Convergence of the generalized-α scheme for constrained mechanical systems

Transcription

1 Convergence of te generalized-α sceme for constrained mecanical systems Martin Arnold, Olivier Brüls To cite tis version: Martin Arnold, Olivier Brüls. Convergence of te generalized-α sceme for constrained mecanical systems. Multibody System Dynamics, Springer Verlag, 27, 85, pp <.7/s >. <al-49825> HAL Id: al ttps://al.arcives-ouvertes.fr/al Submitted on 6 Mar 27 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiues de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés. Distributed under a Creative Commons Attribution 4. International License

2 Convergence of te generalized-α sceme for constrained mecanical systems Martin Arnold Olivier Brüls Abstract A variant of te generalized-α sceme is proposed for constrained mecanical systems represented by index-3 DAEs. Based on te analogy wit linear multistep metods, an elegant convergence analysis is developed for tis algoritm. Second-order convergence is demonstrated bot for te generalized coordinates and te Lagrange multipliers, and tose teoretical results are illustrated by numerical tests. Keywords DAEs Generalized-α metod Introduction Te generalized-α sceme as been initially developed for te simulation of finite element models in structural dynamics. It allows a simple and efficient implementation, as well as an optimal combination of accuracy at low-freuency and numerical damping at ig-freuency. Tis last feature is especially interesting, since it allows to eliminate te contribution of non-pysical ig-freuency modes, wic are generally present in finite element models. Te generalized-α algoritm results from successive contributions by Newmark [8], Hilber, Huges and Taylor [4], and Cung and Hulbert [9]; an overview of its properties in te non-linear regime is also given by Erlicer et al. []. Tis work concerns te employment of te generalized-α sceme for te simulation of constrained mecanical systems. Cardona and Géradin [8] ave sown tat numerical damping is critical to avoid numerical oscillations in te Lagrange multipliers. Teir teoretical investigations are restricted to linear problems, but tey also report consistent results in nonlinear test cases. However, it is well-known tat te order of an integration algoritm can M. Arnold ) NWF III Institute of Matematics, Martin Luter University Halle Wittenberg, 699 Halle Saale), Germany martin.arnold@matematik.uni-alle.de O. Brüls Department of Aerospace and Mecanical Engineering, University of Liège, 4 Liège, Belgium o.bruls@ulg.ac.be

3 be reduced due to te presence of algebraic constraints [4, 3], wic calls for a rigorous convergence analysis of te generalized-α metod in te context of differential-algebraic systems. Recently, Lunk and Simeon [6] and Jay and Negrut [5] ave proven secondorder convergence for regularized index-2 formulations, i.e. algoritms wic make use of kinematic constraints at bot position and velocity level. In contrast, an algoritm based on an index-3 formulation, i.e., solely based on position level constraints, is analysed in tis paper. Tis algoritm can deal wit a non-constant mass matrix and it computes te acceleration variables wit second-order accuracy. It extends te approac of Negrut et al. [7] wo report on positive practical experience wit te Hilber Huges Taylor algoritm applied to te index-3 formulation of te euations of motion in an industrial multibody system simulation tool. Recently, Bottasso et al. [3] proposed a scaling tecniue to reduce te numerical instability effects being typical of suc time integration metods for DAEs of index 3, see [, 2, 4]. In te present paper, te analogy between generalized-α algoritms and linear multistep integrators elps to get a simpler and more intuitive convergence proof, see also te rater tecnical work of Lunk and Simeon [6] or Jay and Negrut [5]. Second-order convergence is demonstrated for bot te generalized coordinates, and te Lagrange multipliers. Te remaining part of te paper is organized as follows: te generalized-α algoritm and its multistep representation are introduced in Sect. 2. Estimates for local and global errors are proven in Sect. 3. Some tecnical parts of te proof are collected separately in Sect. 4. In Sect. 5, te results of te teoretical investigations are illustrated by numerical tests. Te conclusions in Sect. 6 summarize essential parts of te paper. 2 Te generalized-α metod Let us consider te constrained mecanical system M) = f,,t) T λ, ) =,t) 2) were ) represents te dynamics of te mecanical system and 2) represents te kinematic constraints. Te vectors and λ denote te generalized coordinates and te Lagrange multipliers, respectively, M is te symmetric mass matrix, te vector of apparent forces f collects external forces, internal forces and complementary inertia forces, and is te matrix of constraint gradients. Te mass matrix is not necessarily constant, but it may depend on te generalized coordinates, wic allows to cover te case of mecanical systems wit large rotations. We note tat te present developments could be furter extended to mecatronic problems, as suggested by Bruls and Golinval [6, 7]. 2. Description of te algoritm We propose an implementation of te generalized-α metod wic does not rely on a weigted formulation of te residual euation. Instead, te dynamic euilibrium is enforced exactly at every time step, wit tree major advantages: i) te accelerations are computed wit second-order accuracy, ii) te consistency of te algoritm is not affected if te mass matrix is not constant, and iii) te algoritm is closer to te pysics of te problem, wic also simplifies teoretical investigations. 2

