Dynamic analysis of 3D beams with joints in presence of large rotations

Transcription

1 Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 Dynamic analysis of 3D beams with joints in presence of large rotations G. Jelenic, M.A. Cris eld * Department of Aeronautics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 BY, UK Received October 1999; received in revised form 1 August Abstract In this paper we present a way to extend the earlier static master±slave formulation for beams and joints [1] to dynamic problems. The dynamic master±slave approach is capable of i) handling the problems of linear elasticity in a geometrically non-linear environment, ii) accounting for the non-linear kinematics of arbitrary types of joints and iii) performing the numerical time integration while preserving some of the important constants of motion like total energy and the total momenta for Hamiltonian systems in the absence of external loading. The performance of the formulation is demonstrated with the aid of two representative numerical examples. Ó 1 Elsevier Science B.V. All rights reserved. Keywords: Master-slave technique; Multi-body dynamics; Conserving dynamic integrators; 3D rotations; Flexible mechanisms 1. Introduction Structures composed of beams and joints undergoing large overall motion, including large 3D rotations, can be used to model a variety of engineering problems. Among others, these problems include deployment and retraction of large truss roofs and domes [±9], deployment of large satellite aerials and other exible appendages [1±13], opening of folded wings on rockets and missiles during the ight [1], actuation of serial robot arms and hands [14±4], operation of exible mechanisms [14,5±35], analysis of sprung suspension systems in automotive industry [36,37], and actuation of parallel robotic mechanisms and platforms [38±4]. In all of these applications, the axes of kinematic releases translate and rotate along with the structure, which means that the joint kinematics has to be expressed in the body-attached coordinate system. This proves to be a considerable problem in three-dimensional space, especially if the elasticity of the members is to be taken into account. A natural idea is to account for the joint kinematics through the use of Lagrange multipliers [11,9,3,43±51]. This, however, poses the problem of applying a time-integration scheme, which takes into account the joint kinematics without introducing constraint violation [5]. An additional problem in the time-integration of systems of di erential and algebraic equations appears to be the unconditional instability of Newmark's method [43] and, consequently, Cardona and Geradin advocated the introduction of some sort of numerical damping. Furthermore, the Lagrange multipliers pose as additional force) degrees of freedom, which introduce complexities when mixed with the other kinematical) degrees of freedom and also increase the size of a problem. The latter drawback is avoided if the joints are simulated using a penalty-type approach [18,53±57], but such a procedure is dependent on the choice of parameters, lacks robustness [1] and is prone to numerical ill-conditioning. Also, not all of the joint types can be * Corresponding author /1/$ - see front matter Ó 1 Elsevier Science B.V. All rights reserved. PII: S ) 344-3

2 4196 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 Notation d; d m ; q slave, master and released displacements Q; Q m ; Q slave and master triads and the triad associated with released rotation u released rotation in xed coordinate system q i ; q i m columns of Q and Q m i ˆ 1; ; 3) s; u released translation and rotation in coordinate system de ned by Q m w tangent-scaled released rotation w ˆ tan u= u in coordinate system de ned u= by Q m db; db m ; db r spin variables associated with the variation of Q, Q m and Q r u; u m incremental over a time step) slave and master translations u r incremental over a time step) released translation in coordinate system de ned by Q m;n a; a m incremental over a time step) slave and master tangent-scaled rotations a r incremental over a time step) released tangent-scaled rotation given in coordinate system de ned by Q m;n S a r transformation from db r to da r dp variations of slave variables dp mr variations of master and released variables H transformation from variations of master and released variables to variations of slave variables Dp changes of slave variables over a time step Dp mr changes of master and released variables over a time step E transformation from changes of master and released variables to changes of slave variables over a time step A; B; C; D energy conserving matrix coe cients within E t; Dt time, time-step length a; b; c parameter in the HHT method and Newmark's integration parameters g; g mr vectors of slave and master/released dynamic residuals t; q; f vectors of inertial, internal and external loads r; K position vector and orientation of a cross-section in a beam nite element N; M stress and stress-couple resultants in a beam nite element q; E; G; m density of material, Young's modulus, shear modulus, Poisson's coe cient A; A ; A 3 cross-sectional area and shear areas J t ; I ; I 3 torsional and cross-sectional moments of inertia L; x length of a nite element, arc-length coordinate N; M number of nodes on nite elements H; T total and kinetic energies k; p speci c translational and angular momenta L; J total translational and angular momenta I i Lagrangian polynomial shape functions K ij dynamic sti ness matrix relating the unknowns at node j to the residual at node i for the original element K ij H; K ij E dynamic sti ness matrices relating the unknowns at node j to the residual at node i due to the variation of H and E ; _ new derivatives of with respect to time, updated quantity b ; T cross-product operator, transposed quantity ; n at times t and t n n 1; n 1 at times t n 1 and t n 1 i ; i kinematic and force quantities at node i

