Nonlinear dynamics of exible beams in planar motion: formulation and time-stepping scheme for sti problems

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1 PERGAMON Computers and Structures 70 (999) ± Nonlinear dynamics of exible beams in planar motion: formulation and time-stepping scheme for sti problems A. Ibrahimbegovic *, S. Mamouri CompieÁgne University of Technology, UTC, Dept, GSM, Division MNM, BP-59, 6006, CompieÁgne, France Received 9 Month 0; received in revised form 9 Month 0 Abstract In this work we present a systematic development for dealing with nonlinear dynamics of multi-body systems in planar motion. The model problem addressed is the d beam theory of Reissner, featuring a simple quadratic form of the kinetic energy and a convenient spatial form of the strain measures which simpli es the strain energy expression. Special care is dedicated to constructing a time-stepping scheme adequate for multi-body systems with large di erence in sti ness between di erent system components or di erent deformation modes, yet referred to as sti problems. Several numerical simulation are presented to illustrate an excellent performance of the proposed method. # 998 Elsevier Science td. All rights reserved. Keywords: Dynamics; Beams; Time-integration; Sti problems. Introduction In the traditional approach to dynamic analysis of mechanisms and machines, one assumes that the systems are composed of rigid bodies. However, in modern systems operating at high speed one needs to account for a possibility of the systems undergoing severe elastic deformations to the extent that they cannot be ignored. Flexible beams, capable of properly describing large overall motion and strains, become an indispensable part of such a model. In practical applications of mechanism dynamic analysis, the modeling re nement of accounting for system deformation can lead to an additional di culty related to so-called sti problems, which appear as a consequence of large di erences in system exibility to various deformation patterns (e.g. axial vs exural deformation). The same problem might also occur if the agrange multiplier technique is used to impose the constraints between di erent members, resulting in a set of di erential-algebraic equations describing the system motion. * Corresponding author. The main thrust of this work is directed towards an e cient formulation of a exible beam model for nonlinear dynamics, and a development of a robust timestepping scheme which performs well for the sti problems. Traditional approaches to introducing exibility in mechanism analysis (e.g. Ref [3]) considers a separation of the deformation from the rigid body modes. With this approach, which follows the steps of the fundamental works of Fraeijs de Veubeke [] and Kane and evinson [3], a simple quadratic form can be obtained for the potential energy of the beam provided that the strains remain small and that the formulation is set in a oating frame (i.e. a frame which is rigidly rotating with the beam). However, the use of the oating frame triggers the Coriolis and centrifugal e ects, which highly complicate the evaluation of the kinetic energy leading to its nonlinear, coupled form. In conclusion, the oating frame approach is not appropriate for dynamics of highly exible beams, despite several subsequent re nements of this model, such as capturing the motion induced sti ening e ect (e.g. see Refs [, 6], or geometric sti ness contribution to the system response (e.g. see Ref. [5]). More recent works (e.g. [7,, 6]) have turned to using the inertial frame for developing the formu /99/$ - see front matter # 998 Elsevier Science td. All rights reserved. PII: S (98)0050-3

2 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± lations appropriate for the dynamics of beams, as initially suggested by Simo and Vu-Quoc [8]. As opposed to the oating frame approach, in this case one obtains a simple quadratic form for the kinetic energy. The key ingredient of this approach which also leads a straight-forward de nition of the potential energy even for large strains is the geometrically nonlinear theory of beams undergoing large displacements and rotations ([, 7]). The governing equations of motion resulting from this approach lend themselves to standard time-stepping schemes in structural dynamics. In the vast majority of recent works [7,, 6, 8, 33] the classical Newmark time integration scheme is used. However, as pointed out in Ref. [8], such a scheme is not appropriate for the nonlinear dynamics of systems leading to sti problems where the beam is not necessarily highly exible in each deformation pattern. Therefore, we set to develop an alternative time integration scheme, which can handle these more realistic applications. Our main contributions can be stated as follows:. Following the work of Simo and Vu-Quoc [8] we employ the inertial frame to set the dynamic equations of motion, which leads to a simple quadratic form of kinetic energy. However, as opposed to this work, we use the spatial strain measures and the corresponding stress resultants, which considerably simpli es the construction of the internal force vector. Moreover, we show that the chosen spatial representation also leads to a very spare form of the tangent operator. We note in passing that the latter is obtained by using the ie derivative formalism as it applies to the present case (e.g. Ref. []).. A new time-stepping scheme is developed for the exible beam problem. The scheme is based on the mid-point rule approximation, which is recognized (e.g. Refs [0, 4]) to be well suited for dealing with sti problems. The particular modi cation of the standard mid-point scheme, proposed here in order to ensure the energy conservation, was shown to yield an improved performance in sti problems. The latter is a more than su cient justi cation for the increase of computational cost brought about by the resulting non-symmetric form of the tangent operator. 3. Simple extension of the proposed formulation to multi-body dynamics is outlined and illustrated by numerical examples. The di erent joints in a multibody system (e.g. revolute or prismatic) are taken into account by the standard nite element assembly procedure, which results neither in the additional degrees of freedom nor in a set of di erential-algebraic equations. The outline of the paper is as follows. In the next section we present the governing equation for the nonlinear dynamics of a beam undergoing large overall planar motion. The mid point time-stepping scheme is then described, along with its modi cation that enforces energy conservation. A set of selected numerical simulations is described in order to illustrate the performance of the presented scheme. Some concluding remarks are stated at the end.. Dynamics of a beam in planar motion.. Beam kinematics We consider a exible beam undergoing a planar motion (see Fig. ), without restricting the order of displacements and rotations. Initial con guration of the beam is speci ed by the position vector x of the points on neutral axes, and the unit vector t 0 which is normal to the beam cross-section plane. Without loss of generality we choose t 0 as the tangent to the beam axis with t 0 ˆ x 0 where () 0 denotes the partial derivative with respect to the s-coordinate. Following the basic kinematic hypothesis (e.g. Refs [, 7]) that the plane sections remain plane but not necessarily perpendicular to the neutral axis of the beam, we can de ne the beam deformed con guration. fs; z; t ˆjs; t zt s; t where j is the position vector of a point on the neutral axis, whereas t is a unit vector (director) placed in the plane of the cross section. In accordance with the basic kinematic hypothesis, the director vector t and the unit normal t to the cross section are obtained simply by rotating their initial positions t 0 and t 0 with a two dimensional rotation matrix as t ˆ t 0 ; t ˆ t 0 cos c sin c ; ˆ : 3 sin c cos c In general, t can no longer be identi ed with j 0, due to shear deformation. We require that t j 0 r0 4 in order to preclude severe shear deformation... Kinetic energy Using the inertial frame, the kinetic energy of the beam can be cast as a quadratic, uncoupled form. In order to show this, we rst note that the time deriva-