4 Hence, te numerical variables n+, n+, n+, λ n+ satisfy )and2) at time t = t n+, wereas, te vector a of acceleration-like variables is defined by te recurrence relation α m )a n+ + α m a n = α f ) n+ + α f n, a =. 3) We empasize tat a is an auxiliary variable, wic is not eual to te true accelerations. Since M depends on, tis euation cannot be restated as a weigted form of ). Te generalized-α sceme is obtained using a in te Newmark integration formulae n+ = n + n β )a n + 2 βa n+, 4) n+ = n + γ)a n + γ a n+ 5) were is te step-size. Te numerical parameters α m, α f, β and γ can be selected in order to ave suitable accuracy and stability properties. Algoritm solves formulae 3), 4), and 5) togeter wit te dynamic euilibrium at time t n+. Te correction step involves te parameters β = α m 2 β α f ), γ = γ β 6) wic satisfy te properties n+ n+ = Iβ, Moreover, te iteration matrix is given by [ ] Mβ + C t γ + K t ) T S t = n+ n+ = Iγ. 7) 3

5 wit te tangent stiffness matrix K t = M f+ T λ)/ and te tangent damping matrix C t = f)/. Compared to a classical algoritm based on a weigted formulation of ), as described in [9, 4], Algoritm involves similar computational resources. Indeed, it only reuires one additional vector a, and te correction step, wic is te most demanding part of te algoritm, is barely modified. For small time steps, te matrix S t becomes severely ill conditioned. Bottasso et al. [3] ave proposed a scaling metod in order to avoid tis penomenon, see also te closely related scaling approac of Hairer and Wanner [3] in te classical general purpose DAE solver RADAU5. Te linear system S t x = r, 8) wit x T =[ T λ T ] and r T =[r ) T r λ ) T ] is replaced by te euivalent scaled form S t x = r 9) wit S t = D L S t D R, x = D R x and r = D Lr. In te numerical tests presented in Sect. 5, te diagonal left and rigt preconditioners [ ] [ ] Iβ 2 I D L =, D R = ) I I/β 2 are used to acieve an optimal conditioning of te matrix S t, see [3]. 2.2 Multistep representation of te algoritm Assuming tat te mass matrix is non-singular, te dynamic euilibrium is euivalent to an explicit form = g,, λ, t), ) =,t) 2) wit g = M f T λ). We also assume tat te constrained system, 2) as DAE index 3, i.e., tat te matrix g λ 3) is non-singular, were g λ is given by g λ = M T. Since te dynamic euilibrium is enforced at every time step, te algoritm leads to te same solution wen applied to te system, 2), and it is sufficient to analyse tis euivalent system. It is possible to eliminate a from te integration formulae at time steps t n t n and t n t n+, leading to a two-step formulation [] a k n+k + u k n+k = 2 b k g n+k, 4) a k n+k = c k g n+k, 5) = n+,t n+ ) 6) 4

