An Application of the Udwadia-Kalaba Dynamic Formulation to Flexible Multibody Dynamics

ARTICLE IN PRESS

Journal of the Franklin Institute 347 (2010) 173–194

0016-0032/$3

doi:10.1016/j

�CorrespoE-mail ad

www.elsevier.com/locate/jfranklin

An application of the Udwadia–Kalaba dynamicformulation to flexible multibody systems

Ettore Pennestrıa,�, Pier Paolo Valentinia, Domenico de Falcob

aUniversit �a di Roma Tor Vergata - Roma, ItalybII Universit �a di Napoli, Italy

Received 15 July 2009; received in revised form 22 September 2009; accepted 2 October 2009

Abstract

For the transient response analysis of a flexible slider-crank mechanism, the coordinate

partitioning scheme and the Udwadia–Kalaba formulation are herein compared. The flexibility of

the coupler has been modelled by means of a single Timoshenko beam element. The present

analytical treatment can be viewed as an extension of the model discussed by Shabana in his

textbook.

& 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Multibody dynamics; Flexible multibody dynamics; Coordinate partitioning; Timoshenko beam

1. Introduction

The modelling of mechanical systems taking into account the effects of distributedflexibility is required when the influence of deformation on the dynamics is important.

Scientific literature offers several formalisms and analyses for systems with distributedelasticity of the links (e.g. [1–11]).

The modelling of flexible linkages with sliding joints is considered in [12,13]. Linearizedinput–output representation of flexible multibody systems is proposed in [14]. This form ofrepresentation is particularly useful for control purposes, as demonstrated by the examplesdiscussed in the cited reference.

2.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.jfranklin.2009.10.014

nding author.

dress: [email protected] (E. Pennestrı).

www.elsevier.com/locate/jfranklin

dx.doi.org/10.1016/j.jfranklin.2009.10.014

mailto:[email protected]

mailto:[email protected]

ARTICLE IN PRESS

Nomenclature

[_]/[€] dots denote differentiation with respect to timefuig position of point P in the deformed state of body i

fui0g position of point P in the undeformed state of body i

fuif g displacement vector of point P due to elastic deformation of body i

flig Lagrange multipliers appearing in the dynamic equation of body i

fQielg elastic forces on body i

fQieg external forces on body i

fQivg quadratic velocity forces on body i

mi mass of body i

aEI

kG‘ element lengthk shear coefficientl Lagrange multipliers[Ai] transform matrix of body i

[Cqi ] Jacobian matrix with respect to coordinates qi

[Aiy]

@

@yi

½Ai�

[Ki] body i stiffness matrix[mi] mass matrix of body i

[Si] matrix of element shape functions[Cq] Jacobian matrixfdri

cg vector of virtual displacements of forces application pointsfCg vector of constraintsfF i

eg vector of external forces on body i

fQg vector of generalized external forces acting on the mechanismfqig generalized coordinates of body i

fQieg vector of external generalized forces acting on body i

fqg vector of generalized coordinatesT

ikinetic energy of the finite element

Ui

strain energy of the finite elementri density of body i

yi angle of rotation of body i with respect to inertial coordinate systema area of the element sectionab area of the coupler sectionE Young modulusEI flexural rigidityG shear modulusg acceleration gravityIc moment of inertia of the crank with respect to center of massJi0 moment of inertia of the i body with respect to origin oi

k stiffness matrix of the Timoshenko beam elementL crank lengthLb length of the undeformed coupler

E. Pennestrı et al. / Journal of the Franklin Institute 347 (2010) 173–194174

ARTICLE IN PRESS

M mass matrix of the entire systemm mass matrix of the Timoshenko beam elementmb mass of the couplermc mass of the crankoiGx and oiGy Cartesian components of the vector joining oi, origin of local axes,

with G, center of mass of body i.Vi volume of body i

x abscissa measured along the beam axis (see Fig. 1)

E. Pennestrı et al. / Journal of the Franklin Institute 347 (2010) 173–194 175

A widely adopted flexible multibody dynamics formulation, based on the use of floatingframe of reference, has been proposed by Shabana (e.g. [15,16]).

Such formulation defines a differential-algebraic equations (DAE) system where theunknowns are the sets of rigid (qr) and flexible (qf ) coordinates and the Lagrangemultipliers l. Shabana suggests to solve the DAE system through the coordinatepartitioning scheme [17,18]. In this paper a different strategy will be adopted. In particularthe Udawdia–Kalaba formulation [19] will be tested for the dynamic analysis of a flexiblelinkage. This last formulation, deduced from the Gauss’ Principle of Least Action, doesnot require the use of Lagrange multipliers and gives explicitly the vector of accelerationsthrough a sequence of standard linear algebra operations. No previous applications of thisformulation to flexible multibody systems are known to the authors. Thus one of thepurposes of this paper is the comparison of the numerical efficiency between theUdwadia–Kalaba formulation and the coordinate partitioning scheme usually adopted inthese studies.