3 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 3 Fig.. Two-dimensional beam: initial and current con gurations. tive of orthogonal matrix can be written as _ ˆ _cw; 0 W ˆ 0 where ()=@/@t() denotes the partial time derivative. The last result, in relation to Eq. (3), leads to the classical expression for the derivatives of the unit vectors of rotated base as _t ˆ _cwt _t ˆ _cwt ˆ _ct ; ˆ _ ct : Using these results, the time derivative of the position vectors in Eq. () can be obtained as _f ˆ _j z cwt _ : 7 The kinetic energy of the beam can then be written as K ˆ r f _ f _ dv ˆ fa r _j _j J r c _ gds 8 where A r = f A rda and J r = f A rz da are inertia coe cients. Hence, the biggest advantage of the inertial frame is in a resulting simple, quadratic form of the kinetic energy..3. Potential energy: spatial strain measures The strain measure for the beam theory can be written in a vectorial form (see [0]) as E ˆ j 0 t ; k ˆ c 0 e 3 where E contains axial and shear strains and k is the bending strain. We will denote by n and m the stress resultant and couple, which are energy conjugate to E and k, respectively. It is important to note that n and m are spatial objects acting in the deformed con guration, but parameterized by the coordinates set in the initial con guration. (In that respect, they are equivalent to the rst Piola±Kirchho stress tensor in solid mechanics). These stress resultants and strains can be decomposed in the local rotating frame, so that n ˆ N t N t ; m ˆ Me 3 ; N E ˆ E t E t k ˆ Ke 3 0 and the beam potential energy can be written as P ˆ E N E N KM ds P ext : If the linear elastic constitute equations are chosen for stress resultants and coupled with N E ˆ C ; C ˆ diagea ; EA E M ˆ C m K ; C m ˆ EI the potential energy can be written as a quadratic form in nonlinear strain measures. In Refs [7 or 8], the material form of the strain measures is preferred with E ˆ T E; K ˆ k which are energy conjugate to the material form of stress resultants N ˆ T n; M ˆ m: 3 We note in passing that when the material stress resultant and strains are decomposed in the global

4 4 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± frame, the same components are recovered as in Eq. (0), i.e. N ˆ N t 0 N t 0 ; E ˆ E t 0 E t 0 M ˆ Me 0 3 ; K ˆ Ke and the component form of the potential energy remains the same as in Eq. (). For further development, we retain only the spatial representation of the potential energy. It is shown in the foregoing that this form leads to a very e cient numerical implementation..4. Equations of motion Having de ned the kinetic and the potential energy of the beam and thus its total energy as well, the standard application of Hamilton's principle leads to the weak form of the equation of motion of the beam as Gj; c; dj; dc :ˆ dj A r j dcj rc ds d E n d k m ds G ext ˆ 0 5 where G ext is the work of the external forces. In the last expression, the computation of the spatial strain rates should comply with the objectivity requirements under superposed rigid body motion. The latter is automatically enforced by exploiting the ie derivative formalism (e.g. Ref. [4]) where the spatial strain measures are rst taken to the initial con guration (pull-back), their variations are computed and the corresponding results are nally transformed to the deformed con guration (push-forward). In the case under consideration the pull-back and push-forward of the axial and shear strain measures are carried out by (e.g. Ref. []), to get d E ˆ d T E ˆ dj 0 Wj 0 dc 6 where dj and dc are, respectively, the virtual displacements and virtual rotation. Due to the planar nature of the problem, the equivalent result for the virtual bending strain turns out to be much simpler, since the pull-back and push-forward are performed by identity tensor I, to get d k ˆIdIk ˆ dc 0 e 3 : Remark In the case of absence of external forces, with G ext = 0, it can readily be shown from Eq. (5) that the total energy remains constant. First, we can compute the time derivative of material strains in Eq. () to get _E ˆ T v 0 Wj 0 o ; _K ˆ o 0 where v = _j and o = c _ are, respectively, linear and angular velocities. Furthermore, by making use of the relationship between material and spatial forms of stress resultants in Eq. (3) and replacing dj and dc, by v and o, respectively, the weak form of the equations of motion in Eq. (5) can be rewritten as 0 ˆ fv A r _v oj r _ogds f _E N _K Mgds which, in conjunction with expressions in Eqs. (8) and (), implies that d dt K d dt P ˆ 0 ) K P ˆ const:.4.. Remark The ie derivative formalism can also be applied in computing the linearized form of the equations of motion. For example the linearized form of the second term in Eq. (5), which gives rise to the tangent sti ness matrix, can be written as follows dj 0 Wj 0 dc in G P ŠˆG P dc 0 C T 0 0 T C m dj dc Dj 0 Wj 0 Dc Dc 0 ds 0 Wn Dj 0 W n T n j 0 Dc ds where Dj and Dc are, respectively the incremental displacements and rotation. The linearized form of the rst term in Eq. (5), which gives rise to the mass matrix, takes the much simpler form in G K ŠˆG K dj dc.5. Semi-discrete approximation Ar I 0 D j 0 T J r Dc ds: We believe that the proposed weak form of the equations of motion in Eq. (5) provides the most suitable basis for constructing a semi-discrete approximation by the nite element method; on the one hand, the choice of inertial frame and resulting quadratic form of the inertia term lead to the constant mass matrix and a convenient form of the discrete equations of motions which remain nonlinear only in internal force expression. On the other hand, the use of spatial strain measures leads to a signi cant simpli cation and to a very sparse structure of the matrices used in the