6 wit te coefficients a = α m, a = + 2α m, a 2 = α m, u = α m, u = + α m, b = α f /2 β), b = α f )/2 β)+ α f β, b 2 = α f )β, c = α f γ ), c = α f ) γ)+ α f γ, c 2 = α f )γ. In Sect. 3 below, we will see tat te detailed analysis of te error propagation in te difference uotients n := n n t n ) 7) 2 is very useful to study te error propagation for te Lagrange multipliers λ. A two-step recursion for is obtained by te difference uotient of 4) in its original form and 4) wit n being substituted by n. In tis difference uotient, te velocities may be eliminated using 5) resulting finally in te multistep formula wit coefficients a k n+k = b k g 8) b = α f /2 + β γ ), b = α f )/2 + β γ)+ α f /2 2β + γ ), b 2 = α f )/2 2β + γ)+ α f β, b 3 = α f )β. 9) 2.3 Coice of te numerical parameters In te present paper, generalized-α algoritms wit fixed step-sizes are considered. In tat case, te generalized-α algoritm for unconstrained mecanical systems is second-order accurate provided tat [9] γ = 2 + α f α m. 2) Note, owever, tat tis condition is no more valid for variable step-size algoritms and sould be replaced by an update condition for te parameter γ, wic means tat te value of γ sould be adapted at eac time step to guarantee second-order accuracy [5]. Te numerical solution is zero-stable i.e. stable for ) if te polynomial ϱζ ) := a k ζ k 2) satisfies te root condition, i.e., if condition ζ i is satisfied for all roots ζ i, i =, 2 of polynomial ϱ. For multiple roots ζ i, te stronger condition ζ i < as to be enforced [2]. Since te roots of ϱ are ζ = and ζ 2 = α m / α m ), zero stability reuires α m.5. We note tat bot roots are necessarily simple. 5

7 Te algoritm is strictly stable at infinity i.e. strictly stable for,see[3]) if all roots ζ i of te polynomial σζ):= b k ζ k 22) satisfy ζ i <, i =, 2, 3. Lemma see Sect. 4 below) sows tat strict stability at infinity is guaranteed if α m <α f < 2, β> α f α m ) 23) and te order 2 condition 2) is satisfied. For stiff problems, te numerical solution sould be computed accurately only in te low-freuency range, wereas te ig-freuency response sould rater be damped out by te algoritm. Te ig-freuency numerical damping is represented by te spectral radius of te algoritm at infinity ρ : An undamped sceme is caracterized by ρ =, wereas ρ = means asymptotic anniilation of te ig-freuency response. For a given value of ρ [, ], Cung and Hulbert [9] ave proposed optimal algoritmic parameters for second-order ODEs α m = 2ρ ρ +, α f = ρ ρ +, β = 4 γ + 2 ) 2 24) wereas γ is computed according to 2). For ρ [, ), te resulting algoritms are bot zero-stable and strictly stable at infinity. Indeed, ρ < implies α m <α f < 2, γ> 2 and β = γ + ) 2 > γ + ) 2 γ ) 2 = γ = α f α m ), so tat bot conditions in 23) are satisfied. 3 Convergence analysis 3. Local truncation error By definition, te metod is convergent of order 2 in te classical unconstrained case if t n ) n and t n ) n are O 2 ). In contrast, te order 2 condition 2) means tat te local error, i.e., te error after one time step, is O 3 ). It can be demonstrated [2] tat tis last condition is satisfied if l n and l n are O 3 ), were te local truncation errors l n and l n are defined by introducing te exact solution t n ), t n ), λt n ) into 4) and 5) l n = a k t n+k ) + u k t n+k ) 2 b k gt n+k ), 25) l n = a k t n+k ) c k gt n+k ) 26) wit te notation gt n ) = gt n ), t n ), λt n ), t n ). 6

8 For te extension of tese results to te constrained case, see, 2), we consider furtermore, te local truncation error l n in 8) l n = a k t n+k ) t ) b k gt ). 27) Te classical order 2 condition in 2) implies tat te new consistency conditions a k = and j + )! a k k ) j+ k 2) j+) = j )! b k k 2)j j =, 2), are satisfied and te estimate l n =O 3 ) may be sown by Taylor expansion. 3.2 Error propagation in te differential solution components Wen a zero-stable algoritm is used to solve an ODE, te order condition implies global convergence [2]. Te extension of tis classical result from te ODE case to te constrained system in, 2), i.e., to index-3 DAEs, follows te basic lines of te convergence analysis for multistep metods applied to DAEs, see [3]. In a first step, let us analyse te error propagation during te integration process for te differential components, and. Using te multistep formulae 4), 5) and8), te defect 25), 26) and 27) become l n = l n = l a k e n+k + u e k n+k 2 b k e g n+k, 28) a e k n+k c k e g n+k, 29) n = a k e n+k b k eg 3) were e ) n = )t n) ) n represent a global error after n steps and e n := t n) t n ) n n = e n e n, 3) see 7). Inspired by te identical structure of te first terms in te rigt-and side of 28 3), te tree differential components, and and te corresponding local and global errors are summarized in t n ) n vt n ) := t n ), v n := n, t n ) n e n l 32) n e v n :=, l v n := = O 3). e n e n l n l n 7