3 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195± A ; B m ; r ; mr i f ; i / d ; D quantity related to element A and B master variable, released variable and vector containing both force and moment parts of load vector i variation of, change of over a time step modelled using a penalty approach [1]. A third idea is the introduction of the joint kinematics at a node prior to the assembly of the nite element mesh. In [1] this idea was called the ``master±slave'' approach, while in [58] it was referred to as the ``parent±child'' method. The modelling of linear elasticity in 3D beams is not trivial, the main di culties being the non-linear relationship between the deformed geometry and the adopted strain measures [59,6] and the extensive presence of large 3D rotations, which are governed by a speci c non-commutative algebra [61,6]. Another di culty lies in a possible loss of strain invariance when trying to interpolate rotations in a standard additive manner [63±65]. The theory of beams presented in [59,6], which we will refer to as the Reissner±Simo theory, provides a sound base for the dynamic applications. This theory can be regarded as an extension of the classical Kirchho ±Love theory [67] to account for the shear deformation. Alternatively, it can be viewed as a member of the family of Cosserat constrained director theories, where the two directors remain rigid and orthogonal to each other [68±7]. Following the establishment of the fully geometrically exact 3D beam theory [59], the rst nite element implementations of this theory have been reported in [6,71]. In this approach, after facing the complexities of modelling the elasticity, we are free to express the kinetic energy with respect to the inertial frame, thus arriving at a very simple quadratic form which includes all of the inertia e ects [66±76]. Alternative, older, formulations preferred to consider linear relationships between the geometry and the strain measures, which are only possible when i) the total motion is split into rigidbody part and the deformation with respect to a ``shadow'' frame and ii) when the latter is small. Such formulations encountered high non-linearities in the kinetic energy and were often unable to capture some of the important phenomena like the centrifugal sti ening e ect [77,78]. Traditional non-linear nite element formulations usually adopt either the Newmark method [79] or the Hilber et al. a-method [8] which adds a form of numeric damping. However, in recent years much research has demonstrated the potential instability of these methods in a large-rotation environment [43,53,64,66,7,81±83], even when numerical damping is applied [84]. Much work has been dedicated to resolving this issue [3,66,81±9]. In [66], an energy/momentum-conserving algorithm for 3D beams, which is stable in the presence of large rotations, has been developed. Unfortunately, the formulation su ers from the lack of strain invariance, discussed earlier in relation to statics. The authors are working towards a solution to this problem but, so far, without full success. In this paper, we build on our earlier work [1] and apply the master±slave approach to problems of structural dynamics. In the master±slave approach, a beam element with a particular type of end release is assumed to be unconnected to the adjoining elements prior to the nite element assembly. In contrast to the standard procedure, where at this stage the conditions of compatibility of deformations at structural nodes are introduced, in the master±slave approach we now de ne the relationships between the displacements and rotations from both sides of the joint, which depends on the character of the joint in question. In some special cases revolute joint in plane [91] or spherical joint in space [5]) this is not needed and the joint can be easily modelled using linear constraint equations or special assembly procedure. It should be noted that the establishment of the algebraic kinematic relationship which de nes the behaviour of a joint is possible only in case of holonomic joints [9]. Consequently, rotational joints revolute, universal, spherical joints), and translational joints e.g. a prismatic joint) with rigid sliding segments, which fall into this category, can be modelled accurately and ef ciently. In case of non-holonomic joints e.g. a prismatic joint with exible sliding segments [93]), the master±slave approach as presented in this work can not be directly applied. Extending the earlier static formulation [1] to cater for dynamic problems is remarkably straightforward and boils down to applying the linear operator between the variations of slave variables and master/released variables, which originally acted on static residual force vector, to act on the vector of dynamic residuals. In

4 4198 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 this way, traditional approaches like Newmark's [79] and Hilber et al.'s a-method [8] are immediately accommodated. As in [1], it is intended that this procedure should be immediately applicable to all elements having six degrees of freedom at each of the two end-nodes elements with internal nodes can also be handled). The six degrees of freedom should involve three translational and three rotational degrees of freedom, the variations of which are work conjugate to the vector of residuals applied. The developments in this paper are directly applicable to the elements having spin variables as the variations of the rotational degrees of freedom as in [64,7,73,81]. Other rotation variables, as in [74,75], can be used although, in these circumstances, some modi cation will be required to the presented theory. The issues of numerical stability also a ect the formulations for the joints and in this paper we present a procedure which ensures conservation of both energy and the momenta and is stable in the presence of large rotations. This procedure introduces an ``energy-conserving'' transformation which relates the changes of the slave and the master/released variables over a time step rather than the variations of the slave and the master/released variables, utilised in the original formulation. From a family of energy-conserving incremental transformations, a particular transformation is then chosen, which provides the conservation of the total momenta. In order for the method to be fully energy and momentum conserving it is essential that the underlying element formulation is itself energy and momentum conserving. At present, the adopted energy and momentum-conserving technique is applicable to beam elements having three translational and three rotational degrees of freedom at each end-node which utilise the interpolation of incremental tangent-scaled rotations [65,66]. In this paper, the original approach [1], developed for holonomic prismatic sliding) joints, revolute joints hinges), spherical joints and cylindrical joints, will be generalised in order to make it applicable to universal also called Hooke or Cardan) joints, the kinematics of which requires a special treatment. Clearly, the behaviour of joints and hinges will in practice be associated with friction and dissipation, which are issues not addressed in this paper. We believe that the rst important step is to develop formulations that are fully conserving in the absence of such dissipation. These issues could then be addressed by generalising the underlying energy conserving algorithm to provide an amount of energy dissipation in a manner similar to that presented in [9]. The paper will end by describing the application of the developed beam elements with end releases to two problems. The rst example tests a simple problem with a universal joint in relation to results in the literature. The second example considers the dynamic rotation of a exible double pendulum with a variety of joint types. This example is used to demonstrate the conserving properties of the new algorithm.. Basis for the master±slave procedure The kinematics of end releases suitable for the application of the nite element technique has been presented in [1]. In this section, we summarise the main results obtained therein and, in addition, make a number of changes which lead the way for the later dynamic formulation. The underlying philosophy of the master±slave approach is illustrated in Fig. 1 a). A beam element is assumed to be unconnected to the adjoining elements prior to the assembly of the structural equilibrium. In contrast to the standard assembly procedure, which at this stage introduces the conditions of compatibility of deformations at structural nodes, in the master±slave approach we introduce the existing joint kinematics. The node which is initially shared by a number of elements, one of which is not fully connected to the others, is in the deformed con guration no longer completely shared and from Fig. 1 a) we establish the following relations: d ˆ d m q; :1 Q ˆ Q Q m ; : where d m and Q m de ne the displacement vector and rotation matrix of a node taken to be the master node, and d and Q de ne the displacement vector and rotation matrix of the disconnected at least partially) slave node. The rotation matrices Q m and Q are shown in Fig. 1 a) in terms of their three constituent orthonormal vectors. The translational displacements at the slave node, d, di er from those at the master node,

