Bruls and Golinval 2008 - On the Numerical Damping of Time Integrators for Coupled Mechatronic Systems

8/2/2019 Bruls and Golinval 2008 - On the Numerical Damping of Time Integrators for Coupled Mechatronic Systems

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On the numerical damping of time integrators for coupledmechatronic systems

Olivier Bruls *, Jean-Claude Golinval

Department of Aerospace and Mechanical Engineering (LTAS), University of Liege, Chemin des Chevreuils, 1, B52/3, B-4000 Liege, Belgium

Received 16 June 2006; received in revised form 15 August 2007; accepted 19 August 2007Available online 1 September 2007

Abstract

The generalized-a time integrator is considered for the simulation of mechatronic systems. In this context, the fundamental concept ofnumerical damping is analysed for coupled sets of first and second-order differentialalgebraic equations. First, it appears that the alge-braic variables do not influence the spectral properties of the dynamic variables. Second, we demonstrate that the coupling between thedynamic variables does not influence the high-frequency spectral response, so that the numerical damping can be determined as usualfrom elementary characteristic polynomials. Those results are exploited to assess the stability properties of the scheme and to selectan algorithm with optimal damping properties. 2007 Elsevier B.V. All rights reserved.

Keywords: Generalized-a scheme; Coupled problems; Flexible multibody dynamics; Mechatronics; Numerical damping

1. Introduction

Numerical damping refers to the ability of a time inte-grator to filter spurious effects at high-frequency. Forexample, the higher-order modes of a finite element modeldo not represent accurately the physical behaviour of amechanical structure, and their contribution should ratherbe eliminated from the numerical solution. In structuraldynamics, key contributions to the development of dissipa-tive algorithms have been realized by Newmark [1] and Hil-ber et al. [2]. Later, Chung and Hulbert [3] have defined thefamily of the generalized-a methods, which includes as spe-

cial cases the most important algorithms in this field. Formore details, we refer to standard textbooks by Hughes [4]and Geradin and Rixen [5], and also to the nonlinear anal-ysis by Erlicher et al. [6]. For example, the CH-a scheme,presented by Chung and Hulbert [3], combines the advan-tages of unconditional stability (A-stability), second-orderaccuracy, adjustable numerical dissipation at high-fre-

quency, and one-step implementation. Geradin andCardona [7,8] have applied the generalized-a method forthe direct integration of index-3 differentialalgebraic equa-tions (DAEs) in flexible multibody dynamics. The presenceof kinematic constraints then leads to a weak instability inthe numerical solution, which can, however, be eliminatedby a small amount of numerical dissipation. In the samecontext, global convergence analyses have been recentlyrealized by Lunk and Simeon [9] and Jay and Negrut [10]for regularized index-2 formulations, and by Arnold andBruls [11] for an index-3 formulation.

This paper proposes an extension of the generalized-a

method for the simulation of mechatronic systems. Indeed,the development of a concurrent framework for the analy-sis and design of controlled flexible mechanisms is moti-vated by the demand for faster, lighter and moreaccurate machines and robots [12]. Mechatronic systemsare described by coupled sets of first and second-orderDAEs, which cannot be solved in the time domain usingthe original form of the generalized-a method. Negrutet al. [13] have proposed to combine the generalized-ascheme for the mechanical variables with a specific integra-tion formula for the control state variables. However, their

0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.08.007

* Corresponding author. Tel.: +32 4 366 91 84; fax: +32 4 366 48 56.E-mail addresses: [email protected] (O. Bruls), [email protected]

(J.-C. Golinval).

www.elsevier.com/locate/cma

Available online at www.sciencedirect.com

Comput. Methods Appl. Mech. Engrg. 197 (2008) 577588
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approach does not allow to combine second-order accu-racy with numerical damping for the control state vari-ables. In contrast, in previous publications [14,15], wehave shown that a monolithic generalized-a scheme canbe implemented provided a slight reformulation of the cou-pled equations of motion. The present paper demonstrates

that this algorithm, which is based on an index-3 formula-tion for the kinematic constraints, allows the introductionof high-frequency numerical damping in an efficient way.