Our analysis will be focused on the transient response of a flexible slider-crankmechanism. As another element of novelty, the coupler is modelled with a single elementTimoshenko beam whereas the crank is rigid [20]. The first application of the Timoshenkobeam element in flexible multibody dynamics is due to Shabana and Bakr [21]. Assuming aTimoshenko beam model, all the elements of mass, Jacobian matrices and generalizedforces vectors have been deduced in analytical form. A higher number of beam elementscould have been used. Our choice to use only one beam element allows a direct comparisonbetween the symbolic analytical expressions reported by Shabana [16] and those deducedin this paper.

2. Description of the Timoshenko beam element

It is well known that as the ratio depth of the beam to the wavelength of vibrationincreases, the Euler–Bernoulli beam model tends to overestimate natural frequency. In thiscase the Timoshenko beam model, where the effects of shear deformation and rotaryinertia of the beam are included, is more appropriate.

In literature there are different Timoshenko finite element beam models. In this paper wewill adopt the one proposed by Davis et al. [22]. This section outlines the main steps for thededuction of stiffness and mass matrices.

ARTICLE IN PRESS

Fig. 1. Node displacements.


We assume, as shown in Fig. 1, that each node has three degrees-of-freedom. The vectorof node displacements of the i th beam element is as follows:

feig ¼ fuj vj cj ujþ1 vjþ1 cjþ1gT ð1Þ

The displacement Cartesian components and rotation are, respectively, given by theexpressions

uðx; tÞ

vðx; tÞ

( )¼ ½Nv�fe

ig ð2Þ

cðx; tÞ ¼ fNcgT feig ð3Þ

where

x ¼x

‘ð4Þ

f ¼ 12EI

kGa‘ð5Þ

½Nv� ¼

1� x 0

02x3 � 3x2 � f xþ ð1þ f Þ

1þ f

0‘i

1þ fx3 � 2þ

f

2

� �x2 þ 1þ

f

2

� �x

� �x 0

0�2x3 þ 3x2 þ f x

1þ f

0‘i

1þ fx3 � 1�

f

2

� �x2 �

f

2x

� �

2666666666666666664

3777777777777777775

T

ð6Þ

ARTICLE IN PRESSE. Pennestrı et al. / Journal of the Franklin Institute 347 (2010) 173–194 177

and

½Nc� ¼

06

ð1þ f Þ‘ðx2 � xÞ

1

1þ fð3x2 � ð4þ f Þxþ ð1þ f ÞÞ

0

�6

ð1þ f Þ‘ðx2 � xÞ

1

1þ fð3x2 � ð2� f ÞxÞ

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

ð7Þ

The strain energy Uiaccumulated by the element and its kinetic energy T

iare given,

respectively, by the following expressions:

Ui¼

EI

2

Z ‘

0

@c@x

� �2

dxþkaG

2

Z ‘

0

c�@v

@x

� �2

dxþEa

2

Z ‘

0

@u

@x

� �2

dx ð8Þ

Ti¼

ria

2

Z ‘

0

@v

@t

� �2

dxþriI

2

Z ‘

0

@c@t

� �2

dxþria

2

Z ‘

0

@u

@t

� �2

dx ð9Þ

The stiffness and mass matrices are then obtained with the well known approach used infinite element theory. For completeness, these matrices are reported in Appendix A.

3. Brief review of flexible multibody

In this section the main terms of the equations of motion of a flexible multibody systemare reported. The general form of these equations is due to Shabana [15,16].

3.1. Displacements and coordinates

With reference to Fig. 2, the position vector of a generic point P on a body i constrainedto planar motion is defined as

friPg ¼ fR

ig þ ½Ai�fuig ð10Þ

where

fuig ¼ fui0g þ fu

if g ð11Þ

and

½Ai� ¼cosyi �sinyi

sinyi cosyi

" #ð12Þ

The absolute position of the elementary mass concentrated at point P can be expressedin terms of rigid and elastic coordinates.

ARTICLE IN PRESS

X

Y

XiYi

�i

Ri→r i→

P

ui→f

ui→0

P

Fig. 2. Coordinates and displacements in a flexible beam element.


