˜ωj+ ωjωj (1.47)

• Absolute linear accelerations:


αj = Rj,i(qj)¡αi + βi(zi + dij)

¢+ 2 ωjψj qj + ψj qj (1.48)

end.


α∗i = αi + βiψiqi (1.49)

αj = Rj,i(qj)R(sj)³RT (si)α∗i + dij

[E]+ 2 ωE dij

[E]+ βEdij

[E]

´+2 ωjψj qj + ψj qj (1.50)

end.

• end.


1.3.3 Joint equations

Introduction

Particular care is taken to obtain the various equations corresponding to jointdisplacements and body deformations in a unified manner as in [4] by applyingthe virtual power principle to the tree-like structure. This principle can bewritten in the following general form [22] :

NbodyXk=1

ZX∈k

(..x(X)− f) .∆

.x(X) dm = 0 (1.51)

where:

I1

2I

3I

u

p

X2

X1

X3

PX

Figure 1.5: Virtual velocity field

...x(X) denotes the absolute acceleration of a material point X (of body k),

. f is the local force density.

In order to generate the first set of equations of motion related to the N body

joint degrees of freedom, we may choose a field of virtual velocity changes∆.x(X)

such that (see figure 1.5):

. bodies k 6= i are virtually frozen ⇔ ∆ .x(X) = 0, ∀ body k 6= i,

. body i is virtually rigidified⇔ ∆ .x(X) = ∆

.p+∆ωi× u(X), if body k = i,

(P being an arbitrary reference point of body i, see figure 1.5).


It then proves useful to separate the internal and external interactions bywriting: f = fint + fext. Since the chosen virtual velocity field is rigidifying forbody i, the virtual power of the internal interactions fint vanishes and hence,introducing this field into (1.51),µZ

i

..x(X)dm−Fext

¶.∆

.p

+

µZi

u(X) dm× ..p+

Zi

u(X)× ..u(X) dm− LPext

¶.∆ωi = 0

where

. Fext∆=Rifext dm is the resultant of the external forces applied to body i,

. LPext∆=Riu× fext dm is the external torque resultant on body i (sum of

the moments of external forces and pure torques) with respect to point P .

Since this equation must be satisfied for any choice of the virtual velocitychanges ∆

.p and ∆ωi, we finally obtain Newton’s translational equation and

the rotational equation with respect to the arbitrary point P , i.e.:

mi ..xi

= Fext (1.52)

P.

Hi+ miui× ..

p = LPext (1.53)

where:

...xiis the absolute acceleration vector of the center of mass Gi,

. ui is the relative position vector6 of the center of mass Gi with respect topoint P

miui∆=

Zi

u(X) dm (1.54)

. P.

Hi ∆=Riu(X)× ..

u(X) dm is the time derivative of the angular momentumof body i with respect to the arbitrary point P .

Backward dynamics

With the forward kinematic computation having been performed, the backwarddynamical recursion (from body Nbody to body 1) can now take place to generatethe first set of equations of motion relating to the Nbody joint degrees of freedom.In figure 1.6, the following vector quantities have been introduced:

. Fi = [Xi]T F i and Li = [Xi]T Li, respectively the force and torque actingon body i through joint i, evaluated at point Oi,

6which should not be confused with u(si).


I1

2I

3I

Gi

dii

hk

k

j

ix

g

L

F

-F

-FL

-L

-L

i

Fi

i

k

j

k

jj

ext

exti

O0i

Oi

O

O

Figure 1.6: Dynamical notations for rigid bodies

. Fiext = [Xi]T F iext and Liext = [Xi]T Liext, the external interactions on

body i (barring gravity) under the form of an equivalent resultant forceapplied to the center of mass Gi and a pure torque. These external forcesand torques will not be considered when body i is a beam since it turns outthat their introduction is more user-friendly in ROBOTRAN by adding tothe beam a fictitious joint and a massless child-body to which the desiredexternal load is applied.

During the backward recursion, the formulation of the joint equations ofmotion depends on the type of body encountered: rigid or flexible. To clarify thepresent section, the two cases are treated separately, while in the computationalalgorithm, a simple test is of course performed first for each body, allowing aswitch to the appropriate equations.

Rigid body On the basis of figure 1.6, we can write the equations of motion1.52 and 1.53 for body i, the reference point for the rotational equation in thiscase being the center of mass Gi of the body. Then, using the previous kinematiccomputations (see section 1.3.2) as well as the definition of the joint rotationmatrices, we can project these equations into the body-fixed frame Xi and


obtain the Newton-Euler equations in recursive form:

W i ∆= mi (αi + βi diiz )− F iext

F i =Xj ∈ ı

Ri,j(qj)F j + W i (1.55)

Li =Xj ∈ ı

³Ri,j(qj)Lj + dijz Ri,j(qj)F j

´+ diiz W i − Liext

+Ii.ωi+ ωi Ii ωi (1.56)

Apart from the fact that in the present case we do not recursively extract themass matrix, these equations are strictly equivalent to those obtained in section?? (equations ?? and ??).

Flexible beam On the basis of figure 1.7 and now choosing point OE as thereference point for the equation of rotation, we can project 1.52 onto frame Xi(∆= S(si) ).

Gi

XiE

h

j

dii

F

L

i

iOi

O0iS

Sj

SiE

-Fj-L

j

j

i

z

z

Oj

O0j

OE

Figure 1.7: Dynamical notations for flexible beams

Let us first describe the position vector (and its “relative” derivatives) of thecenter of mass of beam i (see figure 1.7):

dii =−−−→O0iGi = ui − u(si) = [E]Tdii

[E](1.57)

o

dii

= [E]T dii[E]

andoo

dii

= [E]T dii[E]

(1.58)


The vector translational equation can thus be written as7 :

Fi −Xj ∈ ı

Fj + mig =mi ..xi

(1.59)

and, thanks to definition 1.35:

αj∆=

..pj+

oo

zj

+2 ωj .o

zj −g

and its consequence (resulting from 1.33):

..xi − g = αi + βi . zi +

..

dii

this vector equation becomes

Fi =Xj ∈ ı

Fj + mi(αi + βi . zi+oo

dii

+2 ωEo

dii

+βE .dii)

In matrix form, and using 1.49, the joint translational equation can thus bewritten as

F i = miα∗i + R(si)

Xj ∈ ı

RT (sj)Ri,j(qj)F j + mi³dii[E]

+ 2 ωE dii[E]

+ βEdii[E]

´(1.60)

The vector rotational equation 1.53 with respect to reference point OE reads

E.