5 residual computations. In that sense, the proposed formulation is as e cient as the formulations of Belytschko and Hsieh [6], Bathe, Ramm and Wilson [3] or Downer, Park and Chiou [], who all use rotating frames to simplify the internal force computations. However, as opposed to earlier formulations, which are limited to -node beam elements, the present formulation can be used for any order of the shape function polynomials and any number of element nodes n en. If we choose a nite element approximation employing the beam elements with n en nodes, the position vector and the rotation angle can written, respectively, as js; t cs; t ˆ Xnen e a ˆ ˆ Xnen e a ˆ N a s j a t ; N a s c a t 8 where N a (s) are element shape functions, e.g agrange polynomials in s of the order n en. By using isoparametric interpolations (e.g. Refs [4, 34]) for the virtual displacements and rotations, we can write the semi-discrete approximation to the weak form of the equations of motion in Eq. (5) as A n el e ˆ ( X n el b ˆ M ab 0 jb t 0 T H ab c b t n at m a t where A n el e ˆ ds ˆ 0! n at m a t 9 denotes the nite element assembly procedure. In the equation above, M ab and H ab are the translational and rotational parts of the mass matrix given as M ab ˆ N a s A r IN b s ds; e H ab ˆ N a s J r N b s ds: e A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 5 n a t ˆ N 0 as n ds l e 0 We note that the mass matrix has constant entries independent of the current values of displacements and rotations, which considerably simpli es the construction of time-stepping schemes. The nodal internal force vector in Eq. (9) can be written explicitly as and m a t ˆ fn 0 as m N a s Wj 0 ngds: l e 3. Time-stepping scheme for sti problem 3.. Motivation 3 As just shown, the choice of the inertial frame for describing the dynamics of exible beams results in a special form of equations of motions with linear inertia term and all nonlinearities hidden in the internal force vector. The biggest advantage of such a form is in simplifying the construction of the appropriate time-stepping schemes to compute the discrete approximation of the semi-discrete equations of motion in Eq. (9). In the vast majority of the recent works (e.g. Refs [7,, 6, 8]) the standard Newmark scheme (e.g. [4]) is used. In particular, the choice of the Newmark coe cients is made (b = /4 and g = /), which corresponds to the trapezoidal rule or average acceleration method (e.g. Ref. [8]). This kind of procedure is well suited only for highly exible beams. In more realistic problems, beams can be considerably less exible in certain modes of deformation than others (e.g. axial vs bending deformations), which leads to the appearance of so-called high vibration modes whose frequencies are signi cantly higher than the frequencies of the fundamental vibration modes. The same type of di erence between higher and lower modes is found in a multi-body dynamic systems, where in order to improve the accuracy of the modeling procedure, the members which are considerably more rigid than the others are not considered as perfectly rigid. In a problem of this kind, the choice of the appropriate timestepping scheme is much more complicated. In was shown in Ref. [8] that the trapezoidal rule is not the most appropriate time-stepping scheme for problems with discontinuous data exciting high modes, since any spurious noise generated will not decay. One way to attack these problems is by resorting to the time-stepping schemes which introduce the numerical dissipation in high modes while not disturbing the lower modes, as recommended by Cardona and Geradin [7] who used the a-method of Hilber, Hughes and Taylor [5]. However, this appears not to be a universal solution as demonstrated by Bauchau, Damilano and Theron [5] when the multi-body dynamics problems are formulated in the manner where the constraints between the bodies are imposed by the agrange multipliers. The application of the dis-