The rigid coordinates of body i are defined as follows:

fqirg ¼ fR

ix Ri

y yigT ð13Þ

The elastic coordinates qif represent the displacements at the nodes of the beam elements

in which the body has been discretized and are related to vector fuif g by the equation

fuif g ¼ ½S

i�fqif g ð14Þ

where the elements of ½Si� ¼ ½Nv� contain the shape functions of the Timoshenko beamelement and fqi

f g ¼ feig. For simplicity we assume that the axis Xi is directed along the

element x direction.Hence the vector fqig of coordinates for flexible body i is partitioned in the form

fqig ¼qi

r

qif

( )ð15Þ

The velocity f_riPg of the elementary mass in P is obtained differentiating with respect to

time (Eq. (10))

f_riPg ¼ ½L

i�f _qig ð16Þ

where

½Li� ¼ ½I Bi AiSi� ð17Þ

with

fBig ¼@

@yi

½Ai�fuig ð18Þ


3.2. The mass matrix

The kinetic energy of body i is

Ti ¼1

2

ZVi

rif_riPg

T f_ripg dV i

þ

Z ‘

0

riI

2

@c@t

� �2

dx ¼1

2f _qigT ½mi�f _qig ð19Þ

where

½mi� ¼

miRR mi

Ry miRf

miyy mi

yf

symmetric miff

2664

3775 ð20Þ

The blocks of this matrix are reported below:Block ½mi

RR�2�2:

½miRR� ¼ mi

1 0

0 1

� �ð21Þ

Block ½miRy�2�1:

½miRy� ¼ ½A

i0 �ðfI i1g þ ½S

i�fqi

f gÞ ð22Þ

where

½Si� ¼ mi

Z 1

0

½Si� dx ð23Þ

and

fI i1g ¼ mi

oiGx

oiGy

( )ð24Þ

vector of static moments of mass mi concentrated in the center of gravity of body i withrespect to origin oi.

Block ½miRf �2�6:

½miRf � ¼ ½A

i�½Si� ð25Þ

Block ½miyy�1�1:

½miyy� ¼

ZVi

rifui0g

T fui0g dV i

þ 2

ZVi

rifui0g

T ½Si� dV ifqi

f g þ fqif g

T

ZVi

ri½Si�T ½Si� dVifqi

f g

ð26Þ

½myy� ¼ ð½myy�Þrr þ ð½myy�Þrf þ ð½myy�Þff ð27Þ

ARTICLE IN PRESSE. Pennestrı et al. / Journal of the Franklin Institute 347 (2010) 173–194180

I ð½myy�Þrr:

ð½myy�Þrr ¼

ZVi

rifui0g

T fui0g dV i

¼ Ji0 ð28Þ

II ð½myy�Þrf :

ð½myy�Þrf ¼ 2

ZV i

rifui0g

T ½Si� dV ifqi

f g ð29Þ

III ð½myy�Þff :

ð½myy�Þff ¼ fqif g

T ½miff �fq

if g ð30Þ

Block ½miyf �1�6:Z

Vi

ri½Bi�T ½Ai�½Si� dVi¼

ZV i

rifui0g

T ½~I �½Si� dViþ fqi

f gT ½ ~S

i� ð31Þ

where

½ ~Si� ¼ mi

Z 1

0

½Si�T ½ ~I �½Si� dx ð32Þ

and

½ ~I � ¼0 1

�1 0

� �ð33Þ

Block ½miff �6�6: This block, whose elements are time invariant, coincides with the mass

matrix of the Timoshenko beam element reported in Appendix A.

3.3. External forces

The generalized forces acting on the flexible body can be divided into the followingcategories:

�
applied forces; � elastic forces; � quadratic velocity forces.
3.3.1. Generalized applied forces

The vector of external generalized forces fQegT is computed through the expression

fQieg

T fdqg ¼ fFieg

T fdrcg ð34Þ


Since

fdriPg ¼ ½I Bi AiS

i�

dRi

dyi

dqif

8><>:

9>=>; ð35Þ

if we let

fQieg ¼

QieR

Qiey

Qief

8><>:

9>=>; ð36Þ

then

fQieRg ¼ fF

ieg ð37aÞ

fQieyg

T ¼ fF ieg

T fBig ð37bÞ

fQief g

T ¼ fF ieg

T ½Ai�½Si� ð37cÞ

3.3.2. Generalized elastic forces

This type of forces follows from the strain energy accumulated within the body.In our case these forces are denoted by

fQielg ¼ �½K

i�fqg ð38Þ

where

½Ki� ¼0 0

0 k

� �ð39Þ

is the body i stiffness matrix. The block k, reported in Appendix A, coincides with thestiffness matrix of the Timoshenko beam element.

3.3.3. Generalized quadratic velocity forces

The presence of these forces is due to the time variance of the mass matrix. In particularthe vector fQi

vg contains both centrifugal and Coriolis forces acting on body element i. Itsgeneral expression is

fQivg ¼ �½ _m

i�f _qig þ1

2

@

@fqigðf _qigT ½mi�f _qigÞ

� �ð40Þ

4. Equations of motion

The equations of motion of a flexible body i can be written in matrix notation as [16]

½mi�f €qig þ ½Cqi �T flig ¼ fQi

eg þ fQivg þ fQ

ielg ð41Þ


In the remaining part of the section, the general expressions of the terms of (41) will bereported for completeness. These terms will be then particularized for the case of a slider-crank coupler.