Hi+ miui× ..

xE

= LEext

= Li +−−−→OE Oi×Fi + miui×g −

Xj ∈ ı

³Lj +

−−−−→OE Oj ×Fj

´where the angular momentum EHi is given by

EHi =

Zi

u× .u dm =

Zi

u .o

u dm−Zi

u . u dm .ωE

Denoting by IE∆= − R

iu . u dm = −[E]T

Riu . u dm[E], the inertia tensor of the

flexible beam i with respect to OE, the rotational equation becomes

Li = IE ..ωE

+.

IE .ωE + ωE . IE .ωE +

Zi

u .oo

u dm + ωE .

Zi

u .o

u dm

+miui .³αi + βi . zi− oo

u (si)− 2 ωE .o

u (si)− βE .u(si)´

−(u(si)− zi) .Fi +Xj ∈ ı

¡Lj + u(sj) .Fj

¢7according to our previous assumptions, there are no external forces acting directly on the

flexible elements.


The rotational equation with respect to reference point OE is then given inmatrix form by

Li = R(si)[IE .ωE

+.IEωE + ωE IE ωE +

Zi

u u dm + ωEZi

u u dm

+miui³RT (si)α∗i − u(si)− 2 ωE u(si)− βE u(si)

´−u(si)RT (si)F i

+Xj ∈ ı

¡RT (sj)Ri,j(qj)Lj + u(sj)RT (sj)Ri,j(qj)F j

¢]+ qi ψiF i

(1.61)

Quantities relating to the beam mass distribution appearing in 1.60 and1.61 must be precomputed, at each time step, concurrently with the kinematiccomputations detailed in section 1.3.2. These quantities are:

1. The mass center relative position vector ui defined in 1.54, which can becomputed by integrating over the entire beam, using 1.9:

miui∆=

Zi

u(X) dm = [E]TZi

ur(X) dm + [E]TZi

ud(X) dm

= mi[E]Tuir + [E]TZi

ud(X) dm (1.62)

2. The body i - inertial quantities (with respect to reference point OE):

IE∆= −

Zi

u(X) u(X) dm, its time derivative.

IE

,

and the integrals:

Zi

u(X) u(X) dm and

Zi

u(X) u(X) dm (1.63)

Quantities 1.62 and 1.63 still need to be expanded using 1.9 and 1.10 with thediscretizations 1.11 and 1.14, first to make the dynamical data uir and Iir appear,respectively the position vector of the center of mass and the inertia tensor withrespect to this point of the undeformed beam, and secondly to isolate integralsover (products of) shape functions. Since the latter are constant with respectto time, they can be computed once and for all and considered as parameters(i.e. unequivocal symbols) within the equations of motion. These substitutionsand the corresponding developments are lengthy and tedious, and they will notbe detailed here: the corresponding integrals over shape functions are given inAppendix B.

Joint equations of motion The joint equations of motion are finally ob-tained from (1.55 - 1.56) and (1.60 - 1.61) in a straightforward manner, by


projection onto the joint axes, using the joint vector definitions 1.17 and 1.18.The final backward recursion can thus be summarized as follows:

For i = Nbody : 1- if body i is rigid:

- if joint i is prismatic, the equation of motion is the product:¡ψi¢T

(1.55rhs)

- if joint i is revolute, the equation of motion is the product:¡ϕi¢T

(1.56rhs)

- if body i is flexible (beam):

- if joint i is prismatic, the equation of motion is the product:¡ψi¢T

(1.60rhs)

- if joint i is revolute, the equation of motion is the product:¡ϕi¢T

(1.61rhs)

end.

One should note at this stage that these joint equations of motion are fullyimplicit with respect to the generalized accelerations qi, qit and qir. Theproblem of time integration of flexible multibody system (for simulation pur-poses) will be addressed in chapter ??.

1.3.4 Deformation equations

In the multibody system being investigated, let us assume that Nbeam(≤ Nbody)bodies represent flexible beams whose main modeling hypotheses were set outin section 1.3.1. The present section will now focus on the corresponding de-formation equations of motion. The following developments are largely inspiredby [19], [5] and applied to each beam i of the tree-structure which undergoes

arbitrary motions (..pi,ωi,

.ωi

– see section 1.3.1) and is submitted to externalforces and torques (g,Fi,Li,Fj∈ı,Lj∈ı).

Beam deformation

Let us focus (figure 1.8) on an arbitrary section S (frame [S(s)] = R(s)[E]) andmore precisely on the material just beyond this section. In particular a sectionS? located at s? = s + ds is considered:

[S(s∗)] = (E − ϑ?)[S(s)] (1.64)

In order to express the local deformation in the material, we first express theposition vector du of an arbitrary point X? of section S? with respect to theposition u(X?

r ) it would have if the material between sections S and S? wereundeformed. In that case, S? = S?r , X? = X?

r and [S(s∗)] = [S(s)] in figure 1.8,and according to equation 1.8, we can write

u(X?r )∆=−−−−→OE X?

r = u(s) + [S(s)]T

100

ds + [S(s)]T r(X?) (1.65)


E

SES

X

X

(s)

ds

u

du

OE

*

*

S( )s *

S( )s

r

S*r

S*

Figure 1.8: Description of the beam deformation

In a limiting process, the actual position of X? when the material between Sand S? is deformed is given by

u(X?)∆=−−−−→OE X? = u(s) +

∂u

∂sds + [S(s∗)]T r(X?)