9 Wit tis compact notation, te error recursion defined by 28), 29) and3) may be written as e v n+ = a e v n a a e v n 2 a + O) 2 e n+k + O) g e ) + O 3 = a e v n a a e v n 2 a + O) ) ) e v + e λ + O 3. 33) 2 Te propagation of te errors e n, e n and e n is dominated by te recursion coefficients a, a and a 2, and it is coupled wit te errors e λ n in te algebraic components λ by O) coupling coefficients. 3.3 Error propagation in te algebraic solution components For te analysis of te error propagation in te algebraic solution components λ, we assume tat trougout integration te numerical solution n, n, λ n ) remains in a small neigbourood of te analytical solution t n ), t n ), λt n )) e n C, e n C, e λ n C n ) 34) wit a constant C> tat is independent of n and. Because of te O 2 ) error bounds for e n, e n, e λ n in 4) below, te additional assumption in 34) is always satisfied if > is sufficiently small and te initial values of te numerical solution are sufficiently close to te analytical solution see also part c) of te proof of Teorem VII.3.5 in [3] for a more detailed discussion). Te conditions on e n, e n, e λ n in 34) allow to get an estimate for k b k eλ from 3). Wit we ave ψϑ) := g + ϑe, + ϑe, λ + ϑe λ,t ) e g = ψ) ψ) = ψ ϑ) dϑ = g λ + ϑe, + ϑe, λ + ϑe λ,t ) e λ dϑ + O) e + e ) = g λ t n )e λ + O) e λ + O) e + e ) 35) since + ϑe = t ) + O) = t n ) + O) etc., see 34). As before, te notation g λ t n ) is used as abbreviation for g λ t n ), t n ), λt n ), t n ). Te matrix product t n ), t n ) g λ t n ) is non-singular by te index-3 assumption, see 3). Terefore, te estimate for k b k eλ may be obtained multiplying 35) by [ g λ ) ]t n ), t n ) from te left and using te error recursion for e g n from 3): 8

10 b k eλ = [ ] ) g λ ) tn ), t n + O) b k eg e + e + e λ ) = O) ) tn ), t n + O) e Wit Lemma 3, see Sect. 4 below, we get a k e + e n+k ) + O l n ) + e λ. 36) ) b k eλ = O,t 2 ) ) l + O n + O) + O) e + e + e λ ) e n+k. 37) Te euilibrium conditions at t = t enforce,t ) = and 37) may be summarized in te compact form e λ n+ = b b 3 e λ n 2 b b 3 e λ n b 2 b 3 e λ n +O) e v +O) e λ +O 2). 38) Te propagation of te errors e λ n is dominated by te recursion coefficients b, b, b 2 and b 3 and it is coupled wit te errors e v n in te differential components,, by O) coupling coefficients. We note tat te terms,t ) / 2 in 37) vanis in te formal convergence analysis, but may cause severe problems in a practical implementation of te metod. Stopping te Newton iteration in Algoritm wit non-zero residuals r, r λ may introduce small errors in,t ) tat are amplified by te large factor / 2 during time integration []. Terefore, te scaling metod of Bottasso et al. [3] was used in te numerical tests of Sect. 5 to keep,t ) as small as possible, see also [3]. 3.4 Syntesis Te error propagation in multistep metods may be studied in compact form writing te multistep metod as one-step metod in a iger dimensional configuration space [2]. Wit te error vectors e v n e λ n E v n =, E λ n =, e v n e λ n e v n 2 e λ n 2 9