6 6 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± sipative HHT a-scheme can lead to the spurious generation of energy, rather than its dissipation. In this work, we depart from the recent works brie y reviewed in two following ways. First, we demonstrate that the multi-body dynamics can be tted within the presented nite element framework without resorting to the agrange multiplier procedure. The standard revolute or prismatic joints can be handled simply by the standard nite element assembly procedure. Second, we focus upon development of the mid-point scheme for the problem on hand. This scheme, which is already identi ed to be very appropriate for dealing with sti problems (e.g. Refs [0, 4]), is further improved by enforcing the total energy conservation. The latter is the discrete approximation equivalent to the property discussed in Remark, which turns out to be very important in the algorithm design. 3.. Mid-point scheme Suppose that the nodal values displacement and rotations are given at time t n, along with the corresponding values of their time derivatives, i.e. velocities and accelerations, which can be written as j a;n ˆ j a t n ; c a;n ˆ c a t n ; v a;n ˆ _j a t n ; o a;n ˆ _c a t n ; a a;n ˆ j a t n ; a a;n ˆ c a t n 4 and that we would like to obtain their values at time t n +, i.e. j a,n + etc. In other words, we address the solution procedure over a typical time increment, denoted as Dt = t n + t n. We rst suppose that incremental displacement Dj a,n + and incremental rotation Dc a,n + (their best iterative values) have been provided by the solution to the linearized problem to be described at the end of this section. The nodal displacements and rotations at time t n + can then be easily computed by simple additive updates j n ˆ j n Dj n ; c n ˆ c n Dc n : 5 6 In Eqs. (5) and (6) and the rest of this section, we omit the subscript ``a'', indicating the nodal point number, in order to alleviate notation. The values of velocities and accelerations at time t n + are computed from mid-point approximation to the weak form of the equations of motion in Eq. (5). To that end, these equations are rst recast as a system of rst order di erential equations. From the rst one of these equations, _j = v, we obtain a mid-point rule approximation for the translational motion velocity at time t n + as Dj n ˆ Dtv n ) v n ˆ v n Dt j n j n : 7 In computing rotational velocity, the following nonlinear update is used with o n ˆ o n 4 Dt tan cn c n 8 which corresponds to the mid-point approximation applied to the derivative of the rotation matrix in Eq. (5), n n ˆ Dt o n o n W n ; n ˆ n n : 9 As shown next, this non-conventional update of angular velocities plays a crucial role in constructing the appropriate procedure to enforce the energy conservation. The mid-point approximation of the accelerations at time t n + leads to. a n ˆ Dt v n v n ; a n ˆ Dt o n o n : 30 With this result in hand, the mid-point approximation of the weak form of the balance equations in Eq. (5) can be written as an explicit function of displacement and rotation parameters at time t n +, j n + and c n +, given as Gj n ; c n ; dj; dc :ˆ dj A r a n ds dcj r a n ds dj 0 dcwj 0 n n n ds dc 0 m n ds G ext;n ˆ 0 3 where j 0 n = ˆ =(j 0 n + j 0 n and G ext;n = is the mid-point approximation for the external virtual work. For the kind of loading considered in this work (e.g dead load with xed direction), one can obtain G ext;n ˆ G ext;n G ext;n 3.3. Enforcing energy conservation As noted by Hughes, Caughy and iu [9], when the standard Newmark scheme is modi ed such that the total energy conservation is enforced, we get a more robust version of that scheme with improved perform-

7 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 7 ance in nonlinear dynamics. The same observation has recently been made by Simo and Tarnow [9], Cris eld and Shi [8] and Geradin and Rixen [3] for di erent versions of mid-point schemes proposed, respectively, for problems of nonlinear elasticity, nonlinear trussbar and rigid body dynamics. In accordance with these ndings, in this section we propose a modi cation of the presented mid-point rule for the dynamics of planar beams in order to enforce the energy conservation. et the incremental displacements and rotation be given as Du :ˆ Dtv n ˆ Dt v n v n ; 3 Dy :ˆ Dto n ˆ Dt o n o n : 33 We rst set to show that by replacing dj and dc by Du and Dy, respectively, from the weak form in Eq. (5) we can get the increase of the total energy of the time step Dt. To that end, Eq. (5) can be rewritten as 0 ˆ fdu A r a n DyJ r a n gds fdu 0 Wj 0 n Dy n n Dy 0 m n gds G ext;n : 34 Using the results in Eqs. (8), (30), (3) and (33), it can readily be shown that the rst term in Eq. (34) above represents the corresponding increment of kinetic energy DK ˆ fdu A r a n DyJ r a n gds ˆ fv n v n A r v n v n o n o n J r o n o n gds ˆ fv n A r v n o n J r o n v n A r v n o n J r o n gds ˆ K n K n : 35 The computation revealing that the second term in Eq. (34), upon a small modi cation can be recast in a form corresponding to the increment in potential energy, is somewhat more involved. We rst obtain two auxiliary results. Using the nonlinear angular velocity approximation in Eq. (8), we can write a consistent mid-point approximation of the time derivative of the orthogonal matrix as in Eq. (9). By using this result along with the result in Eq. (7) and its spatial derivative, it can be shown that T n Du 0 Wj 0 n Dy ˆ T n j 0 n j 0 n T n T n j 0 n ˆ T n j 0 n T n j 0 n ˆ E n E n 36 where E n and E n + are material forms of axial and shear strain measures in Eq. () at time t n and t n +, respectively. Moreover, computing the spatial derivative of the nonlinear velocity update in Eq. (8) one can obtain Dy 0 ˆ tan cn c n ˆ tan cn c n c 0 n c 0 n K n K n 37 where K n and K n + are material bending strains at time t n and t n +. With the last two results on hand, the second term in Eq. (34) can be written as fdu 0 Wj 0 n Dy n n Dy 0 m n gds ˆ fe n E n N n K n K n M n gds 38 where we denoted the corresponding stress resultants N n ˆ n n n ; M n ˆ tan cn c n m n : 39 Following the previous considerations for the computation of the kinetic energy in Eq. (35), we propose that these stress resultant be computed as N n ˆ C E n E n and 40 M n ˆ C m K n K n : 4 It is important to note that the axial and shear stress resultant in Eq. (40) are not the same as those computed by the standard mid-point approximation. Namely, since the corresponding strain expression in Eq. () is nonlinear, the average strain in Eq. (40) is not the same as the strain in the average con guration.