= u(s) +∂u

∂sds + [S(s)]T (E + ϑ

?)r(X?)

where

∂u

∂s∆= [E]T

∂u

∂s

Using the definitions 1.5 of u(s) and of the transformation matrices, we obtain

du∆= u(X?)− u(X?

r ) = [S(s)]T

R(s)∂u

∂s− 1

00

ds + [S(s)]Thϑ?r(X?)

i

In order to obtain the displacement gradient ∂u∂s expressed in the local frame

S(s), we note that

[S(s∗)] = (R(s) + dR) [E] = (E − ϑ?)R(s)[E]

whence

ϑ?

= K(s) ds with K(s)∆= −∂R(s)

∂sRT (s) (1.66)

and finally

∂u

∂s= [S(s)]T

hΓ(s) + K(s) r(X?)

i(1.67)

where:


. Γ(s) = [S(s)]TΓ(s), with Γ(s)∆= R(s)∂u∂s −

100

, is the strain vector

of the centroidal axis (Γ1: extension; Γ2 and Γ3 : shear),

. K(s) = [S(s)]TK(s), is the beam curvature vector (K1: twist ; K2 andK3 : bending).

Deformation velocities

In the context of a virtual principle, the virtual power of the internal elasticforces involves the expression of ”local” velocities when formulating a virtualvelocity field compatible with the beam deformation. For that reason, it isuseful to establish the following time derivatives:

1. The relative derivative of the strain vector Γ:

o

Γ (s)∆= [S(s)]T Γ(s) =

∂o

u

∂s− Ω(s) .

∂u

∂s(1.68)

where:

∂o

u

∂s∆= [E]T

∂u

∂sand Ω(s) = [S]TB(s)ρ(s)qr

This result is straightforward from the previous definition of Γ and therelation Ω = R(s)RT (s).

2. The relative derivative of the curvature vector K(s):

o

K (s)∆= [S(s)]T K(s) =

∂Ω

∂s(1.69)

This results from

Ω(s∗) = Ω+∂Ω

∂sds = [S(s∗)]T R(s∗)RT (s∗) [S(s∗)] (1.70)

which, withR(s∗) = (E − ϑ?)R(s), leads to

Ω(s∗) = Ω+∂Ω

∂sds

= [S(s∗)]T (E − ϑ?)R(s)RT (s)(E + ϑ

?) [S(s∗)]

+[S(s∗)]T (E − ϑ?)R(s)RT (s)

.

ϑ?

[S(s∗)]= [S(s)]T R(s)RT (s) [S(s)]

+[S(s)]T.

ϑ?

[S(s∗)]

= Ω(s) + [S(s)]T.

ϑ?

(E − ϑ?) [S(s)]

and, in view of the differential nature of ϑ?, implies

∂Ω

∂sds = [S(s)]T

.

ϑ?

[S(s)] = [S(s)]T.

K(s)ds [S(s)] (1.71)


Internal stress resultant

S

N( )s

M( )s

Figure 1.9: Internal stress resultant

In a context of linear elasticity, the internal stress resultants8 (viz. a forceN and a pure torque M) are related linearly to the beam strain and curvatures:

N∆= [S(s)]TN (1.72)

M∆= [S(s)]TM (1.73)

with:

N = CNΓ and CN = diag( EYA , k2GshA , k3GshA )

M = CMK and CM = diag( GshJSp , EY IS2 , EY IS3 )

where k2, k3 are the shear correction factors in the S2 and S3 directions.

Local equations of motion

Let us consider (figure 1.10) a portion of the beam on which the internal stressresultants and gravity are acting. In a limiting process, we may write the vectorequations of motion of this element as

dN+ µ g ds = µ..x(s) ds (1.74)

du×N+ dM = I ds.ω + ω . I ds .ω (1.75)

where :

...x(s) =

..pi+..zi − ..

u(si) +..u(s) is the absolute acceleration of the origin of

S(s),. I = [S(s)]T I [S(s)] with I = (µ/A) diag (IS2 + IS3 , IS2 , IS3 ),

. ω = ωE +Ω(s) is the absolute angular velocity of frame S(s).8at the section gravity center.


S

- sN( )

- sM( )M( )+s dM

N( )+s dN

du

g

Figure 1.10: Local equations of motion

Virtual power principle

In section 1.3.3, the chosen virtual velocity field was such that each body k 6= iwas frozen and body i was rigidified: the resulting equation of motion reflectedthe overall motion of body i along the degree of freedom of the preceding joint.In the present case, the goal being to set up the deformation equations of thebeam i, a virtual velocity field is chosen in such a way that:

. bodies k 6= i are virtually frozen ⇔ ∆ .x(X) = 0, ∀ body k 6= i,

. body (i)-frame E is virtually frozen while virtual deformation velocities

are considered ⇔ ∆ .x(X) = ∆

o

u (s) +∆Ω(s)× r(X), if body k = i.

Introducing this velocity field into the virtual power principle 1.51 and inte-grating over the entire beam on the basis of equations of motion 1.74 and 1.75,we find (since external loads arise solely from the joints i and j ∈ ı and areapplied at Oi and Oj respectively)Z L

0

µµ (

..x(s)− g)−∂N

∂s

¶.∆

o

u (s)ds

+

Z L

0

µI .

.ω + ω . I .ω−∂M

∂s− ∂u

∂s×N

¶.∆Ω(s)ds

−Fi .∆ o

u (Oi)− Li .∆Ω(si) +Xj ∈ ı

³Fj .∆

o

u (sj) + Lj .∆Ω(sj)´

= 0

(1.76)

Since, on the basis of 1.68 and 1.69,

∆o

Γ (s) =∂∆

o

u

∂s−∆Ω(s) .

∂u

∂sand ∆

o

K (s) =∂∆Ω

∂s(1.77)


by integrating by parts the terms

∂N

∂s.∆

o

u (s) and∂M

∂s.∆Ω(s)

we obtain after rearranging termsZ L

0

µ (..x(s)− g) .∆

o

u (s)ds +

Z L

0

¡I .