Compared to classical multistep or RungeKutta meth-ods [1619], which require a reformulation of the coupledequations as a larger set of first-order DAEs, our approachrepresents an interesting alternative. Indeed, the general-ized-a method relies on different integration formulae forthe displacement and velocity variables, which directlyexploits the second-order structure of the equations ofmotion. The potential benefits of the CH-a scheme comefrom its remarkable stability properties, its optimalcombination of accuracy at low frequencies and adjustable

high-frequency numerical damping, the simplicity of itsimplementation, and its computational efficiency. How-ever, a careful analysis is required to assess the stabilityand numerical damping properties for coupled mechatron-ic problems.

In linear structural dynamics, the analysis of the stabil-ity and damping properties of an algorithm usually relieson a decomposition of the initial problem in terms ofuncoupled natural modes. Since the integration schemewould lead to an equivalent solution when applied to themodal equations, it is then sufficient to analyse a scalar testproblem

q x2q 0where x represents a particular natural frequency. For thisscalar problem, the eigenvalues of the numerical solutioncan be computed explicitly from an elementary characteris-tic polynomial, such as

f 12 hx2

4f 12 0 1

for the constant average acceleration scheme with step-sizeh. For coupled mechatronic problems, it turns out that thedecomposition into uncoupled modes is not possible, sothat the eigenvalue analysis cannot be restricted to a simplescalar test problem. Even though this fact appears as a leit-motiv throughout this paper, we shall still attempt to relatethe spectral behaviour of the numerical solution to elemen-tary polynomials. First, we show that the numerical re-sponse can be decomposed in terms of dynamic andalgebraic variables, and that the contributions of the alge-braic variables are uncoupled. We then demonstrate thatthe coupling between the dynamic variables does not influ-ence the high-frequency response. Finally, those results areexploited for a stability analysis and for the selection of analgorithm with optimal damping properties.

After the description of the equations of motion and of

the integration algorithm in Sections 2 and 3, the linear

eigenvalue analysis is developed in Section 4. Then, the sta-bility properties are discussed in Section 5, and the selec-tion of optimal algorithmic parameters is addressed inSection 6. The paper ends with an illustrative exampleand some conclusions.

2. Equations of motion

In a mechatronic system, the motion of the mechanismis driven by a control system composed of actuators, sen-sors and control units. The coupled equations of motionhave the general form [14,15,20]

Mqq gq; _q; t UTq k Lqyy;0 Uq; t;_x fq; _q; q; k; x; y; t;y hq; _q; q; k; x; y; t;

2

where q is the n 1 vector of mechanical generalized coor-dinates, k is the m 1 vector of Lagrange multipliers, x isthe s 1 vector of control state variables, and y is thet 1 vector of control output variables.

In Eq. (2), we have successively the dynamic equation forthe mechanism, the kinematic constraints, the state equa-tion and the output equation. M is the mass matrix, whichis not constant for a multibody system with large rotations,g represents the internal, external and complementary iner-tia forces, Lqy is a boolean localization matrix and Lqyydenotes the actuator forces. The control system is influ-enced by input measurements from the mechanical system,which can be positions q, velocities _q, accelerations q orinternal forces k. In our simulation software, the controllerequations are constructed in a modular and systematic wayusing the block diagram language, as illustrated in Fig. 1.Since the output y(i) of the block (i) can be an input variablefor other blocks, the output equation is implicit in y.

Both the kinematic constraints U and the output equa-tion are algebraic in nature. In the uncoupled case (e.g.Lqy = 0), it is well-known that the differentiation indexassociated with the mechanical subsystem is three, see Bre-nan et al. [18] or Hairer and Wanner [19]. In contrast, pro-vided some regularity condition, the control system isrepresented by a set of semi-explicit index-1 DAEs, see [14].

Fig. 1. Block diagram model of a mechatronic system.

578 O. Bruls, J.-C. Golinval / Comput. Methods Appl. Mech. Engrg. 197 (2008) 577588


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Unlike in structural dynamics, the coupled Eq. (2) maybe influenced by strongly nonlinear terms in the accelera-tion variables q. However, if an additional output variableyj is defined for each acceleration measurement qi, with theoutput equation yj qi, the control equations can be refor-mulated as

_x fq; _q; k; x; y; t;y hq; _q; k; x; y; t Lyqq; 3

where Lyq is a boolean localization matrix. The advantageof this reformulation will appear clearly in the following.