5. The flexible slider-crank

In our model the slider crank is composed of three bodies: the crank, the coupler and theframe (see Fig. 3). The crank will be considered rigid whereas the coupler is modelled as asingle Timoshenko beam element. This in full analogy with the analysis based on theEuler–Bernoulli beam element discussed by Shabana [16].The coupler and the frame are connected by means of a pin-in-the-slot kinematic pair.The rigid coordinates are chosen according to the method of constraint equations

presented in several references (e.g. [16,18,23]).In this section will be deduced explicitly the terms involved in the DAE system

M CTq

Cq 0

" #€q

l

� �¼

Q

g

( )ð42Þ

where

fgg ¼ �ð½Cq�f _qgÞqf _qg � 2½Cqt�f _qg � fCttg ð43Þ

5.1. Kinematic constraint equations

With reference to Fig. 3, the vector of generalized coordinates is composed of the rigidcoordinates fqrg � fq1 q2 q3 q4 q5 q6g

T and flexible coordinates fqf g � fq7 q8 q9gT

q1

q2q3

q4

q5

q6

q7

q8

q9

Fig. 3. Generalized coordinates and boundary constraints for the coupler.


The system of constraint equations is as follows:

fCg �

q1 �L

2cosq3

q2 �L

2sinq3

q1 þL

2cosq3 � q4

q2 þL

2sinq3 � q5

q5 þ ðLb þ q8Þsinq6

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

¼ f0g ð44Þ

The first four equations refer to the revolute pairs and the last to the one pin-in-the-slotjoint.

Assuming constant angular speed of the crank, from (43) the vector of quadraticvelocities follows:

fgg ¼ �

L

2_q23cosq3

L

2_q23sinq3

�L

2_q23cosq3

�L

2_q23sinq3

�ðLb þ q8Þ _q26sinq6 þ 2 _q6 _q8cosq6

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

ð45Þ

5.2. Mass matrix

The mass matrix has the following structure:

½M� ¼mð1Þ 0

0 mð2Þ

" #ð46Þ

where

½mð1Þ� ¼

mc 0 0

0 mc 0

0 0 Ic

264

375 ð47Þ

is the mass matrix of the rigid crank and (see Eq. (20))


½mð2Þ� ¼

mð2ÞRR m

ð2ÞRy m

ð2ÞRf

mð2Þyy m

ð2Þyf

symmetric mð2Þff

26664

37775 ð48Þ

the mass matrix of the flexible coupler.The explicit expressions of these matrix blocks will be given in the following. These are

obtained from the general expressions previously reported, but assuming as shapefunctions (6) and, for the coupler, the boundary constraints shown in Fig. 3:

½mð2ÞRR� ¼ mb½I �2�2 ð49Þ

½mð2ÞRy� ¼ mb

Lb

q9 � q7

12cosq6 �

Lb þ q8

2sinq6

Lb þ q8

2cosq6 þ Lb

q9 � q7

12sinq6

2664

3775 ð50Þ

½mð2ÞRf � ¼ mb

�Lbsinq6

12

cosq6

2

Lbsinq6

12Lbcosq6

12

sinq6

2�

Lbcosq6

12

2664

3775 ð51Þ

ð½mð2Þyy �Þrr ¼

mbL2b

3ð52Þ

ð½mð2Þyy �Þrf ¼ 2

mbLb

3q8 ð53Þ

ð½mð2Þyy �Þff ¼

Lbr

840ð1þ f Þ2½abL2

bðq27 þ q2

9Þð8þ 14f þ 7f 2Þ � abL2bð12þ 28f þ 14f 2Þq7q9

þ280abð1þ f Þ2q28 þ Ibðq

27 þ q2

9Þð112þ 140f þ 280f 2Þ

�Ibð56þ 280f � 280f 2Þq7q9� ð54Þ

where

s ¼6aL2

b


z1 ¼ �Lbð2þ sÞ

z2 ¼ Lbð1þ sÞ

z2 ¼ Lbðs� 1Þ

z3 ¼ 3ð1þ 2sÞ

z4 ¼ 4ð1þ 2sÞ

z5 ¼ 5ð1þ 2sÞ

From the expression of Timoshenko beam mass matrix reported in Appendix A oneobtains

½mð2Þff � ¼

mð3;3Þ symmetric

0 mð4;4Þ

mð3;6Þ 0 m6;6

264

375 ð56Þ

5.3. Elastic forces

Since the crank is rigid there is not any elastic force acting on it, therefore

fQð1Þel g ¼ f0g ð57Þ

The elastic forces acting on the coupler are computed by means of the followingexpression:

fQð2Þel g ¼ �

0 0

0 Kb

" #q4

q5

q6

q7

q8

q9

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ð58Þ

where

½Kb� ¼E

1þ f

ð4þ f ÞI

Lb

symmetric

0abð1þ f Þ

Lb

ð2� f ÞI

Lb

0ð4þ f ÞI

Lb

266666664

377777775

ð59Þ

is the stiffness matrix of the coupler.