.ω + ω . I .ω

¢.∆Ω(s)ds

= Fi .∆o

u (Oi) + Li .∆Ω(si)−Xj ∈ ı

³Fj .∆

o


−Z L

0

N .∆o

Γ (s)ds−Z L

0

M .∆o

K (s)ds (1.78)

Finally, the virtual power principle is expressed in terms of virtual velocitychanges in the deformation coordinates ∆qt and ∆qr by substituting into 1.78(and 1.77) the expressions:

. ∆o

u (s) = [E]T τ(s)∆qt (from 1.22b)

. ∆Ω(s) = [S(s)]TB(s)ρ(s)∆qr = [E]TRT (s)B(s)ρ(s)∆qr(from 1.21 and 1.14)

The latter relations justify retaining terms in ϑ ϑ in the expression 1.21 ofΩ(s), since it appears clearly that they lead to first order terms (in ϑ) in thedeformation equations: their presence is therefore indispensable as regards thehypotheses underlying the model.

Deformation equations

The final scalar form of the deformation equations of motion can be obtainedby:

. expressing the kinematic quantities relating to beam i (u, u, u, R,Ω, Ω, ...)as functions of the deformation coordinates qt and qr (see sections 1.3.1and 1.3.2),

. performing the various vector operations in the appropriate frames,

. isolating in 1.78 the coefficients of each virtual generalized velocity ∆qkt ,∆qkr , the latter being assumed independent.

Further to this, the various terms appearing in 1.78 are expanded in such away that the integrals to be evaluated involve only combinations of the shapefunctions τα1[2,3], ρ

β1[2,3] (or their spatial derivatives). These integrals are gathered

in Appendix B.

In the present text, the developments9 of 1.78 will be curtailed, since theyare of little interest.

9as required by a multibody program


• Term 1 :R L0∆

o

u (s) . (..x(s)− g)µds

First of all, we can write on the basis of the definition of vectors αi (see1.35) and α∗i (see 1.38)

..x(s)− g =

..pi+..zi − ..

u(si) +..u(s)− g

= αi + βi . zi − ..u(si) +

..u(s)

= α∗i − ..u(si) +

..u(s) (1.79)

and compute the second time derivative..u(s) on the basis of 1.22 (using 1.44)

..u(s) = βE .u(s) + 2 ωE .

o

u (s)+oo

u (s) (1.80)

Term 1 can thus be written as follows:

(∆qt)TZ L

0

τT (s)µds³RT (si)α∗i −

³u(si) + 2 ωE u(si) + βE u(si)

´´+(∆qt)

TZ L

0

τT (s)³u(s) + 2 ωE u(s) + βE u(s)

´µds (1.81)

where:

. α∗i was computed by 1.49,

. ωE and βE were computed by 1.42 and 1.44 respectively.

On the basis of 1.11 and 1.14, u(s), u(s), u(s), R(s),Ω(s) and Ω(s) must nowbe expressed in terms of the shape functions in such a way that the lattercontribute to the final form through constant integrals (see Appendix B).Expression 1.81 is formally of the form

(∆qt)T ( ... )

in which each element inside the parentheses contributes to the correspondingtranslational deformation equation.

• Term 2 :R L0∆Ω(s) .

¡I .

.ω + ω . I .ω

¢ds

Expressing the absolute angular velocity of the frame S(s) as

ω(s) = ωE +Ω(s) = [S(s)]T³R(s)ωE + B(s)

.

ϑ(s)´

and the absolute angular acceleration successively as

.ω(s) =

.ωE

+o

Ω (s) + ω(s)Ω(s)

=.ωE

+o

Ω (s) + ωE(s)Ω(s)

= [S(s)]T³R(s)

.ωE

+ B(s)..

ϑ(s) +.

B(s).

ϑ(s) +¡R(s)ωE

¢∼B(s)

.

ϑ(s)´


we can expand this second term as follows:

(∆qr)T [Z L

0

ρT (s)BT (s) I R(s) ds.ωE

+

Z L

0

ρT (s)BT (s) I³B(s)

..

ϑ(s) +.B(s)

.

ϑ(s) +¡R(s)ωE

¢∼B(s)

.

ϑ(s)´

ds

+

Z L

0

ρT (s)BT (s)³R(s)ωE + B(s)

.

ϑ(s)´∼

I³R(s)ωE + B(s)

.

ϑ(s)´

ds](1.82)

where ωE and ωE were computed by 1.43 and 1.42.

Each kinematic quantity (ϑ(s),.

ϑ(s), ...) must then be expressed as a functionof the shape functions ρ(s) and the generalized coordinates qr on the basis of1.14. The final resulting expression can be written in the following form:

(∆qr)T ( ... ) (1.83)

in which each element inside the parenthesis contributes to the correspondingrotational deformation equation.

• Terms 3 and 4 : Fi .∆o

u (Oi) + Li .∆Ω(si)

The joint force and torque Fi and Li were computed during the backwarddynamical recursion via equations 1.55, 1.60 and 1.56, 1.61 respectively, andexpressed in the body-frame Xi. Their contribution to the virtual powerprinciple 1.78 results from the virtual velocity changes of point Oi and frameXi induced by the virtual deformation velocities. The relative position of Oi

with respect to OE is given by

u(Oi) =−−−→OE Oi = [E]T

¡u(si)−RT (si)zi

¢whence virtual velocity change reads

∆o

u (Oi) = [E]T¡τ(si)∆qt −RT (si)

¡B(si)ρ(si)∆qr

¢∼zi¢

= [E]T¡τ(si)∆qt + RT (si) ziB(si)ρ(si)∆qr

¢The computation of the virtual power is thus straightforward and can be writtenas h¡

F i¢T

R(si)τ(si)i∆qt +

h¡Li¢T

B(si)ρ(si) +¡F i¢T

ziB(si)ρ(si)i∆qr

or equivalently:

(∆qt)T £τT (si)RT (si)F i

¤+ (∆qr)

ThρT (si)BT (si)

³Li − qi ψ

iF i´i

(1.84)


• Term 5 : − Pj ∈ ı

³Fj .∆

o


The reasoning is similar for joints j such that j ∈ ı. The correspondingvirtual power is given by