5 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195± Fig. 1. a) Master, slave and released nodal variables; b) released rotation in a revolute joint; c) released translation in a prismatic joint. d m, by the relative displacement vector, q. The rotation matrix triad) Q in.) de nes an equivalent multiplicative relative rotation..1. Axes of release When modelling di erent types of joints, the master variables, d m and Q m, are generally not entirely independent of the slave variables, d and Q. Depending on the type of joint, some of the components of the displacement vectors, d m and d, and/or parameters of the rotation matrices, Q m and Q, can be the same. Di erent types of joints are de ned by releasing displacements and/or rotations along/around chosen axes. However, in a geometrically non-linear environment, these axes rotate together with the structure. In the present master±slave approach we de ne the master triad as the one along/around which base vectors q 1 m, q m, q3 m the actual releases take place. For translational joints, this means that the di erence vector, q, between the master and slave variables is, when transformed into coordinates de ned by the master triad, equal to the vector of released displacements

6 4 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 s ˆ Q T m q; :3 where the vector of released displacements, s, has zero components in non-released directions. In a similar fashion, if from the rotation di erence matrix, Q ˆ exp bu, we extract the rotational vector u, and transform it into coordinates de ned by the master triad, the result must be equal to the rotational vector of released rotations u, i.e., u ˆ Q T m u ; where u has zero components in non-released directions. Here and throughout the paper, a hat superimposed onto any 3D vector eld denotes a mapping which generates a skew-symmetric matrix, i.e., < v 1 = v 3 v 8 v ˆ v : ; R3 9 bv ˆ 4 v 3 v 1 5 so 3 : v v 1 v 3 It should be noted that for any two vectors v; w R 3 the relations bvw ˆ v w ˆ wvˆ bwv hold true. For the exponential relationship between skew-symmetric matrices and orthogonal matrices e.g. Q ˆ exp bu ) see [61,6]. For a revolute joint which allows rotation about the rotating vector q 1 m, we would only consider its rst component to be non-zero. This is illustrated in Fig. 1 b). An example of a prismatic joint allowing for a translation along the rotating vector q m is given in Fig. 1 c). More details on the master±slave concept can be found in [1]. Following from this discussion,.1) and.) may be rewritten as d ˆ d m Q m s; Q ˆ Q m exp bu; :4 :5 where, in deriving.5), advantage has been taken of the relationships and d Mv ˆ M bvm T exp d Mv ˆ M exp bvm T ; valid for any orthogonal matrix M and any 3D vector v. Note that the term ``translation along the rotating vector'' in the case of the prismatic joint depicted in Fig. 1 c) only makes sense if the sliding segment is rigid, i.e.,.4) cannot be formulated in the case of prismatic joints or indeed any joints with translational degrees of freedom) with exible sliding segments. In the latter case, the kinematic relationship between the master, the slave and the released degrees of freedom can only be fully de ned in terms of the non-integrable in nitesimal changes of these degrees of freedom, which is an indication of the non-holonomic nature of such constraints [9]. In practice, it is often more convenient to work with tangent-scaled rotational vectors. A tangent-scaled rotational vector w of the unscaled rotational vector u is de ned as [84,87] and w ˆ tan u= u; u ˆkuk u= exp ^u cayw; :6 where exp ^u and cayw are given as [61,6,66] exp ^u ˆ I sin u u 1 cos u ^u ^u ; u ˆkuk; u