11 te propagation relations in 33) and38) get te form wit error amplification matrices E v n+ = AEv n + O) ) ) E v n + E λ n + O 3, 39) E λ n+ = B E λ n + O) E v n + O) E λ n + O 2) 4) a I a I a 2 a 2 A = I and B = I I b b 3 I b b 3 I I. b 2 b 3 I In te unconstrained case, zero stability of te multistep metod implies A = in a suitable norm. and second-order convergence follows by standard arguments from E v n+ + O)) Ev n +O3 ),see[2]. In te constrained case, a similar argument is used to sow tat strict stability at infinity implies B < in a possibly different) norm. for te algebraic solution components [3]. More precisely, te metod is strictly stable at infinity if te roots ζ i,i=, 2, 3, of te polynomial σ in 22) are bounded by ζ max := max i ζ i <, see Lemma below, and B =ρ<may be acieved for any ρ>ζ max using an appropriate norm. for te algebraic components [3]. Taking norms in 39) and 4), we obtain E v ) ) n+ + O) O) E v ) E λ n+ n O 3 ) ) O) ρ + O) E λ n + O 2 ) and we deduce, as in [3], tat te global errors after n steps satisfy E v n E λ n ) = O) E v + O) + O)ρ n ) E λ + O 2 ). 4) Note, tat strict stability at infinity allows to prove second-order convergence for all solution components despite te local error O 2 ) in te algebraic solution components, see 4). Furtermore, ρ<implies also tat errors E λ in te initial values of te Lagrange multipliers are damped out rapidly. Summary In te constrained case, te generalized-α metod wit fixed time step-size as global errors O 2 ) in, and λ if te order 2 condition in 2) and te stability conditions in 23) are satisfied and te errors E v, Eλ in te initial values are O2 ). Te CH-α algoritm wit ρ [, ), see24), may be considered as a typical example of a second-order convergent metod for constrained systems. 4 Tecnical details of te error estimates In te present section, some tecnical details of te convergence analysis are summarized. Readers, wo are mainly interested in te basic steps of te analysis, may skip tis section and continue wit te results of numerical tests in Sect. 5 below.

12 Lemma A generalized-α sceme wit parameters tat satisfy te order 2 condition in 2) and te stability conditions in 23) is strictly stable at infinity, i.e., all roots ζ i of polynomial σζ), see 22), are inside te unit circle: ζ i <, i =, 2, 3. Proof Polynomial σζ) may be written as σζ)= b k ζ k = β ) α f )ζ + α f ζ 2 + wit roots 2 + γ 2β β ζ + ζ = α f, ζ 2,3 α = 2 + γ 2β ± R f 2β 2β ) 2 )) /2 R := 2 + γ 2β 4β 2 γ + β. 2 γ + β β Because of α f < /2 we get ζ <, see 23). If te roots ζ 2, ζ 3 are complex, we ave ζ 2 = ζ 3 and Vieta s Teorem implies ζ 2 = ζ 3 = ζ 2 ζ ) 3 /2 = 2 γ + β ) /2 ) β /2 < = β β since α m <α f γ>/2, see 2). For real roots ζ 2, ζ 3, we observe ) 2 ) 2 R 2 = 2 + γ 4β < 2 + γ R< 2 + γ, )) 2 R 2 = 4β 8β2β γ)< 4β 2 + γ R> 4β )) 2 + γ and 2 + γ )) 2 since 2β γ>and4β 2 + γ)= 2β ) + 2β γ)>,see2) and 23). 4 Terefore, real roots ζ 2, ζ 3 are bounded by ) ζ 2,3 < 2 + γ 2β γ 2β 2β = 2β 2β =, ζ 2,3 > 2 + γ 2β 4β 2 + γ) = 2β 2β 2β 2β = and ζ i <, i =, 2, 3 is guaranteed also in tat case. Lemma 2 For vectors wit e = O), k =,, 2, 3, te terms t n ), t n )e n+k,k=,, 2, satisfy te arguments t in and are omitted for simplicity) tn ) ) e n+k = ) n+k ) ) + O) e n+k + e + e ) e ) =tn) t n) n+k ) ). 42)