−Xj ∈ ı

h¡F j¢T

Rj,i(qj)R(sj)τ(sj)i∆qt −

Xj ∈ ı

h¡Lj¢T

Rj,i(qj)B(sj)ρ(sj)i∆qr

or

−Xj ∈ ı

³(∆qt)

T £τT (sj)RT (sj)Ri,j(qj)F j

¤+ (∆qr)

T £ρT (sj)BT (sj)Ri,j(qj)Lj

¤´(1.85)

• Term 6: − R L0N .∆

o

Γ (s)ds

This term represents the virtual power of the internal elastic forces when thebeam is subjected to axial or shear deformation.In order to compute this term properly, it is convenient to momentarily disregardhow the constitutive equations ofN = [S]TN are formulated (see equation 1.72),and consider the latter as a generic (internal) force which contributes to the

virtual power through a projection along the virtual velocity field ∆o

Γ.Starting from 1.77a :

∆o

Γ (s) =∂∆

o

u

∂s−∆Ω(s) .

∂u

∂s

∆o

Γ can be computed considering that, from 1.22,

∂∆o

u (s)

∂s= [E]T τ 0(s)∆qt with τ 0(s) ∆=

dτ(s)

ds

and

∂u

∂s= [E]T

∂u(s)

∂s= [E]T

1 + τ 01(s)qtτ 02(s)qtτ 03(s)qt

resulting in

∆o

Γ = [S(s)]T (R(s)τ 0(s))∆qt

+[S(s)]T

R(s)

1 + τ 01(s)qtτ 02(s)qtτ 03(s)qt

∼B(s) ρ(s)∆qr


Under the linear kinematic assumption of section 1.3.1, all terms having theform ϑατ

0β(s)qt are neglected and we can write term 6 as:

Z L

0

¡N1 N2 N3

¢ (τ 01 + τ 02 ϑ3 − τ 03 ϑ2)(τ 02 − τ 01 ϑ3 + τ 03 ϑ1)(τ 03 + τ 01 ϑ2 − τ 02 ϑ1)

ds∆qt

+

Z L

0

¡N1 N2 N3

¢ −ρ2 (τ 03qt + ϑ2) + ρ3 (τ 02qt − ϑ3)−ρ3 + (ρ1 τ

03qt − ρ3 τ

01qt)

+ρ2 + (ρ2 τ01qt − ρ1 τ

02qt)

ds∆qr

(1.86)

This expression clearly reveals the coupling between the translational and rota-tional deformation equations. Indeed, with the help of figure 1.11, let us focuson the axial strain force component N1 in a 2D-context (1 − 2 plane). Thisfigure highlights the presence of the term N1 ϑ3 in the lateral deformation equa-tion (...τ 02∆qt) and of the contribution N1 (τ 02qt − ϑ3) to the rotational equation(in the third direction : ... ρ3∆qr).

shear angle :

ˆ

ˆ

E

E

S

S*

'

'

1

N1

N1

N1

2

θ3θ3

θ3

(τ 02qt − )

τ 02qt

Figure 1.11: Coupling of the deformation equations

Now focus on the computation of the internal elastic force N in term 6,whose constitutive equation is given by 1.72:

N∆= [S(s)]TCNΓ

where Γ denotes the components of the local strain vector in frame S(s).Based on its definition (see 1.67), Γ can be expressed using 1.6 as:

Γ(s) = (R(s)−E)

100

+ R(s)

τ 01(s)τ 02(s)τ 03(s)

qt (1.87)


- Γ1 : the axial strain

When computing Γ1, and considering only the first order terms while ne-glecting terms of the form ϑατ

0β(s)qt, we would obtain the following result:

Γ1(s) = τ 01(s)qt (1.88)

As explained below, this approximation is a poor and quite erroneous estimationof the axial strain for the most frequent situation, i.e., beam bending [9]. Indeed,when developing the axial strain Γ1 up to the second order, we obtain

Γ1(s) = τ 01(s)qt + (τ 02(s)qt ϑ3 −(ϑ3)

2

2)− (τ 03(s)qt ϑ2 +

(ϑ2)2

2) (1.89)

To emphasize the physical meaning of this result, consider in figure 1.12 a seg-ment dl of the beam subjected to pure bending in the plane 1, 2 where theaxial strain of the centroidal axis Γ1 equals 0. We may write:

τ 01(s)qt =x

dl

(2dorder)= −(ϑ3)

2

2and τ 02(s)qt ' ϑ3

The first order computation 1.88 yields Γ1 = − (ϑ3)22 , which is absolutely erro-neous; expression 1.89 yields

Γ1 ' −(ϑ3)2

2+

(ϑ3)2

2= 0

as it should be. In other words, relation 1.89 expresses an important fact,namely, that a beam submitted to pure bending does not change its length [9],[21], [13].

ˆ

ˆ

E

E

1

2

θ3

dl

dl

x

Figure 1.12: A beam in pure bending

The previous considerations as well as the kinematic assumption of section1.3.1 lead us to write the following relation of order between the quantitiesτ 0α(s)qt and ϑβ for a beam undergoing bending:

order

µτ 01(s)qt,

(ϑ2)2

2,(ϑ3)

2

2

¶¿ order (τ 02(s)qt, τ

03(s)qt,ϑ1,ϑ2,ϑ3) (1.90)

- Γ2 : The shear deformation (y direction)


Expanding the second component of Γ (see 1.87) and taking into account therelation of order 1.90, we finally obtain the following acceptable approximationfor Γ2 (as in [9]):

Γ2 = τ 02(s)qt − ϑ3

- Γ3 : The shear deformation (z direction)

Similarly, Γ3 can be expressed as

Γ3 = τ 03(s)qt − ϑ2

Term 6 is finally expanded by:

. introducing the strain components Γ1,Γ2 and Γ3 into 1.72,

. introducing the latter into integral 1.86,

. expressing ϑ1, ϑ2, ϑ3 in terms of the shape functions ρ(s) and the gener-alized coordinates qr,

. isolating the integrals over the shape function combinations in the variousterms of 1.86. Note that integrals of products of 2, 3 and 4 shape functionsare involved. They result from the necessity of keeping higher order termsin the computation of the strain vector Γ, as detailed above.

