application of the non-smooth dynamics approach to ...

JOÃO PAULO FLORES FERNANDES

APPLICATION OF THE NON-SMOOTH DYNAMICS APPROACH TO MODEL AND ANALYZE CONTACT-IMPACT EVENTS IN RIGID MULTIBODY SYSTEMS USING THE LINEAR COMPLEMENTARITY PROBLEM FORMULATION Sumário pormenorizado da lição nos termos da alínea c) do artigo 5º do Decreto-Lei n.º 239/2007 de 19 de Junho.

UNIVERSIDADE DO MINHO – 2010

PRELÚDIO O objectivo deste documento é o de apresentar o sumário pormenorizado da lição subordinada ao tema “Application of the non-smooth dynamics approach to model and analyze contact-impact events in rigid multibody systems using the linear complementarity problem formulation”, ao abrigo do disposto no Diário da República, Decreto-Lei n.º 239/2007, artigo 5º, alínea c). Esta lição está integrada no conteúdo programático da unidade curricular de Cálculo Automático de Sistemas Multicorpo do Programa Doutoral em Engenharia Mecânica da Universidade do Minho.

Success consists of going from failure to failure without loss of enthusiasm. Winston Churchill

Table of contents Table of contents..................................................................................................................................i Abstract ............................................................................................................................................. iii Resumo ................................................................................................................................................v Keywords.......................................................................................................................................... vii Palavras-chave...................................................................................................................................ix Notation ..............................................................................................................................................xi Abbreviations....................................................................................................................................xv Terminology................................................................................................................................... xvii 1. Introduction ....................................................................................................................................1 1.1. Motivation and objectives ........................................................................................................1 1.2. Literature review ......................................................................................................................2 1.3. Outline of this report ..............................................................................................................11 2. Basic set-valued elements.............................................................................................................13 2.1. The linear complementarity problem .....................................................................................13 2.2. The unilateral primitive ..........................................................................................................14 2.3. The Sgn-multifunction............................................................................................................15 3. Set-valued force laws for frictional unilateral contacts ............................................................17 3.1. Set-valued normal contact law ...............................................................................................17 3.2. Set-valued tangential contact law ...........................................................................................18 4. Generalized contact kinematics ..................................................................................................21 4.1. General issues in contact ........................................................................................................21 4.2. Kinematic aspects of contact between rigid bodies................................................................22 5. Dynamics of non-smooth rigid multibody systems....................................................................27 5.1. Equations of motion ...............................................................................................................27 5.2. Impact laws.............................................................................................................................30 6. Moreau time-stepping method ....................................................................................................33 6.1. Time discretization based on the Moreau midpoint rule ........................................................33 6.2. Computational strategy to solve the equations of motion ......................................................34 Table of contents

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7. Solving the contact-impact problem as an LCP ........................................................................37 7.1. Formulation of the contact-impact problem as an LCP..........................................................37 7.2. Moreau time-stepping method with an LCP formulation.......................................................40 8. Results and discussion..................................................................................................................43 8.1. Bouncing ball..........................................................................................................................43 8.2. Woodpecker toy......................................................................................................................48 8.3. Slider-crank mechanism with a translational clearance joint .................................................56 8.4. Reciprocating cam with flat-face follower .............................................................................67 9. Concluding remarks.....................................................................................................................73 References .........................................................................................................................................75 Appendices ........................................................................................................................................83 Appendix I – The linear complementarity problem ......................................................................83 Appendix II – Demonstrative example of a differential measure .................................................91 Appendix III – General impact laws for both impact and smooth phases.....................................92

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Application of the non-smooth dynamics approach to model and analyze contact-impact events in rigid multibody systems

It's not that I'm so smart, it's just that I stay with problems longer. Albert Einstein