In order to define a monolithic generalized-a timeintegrator, the state equation is transformed into asecond-order equation by introducing the auxiliarydynamic variables z

zt :

Zt

0

xsds:

The value of z is only meaningful at the velocity level(_z x) and at the acceleration level z _x. The coupledequations of motion become

cMpp gp; _p; t 4with the vector p, the constant matrix cM and the vector g

M

Mq

z

y

p

0

0 0

0 0 0

0 00

0 0

00

f

y h

yTq +

+

where Is denotes the identity matrix of dimension s.

3. Time integration algorithm

The Newmark implicit formulae, which have been pro-posed for the time integration of second-order ODEs, resultfrom a truncated Taylor series expansion of the displace-ments and velocities with respect to the time step size h

pi1 pi h _pi h21

2 b

ai h2bai1;

_pi1 _pi h1 cai hcai1;5

where b and c are numerical parameters. In the originalNewmark scheme, the vector a is simply equal to the accel-eration p, whereas for the generalized-a method, it is de-

fined by the linear recursion

1 amai1 amai 1 afpi1 afpi; a0 p0; 6where am and afare additional numerical parameters. If themass matrix is constant, a multiplication of this last equa-tion by cM leads to the modified dynamic equilibrium1 amc

Ma i1 amcMa i 1 afgi1 afgi: 7

A predictorcorrector procedure can be implemented inorder to solve Eqs. (5) and (7) for p, _p, and a at each timestep. If the mass matrix is not constant, the integrationscheme should be implemented without resorting toEq. (7), see [11] for a detailed discussion.

The formulation of the simulation algorithm in terms ofthe global vector p allows compact and elegant notations,which is very appealing for the purpose of our theoreticalinvestigations. However, a batch implementation of thescheme would involve numerous useless operations sinceseveral components of the vectors p, _p or p are not physicallymeaningful. Indeed, the artificial variables _k, k, z, _y and y do

not influence the coupled equations of motion (3) and theircomputation should be avoided for the sake of efficiency.Therefore, the Newmark integration formulae for k and yshould be omitted, as well as the Newmark formula at posi-tion level for z. Likewise, those artificial contributions willalso be neglected at several points of the following theoret-ical developments.

4. Linear eigenvalue analysis

Following the arguments of Chung and Hulbert [3], thefour algorithmic parameters b, c, am and af should beselected in order to obtain an optimal compromise betweenaccuracy at low frequencies and stability at high frequen-cies. In order to analyse the spectral properties of the gen-eralized-a scheme, we consider the linear and homogeneousmechatronic system without mechanical dampingcMp bC _p bKp 0 8with

M

M

C

0 0 0 0

00

0

0

0

0

0

0 0

0 0000

0 0

0 0

0 0

0

0

0

0

0

0

0

0

00

T

=

=

=

J

The matrix A is the linear state matrix of the control sys-tem, K is the mechanical stiffness matrix, B is the matrix oflinear constraints, whereas Gxq, Gy _q, Gyx, Gxq, Gxy and Gyq

represent coupling terms. The matrix J is different from the

O. Bruls, J.-C. Golinval / Comput. Methods Appl. Mech. Engrg. 197 (2008) 577588 579


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identity if the output equation is implicit in y. Since thecontrol system contributes to the matrix bC, the general-ized-a algorithm should be analysed in the presence of anon-negligible damping matrix. The eigenvalues of thenumerical response are the solutions to the generalizedeigenvalue problem

f

1 afbK 1 afbC 1 amcMI 0 bh2I0 I chI

26643775

vp

v _p

va

26643775

afbK afbC amcM

I hI 12 b h2I

0 I 1 chI

26643775

vp

v _p

va

26643775;

where I is the identity matrix of dimension n + m + s + t.The first line corresponds to the modified dynamic equilib-rium, the second and the third, to the Newmark formulae

at position and velocity levels, respectively. Introducingthe polynomials

Pamf : 1 amf am; Pcf : cf 1 c;Paff : 1 aff af; Pbf : bf 12 b;P1f : f 1;the characteristic equation is reorganized as

det

Paffh2bK PaffhbC PamfcM P1fI I PbfI

0 P1

f

I

Pc

f

I

2664

3775

0:

9Due to the complex structure of the matrices bK, bC andcM, this expression is not convenient for analytical investi-

gations. Instead, fundamental concepts can be introducedif the analysis is first restricted to a scalar problem.