• Term 7: − R L0M .∆

o

K (s)ds

A similar reasoning is followed for this last term by momentarily disregardingthe constitutive equation of the internal torque M = [S(s)]TM (see equation

1.73) and by computing the virtual velocity field ∆o

K (s) given by 1.77 on thebasis of 1.21:

∆o

K =∂∆Ω(s)

∂s=

∂

∂s[S(s)]T (B(s)ρ(s))∆qr

= [E]T∂

∂s

¡RT (s)B(s)ρ(s)

¢∆qr

= [S(s)]T

"R(s)

∂¡RT (s)B(s)

¢∂s

ρ(s) + B(s) ρ0(s)

#∆qr (1.91)

The computation of 1.91 is performed under the kinematic hypotheses relatingto the angular deformation and curvature of the beam (see section 1.3.1). In

concrete terms, assuming both ϑα and ϑ0β∆=

∂ϑβ∂s are small, we may express

∆o

K by dropping terms of second order and higher, and introduce it into term7, leading toZ L

0

¡ M1 M2 M3

¢ ρ01 + ϑ3ρ02 + ϑ02ρ3

ρ02 − ϑ3ρ01 − ϑ01ρ3

ρ03 + ϑ2ρ01 + ϑ01ρ2

∆qr

ds (1.92)


As regards the internal torque M itself (see equation 1.73) and in particular thecurvature vector K, its three components can be computed under the kinematicassumptions and using the previously formulated relation of order, resulting in

K = [S(s)]TK with K =

ϑ01ϑ02ϑ03

(1.93)

As with the 6th term, term 7 is finally obtained by:

. introducing 1.93 into 1.73,

. introducing the latter into the integral 1.92,

. expressing ϑ1,ϑ2,ϑ3 and ϑ01,ϑ02,ϑ

03 in terms of the shape functions and the

generalized coordinates,

. isolating in 1.92 the integrals over the shape functions (see Appendix B).

These developments providing the beam deformation equations must be car-ried out for all the beams (i = 1 : N beam) of the multibody system; the cor-responding equations are then added to the joint equations of section 1.3.3 toproduce the final system of equations of motion:

F(q, q, q) = 0 (1.94)

with:

q =

jointsz |

q1...qNbody

,

beam 1z | (qt, qr), ...

beamNbeamz | (qt, qr)

It should be noted that these equations of motion were set up without wor-

rying about the computation of the mass matrix of the system. In other words,they are fully implicit with respect to the generalized accelerations q. Never-theless, in a simulation context, the knowledge of this matrix is indispensable(or highly desirable) whatever is the integration method. To avoid a pure nu-merical derivation of the latter, which is quite costly, it is preferable to deriveit analytically, as shown in chapter ??, section ??.

1.3.5 Symbolic computation of the equations of motion

Using its own symbolic library and taking advantage of the recursive nature ofthe equations of motion, a symbolic module has been added to ROBOTRAN to


compute the multibody equations 1.94 as well as the associated tangent matricesrequired by numerical integrators (see chapter ??):

K =∂F∂qT

, G =∂F∂qT

and M =∂F∂qT

, the mass matrix

These matrices are obtained by symbolic differentiation according to chapter??, section ??.

To generate the multibody model, ROBOTRAN requires a specific symbolicdata file (see chapter ??, figure ??) related to the flexible beams of the system.Such a file is illustrated in figure 1.13.

3

2

7

9

P 3 4 0 0 0 4

G 1 2 2 1 2 2

P 3 4 0 0 0 4

---- flexible beam rubric ----number of beams:

type of shape functions:

”General” or ”Power series”

number of shape functions

beam i :ndexes

τ1 τ2 τ3 ρ1 ρ2 ρ3

Figure 1.13: the ROBOTRAN data file (part)

For the shape functions, the user can choose between two possibilities:

. the so-called ”general” case (flag G in the data file): ROBOTRAN thenimplements unequivocal strings to represent the shape functions and thecorresponding integrals. The latter must be computed by the user (seeAppendix B),

. the so-called ”power-series” case (flag P in the data file) : ROBOTRANthen evaluates the monomial shape functions and the corresponding in-tegrals so that a fully symbolic computation of the equations of motionis performed. The resulting output file can thus be considered as a blackbox, directly usable by a numerical simulation program without any inter-vention from the user, to represent the beam deformations. An example ofa symbolic output file is given in Appendix A for a planar rotating beamusing, for brevity, only one shape function for u1, u2 and ϑ3.

The scheme illustrated in figure 1.14 represents the main steps of the ROBO-TRAN symbolic algorithm, in relation with the developments of the previoussections.


beam

Output format : Fortran ?, Matlab ?, C ?

Result requested : implicit equations ?, tangent matrices: ?, ?, ?M G K

case 'G' : symbolic representationShape function integrals :

case 'P' : symbolic computation

Computation of kinematic variables

Forward kinematics ( = : ) j N

Backward dynamics ( = : ) j N

Deformation equations ( = : )i N

Reading of the symbolic data file

body

body

Joint vectors and transformation matrices

Elimination of the superfluous equations and Printing

Figure 1.14: The ROBOTRAN algorithm


1.4 Numerical examples

Three typical examples illustrate the formalism. The models were generated byROBOTRAN using monomials as shape functions to represent the beam defor-mations.

Rotating beam no 1

The first example has been investigated in various publications (e.g. [9], [7]) andconsists of a uniform flexible aluminium beam attached to a rotating cylinderof radius Rcyl as depicted in figure 1.15.