8 8 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± With these resultants on hand the second term in Eq. (34) is indeed equal to the increment in the potential energy, since it follows that DP ˆ fdu 0 Wj 0 n Dy n n Dy 0 m n gds ˆ fe n E n CE n E n K n K n C m K n K n gds ˆ fe n CE n K n C m K n E n CE n K n C m K n gds ˆ P n P n : 4 We note again that the key ingredients leading to the last result are the nonlinear angular velocity update in Eq. (8) and non-conventional computation of stress resultants in Eq. (40). By substituting the results in Eqs. (35) and (4) into Eq. (34), we obtain the exact value of the total energy increase over the time step Dt. It then readily follows that in the absence of external forces the total energy remains preserved, which is a discrete approximation equivalent to the corresponding result established in Remark for the continuum problem. A couple of very recent works by Cris eld [9] and Stander and Stein [3] propose two alternative formulations of energy-conserving algorithms for dynamics of planar beams to the one given herein, which are based on corrotational formalism and selective scaling of stress resultants, respectively. Another pertinent work on constructing the energy-conserving algorithms is a recent work of Simo et al. [30] on three-dimensional beams, although the direct application of the three-dimensional setting developed therein would impose additional complexity related to multiplicative updates of rotation parameters, which are unnecessary in the present, planar case Remark 3 It can be shown that the proposed mid-point scheme also ensures the conservation of linear and angular momenta, if the external loading (including eventual support reactions) satis es the global equilibrium equations. To that end, we can rewrite Eq. (3) in the case when the virtual displacement eld is replaced by a constant rigid body translation, i.e. dj j 4 c, accompanied with no rotation, dc j 4 0, which leads to 0 ˆ c A r a n ds ˆ Dt c ˆ Dt c p n p ˆ)p n ˆ p n A r v n v n which con rms the linear momentum conservation. Similarly, the conservation of angular momentum can be proved for the virtual displacement and rotations resulting from the constant rigid body rotation of the complete structure, i.e. dc j 4 c and dj j 4 cwj n + /. The contribution of the internal force disappears for such a virtual displacement eld, so that Eq. (3) can be rewritten as 0 ˆ cwj n A r a n ds cj r a n ds ˆ c Wj Dt n A r j n j n ds c J r o n o n ds Dt ˆ c Dt Wj n p n p n c Dt p n p n : Making use of the following vector identities: Wj n p n p n Wj n p n Wj n p n ˆkj n p n k kj n p n k we can show from the previous expression that the angular momentum is conserved p n kj n p n kˆp n kj n pk: 3.4. Consistent linearization Having obtained the velocity and acceleration update for given values of displacement and rotation increments, as given in Eqs. (7)±(30), we now turn to the computation of the new (iterative) values of displacement and rotation increments. To that end, we rst note that in the summary of the discussion given in the previous section, the weak form of the equation of motion in Eq. (3) can be rewritten as an explicit function of the displacement and rotations values at time t n + as Gj n ; c n ; dj; dc :ˆ dj A r Dt j n j n Dt v n cn c n dcj r 4 Dt tan Dt o n T n dj 0 dcwj 0 n C Š T n j 0 n T n j 0 n j 0 n t0 dc 0 cn tan c n C m c 0 n c 0 n ds G ext;n ˆ 0: 43 For the given value of the displacements and rotations, Eq. (43) may not necessarily be satis ed and

9 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 9 one would like to obtain new (improved) values of the dependent variables. If the Newton method is used for that purpose we compute the consistent linear approximation to Eq. (43) as in G n Š : ˆ G n d de Gj n ;E c n ;E ; dj; dc Š ˆ 0 E ˆ 0 44 where j n+,e = j n+ + EDj n+ and c n+,e = c n+ + EDc n+, with Dj n+ and Dc n+ as the incremental displacement and incremental rotations, respectively. The solution to Eq. (44) provides a new value for incremental displacements and rotation, which further leads to the new values of total displacements and rotations obtained from Eqs. (5) and (6), respectively. After somewhat lengthy but straight forward computations, the explicit form of the second term in Eq. (44) can be written as energy. Hence, even for the Galerkin type of discrete approximation described in the next section, the tangent operator is non-symmetric, which doubles the computational e ort in each iteration. However, we believe that the latter is a reasonable price to pay in order to increase the stability property of the proposed scheme Discrete approximation In this section we construct the discrete approximation leading to explicit forms of the residual vector and generalized tangent sti ness matrix. To that end, we rst construct an alternative form of the weak form of balance equations in Eq. (43) employing the spatial objects and leading to d Šˆ de G n ;E Eˆ0 C T 4 C m dj dc 3 Dt A ri 0! 0 T Dj n T n dc 0 Wj J Dt cos c n c r 7 ds n dc 5 Dc n dc 0 n 3 0 dj 0 T 7 5 n Dj 0 n Wj 0! n Dc n ds dc Dc 0 B C A dc 0 tan cn c n 3 0 W n N n 0 W n N n T 0 j 0 n n N n 0 tan c n c n 6 0 T M tan c n c n n 5 Dj 0 n Dc n Dc 0 n C A ds: 45 Contrary to the linearized form of the continuum problem, discussed in Remark, the linearized form of the discrete problem is not symmetric. This is the consequence of the modi cation of the constitutive equations proposed to enforce the conservation of! dj A r a G n ˆ n ds dc J r a n 0 dj 0 0 n n B dc A Wj 0 n n n C A ds ˆ 0 dc 0 m n 46

10 0 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± where m n ˆ and tan cn c n C m K n K n 47 n n ˆ n C E n E n while a n+/ and a n+/ are given in Eq. (30). The linearized form in Eq. (45) can also be fully recast in spatial version as d Šˆ 0 0 dj 0 Dj 0 n dj Dj de G n ;E HŠ n B C B dc A E GŠ Dc E ˆ 0 dc Dc n Ads n dc 0 Dc 0 n where Dt H ˆ A ri T Dt cos cn c n 3 J r E ˆ 6 4 D D Wj 0 3 n 0 0 T 0 tan cn c n C ; D ˆ 7 n C T n 5 m 5 Wj 0 T D Wj 0 n n T DWj 0 n 0 and 3 0 W~n n 0 G ˆ Wn n T j 0 n ~n n 0 ; T cn c tan n 5 m n 0 ~n n ˆ n N n 5 As opposed to the material formulation in Eqs. (43) and (45), the spatial formulation in Eqs. (46) and (49) results with a very sparse structure of the strain-displacement matrix, which reduces the cost of computing the element arrays. Namely, if the isoparametric interpolation, as those in Eq. (8), are used for virtual and incremental values of displacements and rotations, the latter can be written A r a r a ˆ N n a J r a ds n {z } full int for the residual vector and K ab ˆ N a N b HŠ ds {z } e full int: N 0 I T N e a N 0 a 0 n n Wj 0 B n n n A ds ; a ˆ ; ; 3;...; n en 53 m n {z } reduced int: N 0 N ai 0 0 b I T N e a N 0 E GŠ4 0 T N b 5 ds; a 0 T N 0 b {z } a; b ˆ ; ; 3;...; n en 54 reduced int:

11 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± for the tangent sti ness, respectively. It is indicated above that these expressions are evaluated by the numerical quadrature formulae, where full integration is used in computing the rst term, whereas the reduced integration is used on the second term. The latter is done to avoid the locking phenomena (e.g. [0]). 4. Selected numerical simulations In this section we present several representative numerical simulations in order to illustrate an exceptionally good performance of the proposed scheme in the case of sti problems. All the numerical simulations have been performed with an enhanced version of the computer program FEAP, written by Robert.Taylor of UC Berkeley (e.g. [34]). 4.. Simple pendulum Fig.. Simple pendulum: mechanical and geometric characteristics. The rst example of a sti problem we nd in the dynamics of a simple pendulum (see Fig. ). The pendulum is assumed to be nearly rigid, so that in our model the high values of axial, shear and bending sti ness coe cients for the beam element are selected with EA =0 0, GA =0 0 and EI =0 3. A lumped mass m = kg is attached at the free end of the pendulum. The distributed mass of the beam and its rotary inertia are neglected. One beam element is used for constructing a nite element model of the pendulum. The pendulum, which was in a vertical position at rest under a gravity eld, is set in motion by applying an initial horizontal velocity v =.695 to the attached mass. This is a somewhat di erent initial condition than in Ref. [4], where the pendulum problem was initially proposed, or in Ref. [8], where it was modeled by truss-bar elements neglecting bending deformation completely. In this respect, the proposed example corresponds rather closely to the analysis carried out by Cardona and Geradin [7] by using the HHT a-method. However, all these minor di erences in choosing the initial conditions do not change the basic di culty of this problem in that there is a considerably large di erence in associated energy for low swinging modes (see Fig. 3) and high modes of axial vibrations without swinging. In fact, with this di erence being so large, it is enough to trigger axial vibrations (which are not excited by any of the described choices of initial conditions) with a round-o error to have the solution shift to the completely di erent path from low swinging modes. As shown in the forgoing this is especially the case in the Newmark scheme, which is not able to dissipate (numerically) the energy of these high modes. In Fig. 4 we present the high oscillations in the computed acceleration obtained by the Newmark scheme. The same tendency is shown by the HHT a- scheme although it is much less pronounced due to the positive in uence of numerical damping (see Fig. 5), which led Cardona and Geradin [7] to conclude that the scheme is fully applicable to the sti problems. However, as shown in Fig. 5, it is only a comparison of the accelerations computed by the HHT a- scheme and the proposed mid-point rule which shows the superiority of the latter in terms of its ability to take larger time steps and still produce a very smooth time-history of horizontal acceleration results. The latter applies even for a fairly large time step Dt = 0. s. These results are further corroborated by the total energy computation presented in Fig. 6. We can see that rst the Newmark method and than later the HHT a-method leave the energy level corresponding to free swinging of the pendulum, and switch to the false result corresponding to the much higher energy level. The corresponding buildup of the axial force, shown in Fig. 7, as well as the corresponding deformed shape, readily reveal that the pendulum has jumped into the axial vibration mode with no further swinging which is an entirely wrong solution. It can also be seen in Fig. 7 that not only is the proposed mid-point scheme free from the same pathology, but it also ensures the exact energy conservation in the discrete setting, the property which is shared by neither Newmark's nor by the HHT a- scheme even before their blow-up. 4.. Three-bar swing This example is adopted from Bauchau, Damilano and Theron [5], where it was used to illustrate the performance of their proposed energy conservation time stepping scheme. The three-bar swing (see Fig. 8) consists of a exible beam hinged at its ends into two rigid links. The selected value of Young's modulus is E = 73 GN/m and section A =5 mm. The mass density of the beam is chosen as r = 700 kg/m 3. A concentrated mass m = 0.5 kg is rigidly connected to the beam at its mid-span position. Two rigid links are modeled by the same beam elements with corresponding sti ness coe cients chosen an order of magnitude larger than the corresponding exible beam coe -

12 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± Fig. 3. Simple pendulum: tip displacement components.

13 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 3 Fig. 4. Simple pendulum: tip horizontal acceleration obtained by Newmark's average acceleration scheme. cients. The large di erence of sti ness coe cients places the present test in the category of sti problems. The exible beam is modeled by 4 beam elements and each rigid link is modeled with one element only. The system is initially at rest in the position shown in Fig. 8. It is set in motion by applying a pulse in the horizontal direction to the mid-span mass. The pulse is of the triangular shape, starting at 0, peaking at t = 0.8 to and diminishing at t = After that time, the system undergoes free vibrations, with its total energy being preserved. For the prescribed loading the system undergoes a smooth low mode motion, swinging to the right, in such a manner that the two ends of the beam remain on the circular trajectories rotating in the same sense (see deformed shape in Fig. 9). Roughly at time Fig. 5. Simple pendulum: tip horizontal acceleration obtained by the HHT a-scheme and the present scheme.

14 4 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± Fig. 6. Simple pendulum: time history of total energy obtained by Newmark average acceleration scheme, the HHT a-scheme and the present scheme. Fig. 7. Simple pendulum: time history of axial force obtained by Newmark average acceleration scheme, the HHT a-scheme and the present scheme.

