Use of the floating frame of reference formulation in large deformation analysis: experimental and...

45

Use of the floating frame of reference formulationin large deformation analysis: experimentaland numerical validationA A Nada1, B A Hussein2, S M Megahed3∗, and A A Shabana2

1Benha Institute of Technology, Benha University, Qalubia, Egypt2University of Illinois at Chicago, Chicago, Illinois, USA3Mechanical Design and Production Engineering Department, Faculty of Engineering, Cairo University, Giza, Egypt

The manuscript was received on 17 March 2009 and was accepted after revision for publication on 31 July 2009.

DOI: 10.1243/14644193JMBD208

Abstract: The finite-element floating frame of reference (FFR) formulation is used, for the mostpart, in the small deformation analysis of flexible bodies that undergo large reference displace-ments. This formulation allows for filtering out systematically complex shapes associated withhigh frequencies that have no significant effect on the solution in the case of small deforma-tions. The resulting low-order FFR models have been widely used to obtain efficient and accuratesolutions for many engineering and physics applications. In this investigation, the use of theFFR formulation in the large deformation analysis is examined, and it is demonstrated that largedeformation FFR models can be accurate in applications, where the deformation can be describedusing simple shapes as it is the case in robot system manipulators. In these cases, the standardfinite-element FFR formulation must be used with non-linear strain–displacement relationshipsthat account for the geometric non-linearities. The results obtained using the large deformationFFR models are compared with the results obtained using the large deformation absolute nodalcoordinate formulation (ANCF), which does not allow for the use of linear modes. The ANCFmodels are developed using two different methods for formulating the elastic forces: the basiccontinuum mechanics approach (ANCF-BC) and the elastic line method (ANCF-EL). While theexplicit Adams method can be used to obtain the numerical solution of the FFR model, twoimplicit integration methods are implemented in order to be able to obtain an efficient solutionof the FFR and ANCF models. These implicit integration methods are the RADAU5 method andthe Hilber–Hughes–Taylor (HHT) method. In the case of simple large deformation shapes, thesimulation results obtained in this study show a good agreement between the FFR and the ANCFsolutions. The results also show that, in the case of thin and stiff beams, the coupled deformationmodes that result from the use of the ANCF-BC can be a source of numerical and locking prob-lems, as reported in the literature. These ANCF-BC numerical problems can be circumventedusing the implicit HHT integration method. Nonetheless, the HHT integrator does not capturehigh-frequency FFR axial modes which are necessary in order to obtain accurate solutions forhigh-speed rotating beams. In addition to the comparison with the ANCF solutions, experimentalresults of a forward dynamics model are used in this study to validate the large deformation FFRnumerical solutions. The experimental set-up used in the validation of the numerical solutionsis also described in this investigation.

Keywords: floating frame of reference, absolute nodal coordinate formulation, flexible manipu-lators, coupled deformation modes

∗Corresponding author: Mechanical Design and Production Engineering Department, Faculty of Engineering, Cairo University, 1 AlGamea

St., Giza 12613, Egypt.

email: [email protected]; [email protected]

JMBD208 Proc. IMechE Vol. 224 Part K: J. Multi-body Dynamics

by Ayman Nada on January 8, 2015pik.sagepub.comDownloaded from

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46 A A Nada, B A Hussein, S M Megahed, and A A Shabana

1 INTRODUCTION

For the most part, the finite-element floating frameof reference (FFR) formulation has been used in thesmall deformation analysis of flexible multibody sys-tem applications. In this formulation, which allowsfor systematically filtering out complex shapes asso-ciated with high frequencies that have no significanteffect on the solution, two sets of coordinates are used.The first set has the reference coordinates that definethe location and orientation of the body coordinatesystem, whereas the second set has the elastic coor-dinates that describe the deformation with respect tothe body coordinate system. Using these coordinatesets, a local linear problem can be defined, allow-ing for the use of modal reduction techniques [1, 2].The resulting reduced-order model includes only low-frequency modes that assume simple shapes and canbe significant in the analysis of small deformations[3]. In the case of small deformations, linear strain–displacement relationships can be used, leading toa constant stiffness matrix. The FFR mass matrix,on the other hand, is highly non-linear because ofthe geometric non-linearities resulting from the finiterotation of the body reference. Efficient numericalsolution of small deformation multibody system prob-lems based on the FFR formulation can be obtainedusing well-developed numerical integration methodsand sparse matrix techniques [4].

The FFR formulation has been used in a smallnumber of investigations in the large deformationanalysis of flexible multibody systems. Most of theseinvestigations employ a sub-structuring method inwhich each finite element is treated as a separateflexible body and is assigned its FFR. These finiteelements are connected by non-linear rigid joints inorder to obtain the equations of the flexible body.The use of this approach has several computationaldrawbacks including large dimensionality and thedifficulty in employing modal reduction techniquesthat allow for filtering out high-frequency modes. Itis demonstrated in this investigation that the finite-element FFR formulation can effectively be used inthe large deformation analysis without the need forassigning each element its FFR, thereby allowing forassembling the finite elements of a flexible bodyusing the standard finite-element assembly proce-dure. It is important, however, to point out that theFFR formulation as presented in this investigationcan only be used in the analysis of the large defor-mation when such a deformation can be describedusing simple shapes. Furthermore, non-linear strain–displacement relationships must be used in order tocapture the effect of the geometric non-linearities dueto the large deformation. Therefore, both the massand stiffness matrices in the large deformation FFRformulation used in this investigation are non-linearfunctions of the body coordinates. This non-linear

FFR formulation, however, allows for systematicallyeliminating high-frequency modes because simpledeformation shapes are assumed, and consequently,modal reduction techniques can be applied.

The large deformation FFR solution can be val-idated numerically using other large deformationformulations such as the absolute nodal coordinateformulation (ANCF) [2, 5–8]. The FFR formulation andthe ANCF employ conceptually different methods forthe motion description. While both formulations candescribe correctly arbitrary rigid body displacements[2], ANCF does not use a local frame of reference to for-mulate the inertia forces. Unlike the FFR formulation,ANCF leads to a constant mass matrix and zero Coriolisand centrifugal forces. On the other hand, ANCF leadsto a more complex expression for the elastic forces anddoes not allow for the use of linear modes because thedeformation and the reference motion are not sepa-rated. The large deformation FFR solution obtainedusing modal reduction techniques will be comparedwith the ANCF solution obtained using the nodal coor-dinates instead of modal coordinates used with theFFR formulation.

