A catching-up algorithm for multibody dynamics with impacts and dry friction Alexandre Charles
a,b,∗
a
b
, Fabien Casenave , Christoph Glocker
a Safran, Rue des Jeunes Bois, Châteaufort CS 80112 - 78772 Magny-les-Hameaux Cedex - France b ETH Zürich, Institute for Mechanical Systems, Tannenstrasse 3, 8092 Zürich - Switzerland
Abstract
In the beginning of the 80's, a rigorous mathematical framework was developed for the dynamics of multibody systems with perfect unilateral contacts, particularly due to the contributions of Schatzman and Moreau. E cient numerical methods have been proposed, for instance Moreau's NonSmooth Contact Dynamics (NSCD) [1], which was then extended by Jean to cases with friction [2]. But the algorithm, in the latter case, is no longer the time discretization of an evolution problem. In this work, we derive a new algorithm from the time discretization of an evolution problem for multibody dynamics with contacts and friction. Our algorithm has many points in common with the one of Jean and Moreau, but it converges reliably and xes some energetic inconsistencies. The similarities and dierences between the algorithms are illustrated on three planar archetypal examples.
Keywords:
Time-stepping scheme, NonSmooth Contact Dynamics, Impact, Dry friction, Painlevé's
paradox, Kane's paradox
2010 MSC:
00-01, 99-00
1. Introduction
The NonSmooth Contact Dynamics (NSCD) procedure, developed by Jean and Moreau [1, 2, 3], is an algorithm designed to deal with collections of packed bodies and implemented in LMGC90 [4], SICONOS [5] and other softwares. The NonLinear Gauss-Seidel (NLGS) algorithm is the generic solver for the incremental problem arising from the NSCD. The advantage of the NCSD approach is the possibility to solve numerous simultaneous contacts in one time step.
This allows large time increments to be used, contrary to the
Molecular Dynamics approach [6], where time steps are bounded by the time between collisions. However, NSCD-NLGS has two drawbacks [7].
The rst one is related to a formulation of constraints introduced
by Moreau [1] on the velocity level for the noninterpenetration conditions.
This formulation is exact for
time continuous modeling but may generate parasitic interpenetrations in a numerical scheme.
Recent
attempts to overcome this di culty can be found in [8, 9] where the authors have introduced modi cations of the time-stepping discretization scheme. The second drawback of NSCD is the slow convergence of the NLGS algorithm and even the absence of convergence in some cases.
The present paper deals with this
second topic, which is usually related to the indetermination generated by self-equilibrated force networks in strongly con ned granulates [10, 11]. In fact, the convergence of NLGS is not guaranted even for systems with a very small number of degrees of freedom because the NSCD procedure relies on the resolution of a ill-posed problem at each time step. This is known as Painlevé's paradox, and many examples can be found 1
in the literature
[7, 11, 12, 13, 14, 15]. In other words, this numerical convergence problem is related to
∗ Corresponding
author
[email protected] (Alexandre Charles) 1 Originally, Painlevé did not discussed the ill-posedness of NSCD. But Moreau himself [11] recognized the ill-posedness of the NSCD procedure is related to Painlevé's paradox. Email address:
Preprint submitted to Elsevier
February 2, 2018
the properties of the evolution problem that the algorithm is trying to compute. In order to improve the convergence of the NLGS solver, we have to either adapt the solver to the ill-posedness of the problem, for example with regularization techniques as in [7], or to reformulate the evolution problem in order to get a well-posed problem at each time step. The last strategy is the one adopted in this work. To present this strategy, we need to brie y review the theoretical statement of the evolution problem for nonsmooth systems. We will point out that the described numerical di culties for multibody systems with unilateral contacts and dry friction are the echo of open questions about the evolution problem. Until the late 1970s, the usual practice in rigid body dynamics with contact and friction was to write a collection of ordinary dierential equations to replace the dierential inequations.
One was supposed
to switch from one dierential equation to another using so called discrete equations according to the status (contact or not) of the unilateral constraints. This point of view is referred to as event-driven (this terminology seems to be due to Moreau) in this paper, but the idioms "hybrid systems", "discrete element method" or "molecular dynamics" are also frequently encountered in the literature. The event-driven point of view is convenient neither for theoretical investigations nor for computational purposes because it gives a peculiar status to the instant of impacts in the formulation despite these being unknown of the problems. A new strategy emerged at the end of the 1970s: Schatzman has [16] formulated a global (with respect to the time) evolution problem governing the frictionless bouncing of a point particle against some obstacle. Her formulation was the rst one to encompass the episodes of smooth motion as well as the impacts. However, the problem was no longer governed by an ordinary dierential equation, but by something more complicated involving extensive use of measure theory (in this framework, the percussion, that is the instantaneous reaction force exerted by the obstacle during a collision is represented by a Dirac measure with respect to time).
The paper of Schatzman [16] was restricted to the (frictionless) dynamic motion of a point
particle (or a nite collection of such points). It was rapidly extended by Moreau [17] to include cases with dynamic motion of a nite collection of rigid bodies satisfying frictionless noninterpenetration conditions and possibly bouncing against a nite number of (frictionless) xed external obstacles. The formulation of Moreau suitably modi es the classical Lagrange equations governing the dynamic motion of a nite collection of rigid bodies and allows Lagrange's generalized acceleration to be a measure (with respect to the time) in the line of the setting of Schatzman for point particles.
In particular, the reaction force involved in
Moreau's formulation of frictionless unilateral dynamics is a generalized reaction force, consistent with the lagrangian framework. The work of Moreau [17] has made clear how to consistently formulate the frictionless unilateral dynamics of a nite collection of rigid bodies with enough generality to encompass any practical situation raised by engineering applications.
This opened the way to numerous theoretical investigations
on frictionless multibody dynamics [see 18, for an history of theoretical investigations]. This also lead to the NSCD procedure, a time-stepping (in opposition to event-driven) numerical strategy rst introduced by Moreau for the modeling of granular media [17]. Since then, Jean and Moreau [1, 2, 19] have extended the NSCD procedure to the frictional case (see the top right-hand corner of gure 1). Nevertheless, the resulting algorithm is not the discretization of a continuous in time evolution problem. In this context, the Jean and Moreau algorithm is the model itself. In the present paper, this approach will be referred to as numerical modeling.
This choice was accepted
by Moreau, in the context of the modeling of granular media, where he states Laplace's determinism as irrelevant [11], and indeed, even the de nition of the initial condition of a granular medium is ambiguous. Nowadays, the NSCD is widely used in a large class of engineering problems, including machinery [20, 21], granular media and civil engineering [7, 22, 23], virtual reality and graphics [24], sound generation [25], mechatronics and electricity [26]. Some of them raise bifurcations problems [27, 28, 29] and control problems [30, 31, 32], from there one wishes to have a proper and generic mathematical model. By mathematical modeling we mean writing rst an evolution problem and then discretizing in time.
The mathematical
modeling is a prerequisite for addressing the question of the existence and uniqueness of the solution to the evolution problem and the convergence of the computed solution to the solution to the evolution problem. Therefore, most convergence results are established either in the frictionless case or in the case of trivial mass matrix (point particles) [33, 34, 35, 36] and couldn't be extended so far. The mathematical modeling of many simple systems is possible, as well as the numerical modeling of large systems.
However the mathematical modeling of a generic multi-body system remains challenging. 2
Evolution problem for multibody dynamics with perfect unilateral constraints [16, 19]
Evolution scheme for multibody dynamics with perfect unilateral constraints [1, 2]
Time-stepping discretization
Jean-Moreau's point of view Extension to the frictional case Point of view of the present paper
Jean-Moreau's NSCD [2]
Time-stepping discretization
Catching-up NSCD
Evolution problem for multibody dynamics with unilateral constraints and dry friction [12]
Extension to the frictional case
Evolution scheme for multibody dynamics with unilateral constraints and dry friction
Figure 1: Numerical modeling and mathematical modeling approaches Anitescu
et al
[37] explain that there has been a controversy about the status of rigid body dynamics as a
theory and that a number of time-stepping methods have been developed to overcome these di culties. Stewart recognizes in [38] that time-stepping methods, in the frictional case, is not the discretization of a continuous in time evolution problem: In many ways it is easier to write down a numerical method for rigid-body dynamics than it is to say exactly what the method is trying to compute. The reason is that Coulomb's law, formulated in term of point velocities and forces, is incompatible with Lagrange's equations and generalized forces.
Indeed, combining Lagrange's equations of motion with Coulomb's law gives a
peculiar status to contact points in the equation of motion, despite these being unknowns of the problem. We refer the reader to [12] or to section 2.3 of this work. The mathematical modeling of generic multibody system dynamics with contacts, impacts and friction has been adressed in [12]. In this paper, only generalized forces of reaction [12, page 20], and not reaction forces, are considered, in the line of Moreau's evolution problem in the frictionless case [17].
Moreau's
frictionless evolution problem is completed with a friction law which is a generalization for multibody systems of Coulomb's law for point particles. The evolution problem [12, page 20] is proved to have the following properties:
•
Existence and uniqueness of the solution to the rate problem
2
the evolution problem is Painlevé's
paradox free.
•
Energetic consistency [see 39, for the de nition of this concept] the evolution problem is Kane's paradox free.
2 Given an evolution problem, the rate problem consists in determining the variation of the state variable for a given time t and a given state (q, q) ˙ . In other words, the existence and the uniqueness of the solution to the rate problem is synonymous of Laplace's determinism. The existence and the uniqueness of the solution to the rate problem is a necessary but not su cient condition for the existence and the uniqueness of the solution to the evolution problem (see Schatzman [16] for a counterexample). The existence and the uniqueness of the solution to the rate problem is needed to prove the convergence of NLGS for each time step of NSCD. The existence and the uniqueness of the solution to the evolution problem is the kind of argument one needs to prove that the computed solution of NSCD converges toward the solution to the evolution problem when the time step goes to 0. 3
The present paper investigates whether this reformulation of classical equations provides a new computational strategy.
We apply the time-stepping scheme [17] to the evolution problem of [12] to get an
alternative NSCD (
gure 1 bottom left-hand corner). This new NSCD turns out to be a catching-up
c.f.
i.e.
algorithm, consisting of a succession of well-posed proximation problems [1] (
nding the proximal point
in a convex set of points in a normed vector space). As a consequence, the catching-up NSCD reliably converges and is computationally e cient. Note that the contribution of this paper is about the convergence of the NLGS solved at each iteration of the NSCD procedure. We benchmark the catching-up NSCD against some archetypal systems and we show computed solutions similar to Jean-Moreau's NSCD, except in the case where Jean-Moreau's one is likely to exhibit pathologies. The contribution is restricted to the case of planar systems with a small number of degrees of freedom. The paper is organized as follows. Section 2 is a review of the NSCD procedure, for both the frictionless case and the frictional case. We show how the time-stepping discretization in time (section 2.2) allows one to obtain an incremental problem from Moreau's frictionless nonsmooth evolution problem (section 2.1). For this, we introduce some classical tools in section 2.1: some introductory basics of Lagrange's theory of discrete systems, measure theory and convex analysis for completeness of the presentation.
We then
recall how the NSCD procudure is extended to a larger class of problems by incorporating Coulomb's law (section 2.3) and impact laws (section 2.4). In section 3, we apply the time-stepping discretization to the evolution problem proposed in [12, page 20], accounting for the dynamic evolution of multibody systems with frictional contacts. We derive a catching-up algorithm as an alternative to Jean-Moreau's NSCD procedure. Then, the method is applied to dierent examples.