7 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195± cayw ˆ I ^w 1 ; w ^w w ˆkwk; so that.5) can be re-expressed as Q ˆ Q m cayw: :7 In order to reduce problems associated with excessive magnitudes of w it is useful to work with incremental over a time step) rather than the total released tangent-scaled rotations. The use of incremental quantities is, in any case, essential for the dynamic formulations that will be considered later because the time-integration schemes involve incremental terms. To this end, we write.7) at two time instants t n and t n 1 : Q n ˆ Q m;n cayw n ; Q n 1 ˆ Q m;n 1 cay w n 1 ; :8 :9 which gives Q n 1 ˆ Q m;n 1 cay w n 1 cayw n T Q T m;n Q n; where indices n and n 1 here and throughout the paper denote time instants t n and t n 1. By recognising the rotation matrix associated with the incremental released rotation which we will call a r ) in the product cayw n 1 cayw n T we further obtain Q n 1 ˆ Q m;n 1 cay a r Q T m;n Q n: :1 3. Conventional approach using the principle of virtual work The approach given in this section presents a natural way to extend the static master±slave technique [1] to the problems of structural dynamics of 3D beams with large rotations Relationships between the variations of the master, slave and released variables The rst step in the nite element calculation is the application of the principle of virtual work in order to obtain the element force vector. To this end, we need to express the variations of the slave variables in terms of the master and released variables and their variations. In particular, we will aim to express db associated with the variation of Q n 1 ) in terms of db m associated with the variation of Q m;n 1 )anddb r, which is associated with the variation of caya r. The variation of the rotational vector, db, relates to the variation of the rotation matrix in the following way [6] dq ˆ d b bq: 3:1 The components of db will in future be referred to as the ``spin'' variables. For the purposes of later developments it is worth mentioning that the variation of caya r results in db r caya r ˆ d caya r ; which after some algebraic manipulation gives see [66, Lemma 4.1]) db r ˆ S 1 a r da r ; 3:

8 4 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 where S a r and its inverse are given as S a r ˆI 1 ba r 1 a 4 r a r ; S 1 a r ˆ 1 I 1 ba a r r ; ar ˆka r k: The above equivalence d caya r caya r T ˆ \S 1 a r da r and 3.1) give the variation of.1) as db ˆ db m Q m S 1 a r da r ; 3:3 3:4 where subscript n 1inQ m;n 1 has been dropped for convenience. In contrast to the spin variable db r, da r will be referred to as ``additive'' in nitesimal rotation. This name comes from the observation that the additive in nitesimal rotations are the in nitesimal changes of respective rotational vectors, therefore the update of these vectors is performed in the standard additive fashion. In contrast, spin variables are variations which are not associated with any rotational vector, and they serve to update the rotation matrix directly. As mentioned at the end of Section, here we use incremental tangent-scaled released rotations in contrast to the total unscaled released rotations used in [1]. In order to enable the application of the master± slave approach to all types of joints, we here depart additionally from the original approach presented in [1] and express the slave spin rotation db in terms of the master spin rotation db m and released additive in- nitesimal rotation da r. The reason for opting for the additive in nitesimal released rotations will become apparent and will be commented on) at the end of this section. The relation between the master and the released variations on the right-hand side and the slave variations on the left-hand side follows from Eqs..4), 3.) and 3.4) as dd ˆ Qm I Q d ds >< m s da r db Q m S 1 a r I dd m 8 >: db m 9 >= : 3:5 >; Before proceeding further, it should be re-emphasised that, in practice, not all of the released variables will in fact be ``released''. Instead, some of them will be set to. For each node i on the beam, 3.5) can be rewritten in a compact form as dp i ˆ H i dp mr;i ; 3:6 where dp T i ˆhdd T i db T i i, dpt mr;i ˆhdsT i da T r;i dd T m;i db T m;i i and " # H i ˆ Qm;i I \Q m;i s i : 3:7 Q m;i S 1 a r;i I 3.. Element equilibrium and its linearisation Application of the principle of virtual work to a general N-noded beam element leads to the relationship X N dp i g i XN dp i t i q i f i ˆ ; 3:8 iˆ1 iˆ1 where g i ˆ gi f g i is the dynamic residual or ``out-of-balance'' load vector and t i ; q i and f i are, respectively, the inertial, the internal and the external load vectors at node i. Ing i ; g i f / and g i / denote the nodal dynamic residual force and the nodal dynamic residual moment, respectively. We recall from the introduction to Section the important principle of the master±slave approach: An existing element is taken as it is and subjected to additional processing at the point of assembling the structural equilibrium. This processing depends on the choice of unknowns and their variations and in the present work the formulation has been applied only to particular type of rotational variations ± the spin variables

9 db. Therefore, for present purposes it is not necessary to look closely into the de nition of g i ± any nodal dynamic residual can be processed within the present master±slave methodology provided it is work conjugate to dp T i ˆhdd T i db T i i. In the case of isoparametric beams based on the Reissner±Simo theory [59], the dynamic residuals given in [73] P I on p. 155; note printing errors), [75] coming from 6)), [7] Eqs. 7)± 1)) and [64] Eqs. 4.)± 4.5)), among others, may all be used. Co-rotational formulations, e.g. [81] Eqs. 5), 45) and 5)), are equally applicable so long as the above condition on the structure of dp 1 and dp is adhered to. In all of these elements the integration in time has been performed by employing the Newmark trapezoidal rule [79] or its a-generalisation [8], but other time-stepping schemes are also applicable. The introduction of 3.6) into 3.8) now leads to the virtual work equation X N iˆ1 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 43 dp mr;i H T i gi ˆ : This equation must hold for any virtual variables, dp mr;i i ˆ 1;...; N and hence we can obtain the equilibrium equation at every node i as g i mr H T i gi ˆ ; 3:9 where g i is the dynamic ``out-of-balance'' load vector for the original element. If we expand this equation using a Taylor series around the actual con guration and neglect all the higher order variations, we obtain H T i gi dh T i gi H T i dgi ˆ : 3:1 The term dg i in 3.1) is related to the iterative slave variables, dp j, via the conventional tangential relationship dg i ˆ XN jˆ1 K ij dp j; where K ij de nes the original element's dynamic tangent sti ness matrix coupling nodes i and j. For elements mentioned earlier, K ij is de ned as follows: by 4.7) and 4.8) in [64], in Appendix I in [7], by 4.b) and 4.3) in [73] note printing errors in 4.3c) of [73]) and in Appendix 1 and references therein in [81], while in [75] it has not been given explicitly. If we substitute 3.6) into this equation and return to 3.1) we obtain H T i X N jˆ1 K ij H jdp mr;j dh T i gi ˆ H T i gi : The rst term on the left-hand side in this equation gives the primary contribution to the modi ed tangent sti ness matrix while the term on the right-hand side gives the negative modi ed residual. The second term on the left-hand side can be expressed in the form dh T i gi ˆ XN jˆ1 K ij H dp mr;j; 3:11 with K ij H ˆ for i 6ˆ j and K ii H being derived in Appendix A as 3 Q T m;i bgi f h i K ii H ˆ \ 4 a Q T m;i gi / S T a r;i Q T m;i gi / a r;i S T a r;i Q T m;i bgi / r;i : bg i f Q m;i bg i \ f Q m;i s i