In rotating and spinning structure applications, itis necessary to include the effect of the axial modesin order to be able to obtain an accurate solution.Axial modes can be much stiffer than transversedeformation modes, and as a consequence, the useof explicit integration methods in such cases canbe inefficient as compared to the use of implicitintegration methods. As shown in this study, someimplicit integration methods can fail to capture theeffect of high-frequency axial modes, and therefore,the solution obtained using these implicit integrationmethods must be carefully examined. Explicit inte-gration methods, on the other hand, can be usedto obtain an efficient numerical solution of the FFRmodel when high-frequency axial modes are elimi-nated [9]. If axial modes are considered, the effect ofthese modes appears in many applications as high-frequency oscillations superimposed on a nominalsolution dominated by the transverse vibration modes.These high-frequency oscillations introduce geom-etric stiffness that can have a significant effect onthe solution in the neighbourhood of critical speeds[10, 11]. Implicit integration methods that require theuse of the iterative Newton–Raphson algorithm mayfilter out the effect of the axial modes by eliminatingthe associated high-frequency oscillations. In order toexamine the use of implicit integration methods insolving this type of problems, the two implicit integra-tion methods RADAU5 and the Hilber–Hughes–Taylor(HHT) methods are implemented. As reported in theliterature, implicit integration methods can be usedto obtain an efficient solution for the ANCF models,which do not use linear modes [12, 13]. Such moredetailed models include the effects of coupled defor-mation modes [6, 12, 14], and therefore these ANCF

Proc. IMechE Vol. 224 Part K: J. Multi-body Dynamics JMBD208



Use of the FFR formulation in large deformation analysis 47

models do not suffer from the same problems as theFFR models that are based on linear modes. Nonethe-less, in the case of simple large deformation shapes, agood agreement between the FFR and the ANCF solu-tions can be obtained as demonstrated in this study.In order to numerically validate the large deformationFFR solutions, the ANCF models can be developedusing two different methods for formulating the elas-tic forces: the basic continuum mechanics approach(ANCF-BC) and the elastic line method (ANCF-EL).As reported in the literature, in the case of thin andstiff beams, the coupled deformation modes that resultfrom the use of the ANCF-BC can be a source of numer-ical and locking problems. These ANCF-BC numericalproblems can be circumvented using the implicit HHTintegration method. Nonetheless, as previously men-tioned, the HHT integrator may fail to capture high-frequency FFR axial modes that are necessary in orderto obtain accurate solutions for high-speed rotatingbeams.

In addition to the comparison with the ANCF solu-tions, experimental results of a forward dynamicsmodel are used in this study to validate the largedeformation FFR numerical solutions. To this end, asimple robot arm is used in the experimental studypresented in this article [1]. The rate of change ofthe manipulator rotation makes the effect of cen-trifugal stiffening terms more significant. While formost flexible robotic manipulators, the link axial andtorsional vibration modes can be neglected, in solv-ing rotating beam problems using the FFR formula-tion, an incorrect and unstable solution is obtainedif the axial modes are neglected due to lack of thegeometric stiffening effect, as previously mentioned[10, 11]. In this study, the large transverse deforma-tion of the robot arm is measured, and the measuredexperimental results are compared with the numeri-cal results obtained using the FFR and ANCF models.The experimental set-up used in the validation ofthe numerical solutions, which is described in thisinvestigation, uses a thin beam to represent the flex-ible robot arm. The FFR model that will be used inthis study to obtain the numerical results that willbe compared with the experimental results is theconventional three-dimensional (3D) beam elementthat employs infinitesimal rotations as nodal coor-dinates. While different types of ANCF elements canbe used in the finite-element formulation of largedeformation problems of flexible manipulators, the3D fully parameterized beam element [15, 16] is usedto obtain the large deformation ANCF results thatwill be compared with the FFR and experimentalresults.

This investigation makes three new contributionsthat can be summarized as follows:

(a) numerical validation of the 3D large deformationfinite-element FFR formulation by comparison

with the results obtained using the large deforma-tion ANCF models;

(b) implementation of the implicit numerical inte-gration methods in the large deformation FFRalgorithm in order to examine whether or not suchintegration methods are capable of capturing theeffect of the relatively stiff axial modes;

(c) experimental validation of the 3D large deforma-tion finite-element FFR formulation by comparingthe numerical and experimental results.

2 LARGE DEFORMATION FFR FORMULATION

In the FFR formulation, the deformation shape isdefined in the selected body coordinate system. Sucha description has a clear advantage in the case of smalldeformation problems since a local linear problemcan be defined, thereby allowing for systematicallyapplying modal synthesis techniques that can leadto a significantly lower-dimensional model. While, ingeneral, there is no clear advantage in using the FFRformulation in the large deformation analysis, thereare important applications such as flexible robot arms,helicopter rotors, flexible space structures, and manyothers, in which the large deformation can be rep-resented using simple shapes that can be definedin the body coordinate systems. In these cases, theuse of the finite-element FFR formulation can leadto a much more efficient solution as compared toother large deformation finite-element formulations.There are, however, several fundamental issues relatedto the numerical and experimental validation of theresults and the accuracy of the numerical solutionwhen relatively stiff modes are considered with explicitand implicit numerical integration methods. Thesefundamental issues will be considered in later sec-tions of this article. In this section, the kinematicdescription used in the FFR formulation is brieflyreviewed.

In the FFR formulation, the global position vector ofan arbitrary point P on a flexible body can be written as

r = R + Au (1)

where the vector R defines the location of the ori-gin of the body coordinate system xyz, A = A(θ) is thetransformation matrix that defines the orientation ofthe local coordinate system xyz with respect to theinertial frame XYZ as function of the selected orien-tation parameters θ, and u is the vector that definesthe location of the arbitrary point with respect to thebody coordinate system. In this investigation, the fourEuler parameters are used in order to avoid the sin-gularity problem associated with the three-parameterrepresentation [2].

The position vector u of point P, which is defined inthe local coordinate system xyz, can be expressed in





Fig. 1 Coordinate system: (a) rigid body case, (b) FFRformulation, and (c) ANCF

terms of the elastic coordinates as follows

u = u0 + u f = u0 + Sq f (2)

where u0 = [ux0 uy0 uz0

]Tis the position vector of

point P in the undeformed state (see Figs 1(a) and (b)),S is the shape function matrix, and q f is the vector ofthe generalized elastic coordinates, defined in the localcoordinate system xyz. A kinematic equation similar tothe preceding equation can be defined for each finiteelement of a flexible body. These finite elements byintroducing the concept of the finite-element inter-mediate coordinate system can be assembled usinga standard finite-element assembly procedure. The3D beam element shape function matrix used in thisinvestigation with the FFR formulation is presented inAppendix 1 of this article.

Using the kinematic description of the finite-element FFR formulation, the vector of coordinatesof a flexible body in the multibody system can beexpressed in the following partitioned form

qirf = [

qTr qT

f

]T =[

RT θT qTf

]T(3)

The FFR formulation leads to exact modelling ofthe rigid body dynamics when the structures rotate asrigid bodies. In the case of a rigid body displacement,the elastic coordinates q f are equal to zero, leadingto zero strains in the case of a rigid body motion.