15 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 5 Fig. 8. Swing: mechanical and geometric characteristics. t = 0.64 s, the right end of the beam is forced to reverse the sense of rotation being pulled over by the constrained motion of the left end of the beam, which triggers high mode axial vibrations in the beam. This is immediately apparent from Fig. 0, where time histories of horizontal and vertical displacement components of point B (initially placed at one quarter of the beam length from the left end of the beam) clearly display a sharp increase in high frequency content of motion. A very similar tendency is noted in the time history of the axial force, where the sudden motion reversal of the right hand end of the beam introduces a signi cant local peak (see Fig. ), followed by subsequent high frequency vibrations of reduced amplitude with respect to this peak value. This event does not perturb the predominantly swinging motion. We show in Fig. that the proposed scheme achieves perfect energy conservation despite the described interference of the high energy modes. This is a very signi cant improvement with respect to the results of Bauchau, Damilano and Theron [5], who reported a number of di culties in enforcing energy conservation with both the scheme proposed there and in the HHT a-scheme Multi-body system This example is chosen to illustrate the versatility of the presented approach in dealing with di erent joints in a multi-body system, without increase in the problem complexity. The multi-body system (see Fig. 3) consists of 4 exible members interconnected by either revolute or prismatic joints. Both of these joints can be taken into account through the standard nite element assembly procedure, simply by constraining the corresponding Fig. 9. Swing: deformed shapes from 0 to, in increments of 0..

16 6 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± Fig. 0. Swing: time history of displacement components of point B. dynamic motion and not having to increase the total number of unknowns. The nite element model is constructed by 0, - node beam elements. The system is initially at rest, as shown in Fig. 4. It is put in motion by applying a concentrated moment at its right support. The value of the applied moment varies linearly, peaking at 5 Nm at time 0.5 s, and diminishing at time 0.5 s. As shown by the time history of vertical displacements at points A and B in Fig. 5, the initial motion of the system is relatively smooth. At time 0.5 s (approximately), the high frequency modes related to axial vibrations of Fig.. Swing: time history of beam axial force.

17 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 7 Fig.. Swing: time history of total energy. the system start gradually building up (see Fig. 5). A similar tendency is observed in the time history of axial force computed at the Gauss point to the left of node A (see Fig. 6). However, the total energy computed by the proposed scheme remains constant and una ected by these high frequency vibrations. This is not the case for either the Newmark or the HHT a-scheme, which both lead to divergent behavior manifested in the corresponding blow-up of the total energy (see Fig. 7) Planar motion of a pinned beam The last example is adopted from Ref. []. It considers the free vibrations of a exible beam, pinned at one end and free at the other (See Fig. 8). The chosen properties are: EA =4 0 7, GA = 0 7, EI =.3 0 7, A r = 0.05, J r = and l = 00. The beam is set in motion by applying the initial velocity, distributed linearly, so that it peaks at 600 in/s in the middle of the beam, and decreases to zero at the ends. Subsequently, the beam undergoes a large rotational motion about the pinned end, in which the salient characteristics of the motion, such as the total energy and the angular momentum, remain conserved. The nite element model of the beam is composed of eight, -node beam elements. The computation is performed by using the present scheme, as well as the Fig. 3. Multi-body system: mechanical and geometric characteristics.

18 8 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± Fig. 4. Multi-body system: deformed shapes. Newmark and the HHT a-schemes. The total time is set to T = 0 s, and the selected time step is Dt = 0.0. Selected deformed shapes at each third time step are plotted in Fig. 9, for the rst.4 s of motion. The plots for the time history of the angular momentum, computed by the present scheme as well as the Newmark and HHT a-schemes, are given in Fig. 0. We can see that the present scheme preserves the angular momentum, whereas neither Newmark's nor the HHT a-schemes possess this desirable property. Fig. 5. Multi-body system: time history of vertical displacement at points A and B.

19 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± 9 Fig. 6. Multi-body system: time history of axial force. 5. Conclusions The proposed methodology for solving the problems in nonlinear dynamics applies to the multi-body systems composed of highly exible to almost perfectly rigid members undergoing large planar motion. A number of practical problems can be solved with the presented approach. A signi cant departure from the early works on multi-body system dynamics, based on the oating frame approach and restricted essentially to an in nitesimal deformation case, is the ability to take into Fig. 7. Multi-body system: time history of total energy.

20 0 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± Fig. 8. Pinned beam: mechanical and geometric characteristics. account the truly large deformation and the model in, highly exible members without a signi cant increase in problem complexity. The latter was made possible by using the geometrically nonlinear beam model of Reissner. Several recent works have explored the combination of such a beam model with the classical Newmark scheme, demonstrating their suitability for modeling of multi-body systems composed of highly exible beams. This work con rms the most signi cant nding of very recent research on dynamics of beams in the more realistic cases of multi-body systems, where a large di erence in sti ness appears between di erent members or even for a single member in di erent deformation modes; the Newmark scheme is no longer appropriate and exhibits instabilities due to the undesirable contribution of high frequency modes. It is also shown that one cannot cure these instability problems by introducing the numerical dissipation to damp high modes, such as in the HHT a-scheme, but merely postpones them. In contrast to these developments, the proposed mid-point scheme completely eliminates the instability problem. The choice of the mid-point scheme was made following the recommendations of several lucid works on numerical analysis dealing with sti problems (e.g. Refs [0, 4]). However, the numerical analysis was not su cient to fully master the problems: the proposed modi cation of the standard mid-point scheme enforcing total energy conservation is very motivated by the mechanics of the problem. We nd it quite remarkable that the proposed modi cation is able to control the instability inducing contribution of high modes without actually eliminating them or damping them out. Moreover, the latter is done while retaining a highly coupled nonlinear form of governing equations without the need to actually extract the high modes of vibrations. In conclusion, with all these desirable features and a careful development of the discrete approximation Fig. 9. Pinned beam: deformed shapes from 0 to.4 s, at each third time step.