10 44 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 Consequently, we can write X N jˆ1 K ij dp mr;j ˆ g i mr 3:1 with K ij ˆ H T i K ij H j K ij H ; g i mr ˆ H T i gi : At this point, the actual information on which variables out of those that may be released in fact are released should be introduced into the algorithm. This is done by deleting all the entries in the corresponding rows of g i mr and the rows and columns of K ij and inserting units on the corresponding diagonal positions of K ij for all the non-released variables. In so doing, we actually impose the given kinematics of joints on the variations of the released variables, rather than on the released variables themselves. In the rest of this section, we argue that for most types of joints, non-existence of a particular component in a released variable implies the non-existence of the same component in the variation of the released variable and that the above approach is justi ed. However, there is one joint type where this is not true and which requires a careful treatment. The displacements and their variations always belong to the space which is spanned by the same set of base vectors i.e. a variation of a one-dimensional displacement is co-linear to the displacement itself and a variation of a planar displacement does not ``stick'' out of the plane), so the above approach does not introduce any errors. The same is also true for the one-dimensional released rotations revolute joint), where the variation of the released rotation for one-dimensional rotations, there is no di erence between the spin variable and the additive in nitesimal rotation) is co-linear to the actual released rotation, and for the three-dimensional released rotations spherical joints), where both the released rotation and either of its variations spin or additive) are 3D vectors. For these types of joints, the imposition of constraints in the above way does not spoil the kinematics of joints. However, in the case of universal joints also called Hooke joints or Cardan joints), which enable rotations around two perpendicular rotating axes, only the additive in nitesimal rotation and not the multiplicative ``spin'' variable) belongs to the same twodimensional space as the released rotation itself. Indeed, by analysing the relationship 3.) between da r and db r, it is evident that the structure of the transformation in 3.3) creates a three-dimensional db r from a two-dimensional da r. Therefore, the kinematics of a universal joint requires the use of the additive in nitesimal rotation da r as the solution variable, which remains two-dimensional for any two-dimensional a r. This approach has been consistently followed in all the derivations heretofore, while in earlier work [1], the spin variable db r has been used. Hence the earlier formulation could not correctly handle universal joints. 4. Master±slave approach in energy±momentum conserving dynamics The approach given in this section presents a way to apply the master±slave technique to dynamic algorithms for 3D beams with large rotations which conserve the total energy and total translational and angular momenta of the system. In contrast to traditional approaches, the conserving algorithms do not originate from the principle of virtual work. Instead, the energy conservation implies that the nodal dynamic residual g i should be extracted from the condition that the change in total energy H over a time step remains equal to zero [3,66,81±89], i.e., H n 1 H n XN iˆ1 Dp T i gi ˆ ; where Dp i is the vector of displacement and rotation changes over the time step at node i. In Section 4.1 we formulate the equivalent of 3.6) in an incremental rather than an in nitesimal manner, i.e., we derive the relationship between the incremental over a time step) changes of the master and released variables on one hand and the incremental changes of the slave variables on the other. 4:1