If the shape of the large deformation is assumed tobe simple, the vector of elastic nodal coordinates q f

can be expressed in terms of the modal coordinates,leading to a lower-dimensional model that includesonly significant deformation modes.

3 FFR LARGE DEFORMATION GEOMETRICNON-LINEARITIES

In the case of the large deformation analysis, non-linear strain–displacement relationships must be usedin order to capture the effect of geometric non-linearities. The vector of the elastic forces Qk canbe obtained by differentiating the strain energy Uwith respect to the elastic coordinate vector q f asQk = −(∂U /∂q f )

T , where the strain energy U can bewritten as

U = 12

∫V

εT σ dV = 12

∫V

εT Eε dV (4)

In this equation, V is the volume, E is the matrix ofthe material elastic coefficients, ε is the strain vector,and σ is the stress vector. The displacement of an arbi-trary point on the beam is related to the deformationalong its neutral axis by the following equation (seeFig. 2)

u f =⎡⎣u

vw

⎤⎦ =

⎡⎢⎣u0 − y

∂v0

∂x− z

∂w0

∂xv0 − zϕ0

w0 + yϕ0

⎤⎥⎦ (5)

where u f = [u v w

]Tis the vector of elastic dis-

placements, subscript 0 refers to the centroidal axis,ϕ0 is the twist angle, (x, y, z) are the coordinates of thematerial point in the beam undeformed configurationwith respect to the body coordinate system. The pre-ceding equation defines, respectively, the in- and out-of-plane transverse displacements u and w assumingsmall rotations due to bending and twist angle ϕ0, i.e.the displacement field is linearized with respect torotations due to deformation. The shear deformationresulting from the shear force at the cross-section ofthe beam is considered in this study. For the two-node

Fig. 2 Beam element displacement field





beam finite element used in this investigation, the 12 ×1 vector of element nodal coordinates q f is given by

q f = [uI vI wI θxI θyI θzI uJ vJ wJ θxJ θyJ θzJ

]T

(6)

The strain energy of the beam element can be for-mulated to include the contributions from the axialstrain εxx and shearing strains εxy and εxz , while theeffects of other strain components are neglected byusing the appropriate form of the matrix of elasticcoefficients E as discussed by Nada [1]. Therefore thestrain energy of equation (4) may be written as

U = 12

∫V

[Eε2

xx + 4G(ε2xy + ε2

xz)]

dV = UA + US (7)

where E is the modulus of elasticity, and G is the mod-ulus of rigidity. The first energy term UA representsthe contribution due to the axial strain, while the sec-ond energy term US is due to the shearing strains.By neglecting higher-order terms and those involvingfourth moments of area in the axial strain energy, andretaining only the linear contribution in US, the sim-plified expression of U in the case of a flexible rotatingarm can be written as [1]

U = UA,L + US,L + UG + UB (8)

where

UA,L = EA2

∫l

(∂u0

∂x

)2

dx + EIzz

2

∫l

(∂2v0

∂x2

)2

dx

+ EIyy

2

∫l

(∂2w0

∂x2

)2

dx

US,L = GIxx

2

∫l

(∂ϕ0

∂x

)2

dx

UG = EA2

∫l

[(∂u0

∂x

) (∂v0

∂x

)2

+(

∂u0

∂x

) (∂w0

∂x

)2]

dx

UB = EA8

∫l

[(∂v0

∂x

)4

+(

∂w0

∂x

)4

+ 2(

∂v0

∂x

)2

×(

∂w0

∂x

)2]

dx

(9)

where UA,L, US,L, UG , and UB are the axial, shear, geo-metric, and bending terms.

3.1 Constant stiffness matrix K ff ,L

The linear part of the strain–displacement relationsleads to the following part of the strain energy

UL = (UA,L + US,L)

= EA2

∫l

(∂u0

∂x

)2

dx + EIzz

2

∫l

(∂2v0

∂x2

)2

dx

+ EIyy

2

∫l

(∂2w0

∂x2

)2

dx + GIxx

2

∫l

(∂ϕ0

∂x

)2

dx

(10)

The elastic forces in this special case can be obtainedby using the equation Qk = −(∂UL/∂q f )

T = −K ff ,Lq f .In this equation, K ff ,L is the constant stiffness matrix ofthe 3D straight element of uniform cross-section. Thematrix K ff ,L, which is the same as the conventionalstiffness matrix used in linear structural dynamics,does not contain coupling between the axial andbending deformation of the finite element. Therefore,this matrix does not capture the effect of the geo-metric stiffness and its use alone does not lead to acorrect solution at critical speeds. The stiffness matrixK ff ,L is used in this investigation with the constantmass matrix associated with the elastic coordinates toobtain the system eigen vectors that are used to definethe shapes of the large deformation with respect to thebody coordinate system.

3.2 Geometric stiffness matrix K ff ,G

The geometric stiffness matrix has terms that couplethe axial and transverse displacements, and therefore,accounts for the geometric stiffening. This geometricstiffness matrix can be defined by using the part UG ofthe strain energy as (∂UG/∂q f )

T = K ff ,G(q f ) q f , wherethe geometric stiffness matrix K ff ,G for the 3D beamelement presented in Appendix is defined as

K ff ,G = EA(u1 − u2)

�2

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 00 6/5 0 0 0 �/100 0 6/5 0 −�/10 00 0 0 γ 0 00 0 −1/10� 0 2�2/15 00 �/10 0 0 0 2�2/150 0 0 0 0 00 −6/5 0 0 0 −�/100 0 −6/5 0 �/10 00 0 0 −γ 0 00 0 −�/10 0 0 −1/30�2

0 �/10 0 0 0 −�2/30

0 0 0 0 0 00 −6/5 0 0 0 �/100 0 −6/5 0 −�/10 00 0 0 −γ 0 00 0 1/10 0 0 00 −�/10 0 0 −�2/30 −�2/300 0 0 0 0 00 0 0 0 0 −�/100 0 6/5 0 1/10� 00 6/5 0 γ 0 00 0 �/10 0 2�2/15 00 −�/10 0 0 0 2�2/15

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)





Using this matrix, the non-linear elastic forces of theflexible body can be updated to include the effect ofthe geometric stiffness matrix as

Qk =[

K ff ,L(q f ) + K ff ,G(q f )]

q f (12)

The generalized modal forces associated with themodal coordinates of the flexible body can be obtainedfrom the vector Qk by using the modal transformation.The non-linear elastic force vector of equation (12)accounts for the coupling between the axial and trans-verse deformations. It is known that implicit integra-tion methods lead to efficient solution and smoothresults for stiff systems because such integrationmethods can filter out high frequencies. It is, therefore,important to examine the effect of using these implicitintegration methods with the large deformation FFRformulation that allows for inclusion or exclusion ofvibration modes.