1

2

3

X

L

R

r

ϑ t( )

Figure 1.15: A rotating flexible beam

For this first example, the numerical data are the following : Rcyl = 0,

ρ = 3000 kgm3 , L = 10m, EY = 6.9e10 N

m2 , Gsh = 2.6e10 Nm2 , R = 0.0326m and

r = 0.0306m. The hub starts to rotate from rest up to an angular speed of 6 radsec

according to the following law:

ϑ(t) =25

³t2

2 + ( 7.5π )2 cos( πt7.5 )´− 2

5 (7.5π )2 : if t ≤ 15sec

6 t− 45 : if t > 15sec(1.95)

The simulation is performed over 20 seconds.The beam is modeled using series of 5, 4 and 4 monomials for τ1, τ2 and ρ3 re-spectively. Figure 1.16 represents the lateral and longitudinal tip displacementsrespectively. Note that the ROBOTRAN results are identical to those producedby FEM using 10 beam elements [16]. As regards the CPU time performance

1.4. NUMERICAL EXAMPLES 37

— using an implicit Runge-Kutta scheme [6] — a time reduction factor of 14 isobtained in favor of the monomial discretization for which the simulation re-quired 16.4 seconds on a 31.9 Mflops workstation, i.e. faster than a real-timesimulation.

Time [sec]

monomials ( )ROBOTRAN

FEM MECANO ( )

Time [sec]

y [m

]

x [m

]

Figure 1.16: Simulation of a rotating beam

Rotating beam no 2

The aim of the following experiment is to check how the model behaves when thebeam is subjected to full 3D loading ; in particular, torsion and space-flexionare combined in the proposed simulation. The system consists of a 5 m-longaluminium beam which has the same geometrical and physical characteristicsas the previous one. A 5 kg point mass is connected to the tip of the beam,through a perpendicular massless arm (Larm = 1m), as depicted in figure 1.17.The system is placed in a gravity field, which acts along the negative z direction.Once static equilibrium is reached, the hub (which was at rest at t = 0) startsto rotate according to the law (1.95).A monomial discretization was implemented with four functions for τ1, τ2 andτ3, and three for ρ1, ρ2 and ρ3 respectively. Figure 1.18 shows the evolution ofthe x, y, z coordinates of the beam tip and of the point mass, expressed in aframe rigidly attached to the rotating hub. The ROBOTRAN results comparefavorably with those obtained via FEM using 10 beam elements. A CPU timereduction factor of 38 was observed in this case.


L

Larm

R

r

ϑ t( )

1

2

3

X

Figure 1.17: A rotating flexible beam carrying an off-centered mass

monomials ( )ROBOTRAN FEM ( )MECANO

Time [sec ] Time [sec ] Time [sec ]

X-coord. of mass

X-coord. of tip

Y-coord. of mass

Y-coord. of tip

Z-coord. of mass

Z-coord. of tip

[m]

[m]

[m]

[m]

[m]

[m]

Figure 1.18: Simulation of a rotating beam carrying an off-centered mass


A

y

xsliding block

B

L

M

l

ωω t( )

Figure 1.19: A flexible slider-crank mechanism

Flexible slider-crank mechanism

The third example is a slider-crank mechanism, investigated in [10]. Depictedin figure 1.19, the system consists of a rigid crank with length l = 0.15m, adeformable connecting rod and a rigid sliding block with half the mass of theconnecting rod. The rod is L = 0.3m long and has a circular cross-section withdiameter d = 0.006m. Young’s modulus is EY = 0.2e12 N

m2 and mass density is

7.87e3 kgm3 . The crank is driven at a constant angular velocity ω = 150 radsec . The

system is friction-free and gravity is neglected. This system contains a kine-matic loop which has been taken into account by expressing the correspondingconstraints at position, velocity and acceleration levels, and by using the clas-sical Lagrange multipliers technique to compute the constraint forces. Fromthe simulation point of view, a coordinate partitioning reduction (see chapter??, section ??) was implemented to reduce the DAE system to a minimal ODEsystem before time integration. Figures 1.20 (ROBOTRAN results) and 1.21 (re-sults from [10] using a FEM approach) compare the transverse deflection Y (M)of the midpoint M and half the elongation ∆l of the rod between joints A andB. In the present example, two models were envisaged in ROBOTRAN (figure1.20): the first uses the presented approach with monomial shape functions tomodel the single-segment beam (continuous curve in figure 1.20). The seconduses 20 finite rigid segments to model the beam (dashed curve in figure 1.20).The difference in results is mainly due to the fact that the finite segment modelhas no extensional d.o.f. and that the latter is excited in this simulation, as aresult of inertia forces induced by the sliding block motion.The results clearly illustrate the previous discussion relating to the fact that anytransverse displacement of a point (on the centroidal axis) of a bending beamgives rise to a non negligible axial displacement.


[sec][sec]

Y(M) ∆

[mm

]

[mm

]

Finite segments :

Shape function :Finite segments :

Shape function :

l

Figure 1.20: Flexible slider-crank simulation: ROBOTRAN results

Figure 1.21: Flexible slider-crank simulation: results from [10]


Appendix A: ROBOTRAN symbolic output

% ROBOTRAN - Version 6.3 (build: 8 mai 2002)

function [f] = dirdynaF(q,qd,qdd,d,l,m,In,frc,trq,g,XE,L1,L2,RO,K1,K2,K3,K4,K5,K6,IY,IZ)

% Trigonometric VariablesS1 = sin(q(1));

C1 = cos(q(1));

% Auxiliary Variables (flexible beams)XCG11 = -XE(1)+l(1,1);

IRE122 = In(5,1)+m(1)*XCG11*XCG11;

IRE133 = In(9,1)+m(1)*XCG11*XCG11;

MU1 = (m(1))/(L2(1));

ID11 = RO(1)*(IY(1)+IZ(1));

ID12 = IY(1)*RO(1);

ID13 = IZ(1)*RO(1);

L2I1 = 1.000/L2(1);

si11 = -L2I1*XE(1);

L21 = 0.500*L2(1);

L31 = 0.333*L2(1);

LS31 = 0.333*L2(1)*L2(1);

% Prelimiar Kinematic and Dynamical TermsTI31 = q(4)*si11;

TIP31 = qd(4)*si11;

TIPP31 = qdd(4)*si11;

XI11 = -XE(1)+q(2)*si11;

XI21 = q(3)*si11;

XIP11 = qd(2)*si11;

XIP21 = qd(3)*si11;