4.1. Elementary characteristic polynomials

Elementary characteristic polynomials are defined forthe second-order scalar system

p

r _p

l2p

0;

10

where r and l are, in general, complex parameters. Aftersome developments, Eq. (9) becomes

Pq0 hrPq1 hl2Pq2 0

with the notations Pq0 : P21Pam , Pq1 : P1PafPc andP

q2 : PafPc P1Pb. The polynomials Pq0, Pq1 and Pq2 are

the so-called generating polynomials of the algorithm. Inthe theory of multistep methods for first-order ODEs,two generating polynomials are sufficient for the eigenvalueanalysis. One additional generating polynomial is requiredhere because the Newmark integration formulae are de-fined for second-order systems, and are not equivalent for

the displacement and for the velocity variables.

The scalar undamped mechanical system is a subcase ofEq. (10), with r = 0 and l = jx (x 2 R). The characteristicpolynomial becomes

Pqhx : Pq0 hx2Pq2:

The scalar control system is another subcase of Eq. (10),

with l = 0. This appears more clearly if the standard stateequation _x rx is recasted as z r_z 0. After removingthe fictitious contribution ofz at displacement level, we get

Pxhr : detPaffhr Pamf

P1f Pcf !

Px0 hrPx1:

The polynomials Px0 : P1Pam and Px1 : PafPc are not inde-pendent from Pq0 and P

q1 since we have: P1P

xi Pqi (i= 0,1).

Let us note that the characteristic polynomial for thedisplacement variables Pqhx is of degree three, whereas thecharacteristic polynomial for the state variable Pxrh is ofdegree two. This observation is consistent with the well-known analogy between the generalized-a method and lin-ear multistep methods [4], i.e. the generalized-a scheme isequivalent to a linear three-step formula for the displace-ments and to a linear two-step formula for the velocities(as well as for the state variables since x _z).

Finally, it is convenient to define asymptoticpolynomials:

Pq1f : limhx!1

Pqhxf

hx2 Pq

2f PafPc P1Pb;

Px1f : limhr

!1

Pxhrfhr

Px1f PafPc:

We already note the special role of Paf, which appears inboth Pq1 and P

x1.

4.2. Contribution of the algebraic variables

In order to transform the initial matrix problem (9) intoa scalar problem and refer to the elementary polynomialsdefined above, we would like to diagonalize the initial lin-ear equations of motion using a transformation into modalcoordinates. Observing that the integration algorithmwould lead to an equivalent solution when applied to the

transformed system, the spectral analysis could then beachieved independently for each scalar subsystem. Such adiagonalization is trivial for an explicit state space system

without outputs (bK 0 and cM regular), or for anundamped and unconstrained mechanical system (bC 0and cM regular). In general, it is impossible to diagonalizea system with arbitrary matrices bK, bC and cM. However,due to the special structure of the coupled equations ofmotion (8), the following result, demonstrated in AppendixA, can be used in order to isolate the contribution of thealgebraic variables.

Under some reasonable assumptions, the linear system

(8) can be transformed into

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qr X2qr Crxx 0;qc qk Ccxx 0;qc 0;x Rx Pxrqr Dxr _qr !xrqr Pxcqc

Dxc _qc !xcqc 0;y Cyxx Pyrqr Dyr _qr !yrqr Pycqc

Dyc _qc !ycqc 0

11

with the transformed state variables x 2 Rs and outputvariables y 2 Rt. The coordinates qc 2 Rm are interpretedas constrained coordinates, qk 2 Rm, as Lagrange multipli-ers, and qr 2 R"n, as remaining free dofs ("n : n m). Thematrices Crx, Ccx, Pij, Dij, !ij (ij = xr, xc, yr, yc) representthe coupling between the mechanical and the control sys-tems, whereas the matrices R and X2 are diagonal

R diagr1; . . . ;rs; X2 diagx21; . . . ;x2"n:

Each value xi 2 R is a mechanical natural frequency,and each value ri 2 C is a pole of the control system. A dis-tinction should be made between the dynamic variables qr

and x* and the algebraic variables qc, qk and y*. Let us showthat the contributions of the algebraic variables to the char-acteristic Eq. (9) are uncoupled. If the matrix involved inthis equation is reorganized by block, and if non-physicalcontributions are eliminated, an equivalent characteristicequation is obtained

det

U rr 0 0 U rx 0

0 U

cc

U

c

U

cx

00 U c 0 0 0

U xr U xc 0 U xx 0

U yr U yc 0 U yx U yy

= 0

with the submatrices

Urr :Paffh2X2 0 PamfI"n

P1fI"n I"n PbfI"n

0 P1fI"n PcfI"n

2664

3775;

Urx :PaffhCrx 0

0 0

0 0

26643775;

Ucc :0 0 PamfIm

P1fIm Im PbfIm

0 P1fIm PcfIm

26643775;

Ucx :PaffhCcx 0

0 0

0 0

2

664

3

775;

Uck : Ukc T : Paffh2Im

0

0

264375;

Uxx : PaffhR PamfIs

P1fIs PcfIs !

;

Uij : Paffh2Pij PaffhDij Pamf!ij0 0 0

" #ij xr; xc; yr; yc;

Uyx : PaffhCyx; Uyy : PaffIt:After some algebra, the contribution of the algebraic

variables can be extracted. Defining P1..."n;1...s as the charac-teristic polynomial associated with the dynamic variables

1 n, 1 s () := det (U* ) with U * :=

U rr U rx

U xr U xx, 12

the characteristic polynomial takes the final expression

PaftPafPq1mP1..."n;1...s 0: 13

Clearly, Paf represents the contribution of one outputvariable or of one Lagrange multipliers, whereas Pq1 repre-sents the contribution of one constrained coordinate. Sinceany root ofPaf is also a root ofP

q1, one concludes that the

spectral contribution of the algebraic variables is fully rep-resented by the polynomial Pq1. In contrast, the analysis ofthe eigenvalues associated with the dynamic variablesrequires the development of the polynomial P1..."n;1...s, which

turns out to be a challenging task.

4.3. Contribution of the dynamic variables

Since the matrix U* is not diagonal, the polynomialP1..."n;1...s cannot be decomposed in terms of uncoupledmodes, and the investigations cannot be continued withoutfurther assumptions. More detailed theoretical results canbe obtained for three particular subcases developed hereaf-ter: a scalar coupled system, an uncoupled system and theasymptotic situation of a system with very stiff compo-nents. Using those partial results, practical guidelines willbe proposed for the selection of optimal algorithmicparameters.

First, a scalar coupled system with one mechanical var-iable q and one state variable x is analysed. The character-istic polynomial (12) can be formulated in terms of thegenerating polynomials:

P1;1f PqhxPxhr hCPx1 h2PPq2 hDPq1 !Pq0

:

For a given algorithm, this polynomial depends para-metrically on several variables: hx, hr, h3CP, h2CD andhC. Thus, even for a scalar problem, the properties ofthe numerical response should be investigated in a rather

high-dimensional parameter space.

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Second, if Crx = 0 (equivalent to Urx = 0) orP

xr = Dxr = !xr = 0 (equivalent to Uxr = 0), a block trian-gular structure appears in U*, indicating an uncoupling inthe eigenvalues of both dynamic subsystems

P1..."n;1...s detUrr detUxx:

Since each subsystem has a diagonal structure, we obtain

P1..."n;1...s Y"nj1

Pqhxj

fYsk1

Pxhrkf:

Hence, all eigenvalues of the numerical solution can bedetermined from the fundamental polynomials Pqhx andPxhr.

Third, in the presence of stiff dynamics in the mechanicalsystem and/or in the control system, the characteristicpolynomial satisfies the following properties, which aredemonstrated in Appendix B

limxj!1P1..."n;1...s

hxj2 Pq

1P

1...j..."n;1...s;

limrj!1

P1..."n;1...s

hrj Px1P1..."n;1...j...s;

limh!1

P1..."n;1...s

h2"ns Pq1"nPx1sj:

14

P1...j..."n;1...s (resp. P1..."n;1...j...s) is the determinant of the re-duced system after elimination of the contribution of qj(resp. xj), and j is a scalar constant depending on X, R,C and P. This implies that the coupling does not influencethe asymptotic behaviour of the eigenvalues; we shall referto this result as the asymptotic separation principle. If an

algebraic variable is interpreted as the limit case of a dy-namic variable with a high-frequency, the asymptotic sepa-ration principle confirms that its contribution to thecharacteristic polynomial is uncoupled. For h ! 1, theasymptotic expression of the characteristic polynomial (13)is thus

PaftPafmPq1nPx1s 0;which clearly emphasizes the separate contribution of eachgroup of variables.