3.3 Centrifugal and Coriolis forces

The finite rotation of the body reference in the FFRformulation also introduces geometric non-linearitiesthat result in a highly non-linear mass matrix. As aconsequence, the centrifugal and Coriolis forces arenot equal to zero and they must be included in theFFR equations of motion. Using the definition of thekinetic energy and the mass matrix, the quadraticvelocity centrifugal and Coriolis force vector Qv canbe obtained in a closed form as follows

Qv = [QT

vR QTvθ QT

vf

]T =(

∂T∂q

)T

− Mq (13)

A more explicit form of the components QvR, Qvθ ,and Qvf is given as

QvR = −A( ˜ω2L + 2 ˜ω�(2)

q f

)Qvθ = −2 ˙GT Iθθ ω − 2 ˙GT Iθ f q f − G

T ˙Iθθ ω

Qvf = −∫

VρST

( ˜ω2u + 2 ˜ω ˙u f

)dV

(14)

In this equation, ρ and V are the mass density and vol-ume, A is the transformation matrix that defines theorientation of the FFR, ω is the FFR angular veloc-ity defined in the body coordinate system, ˜ω is theskew symmetric matrix associated with the vector ω,G is the matrix that relates the time derivatives of theorientation parameters to the angular velocity vec-tor ω, Iθθ is the inertia tensor of the flexible bodydefined in the body coordinate system, and ˜L is theskew symmetric matrix associated with the vector L,which can be written in terms of two inertia shape inte-grals as L = �(1) + �(2)

q f , where �(1) = ∫V ρu0 dV and

�(2) = ∫V ρS dV .

4 THREE-DIMENSIONAL NUMERICALVALIDATION OF FFR FORMULATION

One of the main contributions of this investigation isthe experimental and 3D numerical validation of thelarge deformation FFR formulation discussed in thepreceding section. For the numerical validation, theconceptually different large deformation ANCF thatleads to significantly different structure of the equa-tions of motion is used. Unlike the large deformationFFR discussed in the preceding section, ANCF finiteelements lead to a constant mass matrix and zerocentrifugal and Coriolis forces. ANCF finite elementscapture coupled deformation modes and account forthe geometric stiffness in different ways. Figure 1(c)shows the coordinates used to define the ANCF abso-lute position and gradient coordinates of the fullyparameterized beam element used in this numericalvalidation. The global position vector r of the materialpoint P on the beam element can be defined by usingthe element shape function as

r = Se (15)

where S is the element shape function matrix, and e isthe vector of element nodal coordinates, which doesnot contain infinitesimal or finite rotation coordinates.In the preceding equation, the beam is considered asa continuous volume, making it possible to relax theassumption of the rigidity of the cross-sections. There-fore, in large deformation problems the cross-sectionmay deform and change its orientation with respectto the centre-line. For a fully parameterized 3D beamelement (see Fig. 1(c)), 12 nodal coordinates are usedfor each node. The coordinates of node I is denoted aseI and can be written in the following form

eI =[

rTI

∂rI

∂x

T ∂rI

∂y

T ∂rI

∂z

T ]T

(16)

where the vector rI defines the global position ofnode I and the three vectors ∂rI /∂x, ∂rI /∂y, and∂rI /∂z define the position vector gradients (slopes)at node I . The nodal coordinates of the element canthen be given by the vector e = [

eTI eT

J

]T = qe . Usingthe ANCF motion description, one can show that theelement mass matrix is always constant, and as a con-sequence, the vector of centrifugal and Coriolis forcesis identically equal to zero. On the other hand, ANCFallows the use of different methods for formulatingthe elastic forces. In this investigation, the continuummechanics and the elastic line approaches are used.In both cases, one can formulate the strain energyexpression Ue and obtain the elastic forces using theequation Qk = −(∂Ue/∂e)T = −K(e)e, where K is thestiffness matrix that is highly non-linear function ofthe absolute coordinates.





4.1 Continuum mechanics approach (ANCF-BC)

The general continuum mechanics approach can beused to define the non-linear elastic forces of fullyparameterized ANCF beam or plate elements. Thenon-linear Green–Lagrange strain tensor is calculatedfrom the matrix of position vector gradients D, whichcan be written as follows [17]

D = ∂r∂X

= ∂r∂x

∂x∂X

= JJ−10 (17)

where X is the vector of coordinates of an arbitrarymaterial point P0 in the undeformed reference config-uration as shown in Fig. 1(c), x is the vector of the localspatial element coordinates in the reference configu-ration, J = ∂r/∂x, and J0 = ∂X/∂x. Note that the matrixJ0 is constant. The Lagrangian strain tensor ε canbe defined as ε = (DT D − I)/2. Using this definitionof the strain tensor and the constitutive equations,the strain energy can be defined and used to derive theexpressions for the generalized forces. The use of theANCF-BC approach for formulating the elastic forcesleads to a model that includes coupled deformationmodes and accounts for all geometric non-linearities.

4.2 Elastic line approach (ANCF-EL)

An alternative method that can be used for formulat-ing the ANCF elastic forces is the elastic line approach[6]. In this approach, the extension, shear, torsion,and bending are defined at the element elastic line.The curvature expression is used to define the bend-ing strains. The vectors of position gradients along thebeam centre-line are defined as follows

r0x = r,x(x, 0, 0), ry = r,y(x, 0, 0), rz = r,z(x, 0, 0)

(18)

where the vector r0x is tangent to the beam centre-line, and the vectors ry and rz define the shear planeat an arbitrary point x on the beam centre-line. Thestrain components along the beam centre-line canbe defined using the Lagrange strain tensor and thecurvature for the element as

εxx = 12

(rT

0xr0x − 1)

, εyy = 12

(rT

y ry − 1)

εzz = 12

(rT

z rz − 1)

γxy = rT0xry , γxz = rT

0xrz , γyz = rTy rz

κx = 12

(rT

z r0xy − rTy r0xz

), κy = rT

z r0xx

κz = −rTy r0xx

(19)

where r0xx = ∂2r(x, 0, 0)/∂x2, r0xy = ∂2r(x, 0, 0)/∂x∂y,and r0xz = ∂2r(x, 0, 0)/∂x∂z. Using the preceding

equation, the total strain energy of the beam can bewritten as the sum of extension, shear, and bendingstrain energies defined as U = UA + US + UT + UB.The forms of the strain energies are presented bySchwab and Meijaard [6]. As reported in the literature,because of the kinematic definitions presented in thissection, the ANCF-EL approach leads to more efficientsolution for the equation of motion because such anapproach leads to models that do not include coupleddeformation modes that can be the source of lockingproblems when fully parameterized ANCF elementsare used.

5 MULTIBODY EQUATIONS OF MOTION ANDSOLUTION PROCEDURE

As previously mentioned, the numerical validationof the large deformation 3D FFR formulation isaccomplished in this investigation by developing twocomputer models. The first is the FFR model, whilethe second is the ANCF model. The two modelsare based on two conceptually different formulationsand employ different types of coordinates as dis-cussed in the preceding section. The ANCF and FFRformulation also lead to different structures of theequations of motion. Nonetheless, the sparse matrixstructure obtained using both formulations is effec-tively exploited in this study [4]. In this section, theconstrained multibody system equations used in thisinvestigation to study the large deformation robotdynamics are presented.