In section 4, we consider a classical benchmark: the
woodpecker toy [40]. Similar results are obtained for Jean-Moreau's NSCD and our algorithm. The second example in section 5, an idealization of a quarter car, allows us to illustrate the dierences between both approaches in situations of multiple impacts. Our algorithm is well-posed at each time-step whereas JeanMoreau's NSCD diverges for some values of the initial conditions. In section 6, we conclude with a very simple example with two degrees of freedom and a single frictional contact. We show that Jean-Moreau's NSCD can violate the conservation of energy whereas our catching-up NSCD does not.
2. The Jean-Moreau time-stepping approach
2.1. The global-in-time evolution problem with perfect unilateral constraints according to [41, 17, 16] To get a global-in-time evolution problem with perfect unilateral constraints, we need to introduce three ingredients: 1. Lagrange's theory of discrete systems, 2. Measures over time to account for velocity jumps or impacts, 3. Some basic concepts of convex analysis. Lagrange's theory of discrete systems. For a given system consisting of a collection of bodies
we consider the material point displacements of the collection of bodies
p M ∈ ∪b∈B Ωb
Ωb , b ∈ B ,
. If we consider
some constraints on the system, for example the rigid body hypothesis or some idealization of joints between bodies, we de ne the con guration set to these constraints [42, 43].
Q
of a system as the set of admissible displacements with respect
The con guration
kinematically admissible displacement eld
p(M ).
q ∈ Q
denotes, in a synthetic and abstract fashion, a
From there, a modern de nition of Lagrange's discrete
system [42] can be the following: the system is said to be a discrete system of dimension set is a Riemannian manifold of dimension
n, see [19, 41, 44] and [45, section 3.5].
n if the con guration
We explain the implication
of this statement so that the language of dierential geometry is not a prerequisite for understanding the paper. Consider there exists a to
R
n
C 1 -dieomorphism
called a parametrization. The con guration
from a neighborhood of the current con guration
q
is identi ed as the column vector of
Rn
O⊂Q
containing the
so-called generalized coordinates. The knowledge of the generalized coordinates is su cient to determine the position
p(M ) of any material point of the system [42].
The dierential at
q of the parametrization provides Rn . In this setting, a eld of
a bijective linear mapping from the space of admissible velocities elds to
admissible velocities is identi ed as the vector containing the time derivative of the generalized coordinates and is hence called a generalized velocity [42]. 4
Multibody systems E d , Ed
Euclidian a ne and vector spaces Bodies of the system Material point of the system Position of the material point Velocity of the material point A generic force Lagrange's multibody dynamics n Number of degrees of freedom Q ∼ Rn Con guration manifold q = (qi )i=1,··· ,n Con guration (q, u) State u = q˙ Generalized velocity v Virtual generalized velocity Gq (M ) Velocity distributor 2 1 Kinetic energy kuk q 2 M (q) Mass matrix (v1 , v2 )q Riemannian metric f A generic generalized force ∗ Gq (M ) Map forces on generalized forces (f1 , f2 )∗q Scalar product on generalized forces γ Lagrange's generalized acceleration Unilateral constraints j Index for constraints ϕj (q) Constraint function ϕj (q) ≥ 0 Holonomic unilateral constraint J(q) Set of active constraints ∇ϕj (q) nj (q) = k∇ϕj (q)kq Gradient to the constraint (with respect to (, )q ) dϕj (q) ∗ nj (q) = kdϕj (q)k∗ Normal to the constraint (with req spect to ·) V(q) Cone of admissible velocities λj P Lagrange's multiplier r = j λj n∗j (q) Generalized force of reaction in the frictionless setting [17] N∗ (q) Cone of normals
d = 1, 2 or 3 ⊂ Ed ∈ Ω = ∪b∈B Ωb ⊂ Ed ∈ Ed ∈ Ed ∈ Ed
Ωb , b ∈ B M p(M ) ~v (M ) f~
∼ stands for identi cation ∈Q ∈ TQ ∼ Rn × Rn ∈ Tq Q ∼ Rn ∈ Tq Q ∼ Rn ~ v (M ) = Gq (M )v R 1 2 1 kuk := ρkG (M )uk2 dM q q 2 Ω 2 2 1 ∀v, 2 kvkq = 21 t vM (q) v (v1 , v2 )q := t v1 M (q) v2 T∗q Q ∼ Rn ∗ f = Gq (M )f~(M ) −1 (f1 ,f2 )∗q := t f1 [M f2 (q)] 2 1 d ∂ ∂ kukq γi := dt ∂ui − ∂qi 2 ∈N Q→R {j : ϕj (q) ≥ 0} ∈ Tq Q ∈ T∗q Q
v : ∀j ∈ J(q), n∗j (q) · v ≥ 0 R ∈ T∗q Q ∼ Rn N∗ (q) =
P
j∈J(q)
R+ n∗j (q)
Table 1: Glossary
Remark:
In addition, the physics is invariant with respect to changes of generalized coordinates. We
emphasize that we adress the invariance with respect to the group of the change of generalized coordinates and not with respect to the Galilean group of the change of referential, as it is sometimes done by other communities [46]. Indeed, for the sake of simplicity, the unilateral constraints will be supposed holonomic (that is they can written under the form
φ(q) ≥ 0)
in this paper. A classical case of holonomic constraints
is the case of the noninterpenetration condition of the system with an external motionless obstacle. Saying something is motionless underlies a reference frame that we assume Galilean. For further details, see [47] on impact laws in a nonGalilean context.
q xed, the space of generalizedP velocities is a normed vector space. Indeed, the kinetic R 1 1 2 2 kvk := q b∈B Ωb 2 ρ(M )kGq (M )vk dM (k · kq is a quadratic 2 positive form on u provided the volumic mass ρ(M ) is positive). We shall denote Tq Q as the normed vector n space R endowed with k · kq . The polarization identity allows us to de ne the scalar product (·, ·)q on Furthermore, for
energy provides a norm on velocities:
5
Tq Q3 .
We de ne the mass matrix
M(q)
as the symmetric de nite positive matrix whose components are
Rn . For all v, w ∈ Tq Q, we have 1t 1 2 t 2 kvkq := 2 vM(q)v and also (v, w)q = vM(q)w. In practice, discrete systems in mechanics are collections of rigid bodies, possibly articulated with joints.
the cross scalar product between the vectors of the canonical basis of
In the setting of Lagrange's theory of discrete systems [42], the generalized forces are the suitable mathematical description of forces. Generalized forces are cofactors of generalized velocities or Riesz's representants in the virtual power of forces and are vectors of
Rn .
They are deduced from forces in
tual power point of view [43, chapter 4]. With the canonical scalar product
·
of
Rn
Ed
through a vir-
standing for the duality
product between generalized velocities and generalized forces, we can de ne a scalar product on generalized
(f , g)∗q := t fM(q)−1 g and a norm k · k∗q accordingly. We denote T∗q Q the space Rn endowed with ∗ (·, ·)q . The equation of dynamics is obtained by setting the generalized acceleration equal to the generalized forces
forces. The generalized acceleration
γ
is obtained from the Lagrange equation [42], by taking the derivative
of the kinetic energy with respect to the time, the con guration and the generalized velocity. The reader familiar with dierential geometry will recognize an expression of a total derivative.
∀i = 1, · · · , n,
γi =
∂ d ∂ − dt ∂ui ∂qi
1 kuk2q . 2
In practice, the generalized acceleration is always the sum of the time derivative of
u and additional terms.
In
order to simplify expressions, the additional terms are added to the right term of the equation of dynamics, which reads as follows:
M(q)
du = f (t, q, u) . dt
Measures over time to account for impact. This well-known framework is nevertheless too restric-
tive and some engineering applications require an extension of the theory to handle unilateral constraints. Take the example of the bouncing ball, idealized as a point particle falling under its weight to the ground. At the instant of contact, it has a velocity pointed toward the ground. A continuous evolution of the ball would necessarily violate the noninterpenetration condition. To recover existence of the evolution, one has to allow discontinuities in velocities. Those instants of discontinuities are called impacts. In this setting, the ball has a right (with respect to time) velocity dierent from the left velocity. More generally, these impacts happen when combining the hypothesis of rigid bodies together with that of noninterpenetration of bodies. The evolution problem governing the dynamic evolution of a discrete system with contacts is presented in the line of [16, 17]. Because of impacts, the acceleration cannot be de ned as a function of time. Instead, we look for the acceleration, as well as the reaction force, in the space of measures with respect to time
M ([0, tf ]; T∗ Q)
[16]. Indeed, the space of measures with respect to time de ned as the dual space of
the space of continuous in time test functions
C 0 ([0, tf ]; TQ)
contains the Dirac measure whose integral
over a time instant represents the total percussion of contact forces over the impact. The motion is to be looked for in the space
M M A ([0, tf ], Q)
of motions with measure acceleration [16, 41, 44], the motions
whom second time derivative in the sense of Schwartz's distributions are Radon's measures. If the motion
q(t)
belongs to
M M A,
then the generalized velocity
u = q˙
is a function with bounded variations [see 48,
for a very clear monograph on BV functions of time]. This entails in particular, that the velocity has a left limit
u−
and a right limit
u+
at every time instant. The instants of discontinuity are called impacts and
u in the sense of R distribution is a measure u+ (b) − u− (a) = [a,b] du, for all a, b ∈ R. ϕj (q) ≥ 0, which are assumed to be linearly
these impacts are countable. The derivative of called the variation of velocity since we have Now consider, the unilateral constraints
all q in a neighborhood of the initial condition.
denoted
du
and
independent for
In addition, we assume the unilateral constraints are
∗ j λj nj (q). The Lagrange's multipliers satisfy complementary conditions [49] in the case of contact without adhesion where
perfect or frictionless. Classically, this can be modeled through Lagrange's multipliers
r=
P
3 Note that without a scalar product, it is not possible to de ne tangential generalized velocities in view of describing friction. Note in addition the scalar product is intrinsic with respect to change of parametrization. 6
distant interactions are allowed.
We introduce the unit normal to the constraints
n∗j (q) =
dϕj kdϕj k∗ .
The
evolution problem now reads as follows:
P1 . Find q ∈ MMA([0, tf ]; Q), λj ∈ M([0, tf ]; R) such that: q(0) = q0 , u+ (0) = u0 , initial condition ˙ dt-almost everywhere u = q, X − ∗ M(q) · du = f (t; q, u ) + λj nj (q), equation of motion, in j ϕj (q) ≥ 0, λj ≥ 0, λj ϕj (q) = 0. contact law
Problem
(1a) (1b) the sense of distributions
(1c)
(1d)
J(q) =
Following Moreau [17], the set of active constraints is de ned by
j : ϕj (q) ≤ 0
and the
V(q) of admissible velocities as follows: if q is in the interior of the admissible domain A = {q : ∀j, ϕj (q) ≥ 0}, then V(q) is equal to Tq Q, if q is not admissible then V(q) is empty, otherwise V(q) convex cone
is de ned as follow:
V(q) = v ∈ Tq Q : ∀j ∈ J(q), n∗j · v ≥ 0 . Accordingly, we say a velocity is (right) admissible if it belongs to be smooth if admissibility
u− is admissible whereas we + of u . Of course, the choice
P1
to problem
expect an impact to occur if of
u+
in
V(q)
(2)
V(q). We expect the evolution to u− ∈ / V(q) in order to enforce the
is not unique and the uniqueness of the solution
cannot be expected. To recover Laplace's determinism, the uniqueness in the choice of
u+
is
enforced by a constitutive law, called an impact law. Some basics of convex analysis.