11 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 45 The following development guarantees the conservation of the total energy only if the original element's dynamic residual g i is energy conserving, i.e., if 4.1) holds true regardless of the presence of any end releases. Also, the total translational and angular momenta will be conserved only if they are already conserved by the nite element to be embedded within the master±slave procedure. The nite element developed by Simo et al. in [66] g i is given by 94) and 95) therein) ful ls these requirements and the rest of this section along with Appendices B and C is therefore fully applicable to that element. It has recently been reported [63,64] that the standard nite element implementation of the Reissner± Simo beam theory does not automatically preserve the invariance of material strain measures and the nite element given in [66] also su ers from this drawback. An earlier attempt by the present authors to devise an element which would provide strain invariance on top of the conservation properties [65] was not fully successful the strain invariance was preserved only for quasistatic problems) and we are still endeavouring to provide a solution to this problem. However, the element devised in [65] does preserve energy and the momenta and is therefore applicable in the current context g i is given by Eqs. 3.7)± 3.1) of [65]). For the elements in [65,66], the incremental rotational variables within Dp in 4.1) are the tangent-scaled incremental rotations so in the following section we derive the relationship between the tangent-scaled incremental master and released rotations on one hand and the tangent-scaled incremental slave rotations on the other Relationships between the increments of the master, slave and released variables In order to derive the incremental relationship between the slave rotations and the master and released rotations, we rst apply.1) to obtain Q n 1 Q T n ˆ Q m;n 1 caya r Q T m;n ; 4: where indices n and n 1 denote time instants t n and t n 1. We also express the master and the slave triads at time t n 1 in terms of these triads at time t n and the incremental tangent-scaled master and slave rotations a m and a through Q m;n 1 ˆ caya m Q m;n ; Q n 1 ˆ cayaq n : 4:3 Inserting 4.3) into 4.) gives note that Q m;n caya r Q T m;n ˆ cay Q m;na r ) caya ˆ caya m cay Q m;n a r : 4:4 It is worth noting that for in nitesimal rotations between the triads at times t n and t n 1 the above equation turns into I db c ˆ I db d m Q m I d db r Q T m, which after eliminating the higher order term gives db ˆ db m Q m db r. Hence the present incremental relationship is in the limit equal to the in nitesimal relationship given earlier in Section 3 using 3.) in the second equation of 3.5)). Eq. 4.4) can be explicitly expressed as see 16.51) in [84]) 1 a ˆ 1 1 a a m Q 4 m Q m;n a m;n a r 1 r a m Q m;n a r : 4:5 It must be noted that, in contrast to the virtual work approach, in which the second row in 3.5) is linearly dependent on da r and db m and is thus uniquely expressed in these terms, the corresponding expression for the incremental kinematics 4.5) consists of the products of a m and a r. A general expression for the incremental tangent-scaled slave rotations in terms of the incremental tangent-scaled master and released rotations can be written as a ˆ Aa r Ba m ; 4:6 where A and B in this equation are any matrix coe cients that satisfy 4.5). Before addressing an issue of a suitable choice of these coe cients, we also derive the relationship between the slave displacements and the

12 46 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 master and released displacements and rotations in an incremental sense. In order to do so, we apply.4) at two time instants t n and t n 1 and subtract the result at time t n from the result at time t n 1. In this way we obtain d n 1 d n ˆ d m;n 1 d m;n Q m;n 1 s n 1 Q m;n s n : 4:7 By introducing new symbols u ˆ d n 1 d n ; u m ˆ d m;n 1 d m;n ; 4:8 u r ˆ s n 1 s n for the incremental slave, master and released displacements, and after some algebraic manipulation, 4.7) can be rewritten as u ˆ u m 1 Q m;n Q m;n 1 ur Q m;n 1 Q 1 m;n s n s n 1 : 4:9 After introducing the notation Q m;n 1 ˆ 1 Q m;n Q m;n 1 ; s n 1 ˆ 1 s n s n 1 4:1 and recognising see 44a) in [66]) Q m;n 1 Q m;n ˆ ba m Q m;n 1 ; 4:11 4.9) can be expressed as u ˆ u m Q m;n 1 u r \Q m;n 1 s n 1 a m : 4:1 For reasons already mentioned, 4.1) is not a linear expression in u r and a m, and at this stage it is better to keep it in a general form u ˆ u m Cu r Da m : 4:13 The relation between the incremental master and released variables on the right-hand side and the incremental slave variables on the left-hand side can therefore be expressed as 8 9 u ˆ C I D u r >< >= a r ; 4:14 a A B u >: m >; a m where the matrix coe cients A, B, C and D must satisfy 4.5) and 4.1). This equation which plays the same role as 3.5) in the in nitesimal case) can be rewritten for each node i on a beam in a compact form as Dp i ˆ E i Dp mr;i ; 4:15 where Dp i ˆhu T i a T i i; Dp mr;i ˆhu T r;i a T r;i u T m;i a T m;i i and E ˆ C i I D i : A i B i 4.. Element equilibrium and its linearisation For any nodal dynamic residual g i that is energy conserving in the sense that it satis es 4.1), a generalised energy conserving dynamic residual can be expressed as

13 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 47 g i mr ET i gi ˆ : 4:16 This follows from 4.1) and 4.15), where Dp i and Dp mr;i are, respectively, the changes in slave and master/ released variables at node i from the last converged dynamic equilibrium con guration at time t n )tothe current trial con guration at time t n 1. The resulting formulation is energy conserving for any choice of matrix coef cients A i, B i, C i and D i at node i which satisfy 4.5) and 4.1). It is shown in Appendix B.1 that the total translational momentum of the system is also conserved for any choice of these coef cients provided that the original nodal dynamic residual g i is translational-momentum conserving. However, the conservation of the total angular momentum is much more dif cult to attain. In Appendix B., it is shown that this important property is contained within the present algorithm for any momentum conserving g i for the following choice of the matrix coef cients B i ˆ I; D i ˆ 1 \Q m;i;n s i;n Q \ m;i;n 1 s i;n 1 : 4:17 These coe cients and the energy conservation requirements of 4.5) and 4.1) then provide the remaining matrix coe cients see Appendix B.3) 1 A i ˆ 1 1 a S T a m;i Q 4 m;i Q m;i;n a m;i;n ; r;i C i ˆ I 1 4 ba m;i Q m;i;n 1 ; 4:18 with S a m;i being de ned by 3.3) 1. After expanding 4.16) using a Taylor series around the actual con guration and neglecting all the higher order terms we obtain E T i gi de T i gi E T i dgi ˆ ; 4:19 where dg i is related to the variation of the slave variables, dp j, via the conventional tangential relationship dg i ˆ XN jˆ1 K ij dp j; 4: where K ij de nes the original element's equivalent dynamic) tangent sti ness matrix coupling nodes i and j. For element [66], this matrix is given by 113)± 116) of [66] see also Note I. in [7]). For element [65], the equivalent tangent stiffness matrix coupling nodes i and j is given by 5.13) and 5.14) of [65]. If we substitute 3.6) into 4.) and return to 4.19) we obtain E T i X N jˆ1 K ij H jdp mr;j de T i gi ˆ E T i gi : The second term on the left-hand side can be expressed in the form de T i gi ˆ XN jˆ1 K ij Edp mr;j ; 4:1 with K ij E ˆ for i 6ˆ j and K ii E being derived in Appendix C. Matrix K ii E has the following non-zero entries: K ii E 1;4 ˆ 1 QT m;i;n 1 bg i f 1 ba d 4 m;ig i f 1 4 QT m;i;n ba 1 m;i bg i f d ba m;i g i f S a m;i ;