13 Proof To keep notation compact, te proof is given for k = 2 and te argument t in,t) is omitted. Te extension to k =, and to =,t) is straigtforward. For linear time-invariant constraints t)) = Ct) z =, te proposition of te lemma is trivial since C, and e n+ = e n+ e n)/, see 3): Ce n+ = C tn+ ) t n ) ) n+ n = z z n+) + z) n ) + z). In te non-linear case, te identity is used tat follows from ψ) ψ) = n ) = t n ) ) n ) = n + ϑe n) e n dϑ 43) ψ ϑ) dϑ wit ψϑ) := n + ϑe n), ψ ϑ) = n + ϑe n) e n. Because of 43), and te corresponding identity wit n being substituted by n +,weave n ) n+ ) ) = = + n+ + ϑe ) n+ e n+ dϑ n+ + ϑe ) n+ e n+ dϑ n + ϑe n) e n dϑ n+ + ϑe ) )) n+ n + ϑe n e n dϑ 44) since e n+ e n)/ = e n+. Te first term in te rigt-and side of 44) may be written as n+ + ϑe ) n+ e n+ dϑ = tn ) ) e n+ + O) e n+ 45) because n+ + ϑe n+ = t n+) + O) = t n ) + O). Te second term in 44) contains curvature terms. It vanises in te linear timeinvariant case, but needs special care for non-linear constraints. Te term may be expressed as ) ψ; ϑ) ψ; ϑ) dϑ 46) wit ψ ϑ; ϑ):= ϑ; ϑ))e n and generalized coordinates ϑ; ϑ):= n + ϑ n+ n ) + ϑ e n + ϑ e n+ e n)) = tn ) + O). Wit tese notations, we get in 46) ) ψ; ϑ) ψ; ϑ) dϑ = = ) )e n = ϑ;ϑ) ϑ ψ ϑ; ϑ ) d ϑ dϑ ϑ ϑ; ϑ ) d ϑ dϑ 2

14 = = ) )e n = ϑ;ϑ) t n) d ϑ dϑ + O e n ) )e n =tn) t n) d ϑ dϑ + O) e n ) max ϑ, ϑ ϑ ϑ; ϑ ) t n ) + O e ) n max t n+ ) t n ) ϑ)e n+ ϑ, ϑ t n) ) )e n =tn) t n) + O) e n + O) e 47) = since e n =O) by assumption and t n+ ) t n ) = t n ) + O 2 ). Te proof is completed substituting 45 47) in44). Lemma 3 Wit te assumptions of Lemma 2, te first term in te rigt-and side of 36) satisfies te arguments t in and are again omitted): tn ) ) n+ ) a k e n+k = O ) 2 + O) e + e n+k ). 48) Proof Te one-step nature of te generalized-α sceme results in a very special multistep representation of te error recursion in components n because a = + 2α m = α m ) + α m = a 2 a : a k e n+k = α m) e n+ e n e n e + α m Multiplying 49) by t n )) and applying Lemma 2 to n. 49) tn ) ) e n+k e = tn ) ) e n+k tn ) ) e k =, 2), we see tat te estimate in 48) is a straigtforward conseuence of 49), Lemma 2 and ) e ) e n+k 3 t n ) = )e =t n) ) =tn) t n) = O e ) for k =, 2. 5 Numerical tests Te following numerical tests ave been developed in te formalism described by Géradin and Cardona [], wic allows to account for flexible bodies. Hence, te euations of motion are obtained in terms of absolute nodal coordinates wit respect to te inertial frame. 3

15 Fig. Slider crank mecanism Fig. 2 Crank angle θ and position of te additional mass x 4 5. Slider crank mecanism Te first example is a slider crank mecanism wit a spring-mass system attaced to te sliding body see Fig. ). A similar bencmark as been considered in [6]. Tis planar system as two degrees-of-freedom tat can be represented by te crank angle θ and te displacement of te additional mass x 4. Te model involves generalized coordinates: te positions of te centers of mass of body and 2, x,y,x 2,y 2 ), teir orientation θ,θ 2 ), te position of bodies 3 and 4 x 3,x 4 ) and te position of te inge connecting body and 2 x 5,y 5 ). Tis set of coordinates as to satisfy 8 non-linear kinematic constraints. Te spring stiffness is k = N/m, te lengt of bodies and 2 are l =.3 m and l 2 =.6 m, and te masses are m =.36 kg, m 2 =.5 kg, m 3 =. kg and m 4 =.7 kg. Initially, θ = π/2, te spring is undeformed and te mecanism is at rest. A constant torue T = Nm is applied to te crank and te initial conditions are computed so tat te constraints are satisfied at position, velocity and acceleration level. Te parameters of te generalized-α algoritm ave been selected according to 24). Te spectral radius of te algoritm is set to te typical value ρ =.7, wic leads to a sufficient amount of igfreuency numerical dissipation for te problem at and. For a time step = 5.e 3 s, te numerical results are given in Fig. 2. A convergence study as been realized for, and te results are plotted in Fig. 3. Te reference solution as been computed using a smaller time step. Second-order convergence is observed bot for te generalized coordinates and te Lagrange multipliers. 5.2 Andrews mecanism Andrews sueezing mecanism, wic is represented in Fig. 4, consists of seven articulated rigid bodies moving in a plane. Tis standard bencmark, described in details by 4