5.1 Equations of motion

The augmented form of the dynamic equations usedin this investigation allows modelling rigid bodiesas well as flexible bodies modelled using the ANCFor the FFR formulation. Joint constraints betweendifferent bodies are introduced using a set of non-linear algebraic equations, which are adjoined to thesystem differential equations using the technique ofLagrange multipliers. When explicit numerical inte-gration methods are used, the constraint equations aresatisfied at the position, velocity, and acceleration lev-els. The augmented form of the equations of motionthat will be used in this paper can be written as [2]

⎡⎢⎢⎢⎢⎣

Mrr Mrf 0 CTqr

M fr M ff 0 CTq f

0 0 Maa CTqa

CqrCq f

Cqa0

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

qr

q f

qa

λ

⎤⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎣

Qr

Q f

Qa

Qc

⎤⎥⎥⎥⎥⎦ (20)

where subscripts r, f , and a refer, respectively, to refer-ence, elastic, and absolute nodal coordinates, Mrr , Mrf ,M fr , M ff are the inertia sub-matrices that appear in theFFR formulation, Maa is the ANCF constant symmetric





mass matrix, it is the identity matrix when Choleskycoordinates, qa, are used, Cq is the constraint Jaco-bian matrix, λ is the vector of Lagrange multipliers,Qr , Q f , and Qa are the generalized forces associatedwith the reference, elastic, and absolute nodal coordi-nates, respectively, and Qc is a quadratic velocity vectorthat results from the differentiation of the kinematicconstraint equations twice with respect to time. Thegeneralized coordinates, qr and q f , are the FFR for-mulation coordinates used to describe the motion ofrigid bodies as well as flexible bodies that experiencelarge deformations in this investigation. The vectorqa is the same as the vector e if the ANCF Choleskycoordinates are not used [18]. The generalized forcesused in equation (20) also include the elastic forcesthat result from the large deformation of the flexiblebodies.

5.2 Computation algorithm

The second time derivative of the vectors of ref-erence, elastic, and absolute nodal coordinates q =[qT

r qTf qT

a ]T as well as the vector of Lagrange multipli-ers λ can be obtained by solving equation (20) using anLU algorithm. The vector of Lagrange multipliers canbe used with the constraint Jacobian matrix to deter-mine the generalized constraint forces of the joints inthe system. Using the solution for the accelerations,one can define the state vector as Y = [qT qT ]T , andthe equations of motion obtained can be transformedinto the following standard state-space form

Y = [qT qT ]T = F(Y, t) (21)

Given the initial conditions of the coordinates andvelocities and a given integration time step t at timet0, the preceding equation can be integrated forwardin time to determine at time t = t0 + t the coordi-nates and velocities which can be used to reconstructequation (20) using the coordinates and velocities attime t . The solution of equation (20) defines againthe accelerations, which can be used to advance theintegration.

The numerical solution procedure used in thisinvestigation with explicit numerical integrationmethods is designed to ensure that the constraintequations are satisfied at the position, velocity, andacceleration levels. It is clear that equation (20)ensures that the constraint equations are satisfied atthe acceleration level. In order to guarantee that theconstraint equations C(q, t) = 0 are satisfied at theposition and velocity levels, the independent accel-erations are identified and integrated forward in timeto determine the independent coordinates and veloc-ities. Using the independent coordinates, an iterativeNewton–Raphson algorithm can be used to solve theconstraint equations C(q, t) = 0 for the dependent

coordinates. Having determined all the coordinates,the constraint equations at the velocity level Cqq +Ct = 0, where Ct is the partial derivative of the con-straint equations with respect to time, can be solvedusing the generalized coordinate partitioning methodfor the dependent velocities. Determining the depen-dent velocities requires the solution of linear algebraicequations only.

6 AXIAL MODES AND IMPLICIT NUMERICALINTEGRATION METHODS

In the case of stiff systems, explicit integration meth-ods capture high-frequency contents in the solutionby reducing the time step size. For this reason, theseexplicit methods can be very inefficient when stiffaxial modes are included in the model to account forthe geometric stiffness effects. For stiff systems thathave widely separated eigenvalues, the use of implicitintegration methods is recommended. Some implicitintegration methods, which use iterative proceduressuch as Newton–Raphson algorithms and may includeartificial damping, do not trace high-frequency com-ponents of the solution, and therefore, much largertime step size can be used, leading to a much moreefficient solution procedure. Nonetheless, when theFFR formulation is used in the case of rotating beams,the effect of the stiff axial modes can be very sig-nificant particularly in the neighbourhood of criticalspeeds. In order to investigate the effect of filtering outhigh-frequency components of the solution, which is acharacteristic of some implicit methods, the use of twoimplicit numerical integration methods is examinedin the case of the large deformation analysis. Thesetwo implicit methods, which are briefly discussedin this section, are the Runge–Kutta and the HHTmethods.

6.1 Implicit Runge–Kutta method: RADAU5

Runge–Kutta methods are representatives of the classof single-step numerical integration methods. Suchmethods make no use of the solution history. Afterdetermining the value Yn as the numerical approxi-mation of Y(t), the methods proceed to evaluate Yn+1

as an approximation of Y(t + t) according to thefollowing formula

Yn+1 = Yn + ts∑

i=1

biki

ki = Y

⎛⎝ti + cit , Yi + t

s∑j=1

aijkj

⎞⎠, i = 1, 2, . . . , s

(22)





Butcher tableau provides a representation of the s-stage Runge–Kutta methods. This table is given below

c1 a11 a12 · · · a1s

c2 a21 a22 · · · a2s

......

.... . .

...cs as1 as1 · · · ass

b1 b2 · · · bs

⇐⇒ {c} {a}{b}T

For non-linear analysis, a fully implicit third stageand fifth-order accurate scheme, RADAU5 form, willbe used as one step with constant step size t . Inthis case, the corresponding Butcher tableau is givenby [19]

(4 − √6)

10(88 − 7

√6)

360(296 − 169

√6)

1800(−2 + 3

√6)

255(4 + √

6)

10(296 + 169

√6)

1800(88 + 7

√6)

360(−2 − 3

√6)

255

116 − √

636

(16 + √6)

3619

(16 − √6)

36(16 + √

6)

3619

6.2 HHT integrator

The HHT integrator is based on the HHT method(known also as the α-method) [20, 21]. This method iswidely used in solving numerically second-order ini-tial value problems in structural dynamics. The foun-dation of the HHT method is the Newmark methodthat was proposed by Newmark [20]. The Newmarkmethod provides a solution for the structural dynam-ics equations of motion, which has the followingstandard form