5.4. Quadratic velocity forces

The quadratic velocity vector can be splitted as follows:

fQvg ¼ fQð1ÞTv Q

ð2ÞTvR Q

ð2ÞTvy Q

ð2ÞTvf g

T ð60Þ

From (40) one obtains

fQð1Þv g ¼ f0g ð61Þ

fQð2ÞvRg ¼ mb _q

26

q8 þ Lb

2cosq6 � a11q7sinq6 þ a12q9

q8 þ Lb

2sinq6 þ a11q7cosq6 þ a12q9

8>><>>:

9>>=>>;

�mb _q6

2 _q7a11cosq6 þ _q8sinq6 þ 2 _q9a12cosq6

2 _q7a11sinq6 � _q8cosq6 þ 2 _q9a12sinq6

( )ð62Þ

where

a11 ¼Lb

1þ12aL2

b

1

4�

2þ6aL2

b

3þ

1þ6aL2

b

2

0BB@

1CCA

a12 ¼Lb

1þ12aL2

b

1

4�

1�6aL2

b

3�

3aL2

b

0BB@

1CCA

fQð2Þvy g ¼ �2 _q6 ðb11q7 þ b12q9Þ _q7 þ ðb12q7 þ b11q9Þ _q9 þ

mb

3ðq8 þ LbÞ _q8

n oð63Þ

where

b11 ¼ mbL2b

ð7f 2 þ 14f þ 8Þ

840ð1þ f 2Þþ ILbr

ð10f 2 þ 5f þ 4Þ

30ð1þ f 2Þ

b12 ¼ �mbL2b

ð7f 2 þ 14f þ 6Þ

840ð1þ f 2Þþ ILbr

ð5f 2 � 5f þ 1Þ

30ð1þ f 2Þ

fQð2Þvf g ¼ 2mb _q

26

c11 þ c12

c21 þ c22

c31 þ c32

8><>:

9>=>; ð64Þ

where

c11 ¼mb _q

26L

2b

840ð1þ f 2Þ½ð8þ 14f þ 7f 2Þq7 � ð6þ 14f þ 7f 2Þq9�


c21 ¼mb _q

26

3ðq8 þ LbÞ

c32 ¼mb _q

26L

2b

840ð1þ f 2Þ½ð8þ 14f þ 7f 2Þq9 � ð6þ 14f þ 7f 2Þq7�

c12 ¼2mb _q6

1þ12aL2

b

� � Lb

21þ

3aL2

b

� ��

Lb

5�

Lb

31þ

6aL2

b

� �� _q8

c32 ¼2mb _q6

1þ12aL2

b

� � Lb

41�

6aL2

b

� ��

Lb

5þ

2aLb

� �_q8

c22 ¼ �c12 _q7 þ c32 _q9

_q8

5.5. External forces

The vector of external generalized forces is

fQg ¼

Qð1Þe

Qð2ÞeR þQ

ð2ÞvR

Qð2Þey þQ

ð2Þvy

Qð2Þef þQ

ð2Þvf

8>>>>><>>>>>:

9>>>>>=>>>>>;þ

Qð1Þel

Qð2Þel

8<:

9=; ð65Þ

where

fQð1Þe g ¼ f0 �mcg 0gT ð66Þ

fQð2ÞeRg ¼ f0 �mbggT ð67Þ

fQð2Þey g ¼ �

1

2mbgðLb þ q8Þcosq6 þ

mbgLb

1

8þ

3

2

aL2

b

� �

1þ12aL2

b

sinq6ðq7 � q9Þ ð68Þ


fQð2Þef g ¼

�

mbgLb

1

8þ

3

2

aL2

b

� �

1þ12aL2

b

cosq6

�1

2mbgsinq6

mbgLb

1

8þ

3

2

aL2

b

� �

1þ12aL2

b

cosq6

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

ð69Þ

6. The Udwadia–Kalaba formulation

This formulation, proposed Udwadia and Kalaba, is presented in detail in the bookauthored [19] and in a series of papers (e.g. [24–37]).Expressed the equations of dynamics according to the multibody formalism (42), the

accelerations vector at a given time can be computed by means of the following matrixexpression:

f €qg ¼ f €qf g þ ½M��1=2½D�þðfgg � ½Cq�f €qf gÞ ð70Þ

where

f €qf g ¼ ½M��1fQg ð71Þ

are the accelerations when the system is unconstrained and

½M��1 ¼ ½M��1=2½M��1=2 ð72Þ

½D� ¼ ½Cq�½M��1=2 ð73Þ

In (70) the prescribed accelerations due to rheonomic constraints cannot be expressed interms of accelerations of elastic coordinates.The Lagrange multipliers are expressed by the following formula [37]:

flg ¼ ð½Cq�½M��1½Cq�

T Þ�1ðfgg � ½Cq�f €qf gÞ ð74Þ

7. Numerical results

In this section are presented the results of the dynamic analysis of the flexible slider-crank whose coupler has been modelled with a single Timoshenko beam element.In particular a direct numerical comparison will be made with the Euler–Bernoulli’s

model discussed in the textbook of Shabana [16]. The main numerical data of this exampleare the same proposed by Meijaard [20].

�
crank length: L ¼ 0:1524m; � coupler length: Lb ¼ 0:3048m;

ARTICLE IN PRESS

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070.01

0.0080.0060.0040.002

00.0020.0040.0060.0080.01

Time (s)

Mid

poin

t def

lect

ion

EulerBernoulliTimoshenko

Fig. 4. Comparison of dynamic responses computed by means of Timoshenko and Euler–Bernoulli models.

E. Pennestrı et al. / Journal of the Franklin Institute 347 (2010) 173–194 189

�

1

tho

material density r ¼ 7834:56 kg=m3;
� Young’s modulus E ¼ 2:06843� 1011 N=m2; � Poisson’s ratio n ¼ 0:285; � shear factor k ¼ 0:8.
Both the crank and the coupler have a circular section with a diameter of d ¼ 0:00635m.No material damping is assumed.

The angular velocity of the crank is constant and equal to o ¼ 124:8 rad=s. Initiallycrank and coupler links are aligned. The initial deformation and deformation rate of thecoupler are zero. The ode15s has been used for the numerical integration and set value ofrelative error tolerance was RelTol ¼ 10�6.

In Fig. 4 there are the plots of the midpoint deflection divided by the coupler lengthversus time for the Timoshenko and Euler–Bernoulli models.1 The results of the simulationbased on the Euler–Bernoulli model are consistent with those reported by other researchers[20]. It is observed that the peak deformations values of the Euler–Bernoulli’s beam modelare somewhat higher than those obtained with the Timoshenko’s beam model. However,this behavior cannot be generalized for all the simulation.

In Appendix B are included the Matlab listings of the procedures used for computing theaccelerations according to the two formulations herein investigated. To improve theoverall numerical efficiency only array operations are used. Considered the short timeinterval of the simulation, no stabilization technique is applied.

According to the results of The MATLAB program which implements theUdwadia–Kalaba formulation run in about one third the time required by the programbased on the coordinate partitioning scheme (Table 1). In our implementation thecoordinate partitioning is made for each call to the procedure which computes theaccelerations. However, if the coordinate partitioning is executed only at the beginning andmaintained through the simulation the execution times should be comparable.

Note that there are some typos in the analytical expressions reported in [16] which have been eliminated in

se implemented in our program.

ARTICLE IN PRESS

Table 1

Comparison of CPU timea spent in computing accelerations with the Udwadia–Kalaba (UK) and coordinate-

partitioning (CP) formulation.

Formulation Euler–Bernoulli model (s) Timoshenko (s)

CP 2.320 3.770

UK 0.754 1.207

aThese times have been obtained from the Matlab profiler running the programs five times and choosing the

lowest value.


8. Conclusions

In this paper the dynamic formulation proposed by Udwadia and Kalaba has beentested for the simulation of a flexible slider-crank mechanism. For this purpose Shabana’sapproach, which assumes for this linkage an Euler–Bernoulli beam model of the coupler,has been generalized introducing the Timoshenko beam model. All the terms of thedynamic equations have been deduced in analytical form.The flexible multibody formalism is the one proposed by Shabana and based on the

floating frame of reference.The accuracy of the results obtained with the simulation based on the Udwadia–Kalaba

formulation was not affected by the different magnitude of elastic and rigid coordinates.This confirmed the reliability of this formulation also for the solution of flexible multibodydynamics problems. In fact, the numerical testing of the Udwadia–Kalaba dynamicformulation until now was limited to problems where the effects of distributed elasticity ofthe bodies were neglected.The cpu time observed for the simulation based on the Udwadia–Kalaba formulation

compares favorably with the one based on coordinate partitioning.