It is convenient to introduce some elements of convex analysis to
ι : Rn → R ∪ {+∞} be a proper (not identically equal to +∞), lower semi-continuous, convex function. Its domain D(ι) is de ned as the (nonempty convex) n subset of R where ι takes nite values. The subdierential ∂ι [x] of ι at x is de ned by: formulate constitutive laws in a nonsmooth context. Let
∂ι [x] = {y ∈ Rn : ∀ˆ x ∈ Rn , ι(ˆ x) ≥ ι(ˆ x) + y · (ˆ x − x)} . The (Legendre-Fenchel) conjugate function
ι∗
of
ι
de ned by
(3)
ι∗ (y) = sup {y · x − ι(x)}
is a proper lower
x∈Rn semi-continuous convex function. The following equivalence holds [50]:
y ∈ ∂ι[x] ⇔ x ∈ ∂ι∗ [y] ⇔ x · y = ι(x) + ι∗ (y) Now, let
x ∈ C,
C
and
be a nonempty closed convex subset of
IC (x) = +∞
Rn .
(x, y ∈ Rn ).
Its indicatrix function
IC
(4)
is de ned by
IC (x) = 0
if
otherwise.
Moreau [19, proposition 5.1] shows that the contact law is equivalent to the following inclusion during smooth motion episodes:
−r ∈ ∂IV(q) [u] , where
r
(5)
is the generalized reaction force added to the equation of dynamics to account for the interaction
of the systems with the constraints. Indeed,
∂IV(q) [u]
is the exterior normal cone to
V(q)
at point
u
[50].
This entails:
•
The velocity is admissible,
u ∈ V(q),
because (5) implies that
∂IV(q) [u]
is not empty. This condition
readily implies that the con guration over time is admissible.
• •
∂IV(q) [u] = {0}.
If
q
is in the interior of the feasible region, then
If
q
is on the boundary of the feasible region and
u
There is no distant interaction.
is in the interior of
V(q),
then
∂IV(q) [u] = {0}.
There is no adhesion.
•
If
q is on the boundary of the feasible region and u is on the boundary of V(q), r is a linear combination n∗j , the constraints are perfect, then r can only repel the system.
of the
7
Note that the subdierential inclusion reformulates the constraints on the con guration level by constraints on the velocity level. Substituting the right velocity for the velocity in the right term, Moreau extends this inclusion to the case with impacts and obtains an impact law. We introduce Moreau's nonsmooth evolution problem for multibody systems with perfect unilateral constraints:
P2 . Find q ∈ MMA([0, tf ]; Q) u+ (0) = u0 , q(0) = q0 , u = q, ˙
Problem
M(q) · du = f (t; q, u− ) + r, − r ∈ ∂IV(q) u+ . In opposition to problem
and
r ∈ M([0, tf ]; T∗ Q)
such that:
initial condition
(6a)
dt-almost
(6b)
everywhere
equation of motion, in the sense of distributions
(6c)
contact and impact law
(6d)
P1 , problem P2 contains an R impact law. We combine (6c) and (6d) and we integrate {t}. The forces {t} f (t; q, u− )dt vanish because {t} is negligible for the
the subdierential inclusion over Lebesgue's measure
dt.
We get:
M (q(t)) u+ (t) − u− (t) ∈ ∂IV(q(t)) u+ (t) . k · kq , of u− on V(q) [19, page 44]. Denoting ΠC⊂X [x] Banach space X on a closed convex subset C ⊂ X , we have: u+ = ΠV(q) u− .
This characterizes the proximal point, for the metric the projection of the point
x
of some
Hence, the subdierential inclusion (5) contains an impact law, named the standard inelastic shock law (2.1) [19, section 8]. Moreover, the evolution problem
P2
is in fact a sweeping process [1]. Sweeping pro-
cesses were introduced in the 1970's by Moreau for the study of perfect plasticity [51]. The reformulation of the problem as a sweeping process conveys not only a numerical method, the catching-up algorithm renamed time-stepping scheme in the context of contact dynamics, but also techniques for theoretical investigation: arguments of maximal monotony [52] are used to prove existence results for sweeping process [33, 38]. Some discussions about problem
P2
are postponed to the appendix.
2.2. The NSCD procedure for perfect unilateral constraints The directing idea of the nonsmooth contact dynamics approach is that the main object of the computation is the velocity function
u
[1]. Contact dynamics numerical procedures rest on drawing the balance
momentum of the investigated mechanical system over each time-step. No estimation of the acceleration is needed, instead the formulation emphasizes the variation of velocity
∆ui :=
R ]ti ,ti+1 ]
du < +∞.
Time-
stepping algorithms essentially have to determine the evolution of the velocity function by applying the principles of dynamics and the speci ed constitutive laws [1]. The con guration
q
is only updated at each
time-step through adequate integration.
P2 this way: −M (qm ) ui+1 − ul ∈ ∂IV(qm ) [ui+1 ] ,
These points suggest to discretize in time problem
where
i
indexes the time-steps and the loose velocity
(7)
−1
ul = ui + ∆ti M (qm )
f (tm ; qm , ui )
is the nal
velocity of the corresponding contact free evolution problem over the considered time-step. The variation of the geometry over a time-step cannot be taken into account in this setting and Jean and Moreau [1, 2, 17] take a midpoint approximate
qm
for the con guration
q and hold the mass matrix and the cone of admissible
velocities as constant over the considered time-step. The dierential inclusion (7) (similar in many ways to (2.1)) characterizes the proximal point, for the metric
M(qm ),
of
ul
on
V(qm )
[19, page 44]. Hence, the numerical procedure consists in a succession of
well-de ned incremental problems, the proximation of a point on a moving convex set, and the procedure is called a catching-up algorithm [1]. The catching-up algorithm, introduced in the case of the quasi-static 8
elastoplastic evolution of a continuum, is renamed a time-stepping scheme in this context
4
and is summarized
on the left side of algorithm 1. At each time-step, the step 3 is a classic quadratic programming problem [19] solved with the NLGS algorithm [2, 7].
Moreau's time stepping scheme
Jean-Moreau's time stepping scheme
Frictionless case
For a given start time
ti ,
Frictional case
known displacement
qi , and generalized velocities ui , we qi+1 and the generalized velocities
compute an approximation of the displacement
ui+1 Step 1:
at the end of the time interval
]ti , ti+1 ].
Evaluation of midpoint approximants
Compute the midpoint
tm = ti + 12 ∆ti .
tm by qm = qi + 12 ∆ti ui . m Compute the mass matrix Mm := M(t , q ) and the forces f = f (tm , qm , ui ). m m Compute the set J := {j : φj (q ) ≤ 0} and the associated cone of admissible velocm ities V = V(qm ) through equation (2). m Evaluate G = Gqm (Mcm ) and the Delasm sus's matrix G . Compute the midpoint displacement at time
m
Step 2:
Discretization of contactless equation of dynamics
We de ne the loose velocity If Step 3:
m
l
u ∈V
m
ul = ui + ∆t (Mm )
−1 m
.
Standard inelastic shock law
Coulomb and standard inelastic laws
Compute a projection with NLGS:
Try to solve with NLGS the LCP:
Set
ul = ΠVm ul .
−Gm~r − G m ul ∈ ΠT~ (q) ∂IC~ [~r] . Set
Step 4:
f
then go to step 4.
Set
i+1
u
l
=u
ul = ul + (Mm )
−1
∗
(G m ) ~r
.
Compute the end displacement
qi+1 = qm + 21 ∆tui+1 .
Algorithm 1: Jean-Moreau's NSCD procedure
Moreau has introduced the time-stepping scheme to discretize the sweeping process time-stepping scheme can also be applied directly on the evolution problem
P1 .
P2 .
However, the
Emphasis is put on velocities
in the formulation of constraints in the time-stepping scheme. Hence, the holonomic constraints of problem
P1
are linearized to get constraints on the velocity level.
complementary conditions (LCC) The problem
P2
uj ≥ 0, λj ≥ 0, λj uj = 0,
Denoting denoted
uj = n∗j · u, these reads as linear uj ≥ 0 ⊥ λj ≥ 0 for concision.
is reformulated as a linear complementary problem (LCP) [49]. Each LCC de nes a half-
space and the intersection of these half-spaces de ne the polyhedral cone of admissible velocities. Hence, making a projection on
V(q)
and solving the LCP are equivalent.
The equivalent LCP is adapted to practical implementation of step 3 with the NLGS algorithm. Indeed, Loetstedt [49, section 1 and 2] emphasizes that this formulation makes the link between multibody system dynamics with unilateral constraints and the linear complementary theory of mathematical programming. The NonLinear Gauss-Seidel (NLGS) algorithm owes its name to its similarity with the Gauss-Seidel algorithm for solving linear systems. The NLGS algorithm is described in algorithm 2. Note that the problem of the proximation of a point on a polyhedral convex cone is a well-posed problem that can be solved trough LCP. However, LCP are not always equivalent to a proximation problem as in the case of NCSD with friction [2, 19] and they do not always form a well-posed problem [49].
4 In perfect plasticity, the moving convex set depends on the time via the loading. In our context, the moving convex set R depends on the time via q(t) = 0t udt. The time-stepping scheme is a catching-up algorithm on the velocity combined with a midpoint rule on the con guration. 9
u ← ui r←0
not converged
while for
do
j ∈ J(q) do ∗ if nj, q · u > 0 then ∆r ← − r, n∗j, q q n∗j, q else
∆r ← − n∗j, q · u n∗j, q end
r ← r + ∆r u ← u + M(q)−1 ∆r end end
ui+1 ← u
Algorithm 2: NLGS for solving step 3 of the NSCD procedure in the frictionless case
2.3. Extension of the NSCD procedure to unilateral constraints with dry friction Moreau have extended his point of view to the case with dry friction [19].
Nevertheless, to apply
Coulomb's law, the archetypal law for dry friction of point particles, he particularizes the considered system [19, section 12] to write the equations to be discretized. We quote [19, page 47]: "To x ideas, suppose that equality [φ(q) external obstacle
= 0]5
Ω0 ".
expresses that in position
q,
Ω1
some part
of the system touches the unmoving
The algorithm of [19, section 12], which will be presented hereafter, can be extended
to many other circumstances [2, 4] but it is no more clear what are the equations the algorithm is trying to compute [53]. We denote
Mc
as the material point of
a vector of the Euclidean space
Ed ,
identi ed as
Rd
Ω1
obstacle is assumed to be smooth enough to introduce
Mc is Ed . The
experiencing the contact. The velocity of
in practice. An arrow is put on elements of
~n(q, Mc ),
denoted
~n
for conciseness, the normal to
the obstacle. Moreau notes the following points:
•
A force pointed in the geometric normal to the obstacle force. Mathematically speaking,
•
Gq∗ (Mc )~n
and
n∗q
v = V(q) ∩ −V(q) be a tangential generalized velocity. Mc is tangential: Gq (Mc )v · ~n = v · Gq∗ (Mc )~n = 0.
Let of
~n
is associated with a normal generalized
are colinear. Then the associated Euclidean velocity
The rst point allows us to reformulate the frictionless evolution problem by introducing the reaction
~r = rn~n ∈ Ed applied on Mc . We introduce ~u (t, q, u, Mc ) = Gq (Mc )u, denoted ~u for conciseness, un = ~u · ~n. The frictionless contact law is rewritten as −rn ∈ ∂IR+ [un ] and the equation of dynamics − ∗ as M(q) · du = f (t; q, u ) + Gq (Mc )~ r. However, the equation of dynamics contains more information than ∗ what the kinematics convey. The additional information, e.g. Gq (Mc ), make this equation of motion speci c
forces and
to the considered problem: it is not generic to any multibody system. This is the reason why we claim this formulation is not systematic, unlike that of problem
P2 .