Mq + Dq + Kq = F(t) (23)

where q, q, and q are the vectors of generalizedposition coordinates, velocities, and accelerations,respectively, M, D, and K are the symmetric mass,damping, and stiffness matrices, respectively, and Fis a time-dependent external force vector. Newmarkmethod employs the following two equations thatrelate the position and the velocity vectors to theacceleration vector

qn+1 = qn + hqn + h2

2

[(1 − 2β)qn + 2βqn+1

]qn+1 = qn + h

[(1 − γ )qn + γ qn+1

](24)

where h = tn+1 − tn is the step-size, and β and γ are twoassumed parameters. By assuming that equation (23)is satisfied at any time, one can write the equations of

motion at time tn+1 as follows

Mqn+1 + Dqn+1 + Kqn+1 = F(tn+1) (25)

Substituting equation (24) into equation (25) andassuming that qn, qn, and qn are known, one obtains aset of equations that are functions of the accelerationvector qn+1. Therefore, these equations can be usedto determine the vector of accelerations qn+1, whichcan be used in turn to determine qn+1 and qn+1 usingequation (24).

The HHT method is an improvement over the New-mark method in that it has the ability to elimi-nate undesirable high-frequency oscillations, while itremains stable and second-order convergent. TheHHT method can be obtained by replacing equation(25) by

Mqn+1 + (1 + α)Dqn+1 − αDqn + (1 + α)Kqn+1

− αKqn = F(τn+1) (26)

where α is an assumed parameter, and τn+1 = tn +(1 + α)h. In order to obtain a stable solution usingthe implicit HHT method, the parameters α, γ , andβ should be selected to satisfy the following relations

−0.3 � α � 0, γ = 12

− α, β = (1 − α)2

4(27)

The smaller the value of α, the more numericaldamping is added to the system. The choice α = 0leads to the trapezoidal method, which has no numeri-cal damping. Hussein et al. [13] demonstrated that theuse of the HHT method can be very efficient as com-pared to the use of explicit methods when the ANCFfully parameterized finite elements that account forcoupled deformation modes are considered.

7 EXPERIMENTAL VALIDATION

In this section, the test rig used to verify the modalfrequencies and validate the large deformation FFRsimulation results is described. A flexible single robotarm is used in this test rig.

7.1 Verification of modal frequencies

As shown in Fig. 3, the experimental test rig con-sists of a single-link lightweight structural steel robotarm. This test rig is used to obtain experimentallythe eigen frequencies that are compared with numer-ical results obtained using the finite-element method.The PZT (lead zirconate titanate (Pb[ZrxTi1−x])) sen-sor, with a model number ACX QP40W, is placed onthe surface of the link. The output of the sensor issent to the PC through a NI6024E D/A data acquisi-tion card. Because of the difficulties associated withincluding the piezo sensor in the numerical model,it is important to find the minimum number of units





Fig. 3 Test rig

required for accurate description of the model. Thedata of the robot link and the piezo sensor are pre-sented in Table 1. The eigen frequency finite-elementresults reported in this section are obtained using theconstant stiffness matrix K ff , L previously defined inthis article. Table 2 shows a comparison between theexperimental and finite-element results. The finite-element solution is obtained using different numberof elements in order to check the convergence of thesolution. It is clear from the results presented in Table 2that using one element for modelling the flexible beamdoes not lead to accurate convergent solution. Forthe large deformation FFR model used in this inves-tigation, the shapes of the first two modes associatedwith frequencies 2.51 and 14.83 Hz are the most sig-nificant. Based on the results presented in Table 2,one can conclude that two finite elements are suffi-cient for obtaining accurate eigen frequencies of thebeam. Figure 4 shows the mode shapes and their asso-ciated frequencies. The results presented in this figure,which show the numerical solution, indicate that theninth mode is an axial mode. The inclusion of theshape of this mode in the large deformation FFR modelis necessary in order to capture important geometricstiffness effects and obtain an accurate solution.

Table 1 Data used for the modal analysis experiment

Beam Piezo Dimension

Length 0.6 0.101 MWidth 0.04 0.0381 MThickness 0.001 0.000 76 MYoung’s modulus 200 63 GPaDensity 7800 7600 kg/m3

Table 2 Modal frequencies (Hz)

Mode 1-Element 2-Elements 6-Elements Experimental

1 2.283 2.2733 2.2722 2.512 22.493 14.36 14.243 14.833 91.239 48.569 39.944 41.254 890.11 90.853 78.6345 1454 140.96 90.816 2326.5 570.93 131.17 1352.8 196.158 1910.5 302.449 2164.4 415.4

10 4725.8 566.3311 5330.8 567.6412 7561.2 767.79

7.2 Verification of the large deformation FFRresults

In order to verify the large deformation FFR results,the experimental set-up shown in Fig. 5 is used. Thisexperiment consists of a single-link lightweight steelmanipulator driven by a DC brushless servo motorwith a built-in incremental encoder to indicate itsangular velocity. The hub is directly attached to themotor shaft, and therefore, the link is directly drivenby the motor. A digital camera is fixed on the hub(along the frame of the hub-body), which can cap-ture the trace of an installed light-coloured tip mass;

Fig. 4 Mode shapes of the two-element beam





Fig. 5 Experimental set-up

therefore, the tip point position can be recorded withrespect to the frame of the camera. In Fig. 5, thelaptop computer is the host PC, while the desktopcomputer is the target PC. In the real experiment, thereal-time workshop within the XPC (registered trad-marks of the Mathworks, Inc.) target is used. The XPC isa host-target PC solution for prototyping, testing, anddeploying real-time systems [22]. The XPC is used toavoid any delay in data acquisition. A significant delayis found if the traditional real-time workshop withinthe MATLAB is used. The test rig data are as follows:the hub mass is m1 = 0.515 kg and mass momentsof inertia are I 1

xx = 9.9912 × 10−4 kg m2, I 1yy = 9.9686 ×

10−4 kg m2, and I 1zz = 4.9394 × 10−5 kg m2. The tor-

sional damping is identified as 0.015 N m s/rad andthe effect of the torsional spring is not included in themodel considered in this investigation.

7.3 Model construction

Figure 6 is a schematic representation of a two linksystem that is equivalent to the test rig describedin this section. The first link represents the motorrotor and its attached hub, and the second link rep-resents the rotating flexible arm. A revolute joint isused between the hub and the ground and a rigid jointis used between the hub and the flexible link. In theFFR model, two beam elements are used. In order to

Fig. 6 Schematic diagram of the rotating flexible manip-ulator

increase the efficiency of the experiment, the camerais used to record a movie for the tip point and themovie is processed to determine the tip point posi-tion. It is difficult to use the camera as a real-timemeasuring device because of two reasons: low shut-tering speed of the camera and the unavailability ofreal-time video capture devices with the XPC targettoolbox within MATLAB. In this experiment, a con-stant torque of Tθ = 0.1 N m is fed forward from thehost PC to the motor through the digital electronicdevice attached to the motor with closed-loop feed-back to ensure the correct value of the torque. Sincethe critical speed is reached at time 3.5 s, the simula-tion and the experiment time is chosen to be 5 s. Therecorded data of the tip point can be considered asthe local transverse deflection of the tip point of theflexible arm.