14 48 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 A T K ii i g i / a m;i Q m;i;n E ; ˆ ; 4 a m;i Q m;i;n a r;i K ii E ;4 h i Q T m;i;n bg i / a m;i g i / I a m;i g i / A T i g i / a r;i Q T m;i;n ˆ S a m;i ; 4 a m;i Q m;i;n a r;i K ii E 4;1 ˆ 1 bgi f Q m;i;n 1; K ii E 4;4 ˆ 1 bgi f\ Q m;i;n 1s i;n 1 : Consequently, the tangent operator in the linearised equilibrium 3.1) now reads K ij ˆ E T i K ij H j K ij E: 5. Solution process One could solve the element equilibrium in a standard manner to give an iterative Newton±Raphson change at the structural level of the form 1 dp mr;j ˆ K ij g mr;i for every node j of every element in the structure. However, the released degrees of freedom within dp mr;j are only related to the individual elements and so can be solved for at the element level. The master translations are updated in the standard manner using d m;j;new ˆ d m;j dd m;j ; while the master rotation matrices are updated using Rodrigues's formula which is related to the exponentiation of a skew-symmetric matrix: Q m;j;new ˆ exp db d m;j Q m;j ; where db m;j is the incremental rotational vector ``spin'' variable) obtained directly from the nite element solver as part of dp mr;j. In the conserving algorithm, we need to update the incremental tangent-scaled master rotations, which is performed via 1 a m;j;new ˆ 1 1 a a m;j db 4 m;j db m;j 1 m;j a m;j db m;j ; which enables an alternative update of the master rotation matrices via Q m;j;new ˆ caya m;j;new Q m;j;n : The released variables are treated in a similar way, while the released translations are updated through s j;new ˆ s j ds j and the released incremental tangent-scaled rotations are updated through a r;j;new ˆ a r;j da r;j :

15 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 49 In these equations the quantities ds j and da r;j are obtained from the incremental released degrees of freedom within dp mr;j. Since the released tangent-scaled rotations are incremental, a r;j is reset to at the beginning of every new increment.the slave variables in the conventional algorithm are updated using.4) and.1) so that d j;new ˆ d m;j;new Q m;j;new s j;new ; Q j;new ˆ Q m;j;new caya r;j;new Q T m;j;n Q j;n: In the conserving algorithm, the incremental tangent-scaled slave rotations are updated using 4.5) as a j;new ˆ am;j;new Q m;j;n a r;j;new 1 a m;j;new Q m;j;n a r;j;new a m;j;new Q m;j;n a r;j;new ; which enables an alternative update of the slave rotation matrices via Q j;new ˆ cay a j;new Q j;n : 6. Applications Both the conventional and the conserving dynamic master±slave procedures for joints have been implemented within a research version of the nite element program LUSAS [94]. In the rst example, we test the conventional dynamic master±slave approach against results in the literature. In the second example, the conserving master±slave approach is demonstrated and assessed. In both examples we use the following two convergence criteria: i) the displacement norm, de ned as the square root of the sum of the squares of the nodal iterative displacements dd i over all the nodes in the structure as percentage of the square root of the sum of the squares of the total nodal displacements d i over all the nodes in the structure, must be less than 1 9, and ii) the residual norm, de ned as the square root of the sum of the squares of the nodal residual forces g i f the force part of the nodal residual vector g i ) over all the nodes in the structure as percentage of the square root of the sum of the squares of the nodal forces and reactions f i f the force part of the nodal vector of applied loads f i ) over all the nodes in the structure, must be less than 1 6. It is worth noting that end-releases can often be modelled without resorting to any speci cally designed techniques such as Lagrange multipliers, penalty approaches or the present master±slave method. Typical examples include modelling joints and sliders at support nodes, where any release is taken into account at the stage of the introduction of boundary conditions. However, this step is performed prior to the solution of the system of linear equations and since in non-linear analyses these equations give iterative changes to the unknown kinematic quantities, such a method necessarily implies the imposition of constraints on these iterative changes. As argued at the end of Section 3, if a formulation is based on the use of rotational spin variables this technique cannot be applied to universal joints, due to their speci c kinematic properties. Other examples where end-releases can be modelled without using speci c techniques are revolute joints in D problems and spherical joints in 3D problems. In these cases, it is su cient to take the necessary steps at the point of assembling the mesh of nite elements and to introduce the compatibility of only the translational degrees of freedom at nodes associated with the joint. In nite element codes, this is usually achieved by specifying linear constraint equations Example 1 ± Articulated exible beam In this example a beam attached to the ground via a universal joint with the two rotations around the principal axes of inertia of the cross-section being released) has been subjected to two concentrated forces at its free end as shown in Fig.. The forces follow the loading histories given in Fig. 3. This example has been solved in [9,3] for the following geometric and material properties: crosssectional and shear areas A ˆ A ˆ A 3 ˆ 9, moments of inertia around principal axes I ˆ I 3 ˆ 6:75, tor-