Once the frictionless problem has been particularized to make explicit the local force applied. As far as the normal component
rn
~r, Coulomb's law is
is known, Coulomb's law can be derived from a pseudo-potential
of dissipation [51, 54].
n Let C be a nonempty closed convex subset of R . Its support function SC supy∈C {x · y} and is a proper lower semi-continuous convex function [50]. The
is de ned by
SC (x) =
support function is the
conjugate of the indicatrix function and the reciprocate is also true due to nite dimension [50]. Now, we consider As far as
rn
~ B
the unit ball of the plane tangent to the obstacle
T~ (q, Mc ),
5 Note there exists only one unilateral constraint. 10
~n. ψ (~v ) = µrn k~vt k =
and orthogonal to
is known, Coulomb's law derives from the pseudo-potential of dissipation
Sµrn B~ (~vt )
[51, 54].
The inclusion
(because of equivalences (4)).
−~rt ∈ ∂Sµrn B~ [~vt ]
holds, as well as the conjugate
~vt ∈ ∂Iµrn B~ [−~rt ]
They account for sticking episodes of motion as well as sliding episodes.
However, the Euclidean tangential and normal parts are mixed together in the equation of motion so that it is more convenient to work with the total force of reaction.
~ µ = {~r : k~rt k ≤ µrn }. During smooth episodes of motions, C Coulomb's law is equivalent to : −~ u ∈ ΠT~ (q) ∂IC~ [~r] [19, proposition 12.1]. Since T~ (q) is in Ed , the projection is made with respect to the canonical scalar product of Ed . ~ (q), as in the standard inelastic shock law, and is extended This formulation contains a projection on T Hence, we introduce Coulomb's cone
to nonsmooth motions:
−~u+ ∈ ΠT~ (q) ∂IC~ [~r] .
(8)
We obtain the following problem:
P . Find q ∈ MMA([0, tf ]; Q), ~r ∈ M([0, tf ]; Ed ), such that: 3 q(0) = q0 , u+ (0) = u0 , initial condition u = q, ˙ dt-almost everywhere − ∗ M(q) · du = f (t; q, u ) + Gq (Mc )~r, equation of motion ~u = Gq (Mc )u, velocity of the contact − ~u ∈ Π ~ ∂I ~ [~r]. Coulomb's law T (q) C
Problem
(9a) (9b) (9c) (9d)
point
(9e)
The problem is free of nonexistence pathology because Moreau applies the standard inelastic shock law after the friction law and the right velocity is necessarily admissible (we recall the existence proof of Stewart [38]). However, the price for recovering existence in the Painlevé's paradox is the introduction of frictional impact or frictional paroxysm [19], whose physical meaning has been thoroughly discussed [11, 12, 15, 19, 55]. The resulting algorithm of the discretization in time of problem
P3
is presented on algorithm 1. The
algorithm diers from the frictionless one at step 3, despite the fact that in practice both rely on the resolution of a LCP with the NLGS algorithm. For the 2D case, Coulomb's cone
~ C
is a polyhedral cone and
step 3 can be reformulated by a LCP problem (provided the introduction of additional auxiliary variables [56, section 1.3.4]). For the 3D case, Coulomb's cone is often approximated by a polyhedral cone [38, 57, 58] and step 3 can be reformulated as a LCP as well. As a consequence, step 3 is solved by a slight adaptation of the NLGS algorithm [2] consisting in adding the LCC related to friction in the loop on However, in opposition to problem
P2 ,
problem
P3
j
in algorithm 2.
is not a sweeping process. Jean-Moreau's NSCD is a
time-stepping scheme that does not consist in a catching-up algorithm on velocity. The NSCD procedures in the frictionless case and in the frictional case have very dierent properties although they have similar implementations.
Because problem
P3
is not a sweeping process, the step 3 is no longer a well-posed
proximation problem in general [49], leading to poor convergence properties of the NLGS trying to solve the LCP.
2.4. Extension of the NSCD procedure to other impact laws Impacts occur when the left velocity is not (right) admissible with respect to the unilateral constraints. If so, a velocity jump occurs so that the right velocity is admissible. After Newton, people have used impact laws to pick one right velocity in the set of admissible velocities. Impact laws are constitutive laws postulated to recover Laplace's determinism [59]. In the case of multicontact collisions or discrete systems that do not reduce to particle points, the Newtonian impact law has to be extended whereas the classic de nition of the restitution coe cient as a dissipation parameter can still be kept [59]. In all generality, an impact law is a mapping
F
F
which maps the preimpact state
(q, u− )
to the right velocities
u+ .
The choice of the mapping
is arbitrary as long as it respects three conditions [12, 59, 60]. 1.
F
maps
TQ
onto the cone of admissible velocities
velocity is admissible: the restriction of
F (q, ·)
to
11
V(q). Moreover, no impact is V(q) is equal to the identity.
expected if the left
2. The impact law has to be compatible with the contact law. Denoting
P R u− = j∈J(q) Λj n∗j, q with Λj := {t} λj , and j ∈ / J(q) ⇒ Λj = 0. 3. The impact law is energetically consistent:
this compatibility in the
r = M(q) F (q, u− ) − frictionless case reads as Λj ≥ 0 R :=
R
{t}
kF (q, u− ) kq ≤ ku− kq .
A geometrical representation of these conditions is to be found in [59, gure 8]. We have already encountered an impact law: the standard inelastic shock law which minimizes the right kinetic energy while satisfying conditions one and two [59]. The mapping
F can be explicitly or implicitly de ned. In F . The impact law is particularized based
impact law refers to the conditions de ning
the latter case, the on heuristics up to
experimentally determined constitutive parameters.
2.4.1. Newton law We introduce a restitution coe cient to generalize the standard inelastic shock law:
u+ = F q, u− = u− + (1 + ) ΠV(q) u− − u− . The three conditions on
ku− − ΠV(q) [u− ]kq
F
are ful lled if
so that
Equation (10) is equivalent to
0 ≤ ≤ 1.
(10)
ku+ − ΠV(q) [u− ]kq = u+ −u− 0 we de ne u = 1+ .
Note, in addition, this leads to
quanti es a restitution of energy. Following [61], u0 = ΠV(q) [u− ]. In the present work, we speak of a Newton impact law when
the restitution coe cient is applied to the velocity. The analogy with the standard inelastic case shows that
u0
satis es the following LCC:
n∗j, q · u0 ≥ 0 ⊥ Λj ≥ 0 .
The latter form is often extended to the case where each constraint has a dierent restitution coe cient. De ne
vj = n∗j, q · v
for
v ∈ Tq Q
and de ne
u0j =
− u+ j −j uj , where 1+j
j ∈ [0, 1].
Newton's law then reads as
follows [61]:
u0j ≥ 0 ⊥ Λj ≥ 0 .
(11)
This reformulation of Newton's impact law as a LCC allows us to handle the rocking rod case [61, gure 10].
− u+ j experiences a jump despite uj ≥ 0 because 6= 0) such that u− j 0 < 0. The rocking rod example
The rocking rod case is the archetypal example where some of the coupling with another component
j 0 (n∗j 0 , q · nj, q
shows inequalities are better suited than equalities to state multicontact impact laws.
2.4.2. Poisson's law
− ∗ − − . This is the minimal percussion (for j∈J(q) Λj nj, q := M(q) ΠV(q) [u ] − u kq ) making the right velocity admissible. We multiply equation (10) by the mass matrix and obtain Now, we de ne
k·
R− =
P
Poisson's law, where the restitution coe cient is applied on the percussion:
R = (1 + )R− .
(12)
Poisson's law can be easily adapted to the case where each constraint has a dierent restitution coe cient:
R=
P
j∈J(q) (1
∗ + j )Λ− j nj, q .
However, the example of the rocking rod [61, gure 10] shows that equalities
are too restrictive to formulate impact laws so Poisson's law is generalized using LCC. De ning
− −Λ+ j + εj Λj
Λ0j =
[40, 61], Poisson's law reads as follows:
0 u+ j ≥ 0 ⊥ Λj ≥ 0 .
(13)
Newton's and Poisson's impact laws are equivalent for frictionless impacts if all coe cients of restitution are equal [61]. Conversely, they are not equal if all coe cients of restitution are not equal. In particular, Newton's impact law does not de ne a mapping
F
satisfying all the conditions whereas Poisson's law
does [39]. For impacts with friction, one contact point is already su cient to demonstrate signi cant dierences in the results [61]. Newton's law remains very popular because it can be easily be combined with Coulomb's 12
law and implemented within a NSCD procedure. Some authors [62, 63] look for conditions on the restitution coe cient such that Newton's law has the desired properties, in particular energetic consistency. This is however not discussed here since we will implement Poisson's law in the catching-up NSCD. Indeed, Poisson's law always satis es the aforementioned conditions on and
F
(provided the restitution coe cient are between
0
1).
3. A convergent time stepping scheme for multibody problem with contact and frictions
3.1. The evolution problem in the case with friction We consider the evolution problem governing the evolution of a discrete system submitted to (holonomic) constraints and possibly friction by putting emphasis on the generalized force of reaction, in the line of [12]. Because the contact law in the frictionless case de nes the reaction as an element of the normal cone we de ne the normal reaction as the projection of the generalized force of reaction on the tangential reaction, denoted the orthogonal cone (for
rτ
M(q)−1 )
∗
N (q).
N∗ (q),
Accordingly,
is de ned as the projection on the tangential cone. The latter cone is
of
N∗ (q).
The tangential reaction is assumed to obey a friction law on generalized velocities and generalized forces. Extending Moreau's point of view on Coulomb's law for a point particle [51, 54], we suppose the friction law derives from a pseudo-potential of dissipation. Remember we have, if the constraints model contacts with
v ∈ V(q) ∩ −V(q) then ~u = Gq (Mc, j )v vt as dissipation pseudo-potential, where vt V(q) ∩ −V(q).
smooth external obstacles and if contact points are identi ed: if
is
tangential to obstacles. This suggests to take a function of
is
the projection (for
M(q))
of the generalized velocity
v
on
To model dry friction, the dissipation pseudo-potential is taken as a nonnegative homogeneous function of order 1 in
vt
and as a linear function of the Lagrange's multiplier
∗ Cj,q
λj .
Using [50, theorem 13.2], there
∗ j is of the form λj SCj,q [vt ] ∗ ∗ ∗ Cj,q , nj, q q = 0. Coulomb's law for a point particle [51, 54] is then a special case of this general framework. Indeed, choosing Coulomb's ~ ⊂ Rd as C ∗ allows to recover Coulomb's law. In the case of several contacts, we assume the total discs µB j,q
exists some convex set
such that the pseudo-potential associated with contact
∗ (support functions were introduced in section 2.3). The sets Cj,q are such that
energy dissipation is the sum of the dissipation over each constraint. In order to extend this friction law to a nonsmooth context, a weak formulation is adopted in the line of the weak formulation of Coulomb's law presented in [64, 65]. The pseudo-potential of dissipation involved in the weak formulation is because
rτ
∗ λj SCj,q [vt ].
Note, however, that the set of test velocities is the all space
Tq Q ,
is a projection on a cone and not on a space [12, p. 17-18].
Finally, the general evolution problem that governs the dynamics of the discrete mechanical systems with
n
unilateral frictional constraints reads as proposed in [12] :
Problem
P4 .