Appendix A. Stiffness and mass matrices of the Timoshenko beam element

The stiffness matrix of the Timoshenko beam element is as follows:

½k� ¼E

1þ f

að1þ f Þ

‘0 0 �

að1þ f Þ

‘0 0

012I

‘36I

‘20 �

12I

‘36I

‘2

06I

‘2ð4þ f ÞI

‘0 �

6I

‘2ð2� f ÞI

‘

�að1þ f Þ

‘0 0

að1þ f Þ

‘0 0

0 �12I

‘3�6I

‘20

12I

‘3�6I

‘2

06I

‘2ð2� f ÞI

‘0 �

6I

‘2ð4þ f Þ

‘

266666666666666666664

377777777777777777775

ð75Þ


The mass matrix is symmetric and the expressions of the non-zero elements are listedbelow

mð1;1Þ ¼ra‘

3

mð1;4Þ ¼ra‘

6

mð2;2Þ ¼ra‘ð70f 2 þ 147f þ 78Þ

210ð1þ f Þ2þ

36rI

30ð1þ f Þ2‘

mð2;3Þ ¼ra‘2ð35f 2 þ 77f þ 44Þ

840ð1þ f Þ2�

rIð15f � 3Þ

30ð1þ f Þ2

mð2;5Þ ¼ra‘ð35f 2 þ 63f þ 27Þ

210ð1þ f Þ2�

36rI

30ð1þ f Þ2‘

mð2;6Þ ¼ �ra‘2ð35f 2 þ 63f þ 26Þ

840ð1þ f Þ2�

rIð15f � 3Þ

30ð1þ f Þ2

mð3;3Þ ¼ra‘3ð7f 2 þ 14f þ 8Þ

840ð1þ f Þ2þ

rIð10f 2 þ 5f þ 4Þ‘

30ð1þ f Þ2

mð3;5Þ ¼ra‘2ð35f 2 þ 63f þ 26Þ

840ð1þ f Þ2þ

rIð15f � 3Þ

30ð1þ f Þ2

mð3;6Þ ¼ �ra‘3ð7f 2 þ 14f þ 6Þ

840ð1þ f Þ2þ

rIð5f 2 � 5f � 1Þ‘

30ð1þ f Þ2

mð4;4Þ ¼ra‘

3

mð5;5Þ ¼ra‘ð70f 2 þ 147f þ 78Þ

210ð1þ f Þ2þ

36rI

30ð1þ f Þ2‘

mð5;6Þ ¼ �ra‘2ð35f 2 þ 77f þ 44Þ

840ð1þ f Þ2þ

rIð15f � 3Þ

30ð1þ f Þ2

mð6;6Þ ¼ra‘3ð7f 2 þ 14f þ 8Þ

840ð1þ f Þ2þ

rIð10f 2 þ 5f þ 4Þ‘

30ð1þ f Þ2


Appendix B. Matlab code

Matlab code for the coordinate partitioning scheme:


Matlab code for the Udwadia–Kalaba formulation:

References

[1] J. Cuadrado, J. Cardenal, J. Garcia de Jalon, Flexible mechanisms through natural coordinates and

component synthesis: an approach fully compatible with the rigid case, International Journal of Numerical

Methods in Engineering 39 (1996) 3535–3551.

[2] J. Ambrosio, Dynamics of structures undergoing gross motion and nonlinear deformations: a multibody

approach, Computers and Structures 59 (1996) 1001–1012.

[3] D. Bae, J. Han, J. Choi, An implementation method for constrained flexible multibody dynamics using a

virtual body and joint, Multibody System Dynamics 4 (2000) 297–315.

[4] D. Bae, J. Han, J. Choi, S. Yang, A generalized recursive formulation for constrained flexible multibody

dynamics, International Journal of Numerical Methods in Engineering 50 (2001) 1841–1859.

[5] M. Geradin, A. Cardona, Flexible Multibody Dynamics—A Finite Element Approach, Wiley, New York,

2001.

[6] J. Jonker, R. Arts, Dynamic simulation of planar flexible link manipulators using adaptive modal

integration, Multibody System Dynamics 7 (2002) 31–50.

[7] A. Schwab, J. Meijaard, Dynamics of flexible multibody systems with non-holonomic constraints: a finite

element approach, Multibody System Dynamics 10 (2003) 107–123.

[8] P. Eberhard, M. Lehner, A two-step approach for model reduction in flexible multibody dynamics,

Multibody System Dynamics 17 (2007) 157–176.

[9] U. Lugris, M. Naya, J. P�erez, J. Cuadrado, Implementation and efficiency of two geometric stiffening

approaches, Multibody System Dynamics 20 (2008) 147–161.

[10] T. Wu, J. Lee, E. Ting, Motion analysis of structures (MAS) for flexible multibody systems: planar motion of

solids, Multibody System Dynamics 20 (2008) 197–221.