XIPP11 = qdd(2)*si11;

XIPP21 = qdd(3)*si11;

MG11 = q(2)*L21*MU1+m(1)*XCG11;

MG21 = q(3)*L21*MU1;

GX1 1 = L21*MU1-m(1)*si11;

GY2 1 = L21*MU1-m(1)*si11;

LI11 = q(2)*GX1 1+m(1)*l(1,1);

LI21 = q(3)*GY2 1;

LIP11 = qd(2)*GX1 1;

LIP21 = qd(3)*GY2 1;

LIPP11 = qdd(2)*GX1 1;

LIPP21 = qdd(3)*GY2 1;

IE91 = IRE133+MU1*(2.000*q(2)*LS31+q(2)*q(2)*L31+q(3)*q(3)*L31);

IEP91 = MU1*(2.000*qd(2)*LS31+2.000*(q(2)*qd(2)*L31+q(3)*qd(3)*L31));

IXPP31 = qdd(4)*L21*IZ(1)*RO(1)+MU1*(q(2)*qdd(3)*L31-q(3)*qdd(2)*L31+qdd(3)*LS31);


% Forward KinematicsOE31 = qd(1)-TIP31;

OEP31 = qdd(1)-TIPP31;

BSE11 = -OE31*OE31;

BSE51 = -OE31*OE31;

% Backward DynamicsFO11 = LIPP11+BSE11*LI11-LI21*OEP31-2.000*LIP21*OE31...

+TI31*(LIPP21+BSE51*LI21+LI11*OEP31+2.000*LIP11*OE31);

FO21 = LIPP21+BSE51*LI21+LI11*OEP31+2.000*LIP11*OE31...

-TI31*(LIPP11+BSE11*LI11-LI21*OEP31-2.000*LIP21*OE31);

CO31 = IXPP31+IE91*OEP31+IEP91*OE31+...

MG11*(-XIPP21-BSE51*XI21-2.000*OE31*XIP11-OEP31*XI11)...

-MG21*(-XIPP11-BSE11*XI11+2.000*OE31*XIP21+OEP31*XI21)...

-XI11*(FO21+FO11*TI31)+XI21*(FO11-FO21*TI31);

% Flexible Beams DynamicsFIR11 = FO11-FO21*TI31;

FIR21 = FO21+FO11*TI31;

T11 1 = -FIR11*si11;

T12 1 = -FIR21*si11;

XAR11 = XIPP11+BSE11*XI11-2.000*OE31*XIP21-OEP31*XI21;

XAR21 = XIPP21+BSE51*XI21+2.000*OE31*XIP11+OEP31*XI11;

T31 1 = T11 1-MU1*(-BSE11*LS31+L21*XAR11);

T32 1 = T12 1-MU1*(L21*XAR21-LS31*OEP31);

T41 1 = T31 1+qdd(2)*L31*MU1;

T42 1 = T32 1+qdd(3)*L31*MU1;

T51 1 = T41 1-2.000*qd(3)*L31*MU1*OE31;

T52 1 = T42 1+2.000*qd(2)*L31*MU1*OE31;

T61 1 = T51 1+q(2)*L2I1*K1(1)+MU1*(q(2)*BSE11*L31-q(3)*L31*OEP31);

T62 1 = T52 1+q(3)*L2I1*K2(1)+MU1*(q(2)*L31*OEP31+q(3)*BSE51*L31);

T72 1 = T62 1-0.500*q(4)*K2(1);

R11 1 = -CO31*si11;

OMI31 = OEP31*ID13;

R31 1 = R11 1+L21*OMI31;

R41 1 = R31 1+qdd(4)*L31*ID13;

R61 1 = R41 1+q(4)*L31*K2(1);

R71 1 = R61 1+q(4)*L2I1*K6(1);

R81 1 = R71 1-0.500*q(3)*K2(1);

% Symbolic Outputsf(1) = CO31;

f(2) = T61 1;

f(3) = T72 1;

f(4) = R81 1;


Appendix B: Shape function integrals

The table summarizes the shape function integrals required by the symbolicformalism. They are formulated in the first column where τα=1,2,3 and ρβ=1,2,3represent a ”set” of shape functions as defined in 1.11 and 1.14 respectively.The second column gives their analytical expressions when using power series:

τα(k) = ρβ(k) =³ s

L

´kwith k = 1 : ntα and k = 1 : nrβ respectively.

The third column gives the equations in which the corresponding integrals ap-pear, when expanded.R L

0τα(k) ds

R L0ρβ(k) dsR L

0sτα(k) ds

Lk+1

Lk+1

L2

k+2

(1.57) ( 1.60) (1.61 )

(1.62) (1.81)

(1.61) (1.63) (1.82)

(1.61) (1.63)R L0τα(k) τβ(l) dsR L

0ρα(k) ρβ(l) ds

R L0ρα(k) τ 0β(l) dsR L

0τ 0α(k) τ 0β(l) dsR L

0ρ0α(k) ρ0β(l) ds

Lk+l+1

Lk+l+1

lk+l

klk+l−1

1L

klk+l−1

1L

(1.61) (1.63) (1.81)

(1.61) (1.63)

(1.82) (1.86)

(1.78) (1.86)

(1.78) (1.86)

(1.78 ) (1.92)R L0ρα(k) ρβ(l) ργ(m) dsR L

0τ 0α(k) ρβ(l) ργ(m) dsR L

0ρα(k) τ 0β(l) τ 0γ(m) dsR L

0ρα(k) ρ0β(l) ρ0γ(m) ds

Lk+l+m+1

kk+l+m

lmk+l+m−1

1L

lmk+l+m−1

1L

(1.82)

(1.78) (1.86)

(1.78 ) (1.86)

(1.78) (1.92)R L0ρα(k) ρβ(l) ργ(m) τ 0δ(n) dsR L

0ρα(k) ρβ(l) ργ(m) ρδ(n) ds

nk+l+m+n

Lk+l+m+n+1

(1.78) (1.86)

(1.78) (1.86)

Shape function integrals - Analytical values (monomials) - Equations involved

Bibliography

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