Find
q ∈ MMA([0, tf ]; Q), r ∈ M([0, tf ]; T∗ Q), λj ∈ M([0, tf ]; R)
13
such that:
u+ (0) = u0 , q(0) = q0 , u = q, ˙ M(q) · du = f (t; q, u− ) + r, + u+ = u+ n + ut , + with: un ∈ Span (nj,q ) j∈J(q) , + ∗ and ∀j ∈ J(q), nj,q · ut = 0, X r = rτ + λj n∗j,q ,
initial condition
(14a)
dt-almost
(14b)
everywhere
equation of motion
(14c) (14d)
(14e)
j∈J(q)
X λj n∗j,q = 0, with: rτ · j∈J(q) and ∀j ∈ J(q), λj ≥ 0, rτ · nj,q ≤ 0, ∀j, ϕj (q) ≥ 0, λj ϕj (q) = 0, ∀ˆ v ∈ C 0 ([0, T ]; Tq Q), Z X + ∗ (−ˆ ∗ (−u ) ≥ 0, rτ · v ˆ − u+ + λij SCj,q v) − SCj,q t t [0,T ] j∈J(q) u+ = F(q, u− ). n
contact conditions
(14f )
friction law
(14g)
impact law
(14h)
3.2. The catching-up NSCD procedure The evolution problem is discretized according to the Jean-Moreau's time-stepping scheme. As in the frictionless case (section 2.2), the holonomic constraints on the con guration are reformulated as LCC on generalized velocities. Since the operation has already been done for the equation of motion, the contact law and the impact law by Moreau and others in the frictionless case, we will only detail the friction law discretization (14g), which is the novelty of problem
P4 .
The philosophy of the time-stepping scheme suggests
to take step functions as test functions in the weak friction law. Accordingly, the term where where
v ˆ ∈ C 0 ([0, T ]; Tq Q) is modeled at the current time step with Ri is meant to be an approximation of the total integral over
ˆ − ui+1 Riτ · v t
R
r [0,T ] τ
where
· v ˆ − u+ t
ˆ ∈ Tqm Q v
,
and
a time step of the generalized reaction
force. The discretization in time of the weak friction law now reads:
∀ˆ v ∈ Tqm Q,
X
Riτ · v ˆ − ui+1 + t
∗ ∗ Λij SCj,q (−ˆ v) − SCj,q (−uti+1 ) ≥ 0. m m
(15)
j∈J(qm ) The tangential part is de ned according to the midpoint approximant
qm ,
assuming that the geometry of
the contacting bodies is constant over the time-step, in accordance with Jean and Moreau. We recognize in (15) the de nition of a subdierential inclusion (3). We have:
Riτ ∈ ∂
X
−ui+1 ∗ Λij SCj,q . t m
(16)
j∈J(qm ) Although the argument of the support functions is the tangential velocity, the subdierential is taken with respect to
Tqm Q.
Now, we tackle the question of the impact law. normal to the normal velocity, a mapping
F
stated in section 2.4 is compatible with problem sets
∗ Cj, q
Since the tangential generalized reaction force
Rτ
is
introduced for the frictionless case and satisfying the conditions
P4 .
However, in the case where some unbounded friction
are introduced (in order to account for locking eects [12]), we have to de ne the concept of
admissible velocities with respect to the friction law. De ning the support set of a proper convex function 14
as
supp χ = {v ∈ Tq Q : χ(v) < +∞}, we expect the output of the n∗j (q) · F(q, u− ) = 0 implies F(q, u− ) ∈ supp SCj,∗ q .
friction law:
mapping
F
to be compatible with the
In the case of Poisson's law, we have
0 ≤ ε ≤ 1. ∗ Cj, q.
to make an assumption on the restitution coe cient more restrictive than
This is not further
discussed because the examples presented in the paper do not exhibit unbounded Hence, at the
i-th
time-step, the incremental problem reads as follows:
P5 . Find q ∈ Q, vi+1 ∈ Tqm Q, Ri ∈ T∗qm Q, Λij ∈ R m i M(q ) · ∆u = ∆ti f (tm ; qm , ui ) + Ri , ui+1 = ui+1 + ui+1 t , n i+1 with: un ∈ Span (nj,qm )j∈J(qm ) , i+1 m ∗ and ∀j ∈ J(q ), nj,qm · ut = 0, X i i i ∗ R = Rτ + Λj nj,qm , j∈J(qm ) X i with: Rτ · Λij nj,qm = 0, i+1
Problem
such that: equation of motion
(17a) (17b)
(17c)
m
j∈J(q ) m and ∀j ∈ J(q ), Λij ≥ 0, Riτ · nj,qm ≤ 0, ∀i ∈ J(qm ), n∗j,qm · ui+1 ≥ 0, Λij n∗j,qm · ui+1 = 0, X −ui+1 ∗ Riτ ∈ ∂ Λj SCj,q , t m m) j∈J(q i −i Λij = Λ+ j + Λj , ui+1 ≥ 0 ⊥ −Λ+ i + εj Λ− i ≥ 0 . j
j
We recall that the
i Λ− j
contact conditions
(17d)
friction law
(17e)
Poisson's law
(17f ) (17g)
j
are obtained by solving a frictionless problem with the standard inelastic shock
law. For the purpose of the implementation, we reformulate the generalized friction law as a proximation problem. The generalized friction law is a subdierential inclusion involving a nonnegative and homogeneous of order
1
dissipation pseudo-potential in the velocity. This implies the existence of a convex set such that
the dissipation pseudo-potential is the support function of a convex set [50, theorem 13.2]. The convex set is
n o P r : ∀v, r · v ≤ j∈J(qm ) Λij SCj, qm
[50, corollary 13.2.1] and is equal to
P
j∈J(qm )
Λij Cj, qm .
We now are able to apply a Legendre-Fenchel transformation on (17e):
i ∗ −ui+1 ∈ ∂IPj∈J(qm ) Λij Cj,q Rτ . t m We denote
T∗ (qm ) = −M(qm )V(qm ) Y ∗ (qm , Λij ) =
as the tangent cone and we de ne the yield domain as:
⊥ X
Λj n∗j (qm ) ∩ T∗ (qm ) ∩
j∈J(qm ) We look for
ui+1 ∈ Tq Q
and
(18)
Riτ ∈ T∗q Q (Riτ
X
∗ Λij Cj, qm .
(19)
j∈J(q) is now a formal unknown) solution of the following subdier-
ential inclusion:
−uti+1 ∈ ∂IY ∗ Riτ .
(20)
i+1
i i+1 i i If u and Rτ are solutions of (20), then u and Rτ are solutions of (18). In addition, Rτ is the projection P i i ∗ i on the tangent cone of Rτ + j∈J(q) Λj nj (q). We inject (17a) into (20), and the result characterizes Rτ as a proximal point:
" Riτ
= ΠY ∗
# m i i m m i − M(q )u + ∆t f (t ; q , u ) t . 15
(21)
Finally, the incremental problem
P5
can be solved by the procedure presented in algorithm 3, which
consists in a succession of well-posed proximation problems. The third step is common with the NSCD in the frictionless case with standard inelastic shocks. Step 4 is an additional NLGS to account for bouncing with Poisson's law. Step 5 is the resolution of the friction law, formulated as a proximation problem. In fact, the algorithm combines a catching-up algorithm on the normal part of velocities and a catching-up algorithm on the tangential part of the generalized reaction force, together with a mid-point rule for the con guration. We will now refer to the algorithm as the catching-up NSCD in the following.
Time stepping scheme with Poisson's impacts and friction.
For a given start time
ti ,
known displacement
compute an approximation of the displacement
ui+1 Step 1:
at the end of the time interval
qi and generalized velocities ui , we qi+1 and of the generalized velocities
]ti , ti+1 ].
Evaluation of midpoint approximants.
Compute the midpoint
tm = ti + 21 ∆ti .
tm by: qm = qi + 12 ∆ti ui . m Compute the mass matrix Mm := M(t , q ) and the forces f = f (tm , qm , ui ). m m Compute the set J := {j : φj (q ) ≤ 0} and the associated cone of admissible velocm ities V = V(qm ) through (2). Compute the midpoint displacement at time
m
Step 2:
Discretization of contactless equation of dynamics.
We de ne the loose velocity If Step 3:
ul ∈ Vm
ul = ui + ∆t (Mm )
−1 m
f
.
then go to step 6.
Impact law 1/2: Standard inelastic shock law. Compute
ΠVm ui
NGLS provides the Step 4:
m
with NLGS.
(Λ− j )j∈J
such that:
P
j∈J
i ∗ l Λ− j nj,qm = M ΠVm u − u .
Impact law 2/2: Poisson's law. Solve with NLGS the LCP on the
∀j ∈ J,
nj · ul +
X
Λ+ j :
− ∗ ∗ (Λ+ k + Λk ) nj,qm , nk,qm
∗ f
≥0
⊥
− −Λ+ j + ej Λ j ≥ 0 .
k∈J Set
+ Λj = Λ− j + Λj .
Update the loose velocity: Step 5:
ul = ul + M−1
P
j∈J
Λj n∗j,qm .
Friction law.
J˜ = {j ∈ J : Λj > 0}. Riτ = ΠY ∗ − M(qm )ui + ∆ti f (tm ; qm , ui ) t
De ne the set Compute
∗
is de ned by (19))
Set
ul = ul + M−1 Rτ . ue = ul .
(Y Step 6:
Set
Calculate the end displacement:
with NLGS.
qe = qm + 21 ∆tue .
Algorithm 3: Applying Moreau's time-stepping scheme on the evolution problem of [12, page 20], we obtain the catching-up NSCD
4. Illustration of the methodology on the woodpecker toy benchmark
To support the discussion, we take as an example the woodpecker toy. The woodpecker toy was introduced by Glocker and Pfeier [40] as an archetypal example for multibody dynamics with (several) unilateral 16
constraints and it is a well known benchmark problem in contact dynamics.
After presenting the linear
kinematic description of the woodpecker, we identify the constitutive parameters of the evolution problem
P4 .
We model the contacts with unilateral constraints and then design the
∗ Cj, q
in equation (14g) based on
heuristics. The algorithm 3 is implemented on this example and the results are compared with Moeller's implementation of the Jean-Moreau's NSCD procedure [66].
4.1. Modeling the woodpecker toy The woodpecker toy is a mechanical system that contains both impacts and friction. We use the woodpecker model based on linear kinematics as formulated by Glocker [40, 66, 67]. The woodpecker toy consists of a vertical pole, a sleeve with a hole slightly larger than the diameter of the pole, a spring and the woodpecker. When correctly started, the woodpecker moves down the pole, performing a pitching motion interrupted by impacts. The mechanical model of the woodpecker toy is shown in gure 2. freedom and is parametrized by
y,
the ordinate of the sleeve;
α
The system has three degrees of
the rotation angle of the sleeve; and by
the rotation angle of the woodpecker. We identify the con guration
q
with
t
φ,
(y, α, φ).
g A1
A4
A5
A3
A2
Figure 2: Linear kinematics of the woodpecker toy Based on the linear kinematics, the mass matrix is independent of
mS + mM M = mM lM m S lG
mM lM 2 JM + mM lM mS lM lG
q
and reads as follows [40, 67]:
m S lG m S lM lG . 2 JS + mS lG
There are three unilateral constraints. The rst one is the contact of the material point the second and the third ones are limitations on the rotation angle
α.
A1
with the pole,
The second and the third constraints
are not realized by a single contact point, a situation not covered by [19, section 12]. The constraint functions can be de ned in this situation as the distance between contact points and obstacles, sometimes called gap functions:
ϕ1 (q) = = lM + lG − lS − r0 − hS φ, ϕ2 (q) = = rM − r0 + hM α, ϕ3 (q) = = rM − r0 − hM α.