To summarize the linear eigenvalue analysis, the polyno-mials Pqhxf and Pxhrf fully characterize the numericalsolution in the special case of uncoupled problems. Forcoupled problems, they still allow to characterize theasymptotic behaviour of the eigenvalues. Therefore, thenumerical damping at high-frequencies, which is requiredto stabilize the stiff and the algebraic components duringthe simulation, can be analysed from those fundamentalpolynomials.

5. Stability analysis

In the linear regime, the numerical solution is stable if allthe roots of the characteristic polynomial are strictly insidethe unit circle (

jfj

j< 1), and it is unstable if one root is

strictly out of the unit circle (jfjj > 1). If some roots are

on the unit circle, the numerical solution may be stableor unstable. An algorithm is unconditionally stable if itleads to a stable numerical solution when applied to anystable dynamic system.

Due to the difficulty to decompose the characteristicpolynomial in the general case, we are only able to guaran-

tee unconditional stability for uncoupled problems. Asshown in the preceding section, it is then sufficient to ana-lyse the roots fq1;2;3hx ofPqhxf and the roots fx1;2hr ofPxhrf. Let us define the spectral radius associated withthe mechanical variables and with the state variables:

qqhx : maxfjfq1hxj; jfq2hxj; jfq3hxjg;qxhr : maxfjfx1hrj; jfx2hrjg:The algorithm is unconditionally stable for uncoupledproblems if (sufficient conditions)

qqhx < 1 8x 2 R; 15qx

hr

< 1

8r

2C

:

16

We also define the spectral radii at infinity

qq1 : limhx!1

qqhx;qx1 : lim

hr!1qxhr;

q1 : maxfqq1;qx1g:qq1, q

x1 and q1 can be interpreted as amplification factors

which are related with the numerical damping of the algo-rithm in the high-frequency range. A spectral radius equalto 1 indicates the absence of numerical damping, whereas avalue of 0 means asymptotic annihilation of the high-fre-

quency response.For the undamped Newmark scheme, which is charac-

terized by qq1 qx1 q1 1, several eigenvalues lie onthe unit circle. According to the standard stability theory,the numerical solution is marginally stable if the multiplic-ity of the roots on the unit circle is equal to the dimensionof the corresponding eigensubspace. This multiplicity con-dition is usually satisfied in the unconstrained case (m = 0),but Cardona and Geradin [7] have shown that it is not ver-ified for constrained systems (m > 0). Therefore, a reason-able amount of numerical dissipation is necessary toavoid a weakly unstable solution in the presence of kine-matic constraints.

For coupled mechatronic problems, we deduce from theasymptotic separation principle that the high-frequencynumerical damping of the algorithm is not affected by thecoupling, and that stability with respect to any stiff compo-nent is still guaranteed by the conditions (15) and (16). Inthe low-frequency range, stability is also expected as a con-sequence of the convergence of the numerical solution tothe exact solution, which is supposed to be stable.

6. Optimal choice of the algorithmic parameters

Four parameters (c, b, afand am) have to be selected in

order to optimize the accuracy and stability properties of

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7/12

the integration algorithm. From an analysis of the localerror, one obtains the condition for second-order accuracy

c 12

afm

with afm = af am. Moreover, ifb satisfies

b 14

c 12

2; 17

all roots ofPqhxf become real at infinity, which is optimalfor numerical dissipation, as explained by Chung and Hul-bert [3]. We note that the roots of Pxhrf become real atinfinity whatever the choice ofc and b.

The remaining part of this section is dedicated to theselection of the parameters af and am. For this purpose,we focus on the uncoupled case, so that it is sufficient toanalyse the polynomials Pqhxf and Pxhrf. At infinite fre-quencies, their roots are given by

fq11 fx11 f11

af1 af ;

fq

2;31 1 afm1 afm ; f

x21

1 2afm1 2afm :

For stability, the condition q1

< 1 is equivalent to

am < af

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Bruls and Golinval 2008 - On the Numerical Damping of Time Integrators for Coupled Mechatronic Systems