16 41 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 Fig.. Articulated exible beam with universal joint: geometry and loading. Fig. 3. Loading history of the articulated exible beam. sional moment of inertia J t ˆ 13:5, density q ˆ :78, Poisson's coe cient m ˆ :3, Young's modulus E ˆ 1. A nite element mesh of ve equal elements [74] was used and the analysis was run for the total response time t ˆ 1:5 using 6 equal time steps with Dt ˆ :5. The time integration was performed using the a-method [8] with the numerical damping coe cient a ˆ :1 which implied b ˆ :35 and c ˆ :6). The joints were processed using a Lagrange multiplier technique [74]. Here, we use our conventional dynamic master±slave approach and apply it to the linear beam elements given in [6,73]. We apply the a-method with the same amount of numerical damping using the method presented in [7] and run the problem with the same time-step length. Fig. 4 shows the components of the tip displacement along coordinate axes Y and Z. The present results are drawn as lines, while the results read from Figs. 3 and 4 in [9] are reproduced as point symbols. Fig. 5 shows the kinetic and the strain energies. Again, the present results are drawn as lines, while the results read from Figs. 1 and 11 in [3] are reproduced as point symbols. A close agreement between the present results and those in [9,3] is observed. Any discrepancies are attributed to the use of di erent elements, to the di erent application of the a-method, and most of all, to the di erent treatment of the joints. It was argued in [43] that ``in order to time integrate constrained systems, the algorithmic damping at in nite frequency is of utmost importance'' quoted from the abstract of [43]). This importance does not appear to be so big in the present master±slave approach, where the constraints are not introduced via the

17 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195± Fig. 4. Tip displacement of the articulated exible beam. Present formulation: Y-displacement solid line), Z-displacement dashed line), Ref. [9]: Y-displacement diamonds), Z-displacement circles). Fig. 5. Kinetic and strain energies of the articulated exible beam. Present formulation: kinetic energy solid line), strain energy dashed line), Ref. [3]: kinetic energy circles), strain energy diamonds). Lagrange multipliers. If the analysis is re-run using the Newmark trapezoidal rule i.e. with a ˆ ) the results drawn in Figs. 4 and 5 are almost entirely reproduced. 6.. Example ± Flexible double pendulum with di erent joint types We analyse the motion of a double exible pendulum under the impact of the linearly varying initial velocities given in Fig. 6. This example with a spherical joint which releases all three of the components of rotation) between the two beams was solved in [49,5] by using a staggered explicit time-integration scheme with good energy-conserving properties. The complexities associated with conservation of angular momentum in the presence of di erent types of joints were clearly recognised and in [5] it was suggested that for complex exible multi-body systems with di erent types of joints ``the most one can realistically hope

18 41 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 Fig. 6. Flexible double pendulum: properties and initial conditions. for is the preservation of the system energy''. We have to challenge this statement and contend that the conservation of both the energy and the momenta for such systems is possible. Furthermore, we will also solve the problem for the case of kinematically more demanding revolute and prismatic joints between the two beams of the pendulum. The example was originally solved as a exible spatial double pendulum, i.e., with a spherical joint between the two beams. Each beam is modelled using eight two-noded beam elements and we have adopted a time-step length Dt ˆ :8, which is approximately ve times smaller than the period of the second bending mode of natural vibration. No details were given in [49,5] as to the time-step size applied therein, but due to the explicit nature of the integration applied to kinematic quantities implicit integration was used for the Lagrange multipliers) it had to be much smaller. The analysis is run for a total response time t ˆ 8. For the given geometric and material parameters, the initial energy H which is equal to the initial kinetic energy T ) and the components of the initial angular momentum J around the coordinate axes Y and Z can be computed as H T ˆ 1 Z 6 Z Aqv x 6 y 6 dx Aqv x z 6 dx ˆ 891; Fig. 7. Total energy of the exible double pendulum with spherical joint.

19 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195± Fig. 8. Components of the total angular momentum for the exible double pendulum with spherical joint. L J y ˆ Aqv z L L ˆ 6415; L L J z ˆ Aqv y ˆ 169:

20 414 G. Jelenic, M.A. Cris eld / Comput. Methods Appl. Mech. Engrg. 19 1) 4195±43 The following results have been obtained by utilising isoparametric strain-invariant elements [64] in conjunction with the Newmark's trapezoidal rule [79] and Hilber et al.'s a-method [8] with a ˆ :5) and also with Simo et al.'s energy±momentum method [66] applied to the conserving master±slave technique. We rst analyse the original problem with the spherical joint in the middle. Newmark's trapezoidal rule experiences a well-known numerical instability which results in the sudden growth of the total energy near the completion of the rst revolution of the pendulum see Fig. 7). Fig. 7 also shows that the a-method successfully solves the problem, but in so doing dissipates almost a third of the total energy of the system, despite the fact that only 5% of numerical damping has been applied. In contrast, the proposed conserving methodology for joints successfully solves the problem and by design conserves the total energy of the Fig. 9. Deformed con gurations of the exible double pendulum with spherical joint during the rst and the second revolution conserving formulation).