16 Fig. 3 Convergence of relative errors at final time t = s) for a generalized coordinate θ ) and a Lagrange multiplier λ ) Fig. 4 Andrews sueezing mecanism Scielen [9], as been largely exploited to demonstrate te performance of DAE timeintegration scemes. Te mecanism as only one degree-of-freedom, and a constant torue is applied at point O. In te original bencmark, te euations of motion are explicitly given in terms of relative coordinates. In tis work, one absolute rotation is defined for eac body, wereas two translation coordinates are defined for eac moving joint and eac center of mass. Hence, te model involves a total of 3 generalized coordinates and 3 kinematic constraints. Initially, te mecanism is at rest, and te initial conditions are computed so tat te constraints are satisfied at position, velocity and acceleration level. As in te previous example, te parameters of te generalized-α algoritm ave been selected according to 24) wit a spectral radius ρ =.7. For a time step = 3.e 4 s, te numerical results are illustrated in Figs. 5, 6 and 7. Numerical damping is uite important to ensure a stable numerical solution, and a stable error propagation. For instance, Fig. 8 gives some results for te undamped algoritm ρ = ), wic are strongly affected by numerical oscillations. Te results of a convergence analysis are plotted in Fig. 9. In te publised bencmark, a reference solution is given for te body angles at final time. For te multipliers, we ave computed a reference solution using a smaller time step. Second-order convergence is observed bot for te generalized coordinates and te Lagrange multipliers. However, for very small, te Lagrange multipliers are more sensitive to numerical errors. 5

17 Fig. 5 Motion snapsots of te sueezing mecanism Fig. 6 x,y)-position of point P and angle β Fig. 7 x,y)-accelerations of point P and Lagrange multipliers 6 Conclusions Tis paper analyses te accuracy of te generalized-α metod for constrained mecanical systems. We note tat te proposed algoritm, wic is a variant of te original generalized-α algoritm, can deal wit a non-constant mass matrix and tat te accelerations are computed wit second-order accuracy. 6

18 Fig. 8 Witout numerical damping: x,y)-accelerations of point P and Lagrange multipliers associated wit te x,y) internal forces in body Fig. 9 Convergence of relative errors at final time t =.3 s) for a generalized coordinate β) and a Lagrange multiplier λ ) Using te analogy wit multistep algoritms, global second-order convergence as been proven bot for te generalized coordinates and te Lagrange multipliers. Tose properties ave been illustrated by numerical tests. Acknowledgements O. Brüls is supported by a grant from te Belgian National Fund for Scientific Researc FNRS) wic is gratefully acknowledged. Tis work also presents researc results of te Belgian Program on Inter-University Poles of Attraction initiated by te Belgian state, Prime Minister s office, Science Policy Programming. Te scientific responsibility rests wit its autors. References. Arnold, M.: A perturbation analysis for te dynamical simulation of mecanical multibody systems. Appl. Numer. Mat. 8, ) 2. Arnold, M.: Simulation algoritms and software tools. In: Mastinu, G., Plöcl, M. eds.) Road and Offroad Veicle System Dynamics Handbook. Taylor & Francis, London 27, in preparation) 3. Bottasso, C., Dopico, D., Trainelli, L.: On te optimal scaling of index tree DAEs in multibody dynamics. In: Proc. of te European Conference on Computational Mecanics ECCOMAS-ECCM), Lisbon, Portugal 26) 7

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