8 COMPARISON BETWEEN SIMULATION ANDEXPERIMENTAL RESULTS

The comparison between the results obtained usingthe non-linear FFR formulation and the ANCF-BC,ANCF-EL, and the experimental results are shown inFigs 7 and 8. In these figures, the transverse deforma-tion of the tip point is shown as a function of time.The errors between the FFR model, ANCF-EL model,and the experimental results are shown in Fig. 9. In theFFR model, the first modal frequency that defines thecritical speed of the flexible link if the effect of the geo-metric stiffening is not taken into account is 2.2733 Hzas shown in Table 2 for the two-beam element model.Because of the form of the applied torque, it is expectedto reach the critical speed at time 3.5 s, as previouslymentioned. In order to obtain correct results, the axialmode, which is the ninth mode as shown in Fig. 4,should be taken into account. Two approaches canbe followed to select the necessary modal transfor-mation of the model. The first approach is to select

0 1 2 3 4 5-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Tra

nsve

rse

Def

lect

ion

(m)

Time (s)

Fig. 7 Transverse deflection (———� FFR, ———• ANCF-BC, ———� Experimental)





0 1 2 3 4 5-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Tra

nsve

rse

Def

lect

ion

(m)

Time (s)

Fig. 8 Transverse deflection (———� FFR, ———• ANCF-EL, ———� Experimental)

0 1 2 3 4 5-0.001

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Dif

fere

nce

in T

rans

vers

e D

efle

ctio

n (m

)

Time (s)

Fig. 9 Differences in transverse deflection from experi-mental results (———� FFR, ———� ANCF-EL)

0 0.2 0.4 0.6 0.8 1-0.03

-0.02

-0.01

0

0.01

Time (s)

Tra

nsve

rse

Def

lect

ion

(m)

Fig. 10 Effect of modal reduction in the FFR model(———� 9 modes, ———� 3 modes)

all the first nine modes by taking into account thefirst nine eigenvectors. The second approach is toselect only the necessary modes, which are the first,the second, and the ninth modes. Figure 10 showsthe transverse deflection of the rotating manipula-tor with and without modal reduction, a difference of0.003 374 m is found at time 0.023 s. This differencedisappears after 0.2 s.

8.1 Axial modes and implicit integration

The inclusion of high-frequency axial modes in theFFR dynamic models may be necessary in someapplications in order to capture geometric stiffnessthat can have significant effects on the accuracy ofthe solution. Some implicit stiff numerical integrationmethods may fail to capture high-frequency geometriceffects. Two implicit numerical integration methodsare used in this study to obtain the numerical solutionsof the models used. It is important to mention thatthe explicit fourth RK method failed from the start.The implicit integrator RADAU5 was successfully usedto obtain the numerical results of the non-linear FFRmodel [1]. While numerical experimentation in thisstudy demonstrated the robustness of RADAU5 toobtain the solution of the fast flexible link models, thecomputer time is much longer than the time requiredwhen the HHT integrator is used. The HHT integra-tion method, however, failed to capture the geometricstiffness effect. It was found that, in the case of theundamped joint model, the solution does not con-verge when the critical speed is reached, as shown inFigs 11 and 12. For this reason, RADAU5 is used in thecase of the FFR models.

8.2 ANCF-coupled deformation modes

Figure 13 shows the transverse deflection of the rotat-ing manipulator models obtained using the ANCFfinite-element model. Discrepancies between theANCF-BC and ANCF-EL results can be noticed in thetransient region, as shown in Fig. 14. It is clear thatthe integration of ANCF-coupled deformation modesincluded in the ANCF-BC solution is the source ofthe differences, and this difference becomes largerwhen the results are compared with experimentalresults. As it is known, for stiff and thin structure,fully parameterized ANCF finite elements can suffer

0 1 2 3 4 5-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Glo

bal P

ositi

on (

m)

Time (s)

Fig. 11 Effect of integrator (FFR model – undampedjoint) (———� X RADAU5, ———• Y RADAU5,———� X HHT, ———� Y HHT)





0 1 2 3 4 5-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8G

loba

l Pos

ition

(m

)

Time (s)

Fig. 12 Effect of damping (FFR model – damped joint)(———� X, ———� Y)

0 1 2 3 4 5-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Tra

nsve

rse

Def

lect

ion

(m)

Time (s)

Fig. 13 Transverse deflection of ANC models of the rot-ating arm (———� ANCF-BC, ———� ANCF-EL)

0.0 0.5 1.0 1.5 2.0-0.002

0.000

0.002

0.004

0.006

0.008

0.010

Dif

fere

nce

in T

rans

vers

e D

efle

ctio

n (m

)

Time (s)

Fig. 14 Difference between ANCF-BC and ANCF-ELmodels

from locking problems. In the ANCF-EL, coupleddeformation modes are eliminated, and therefore, thebeam element model leads to a better agreement withthe experimental results.

9 SUMMARY AND CONCLUSIONS

The FFR formulation has been widely used in the smalldeformation analysis of flexible multibody systems.This formulation allows for the use of modal repre-sentation, and therefore, a significant reduction of themodel dimensionality can be achieved. The goal ofthis investigation is to demonstrate that the 3D FFRformulation can also be used in the large deforma-tion analysis if such a deformation can be describedusing simple shapes. In this case, the inclusion ofnon-linear strain–displacement relationships is nec-essary in order to be able to capture the geometricstiffness effects which can be significant at certainspeeds. Numerical and experimental validation wasalso another goal of this investigation. In order tonumerically validate the non-linear FFR models, thesolution of the rotating beam model considered in thisinvestigation was obtained using the ANCF finite ele-ments. The ANCF was used with two different elasticforce formulations: the ANCF-BC and the ANCF-EL. Itwas shown that the solution of the ANCF-BC model,which includes high-frequency coupled deformationmodes, does not agree with the experimental results.As it is known, in the case of thin and stiff structures,ANCF fully parameterized elements can suffer fromlocking which has an adverse effect on the accuracy ofthe solution. Good agreement, however, was observedbetween the FFR, ANCF-EL, and experimental results.