[11] I. Romero, A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and

geometrically exact formulations, Multibody System Dynamics 20 (2008) 51–68.

[12] O. Bauchau, C. Bottasso, Contact conditions for cylindrical, prismatic, and screw joints in flexible multibody

systems, Multibody System Dynamics 5 (2001) 251–278.


[13] S. Lee, T. Park, J.H. Seo, J. Yoon, K. Jun, The development of a sliding joint for very flexible multibody

dynamics using absolute nodal coordinate formulation, Multibody System Dynamics 20 (2008) 223–237.

[14] J. Jonker, R. Aarts, J. van Dijk, A linearized input–output representation of flexible multibody systems for

control synthesis, Multibody System Dynamics 21 (2009) 99–122.

[15] A. Shabana, Dynamic analysis of large scale inertia-variant flexible systems, Ph.D. Thesis, The University of

Iowa, 1982.

[16] A. Shabana, Dynamics of Multibody Systems, Cambridge University Press, Cambridge, 2005.

[17] R. Wehage, E. Haug, Generalized coordinate partitioning for dimension reduction in analysis of constrained

dynamic systems, ASME Journal of Mechanical Design 104 (1982) 247–266.

[18] E. Haug, Computer-Aided Kinematics and Dynamics of Mechanical Systems, Allyn and Bacon, 1989.

[19] F. Udwadia, R. Kalaba, Analytical Dynamics: A New Approach, Cambridge University Press, Cambridge,

1996.

[20] J. Meijaard, Validation of flexible beam elements in dynamics programs, in: M. Pereira, J. Ambrosio (Eds.),

Computer Aided Analysis of Flexible and Mechanical Systems, vol. II, NATO-Advanced Study Institute,

1993, pp. 229–245.

[21] E. Bakr, A. Shabana, Timoshenko beams and flexible multibody system dynamics, Journal of Sound and

Vibration 116 (1987) 89–107.

[22] R. Davis, R. Heshell, G. Warburton, A Timoshenko beam element, Journal of Sound and Vibration 22

(1972) 475–487.

[23] P. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1988.

[24] F. Udwadia, R. Kalaba, A new perspective on constrained motion, Proceedings of the Royal Society London

(A) 439 (1992) 407–410.

[25] F. Udwadia, R. Kalaba, An alternate proof for the equation of motion for constrained mechanical systems,

Applied Mathematics and Computation 70 (1995) 339–342.

[26] F. Udwadia, R. Kalaba, Equations of motion for mechanical systems, Journal of Aerospace Engineering 9

(1996) 64–69.

[27] F. Udwadia, Equations of motion for mechanical systems: a unified approach, Journal of Non-Linear

Mechanics 31 (1996) 951–958.

[28] F. Udwadia, R. Kalaba, The explicit Gibbs–Appell equation and generalized inverse forms, Quarterly of

Applied Mathematics LVI (1998) 277–288.

[29] A. Arabyan, W. Fu, An improved formulation for constrained mechanical systems, Multibody System

Dynamics 2 (1998) 49–69.

[30] F. Udwadia, Fundamental principles of Lagrangian dynamics: mechanical systems with non-ideal,

holonomic and non-holonomic constraints, Journal of Mathematical Analysis and Applications 251

(2000) 341–355.

[31] F. Udwadia, R. Kalaba, Explicit equations of motion for mechanical systems with non ideal constraints,

ASME Journal of Applied Mechanics 68 (2001) 462–467.

[32] F. Udwadia, R. Kalaba, Analytical dynamics with constraint forces that do work in virtual displacements,

Applied Mathematics and Computation 121 (2001) 211–217.

[33] F. Udwadia, R. Kalaba, On the foundations of analytical dynamics, International Journal Non-Linear

Mechanics 37 (2002) 1079–1090.

[34] F. Udwadia, R. Kalaba, What is the general form of the explicit equations of motion for constrained

mechanical systems, ASME Journal of Applied Mechanics 69 (2002) 335–339.

[35] F. Udwadia, On constrained motion, Applied Mathematics and Computation 164 (2005) 313–320.

[36] F. Udwadia, Equations of motion for constrained mechanical systems and their control, Journal of

Optimization Theory and Applications 127 (2005) 627–638.

[37] D. de Falco, E. Pennestrı, L. Vita, Investigation of the influence of pseudoinverse matrix calculations on

multibody dynamics simulations by means of the Udwadia–Kalaba formulation, Journal of Aerospace

Engineering 22 (2009) 365–372.

An Application of the Udwadia-Kalaba Dynamic Formulation to Flexible Multibody Dynamics

Documents

Transcript of An Application of the Udwadia-Kalaba Dynamic Formulation to Flexible Multibody Dynamics