21 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± Acknowledgements The nancial support of ABONDEMENT ANVAR is gratefully acknowledged. SM was supported by French-Algerian Cooperation BGFA program. References Fig. 0. Pinned beam: time history of angular momentum. leading to a sparse structure of the tangent operator, we believe we have developed a very e cient approach to dealing with practical multi-body systems. [] Antman S. Qualitative theory ordinary of di erential equations of nonlinear elasticity. Mech Today 974;:58± 0. [] Banerjee AK, emak ME. Multi- exible body dynamics capturing motion induced sti ness. ASME J Appl Mech 99;58:767±75. [3] Bathe KJ, Ramm E, Wilson E. Finite element formulations for large deformation dynamic analysis. Int J. Numer Methods Engng 976;9:353±86. [4] Bathe K.J.. Finite element procedures in engineering analysis. Prentice-Hall, Englewood Cli s, 995. [5] Bauchau OA, Damilano G, Theron NJ. Numerical integration of nonlinear elastic multi-body systems. Int J. Numer Methods Engng 995;38:77±5. [6] Belytschko T, Hsieh BJ. Nonlinear transient nite element analysis with convected coordinates. Int J. Numer Methods Engng 973;7:55±7. [7] Cardona A, Geradin M. Time integration of equations of motion in mechanism analysis. Comput Struct 989;33:80±0. [8] Cris eld M, Shi J. A co-rotational element/time-integration strategy for non-linear dynamics. Int J. Numer Methods Engng 994;37:897±93. [9] Cris eld M. Non-linear nite element analysis of solids and structures. Wiley, Chichester, 997;vol : advanced topics. [0] Dahlquist G, Bjorck A. Numerical methods. Prentice Hall, Englewood Cli s, 974. [] Downer JD, Park KC, Chiou JC. Dynamics of exible beams for multibody systems: a computational procedure. Comp Meth Appl Mech Engng 99;96:373±408. [] Fraeijs de Veubeke BM. The dynamics of Flexible bodies. Int J Engng Sci 976;895±93. [3] Geradin M, Rixen D. Parameterization of nite rotations in computational dynamics: a review. In: Finite rotations in solid and structural mechanics, Eur J Finite Elem. Ibrahimbegovic CA, Geradin M, editors. vol ±554, 995. [4] Hairer E, Wanner G. Solving ordinary di erential equations IIÐsti and di erentialðalgebraic problems. Springer, Berlin, 99. [5] Hilber HM, Hughes TJR, Taylor R. Improved numerical dissipation for time integration algorithms in structural dynamics. Earth Engng Struc Dyn 977;5:83±9. [6] Huston R, Wang Y. Flexibility e ect in multi-body systems, in Computer-aided analysis of rigid and exible mechanical systems. Pareira MFOS, Ambrosio JAC, editors. 35±76, 994. [7] Hsiao KM, Yang RT. A co-rotational formulation for nonlinear dynamic analysis of curved Euler beam. Comput Struct 995;54:09±7.

22 A. IbrahimbegovicÂ, S. Mamouri / Computers and Structures 70 (999) ± [8] Hughes TJR. Stability, convergence, growth and decay of energy of the average acceleration method in nonlinear structural dynamics. Comput Struct 976;6:33±4. [9] Hughes TJR, Caughy TK, iu WK. Finite element methods for nonlinear elastodynamics which conserve energy. ASME J Appl Mech 978;45:366±70. [0] Ibrahimbegovic A, Frey F. Finite element analysis of linear and nonlinear planar deformations of elastic initially curved beams. Int J Numer Meth Engng 993;36:339± 58. [] Ibrahimbegovic A, Frey F, Kozar I. Computational aspects of vector-like parameterization of three-dimensional nite rotations. Int J. Numer Meth Engng 995;38:3653±73. [] Iura M, Atluri SM. Dynamic analysis of planar exible beams with nite rotations by using inertial and rotating frames. Comput Struct 995;55:453±6. [3] Kane TR, evinson DA. Simulations of large motions of non-uniform beams in orbit. Part IIÐthe unrestrained beam. J Astronaut Sci 98;9:3±44. [4] Marsden JE, Hughes TJR. Mathematical foundations of elasticity. Prentice-Hall, Englewoods Cli s, 983. [5] Mayo O, Dominguez J, Shabana AA. Geometrically nonlinear formulations of beams in exible multi-body dynamics. ASME Vibr Acoust 995;7:50±9. [6] Meek J, iu H. Nonlinear dynamic analysis of exible beams under large overall motions and the exible manipulator simulation. Comput Struct 996;56:±4. [7] Riessner E. On one-dimensional nite strain theory: the plane problem. J Appl Math Phys 97;3:795±804. [8] Simo JC, Vu-Quoc. On the dynamics of the exible beams under large overall motionsðthe plane case: part I and part II. ASME J Appl Mech 986;53:849±63. [9] Simo JC, Tarnow N. The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. ZAMP 99;43:757±93. [30] Simo JC, Tarnow N, Doblare M. Nonlinear dynamics of 3 d rods: exact energy and momentum conserving algorithms. Int J. Numer Methods Engng 995;38:43±74. [3] Song JQ, Haug EJ. Dynamic analysis of planar exible mechanisms. Comp Meth Appl Mech Engng 980;4:359±8. [3] Stander N, Stein E. An energy-conserving planar nite beam element for dynamics of exible mechanism. Engng Comput 996;3:60±85. [33] Wasfy TM. A torsional spring-like beam element for the dynamic analysis of exible multi-body systems. Int J. Numer Methods Engng 996;39:079±96. [34] Zienkiewicz OC, Taylor R. Finite element method. McGraw-Hill, ondon, 99.