As discussed in this article, the inclusion of high-frequency axial modes in the FFR models may benecessary in order to capture the geometric stiffnessthat can have a significant effect on the solution.Nonetheless, some implicit integration methods filterout high frequencies and may fail to capture the effectof the high-frequency axial modes. Some of these inte-gration methods use artificial damping in order todamp out high-frequency oscillations. Two differentimplicit integrators were used in this investigation toobtain the numerical solution of the finite-elementmodels: the implicit RADAU5 and the HHT integra-tors. It was observed that the HHT integrator does notcapture the effect of the axial mode in the non-linearFFR model. Nonetheless, the locking problems asso-ciated with the ANCF-BC models as the result of thecoupled deformation modes can be avoided when theHHT method is used. The experimental and numericalvalidations of the non-linear FFR model presented inthis study demonstrate that the FFR formulation canbe effectively used in the large deformation analysis ifthe deformation assumes simple shapes.

ACKNOWLEDGEMENT

This article has been partially supported by the US–Egypt Science and Technology Joint Fund Program(National Science Foundation) through the research





project ‘Modelling and Control of Robotic Manip-ulators with Flexible Links and Joints’, Grant no.OTH7-004-005 and NSF Grant no. 0314378.

© Authors 2010

REFERENCES

1 Nada, A. A. Flexible robotic manipulators: modeling, sim-ulation and control with experimentation. PhD Thesis,Mechanical Design and Production Engineering Depart-ment, Cairo University, Cairo, Egypt, 2007.

2 Shabana, A. A. Dynamics of multibody systems, 3rdedition, 2005 (Cambridge University Press, New York).

3 Gerstmayr, J. and Schoberl, J. A 3D finite elementmethod for flexible multibody systems. Multibody Syst.Dyn., 2006, 15, 309–324.

4 Orlandea, N. V. From Newtonian dynamics to sparsetableaux formulation and multibody dynamics. Proc.IMechE, Part K: J. Multi-body Dynamics, 2008, 222,301–314. DOI: 10.1243/14644193JMBD153.

5 Mikkola, A. M. and Matikainen, M. K. Development ofelastic forces for the large deformation plate elementbased on the absolute nodal coordinate formulation.ASME J. Comput. Nonlinear Dyn., 2006, 1(2), 103–108.

6 Schwab, A. L. and Meijaard, J. P. Comparison of three-dimensional flexible beam elements for dynamic ana-lysis: finite element method and absolute nodal coor-dinate formulation. In Proceedings of the ASME Inter-national Design Engineering Technical Conferences andComputer and Information in Engineering Conference(DETC2005/MSNDC-85104), Long Beach, USA, 2005.

7 Sopanen, J. T. and Mikkola, A. M. Description of elas-tic forces in absolute nodal coordinate formulation.Nonlinear Dyn., 2003, 34(1–2), 53–74.

8 Yoo, W. S., Lee, J. H., Park, S. J., Sohn, J. H., Pogorelov,D., and Dimitrochenko, O. Large deflection analysisof a thin plate: computer simulation and experiment.Multibody Syst. Dyn., 2004, 11(2), 185–208.

9 Shampine, L. and Gordon, M. Computer solution of ODE:the initial value problem, 1975 (W. H. Freeman, SanFrancisco).

10 Garcia-Vallejo, D., Sugiyama, H., and Shabana, A. Finiteelement analysis of the geometric stiffening effect: a cor-rection in the floating frame of reference formulation.Proc. IMechE, Part K: J. Multi-body Dynamics, 2005, 219,187–202. DOI: 10.1243/146441905X10041.

11 Garcia-Vallejo, D., Sugiyama, H., and Shabana, A. Finiteelement analysis of the geometric stiffening effect: non-linear elasticity. Proc. IMechE, Part K: J. Multi-bodyDynamics, 2005, 219, 203–211. DOI: 10.1243/146441905X10050.

12 Dmitrochenko, O. N., Hussein, B., and Shabana, A. A.Coupled deformation modes and their effect on theaccuracy and efficiency of the large displacement finiteelement solutions: generalization. ASME J. Comput. Non-linear Dyn., 2009, 4(2), 021002, 8 p.

13 Hussein, B. A., Negrut, D., and Shabana, A. A. Implicitand explicit integration in the solution of the absolute

nodal coordinate differential/algebraic equations. Non-linear Dyn., 2008, 54(4), 283–296.

14 Hussein, B. A., Sugiyama, H., and Shabana, A. A. Abso-lute nodal coordinate formulation coupled deformationmodes: problem definition. ASME J. Comput. NonlinearDyn., 2007, 2(2), 146–154.

15 Shabana, A. A. and Yakoub, R. Y. Three dimensionalabsolute nodal coordinate formulation for beam ele-ments: theory. ASME J. Mech. Des., 2001, 123(4), 606–613.

16 Yakoub, Y. R. and Shabana, A. A. Three dimensionalabsolute nodal coordinate formulation for beam ele-ments: implementation and applications. ASME J. Mech.Des., 2001, 123(4), 614–621.

17 García-Vallejo, D., Mayo, J., Escalona, J. L., andDominguez, J. Efficient evaluation of the elastic forcesand the Jacobian in the absolute nodal coordinateformulation. Nonlinear Dyn., 2004, 35(4), 313–329.

18 Shabana, A. A. Computational continuum mechanics,2008 (Cambridge University Press, New York).

19 Hairer, E. and Wanner, G. Stiff differential equationssolved by radau methods. In Proceedings of the Numer-ical Methods for Differential Equations, Coimbra, Portu-gal, 1999 (Elsevier North Holland, New York).

20 Newmark, N. M. A method of computation for struc-tural dynamics. J. Eng. Mech. Div., ASCE, 1959, 85,67–94.

21 Hilber, H. M., Hughes, T. J. R., and Taylor, R. L. Improvednumerical dissipation for time integration algorithms instructural dynamics. Earthq. Eng. Struct. Dyn., 1977, 5,283–292.

22 Mosterman, P. J., Prabhu, S., Dowd, A., Glass, J., Erkki-nen, T., Kluza, J., and Shenoy, R. Embedded real-timecontrol via MATLAB, simulink, and XPC target – bookchapter, 2005 (Birkhäuser Boston, Boston).

APPENDIX

Shape function matrix

The conventional beam element used in the FFRmodel has 12 degrees of freedom. The shape functionof this element is given as

ST =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 − ξ 0 0

6(ξ − ξ2)η 1 − 3ξ2 + 2ξ3 0

6(ξ − ξ2)ζ 0 1 − 3ξ2 + 2ξ3

0 −(1 − ξ)lζ (1 − ξ)lη

(1 − 4ξ + 3ξ2)lζ 0 (−ξ + 2ξ2 − ξ3)l

(−1 + 4ξ − 3ξ2)lη (ξ − 2ξ2 + ξ3)l 0ξ 0 0

6(−ξ + ξ2)η 3ξ2 − 2ξ3 0

6(−ξ + ξ2)ζ 0 3ξ2 − 2ξ3

0 −lξζ lξη

(−2ξ + 3ξ2)lζ 0 (ξ2 − ξ3)l

(2ξ − 3ξ2)lη (−ξ2 + ξ3)l 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

In this equation, ξ = x/l, η = y/l, ζ = z/l, and l is thelength of the element.




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