17
(distance from (distance from (distance from
A2 A3
and and
A1 A4 A5
to the pole) to the pole) to the pole)
The unit normals to the constraints are:
2 m m + m m l2 J + (m J M lG + m )J J M S M S M S M S M S = 2 m m + (m lM M S M + mS )JM s 2 m m + m m l2 J + (m J M lG + m )J J M S M S M S M S M S = 2 m m + (m lG M S M + mS )JS s
n∗1
n∗2 n∗3
0 0 −1 0 1 0
= −n∗2 .
Then, we need to make a heuristic guess for the constitutive parameters of the evolution problem, namely by computing the yield domains. The friction law in problem
P4
is de ned up to the sets
∗ Cj, q
which contain information about the physical
reality of the system: information about the geometry of contacts that are not carried by about Coulomb's friction coe cient
µ.
q and information
As part of the modeling process of the woodpecker, we need to make
∗ Cj, q of the friction law. Experimental evaluation is possible for this ∗ ∗ speci c system because the Cj do not depend on q but this is not an option in cases where Cj, q are actually ∗ depending on q. Nevertheless, we provide hereafter a static identi cation procedure where we derive the Cj a choice for the constitutive parameters
from Coulomb's friction coe cient. The implementation of this rule is the topic of sections 4.2 and 4.3. The usual point of view depicted in section 2.3 has provided good predictions so far, except in paradoxical situations. We are then led to tune the
∗ Cj, q
so that both points of view match as close as possible. This
tuning is made by looking for static equilibrium in the case of a single contact point. This static identi cation procedure is detailed here for sake of pedagogy but the resolution is much more simple than it appears. The identi cation procedure for the
∗ Cj, q
will be greatly simpli ed in next sections.
This section and the following answer the question: Is it always possible to rewrite a classically formulated multicontact friction problem in this way and obtain the same results? Except for speci c cases (for example in section 5.2), it is hard to de nitively answer this question because the classical point of view is not suitable for theoretical investigations (see section 2.3). However, our simulations show that we can tune the
∗ Cj, q
so
that both points of view agree, except in pathological situations. Remember that, in the case of several contacts, the historical point of view consists in considering one reaction force per contact.
In some cases, for instance granular media, we can have more reactions
than degrees of freedom, which necessarily leads to indeterminations.
The point of view adopted in [12]
consists in considering one generalized reaction force, regardless of the number of active constraints.
As
a consequence, this reaction force obeys a speci c constitutive law in the case where the con guration lies at the intersection of several constraints. This constitutive law was theoretically constructed from the constitutive laws governing single constraints. The construction relies, on the one hand, on the de nition of the tangential reaction force as the projection on the tangential cone and, on the other hand, on the sum of the pseudo-potentials of dissipation in problem
P4 .
However, for algorithmic purposes, we put emphasis not
on pseudo-potentials of dissipation but on indicatrix functions, so that the algorithm reduces to a catchingup algorithm (equation (21)). We are led to construct the yield domain
Y∗
according to equation (20) or
step 5 of algorithm 3. This is discussed in section 4.4.
4.2. Modeling friction with respect to the rst constraint The historical point of view, restricted to the static case and to one contact point, reads as follows: Problem
P6 .
1 ( ~r ∈ Ed , such that: 0 = f (0, q, 0) + Gq∗ (A1 )~r1 , ~ µ1 . ~r1 ∈ C
Find
equation of motion contact and friction law
P4 to the static case with one constraint. We de ne T1 ⊂ Tq Q T1 = Span t (1, 0, 0), t (0, 1, 0) . We also de ne T∗1 = MT1 . The
We also consider the restriction of problem as the orthogonal to
n∗1
for ·. We have
tangential part of the reaction was de ned as the projection on a cone but in this setting, it is also a projection on the space
T∗1 .
The restriction of problem 18
P4
now reads as follows:
Problem
P7 .
∈ T∗q Q, such that: 0 = f (0; q, 0) + r, ∗ r = rτ + λ1 n1,q , ∗ with: rτ ∈ T1 and λ1 ≥ 0, ∗ rτ ∈ λ1 C1,q .
Find r
Note that for problem
P6
and
P7 , q
simpli ed because we only consider
equation of motion normal and tangential parts
contact law friction law
q is ϕ1 (q) = 0.
the con guration
a datum of the problem. The contact law is
such that
The problem
P6
is paradox free: neither the
uniqueness of the equilibrium con guration nor its existence are expected.
∗ P7 are equivalent. Thus, we look for C1∗ under the form C−1 so+ that problems − P6+ and + ∗ ∗ x t1 : x ∈ −µ1 , µ1 ⊂ R = −µ1 , µ1 t1 , with µ− 1 , µ 1 ≥ 0. 1 A velocity v in T1 is said to be tangential to the rst constraint because G(A )v is tangential to the 1 contact surface. This corresponds to the combination of two cases: the sliding of the material point A along We design
the vertical pole and the rotation of the sleeve while the beak stands still. We expect energy dissipation
t∗1 to be orthogonal ∗ (0, 1, 0)). The vector t1 is de ned by the following
only if the beak of the woodpecker slides along the pole. to the kernel of conditions:
1
1
t
Hence, we require the set
G(A ), which is Ker G(A ) = Span ( ∗ ∗ ∗ (requirement stated by the theory) (t1 , n1 ) = 0, ∗ kt∗1 k = 1 (requirement stated by the theory) ∗ t t1 · (0, 1, 0) = 0. (information on the physical reality
We have:
s t∗1 =
To determine the constitutive coe cients the contact point
A
1
1
lM 2 lG lM mM mS +JM lG mS 2 m m +J (m +m ) (lM M S M M S )
.
2 m m + J (m lM M S M M 2 m lM M + JM
µ− 1
+ mS )
and
µ+ 1,
of the system)
we now consider a local reaction force
which satis es the contact law and Coulomb's law. This local force
with a generalized force
~r
1
~r1
applied at
is associated
r: ry1 . lM ry1 r = G ∗ (A1 )~r1 = −hS rx1 + (lG − lS )ry1
~ µ1 so that the generalized force is in the cone C ∗ ∗ is included in Span (n1 , t1 ) and is drawn on gure 3. The local force
~r1
is in Coulomb's cone
~r1
C1∗ r
~ex G ∗ (A1 ) ~ez
~ µ1 . G ∗ (A1 )C
The last
n∗1
t∗1 µ+ 1
−µ− 1
Figure 3: Coulomb's law in static to design the convex set C1∗ On one side, we note that
µ+ 1 =
∗ ∗ 1 t∗ ex +µ~ ey ) 1 , G (A )(~ ∗ ∗ (A1 )(~ n∗ ,G e +µ~ e ) x y 1
( (
) > 0. )
G ∗ (A1 )
~ ex +µ~ ey ∗ ∗ 1 e +µ~ n∗ ey ) x 1 ,G (A )(~
(
)
∗
= n∗1 +
(t∗1 , G ∗ (A1 )(~ex +µ~ey )) ∗ ∗t . (n∗1 ,G ∗ (A1 )(~ex +µ~ey )) 1
Hence, we set
The analytical expression (obtained with the program Sage [68]) is too large
to be written here yet. The numerical evaluation provides
19
µ+ 1 = 0.1443.
On the other side, we note that
G ∗ (A1 )
~ ex −µ~ ey ∗ ∗ 1 e −µ~ n∗ ey ) x 1 ,G (A )(~
(
)
∗
= n∗1 +
∗
(t∗1 , G ∗ (A1 )(~ex −µ~ey )) ∗ ∗t . (n∗1 ,G ∗ (A1 )(~ex −µ~ey )) 1
We take
(t∗1 , G ∗ (A1 )(~ex −µ~ey )) ∗ > 0. The (n∗1 ,G ∗ (A1 )(~ex −µ~ey )) − evaluation gives µ1 = 0.2676.
µ− 1 = −
analytical expression is obtained with Sage [68] as well and the numerical Finally, we model the friction law along the rst constraint as follows:
rτ ∈ ∂Sλ1 [−µ− ,µ+ ]t∗ −u+ t . 1
1
1
4.3. Modeling friction with respect to the second and third constraints Two material points are at contact if for the classical equation and Glocker which is applied on the point
A2
ϕ2 (q) = 0
et al
(or the contact
ϕ3 (q) = 0).
for constraint
2
and on the point
this benchmark, we will use the same assumption and infer the set
A3
C2∗
C2∗ .
Because
~2 G(A2 )C µ
the case where Hence,
t∗2
is 2-dimensional, we look for
C2∗
3. For the sake of ~ µ2 . However, this G(A2 )C 2 ~2 4 ~4 set G(A )C µ + G(A )Cµ to
for constraint
from the set
hypothesis could easily be removed within our point of view, by considering the design
The case is pathological
[40, 66, 67] assume there is only one reaction force per contact,
under the form
− + ∗ −µ2 , µ2 t2 ,
t
with
+ µ− 2 , µ2 ≥ 0.
t
2
In
u = (0, 0, 1), the woodpecker moves but the sleeve is at rest, (0, 0, 1) belongs to Ker G(A )
.
is determined by the following conditions:
∗ ∗ ∗ (t2 , n2 ) = 0, kt∗ k = 1, ∗2 t t2 · (0, 0, 1) = 0.
(requirement stated by the theory) (requirement stated by the theory) (information on the physical reality of the system)
We have:
s t∗2 =
2 m m + J (m lG M S S M 2m +J lG S S
To determine the constitutive coe cients on the material point
A
2
µ− 2
1
+ mS ) JS lM mS 2 m m +J (m +m ) . (lG M S S M S ) 0 and
µ+ 2,
we consider a local reaction force
~r2
applied only
. We emphasize once again this hypothesis is made to respect classical equations
but is not mandatory in our setting. With
~r2 = rx2 ~ex + ry2~ey ,
we have:
ry2 G ∗ (A2 )~r2 = hM rx2 + rM ry2 . 0
As in the previous section, note that set
µ+ 2 =
∗ ∗ 2 t∗ ex +µ~ ey ) 1 , G (A )(~ ∗ ∗ ∗ 2 n1 ,G (A )(~ ex +µ~ ey )
( (
evaluation gives
) > 0. ) µ+ 2 = 0.2742. We
G ∗ (A2 )
(
~ ex +µ~ ey ∗ ∗ 2 e +µ~ n∗ ey ) x 2 ,G (A )(~
)
∗
= n∗2 +
(t∗2 , G ∗ (A2 )(~ex +µ~ey )) ∗ ∗t . (n∗2 ,G ∗ (A2 )(~ex +µ~ey )) 1
Hence, we
An analytical expression is again obtained with Sage [68]. The numerical do the same procedure for
~r2 = ~ex + µ~ey
to get
µ− 2 = 0.1528.
In the case where the second constraint is the only active constraint, the friction law reads as:
rτ ∈ ∂Sλ2 [−µ− ,µ+ ]t∗ −u+ t . 2
2
2
The case of the third constraint can be inferred from the second one by symmetry. We de ne
+ and µ3
=
µ− 2,
µ− 3
=
µ+ 2.
t∗3 = t∗2 ,
4.4. The case with two active constraints Given the assumptions on the kinematics, the constraints 2 and 3 are disjoint: the set of active constraints is either empty, or a singleton, or of the form
Y∗
is given by an explicit rule (20) from the
J = {1, j} with j = 2 or 3. Cj∗ . 20
In the latter case, the yield domain
We de ne the set
J˜ = {i : λi > 0}.
If
J˜ = {1, j}
tJ˜ = We have
t∗J˜, n∗j
∗
j=2
or
3,
we de ne:
0 0
and
t∗J˜ = MtJ˜.
∗ j ∈ J˜ and t∗J˜ = 1. ∗ domain is: Y = λ1 a1 t∗1 + λj aj t∗j · tJ˜ t∗J˜
= 0,
Then, the yield
with
1 √ mM +mS
for
of the rst component of
t∗1
and
t∗j ,
de ning
+ ak ∈ −µ− , µ . Given the sign k k h i ± ∗ ± ∗ − + ∗ µ± = λ µ t · t + λ µ t · t yields Y = −µ , µ t∗J˜. ˜ ˜ 1 j 1 1 j j J J J˜ J˜ J˜ :
4.5. Implementation and results In this section, we detail on algorithm 4 how step 5 of the catching-up NSCD (algorithm 3) is implemented in this concrete situation. The algorithm is implemented in MATLAB. The catching-up NSCD is compared with an implementation in MATLAB of the Jean-Moreau's NSCD by Moeller [66]. The two implementations are very much alike so that the dierences of results only arise from dierences in the algorithms and not from the technical implementations. Regarding the impact law, we use Poisson's impact law (13) to particularize the impact mapping
F
introduced in
P4
whereas Newton's law (11) is used in Moeller's implementation [66].
Practical implementation of step 5 Step 5:
Friction law.
Set
J˜ = {j ∈ J : Λj > 0}. • If J˜ = {j} then solve
on
uSj ∈ R:
h ∗ i . uSj ∈ ∂IΛi [−µ− ,µ+ ] −uSj + t∗j · ui + f m , t∗j j
j
Set
•
ul = ul + uSj − t∗j · ul M−1 t∗j .
J˜ = {1, j} then ± ± ∗ ± ∗ Set µ ˜ = λ1 µ1 t1 · tJ˜ + λj µj tj · tJ˜. J S Solve on u ˜ ∈ R: J If
h ∗ i . uSJ˜ ∈ ∂I[−µ− ,µ+ ] −uSJ˜ + t∗J˜ · ui + f m , t∗J˜ J˜
Set
J˜
i h ul = ul + uSJ˜ − t∗J˜ · ul M−1 t∗J˜.
Algorithm 4: Practical implementation of step 5 of algorithm 3 in the woodpecker case Table 2 gathers all the parameters needed to run the presented simulation. We observe similar predictions for Jean-Moreau's NSCD and the catching-up NSCD (
c.f.
gure 4), and the catching-up NSCD runs
2.7
time faster than Jean-Moreau's one. Figure 4 plots the total mechanical energy with respect to time for the two considered algorithms.
5. Going back to classical equations
5.1. A procedure for automatic identi cation of the friction law Making our point of view agrees with the classical equations relies on a suitable design for the sets
∗ Cj, q.
In section 4, the sets
∗ Cj, q
do not depend on the con guration 21
q
because the kinematics, including the
r0 lM lS JM ε1 µ1 y0 y˙ 0 ∆t
2.5 mm 10 mm 20.1 mm 5 g · mm2 0.5 0.3 0 mm 0 mm/s 10−4 s
rM lG mM JS ε2 µ2 ϕ ϕ˙ 0 RelTol on
Λj
in NLGS
3.1 mm 15 mm 0.3 g 700 g · mm2 0 0.3 −0.1036 rad 0 rad · s−1 10−15
hm hS mS c ε3 µ3 α α˙ tsimu
5.8 mm 20 mm 4.5 g 5.6 · 106 N m/rad 0 0.3 −0.2788 rad −7.4583 rad · s−1 2s
Table 2: Parameter values for the presented simulation
·10−2
·106 0.08
0.25 0.00
energy [J]
relative error
−0.25 −0.50 −0.75 −1.00 −1.25
0.06
0.04
0.02
−1.50 −1.75
0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.00
0.30
0.05
0.15
0.20
0.25
0.30
(b) Mechanical energy relative error over time
(a) Mechanical energy over time 3
1.5
1.0 [rad/s]
2
1
φ˙
α ˙
[rad/s]
0.10
time [s]
time [s]
0
0.5
0.0
−0.5
-1
−1.0 -0.10
-0.05
0.00
0.05
−0.5 −0.4 −0.3 −0.2 −0.1 φ [rad]
0.10
α [rad]
(c) Phase diagram on α
0.0
0.1
(d) Phase diagram on φ
Figure 4: Comparison between Moeller's implementation of Jean-Moreau's NSCD with the catching-up NSCD. NSCD Jean-Moreau's NSCD
constraints, are linear and we found analytical expressions for the sets
22
Cj∗ .
Catching-up
However, in practical situations,
the sets
∗ Cj, q
must be constructed algorithmically at the con guration corresponding to the current time-
step. This is consistent with the fact that the state of the art algorithms construct the constraint functions at each time step in the vicinity of the current con guration through a collision detection algorithm. As in section 2.3 and [19], we consider the following situation: mid-point approximate
q
m
point experiencing contact normal
~nj ∈ Ed .
the equality
ϕj (qm ) = 0
expresses that at 6
, some part of the system touches an unmoving external obstacle ; the material
A
j
is known and the obstacle is smooth enough to de ne the geometrical outward
Under these hypotheses, which are not stronger than those of the state of the art algorithms,
we are able to preprocess the
∗ Cj, qm
before the resolution of the incremental step.
~ µj . The paper is ~rj ∈ C limited to the case of 2-D systems. We complete ~ nj to de ne an orthonormal basis ~nj , ~tj of E2 . The ~ µj is delimited by the lines with endpoints ~nj + µ~tj and ~nj − µ~tj . The set Gqm (Aj )C ~ µj is a cone cone C j included in Span Gq (A )~ nj , Gqm (Aj )~tj and delimited by the lines pointed by Gqm (Aj ) ~nj + µ~tj and ∗ ∗ ∗ Gqm (Aj ) ~nj − µ~tj (c.f. gure 3). We require that Cj, = 0 (requirement stated by the theory) qm , nj qm j j j j ∗ ~ n , Gqm (A )t (information on the physical reality of the system). To de ne and that Cj qm ⊂ Span Gqm (A )~ ∗ the Cj, qm , we need: ∗ j 1. To complete nj, q to get an orthonormal basis of Span Gqm (A )~ nj , Gqm (Aj )~tj . j nj + µ~tj and Gqm (Aj ) ~nj − µ~tj to de ne the 2. To express the slopes of the lines pointed by Gqm (A ) ~ ± directional friction coe cients µj . Consider the
j − th
contact, with one contact point.
Coulomb's law implies
These steps are readily implementable and the details are not reproduced here. Note however that there is
∗ Gq∗m (Aj ) ~nj ± µ~tj , n∗j < 0.
a special case that needs to be handled: when
This characterizes possible
locking eect / frictional paroxysm / tangential impact and this situation is taken into account by having an unbounded set
∗ Cj, q,
which is with
µ± j = +∞
in the theory. An example of such a case can be found
in [12]. However, for algorithmic purposes, there is no need to set time step a
Λj
µ± j
µ± j
to
+∞,
instead we set at the current
large enough to cancel sliding at the end of the time-step, for example
t∗ ·ul 2 jΛj
with
ul
and
as de ned in algorithm 3.
5.2. A proof in a static setting By design of the
Cj∗ q ,
in the case of one single contact, if a static equilibrium state is admissible with
respect to generalized friction, then it is admissible with respect to Coulomb. for any number of active contacts. More precisely, let restriction of problem
P4
r
This turns out to be true
be a generalized force of reaction solution to the
to static equilibrium. It can be rewritten:
r=
X
λj∈J(q) n∗j, q + rτ ,
(24)
j∈J(q) with
Λj
satisfying LCC and
rτ
obeying to the friction law, that is
rτ ∈ Y ∗ .
Note that, for equilibrium, the
problem is no longer required to be variational and the friction law can be written under a strong form like in (18) and (19). Using (19), for all
j ∈ J(q),
there exists
rt =
j∈J(q)
rjt ∈
such that:
⊥
X
rjt ∈ Λj Cj∗ q
X
Λj n∗j, q ∩ T∗ (q)
(25)
j∈J(q)
Cj∗ q , for all j ∈ J(q), there exists forces ~rj applied at the contact point Aj and in Coulomb's j ~ µ such that Gq∗ (Aj )~rj = Λj n∗ (q) + rjt . Combining this with (24) and (25), we get: cone C j By design of the
6 The case where the constraint describes the contact of a part of the system with another is not more complicated. It involves a pair of material points, with the same location, and a pair of normals, one being the opposite of the other, making the writing of equations more fastidious. 23
mw mc ρ R k d Cw µ
20 kg 255 kg 0.22 m 0.15 m 12500 N · m−1 15 N · s · m−1 −300N · m 0.8
Mass of the wheel Mass of the chassis Radius of the wheel Radius of obstacles on the road Stiness of the spring Damping Momentum applied on the wheel Coulomb's friction coe cient
Table 3: Constitutive parameter for numerical experiments
r=
X
λj∈J(q) n∗j, q +
j∈J(q)
X
X Gq∗ (Aj )~rj − Λj n∗j, q = Gq∗ (Aj )~rj .
j∈J(q)
(26)
j∈J(q)
Reciprocally, if an equilibrium state is Coulomb admissible, it is admissible for the generalized friction law.
5.3. An example: a quarter car mw and mc and we assume that the disk and the car are linked by a damper and a spring. The system is parametrized by x, the abscissa of the wheel and the car, by y , the ordinate of the wheel, by θ , the rotation of the wheel, and by z , the ordinate of the car. The con guration of the system 1 2 t is identi ed by the vector q = (x, y, θ, z) with the mass matrix M = diag(mw + mc , mw , mw ρ , mc ). 2 We now consider a quarter car illustrated in gure 5: the wheel is modeled by a rigid disk of mass
radius
ρ.
The chassis is modeled by a mass
z
mc
d
k θ
x
y
mw ρ
Figure 5: Kinematics of the quarter car The constitutive parameters used in the simulations are given in table 3. contact with the following unilateral constraints, indexed by
The wheel is in unilateral
i ∈ Z:
p ((x − 2iR)2 + y 2 ) − ρ − R ≥ 0, p ϕi,2 (q) = ((2(i + 1)R − x)2 + y 2 ) − ρ − R ≥ 0.
ϕi,1 (q) =
The system is simple enough to avoid the use of a collision detection algorithm to identify the contact points but complex enough so that the design of the
∗ Cj, qm
has to be done with the numerical procedure of
section 5.1. In fact, it is possible, with Sage for example, to determine an analytical expression for the
Cj∗ qm
but the numerical evaluation of this analytical expression is time consuming.
5.4. Results of numerical experiments The dynamical evolution is simulated on MATLAB with Moeller's implementation of Jean-Moreau's NSCD procedure [66] and with the catching-up NSCD procedure developed in this work.
The wheel is
initially placed at rest (or with a very small velocity) at the bottom of a hole on a rough road. We apply a 24
momentum of
−300 N · m (standard sign convention) on the wheel and we check if the car starts moving and
climbs the obstacle. This example is a variant of a classical example of non-convergence of Jean-Moreau's NSCD, a disk in the plan jammed in the corner de ned by two straight obstacles. experiments are given in table 4.
For experiment
#1,
Results of numerical
the initial condition is not at the corner but very
close. The two algorithms provide reasonable and similar dynamics, plotted on gure 7. The experiment
#1 shows the catching-up NSCD does as good as Jean-Moreau's NSCD with a speed-up 1.5 and 3.5. However the speed-up depends a lot on the simulation parameters. #
Spring
1
x [m] 149
y [mm] 338
ϕ1, 1 [mm] −0.615
ϕ1, 2 [mm] 0, 196
x˙ [ mm ] s 0
y˙ [ mm ] s 0
∆l [mm] −200, 1
Cw [N · m] −300
2
150
338
−0.211
−0, 211
0
0
−200, 1
−300
3
150
338
−0.211
−0, 211
0
0
0
−300
4
150
338
−0.211
−0, 211
−1
−1
−200, 1
−300
5
151
338
0.196
−0, 615
0
0
−200, 1
−200
NSCD J-M C-up J-M C-up J-M C-up J-M C-up J-M C-up
factor of between
Computed solution Figure 6a Figure 6a Failure at t = 0s Figure 6b Figure 6c Figure 6d Failure at t = 0s Figure 6e Failure at t = 28ms Figure 6b
Table 4: Numerical experiments to compare Jean-Moreau's and the catching-up NSCD
(a) #1
(b) #2 with catching-up NSCD
(d) #3 with catching-up NSCD
(c) #3 with Jean-Moreau's NSCD
(e) #4 with catching-up NSCD
Figure 6: Results of numerical experiments described in table 4, the color indicates the time However, by slightly changing the initial condition, we observe the nonconvergence of Jean-Moreau's NSCD in experiment
#2 whereas the catching-up NSCD does converge.
The solution of the latter is not the
wheel getting out the hole as could be expected, but the wheel staying still in contact with both constraints. This equilibrium state is also an admissible equilibrium state for the classical equation (recall section 5.2) although Jean-Moreau's NSCD does not converge to it.
Here, Jean-Moreau's NSCD does not converge
because the incremental problem has at least two solutions corresponding to the wheel locked in the corner 25
·10−12 Catching-up NSCD
1.6
1.2
Jean-Moreau NSCD 1.4
1.0 0.8 x [m]
1.0 0.8
0.6
∆
x [m]
1.2
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
time [s]
0.4
0.6
0.8
1.0
time [s]
(b) Dierence of abscissa between the two algorithm
(a) Car abscissa x
Figure 7: Results of numerical experiment #1 of table 2
and to the car which starts moving. In simulation
#3,
we change the initial ordinate of the chassis, everything else remaining the same.
The wheel is in the same position and the force applied on the wheel by the spring increases with time as the chassis falls under its own weight. The simulations
#2
and
#3
show one cannot predict failure of
Jean-Moreau's NSCD, one can only detect cases likely to induce failure. These cases are, in our simulations, these where Jean-Moreau's NSCD and the catching-up NSCD diers. In simulation
#4,
a very small perturbation on the initial velocity is added to simulation
#2.
Jean-
Moreau's NSCD again does not converge but the catching-up NSCD predicts the motion of the car. For the sake of the demonstration, we emphasize pathological situations where the physical relevance of the model is questionable. As discussed in [12], all these troubles are related to the poverty of the kinematics. As in any mechanical system, the kinematics must be consistent with the expected precision of prediction. This is formalized in our point of view by writing a problem
P4
in line with the virtual power point of
view [43]. The kinetics presented in gure 5 are a bit unusual and it is unlikely to start the motion from the corner. A better option would be to penalize constraints
1
and
2
(for example by adding as generalized
coordinate the volume of the air in the tire and by equilibrating contact forces with pressure variation), in order to, on the one hand, recover convergence in Jean-Moreau's NSCD and, on the other hand, to widen the space
T∗ (q)
so that the catching-up NSCD would coincide with Jean-Moreau's NSCD. The kinematics
in gure 5 is suitable for studying the evolution of the running car when contacts are activated one at a time. However not likely, double contacts can happen along the dynamic evolution and only the catching-up NSCD can pass beyond these events.
6. Energetic consistency of the algorithm
We consider now the simulation of a rigid bar, as introduced in [12, section 6.2], which is the most simple example which exhibits Painlevé's paradox. In addition, this example shows that Jean-Moreau's NSCD can break the conservation of energy principle. The generalized coordinates are
x,
the abscissa of the center of mass of the bar, and
θ,
the rotation of
the bar. The ordinate of the center of mass of the bar is constrained to be constant. The con guration is
t
(x, θ)
and the mass matrix
M = diag (m, J),
with
m
the mass of the bar and
J
around the center of mass. We apply a generalized force derived from the potential The system is conservative. 26
q
the inertia momentum
V (q) =
1 2c
θ+
π 2 . 2
A of the bar is constrained to be above the ground and we take as constraint function √ θ0 ∈ − π2 , 0 . We also have n∗q = n∗ = t 0, J . Note that the kinematics are linear
Now, the right end
ϕ(q) = θ − θc ,
with
and at contact, friction occurs making the system dissipative.
y G θ x A
x
Figure 8: A simple discrete mechanical system with two d.o.f. Assuming
1 µ < − tan(θ , c)
it is shown in [12] that
C∗
is bounded and of the form:
√ √ J µ µ m C∗ = √ , . − 0 l(cos(θc ) + µ sin(θc ) l(cos(θc ) − µ sin(θc ) m The implementation of this system is a special case of what we did before and it is not detailed here. We provide on gure 9 the result of the simulation for the numerical values given on table 5. We observe that Jean-Moreau's NSCD leads to a pathological increase of the mechanical energy over time, whereas the proposed algorithm predicts a constant mechanical energy. Indeed, in our framework, the velocity is purely normal (x ˙
= 0)
and the system experiences a succession of elastic impact without friction.
A. = 0) and friction point A changes of
With classical equations, normal and tangential velocities are de ned with respect to the velocity of point At the rst instant of impact, the velocity of point
A
has a tangential part although (x ˙
occurs. However, this is a case of slip reversal at the impact: the tangential velocity of
−
side during the impact. In this case, it is questionable to write Coulomb's law on the post-impact velocity
Gq (A)u+
since this leads to pathologies. In our framework, the notion of slip reversal vanishes because the
generalized tangential velocity is orthogonal to the normal velocity for
M(q)
and the impact law does not
aect the tangential velocity.
m J l c dt reltol
2.45 kg 0.204 kg · m2 0.5 m 10 N · rad−1 10−5 s 10−15
θc ε µ θ(0) x(0) ˙ ˙ θ(0)
− π4 1 0.5 − 3π 16 0 m · s−1 0 s−1
Table 5: Numerical values for the simulation
7. Conclusion and perspective
We saw that the NSCD with frictional contacts is not the time discretization of an evolution problem. If this modeling choice is accepted by Jean and Moreau in the context of granular media [11], the lack of mathematical foundation of the NSCD in the frictional case, remains the source of a series of defects that are listed in [7]. 27
0 −0.6 −0.5
θ
[m]
−1
x
[rad]
−0.65
−0.7
−1.5 −0.75 −2 0
0.5
1
1.5
0
2
0.5
1
1.5
2
time [s]
time [s]
(b) Abscisse of the center of mass of the bar x over time
(a) Angle θ of the bar over time
25
energy [J]
20
15
10
5 0
0.5
1
1.5
2
time [s]
(c) Total mechanical energy of the system over time Figure 9: Results of numerical simulation of the dissipative system sketched on gure 8, Moreau's NSCD
Catching-up NSCD
Jean-
In this work, the frictionless evolution problem [17] is rst extended to the case with friction, as in [12]. Then, the time stepping scheme [1] is applied, see gure 1. This leads to a new algorithm from the time discretization of an evolution problem for multibody dynamics with contacts and friction (c.f. section 3). Our algorithm has similarities to Jean-Moreau's one (c.f. section 4) but, at each time step, the convergence
c.f.
of the NLGS solver is guaranteed (
section 5), and energetic inconsistencies are xed (
A natural extension of this work would be take the limit on
∆t → 0
c.f.
section 6).
in the time-stepping, in order
to prove the convergence of the computed solution as well as the existence of a solution to the evolution problem [12, p. 20]. The work of Monteiro Marques and Dzonou [33, 69] should be of great help in this task. Our contribution diers from Jean-Moreau's NSCD by the (in)equations that are discretized in time, the discretization scheme being the same as Moreau's. As a consequence, our algorithm is subjected to drift-o like Jean-Moreau's NSCD [7]. Some recent improvement of Moreau's time-stepping scheme [8, 9, 70] could 28
solve this issue. It should be interesting to examine if other time-stepping scheme [34, 35, 36] can be applied on problem
P4 .
In order to simulate engineering systems, an extension of this work to nonplanar systems has also to be done. This extension was not trivial in the case of Jean-Moreau's NSCD [37]. Systems with a large number of degrees of freedom,
e.g.
granular media, is also a case of interest. The catching-up NSCD requires a rst
static identi cation procedure (
c.f.
section 5.1) but the computational complexity is linear with respect of
the number of contacts. Although further investigations are needed, the overall procedure is expected to be faster than Jean-Moreau's since the improved convergence of the catching-up NSCD entails fewer iterations. However, is it meaningful to systematically rule out the self-equilibrated force networks in strongly con ned granulates at the origin of NLGS slow convergence [7]?
Acknowledgement
This work has started after the 2nd european network for nonsmooth dynamics workshop in Grenoble. The authors would like to acknowledge participants of this workshop for fruitful discussions. The authors would like to thank Patrick Ballard, Simon Eugster and Remco Leine for interesting ideas, as well as Christian Rey and Arjen Roos from Safran for supporting the redaction of the manuscript and Tonya Rose for reviewing.
Comments on the problem
Given that the problem
P2
P2
is not weak and that the variation of velocity
clear how to understand problem
P2 .
du is a measure of time, it is not
Indeed, it is not possible to de ne the value of a measure at a given
time instant: what does it mean that the reaction force belongs to a moving convex set? The formulation of Moreau [17] underlies a convention of notation using that the set of admissible velocities is a cone and that nite dimension spaces have the Radon-Nikodym property [71]. According to the Radon-Nikodym theorem [48, p 53], there exists a measure example the modulus measure [48, section 7]) and a density function
˜rν ,
dν ∈ M ([0, T ] ; R)
(for
which is absolutely continuous
r = ˜rν dν . The strong form of the standard inelastic shock has then to be − ˜rν ∈ ∂IV(q) [u] holds dν -almost everywhere. The latter subdierential inclusion is meaningful because ˜ r is a function. The decomposition r = ˜rdν is not unique however V(q) is a cone and the convention of notation is independent of the choice of dν [19, proposition 8.2]. with respect to
dν ,
such that :
understood as follows:
However, we will try to avoid this trouble and we will put emphasizes on weak formulations.
In the
context of a sub-dierential inclusion, a weak formulation can be obtained by extending the de nition of the subdierential (equation 3) from the duality between and
∗
M ([0, T ], T Q).
episodes.
Tq Q
and
T∗q Q
to the duality between
C 0 ([0, T ], TQ)
Variational formulations are well de ned for smooth episodes as well as nonsmooth
This procedure turns out to be useful to extend friction laws from the impactfree case to the
case with impacts in section 3.1. In the impactfree case, friction laws are naturally formulated in terms of subdierential inclusions [51, 54]. Moreau extends them to the case with impacts with the Radon-Nikodym theorem [19] while we use weak formulations, in line with [12, 64, 65]. As a consequence, we derive the time-stepping from a weak-problem, in section 3, what is a bit different from Moreau's presentation. Of course, it is a also possible to follow Moreau's presentation and to derive the time-stepping from a strong problem.
It is then possible to apply the step corresponding to
equations (18), (19) and (20) before the discretization in time. The continuous in time nonsmooth friction law reads then as a subdierential inclusion:
−u+ t ∈ ∂IY ∗ [rτ ].
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