Abstract The primary goal of the present study was to study general methodologies to deal with contactimpact analysis of rigid multibody systems. In order to carry out this work, the dynamic modeling and analysis of planar multibody systems that experience contact-impact events was presented and discussed. The methodology was based on the non-smooth dynamics approach, in which the interaction of the colliding bodies is modeled with multiple frictional unilateral constraints. The dynamics of rigid multibody systems were stated as an equality of measures, which were formulated at the velocity-impulse level. The equations of motion were complemented with constitutive laws for the forces and impulses in the normal and tangential directions. The formulation of the generalized contact-impact kinematics in the normal and tangential directions was performed by obtaining a geometric relation for the gaps of the potential contact points. The gaps were expressed as functions of the generalized coordinates. The potential contact points were modeled as hard contacts, being the normal and tangential contact laws formulated as set-valued force laws for frictional unilateral constraints. In this study, the unilateral constraints were described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts were characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem was formulated and solved as a linear complementarity problem, which was embedded in the Moreau time-stepping method. Finally, elementary multibody mechanical systems were used to discuss the mains assumptions and procedures adopted throughout this work. The main results obtained from the present study showed that the effect of the contact-impact phenomena can have a predictable nonlinear behavior. This feature plays a crucial role in the dynamics, design and control of general multibody systems of common application.

Abstract

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The road to success is always under construction. Anonymous

Resumo O presente trabalho teve como objectivo principal estudar metodologias genéricas para análise de sistemas rígidos de corpos múltiplos envolvendo situações de contacto-impacto. Para o efeito, foram apresentadas e discutidas várias abordagens que permitem modelar e analisar o comportamento dinâmico de sistemas de corpos múltiplos em que ocorrem cenários de contacto e impacto. A metodologia foi baseada na abordagem dinâmica de sistemas não-suaves, em que a interacção dos corpos em contacto é modelada através de constrangimentos unilaterais com atrito. A dinâmica dos sistemas rígidos multicorpo é formulada recorrendo a equality of measures, em que as equações do movimento são desenvolvidas ao nível da velocidade e do impulso. Por seu lado, as equações do movimento foram complementadas com leis constitutivas para as forças e os impulsos que actuam nas direcções normais e tangenciais. A formulação cinemática generalizada do contacto-impacto na direcção normal e tangencial foi realizada pela obtenção de uma relação geométrica para as distâncias dos potenciais pontos a contacto. Estas distâncias foram expressas como funções das coordenadas generalizadas. Os potenciais pontos de contacto foram modelados como contactos rígidos, sendo aplicados os conjuntos admissíveis para as leis das forças nas direcções normais e tangenciais. Neste trabalho, os constrangimentos unilaterais foram descritos pela lei de Signorini, enquanto que o atrito foi caracterizado pela lei de Coulomb para o atrito seco. O problema matemático daqui resultante foi resolvido usando a complementaridade linear, a qual foi incorporado no método de Moreau. Finalmente, sistemas mecânicos elementares foram considerados como exemplos de aplicação para discutir os principais pressupostos e procedimentos adoptados neste trabalho. Os principais resultados obtidos com este trabalho mostraram que os fenómenos provocados por situações de contacto-impacto são altamente não lineares. Esta característica desempenha um papel preponderante na análise, no projecto e no controlo dos sistemas mecânicos em geral.

Resumo

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The longer I live, the more beautiful life becomes. Frank Lloyd Wright

Keywords Multibody systems Contact-impact mechanics Friction phenomena Unilateral contacts Linear complementarity problem Non-smooth dynamic systems Moreau midpoint rule Set-valued force laws

Keywords

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Everyone thinks of changing the world, but no one thinks of changing himself. Leo Tolstoy

Palavras-chave Sistemas de corpos múltiplos Mecânica do contacto-impacto Fenómenos de atrito Contactos unilaterais Problemas de complementaridade linear Sistemas mecânicos não-suaves Regra do ponto médio de Moreau Conjunto admissível para as leis das forças

Palavras-chave

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In language, clarity is everything. Confucius

Notation All matrices and vectors are written in boldface.

Latin symbols C

Set

c

Clearance size

dP

Measure for the percussions

dt

Lebesgue-measure

du

Measure for the generalized velocities

dη

Sum of the Dirac pulses at the impact instants

f

System’s degrees of freedom

g

Gap function vector

gN

Normal gap function of the unilateral constraints

gT

Tangential gap function of the unilateral constraints

H

Set of active contacts

h

Height of the bodies’ center of mass

h

Vector of gyroscopic and external forces (including spring and damper forces)

i

Generic contact point

I

Identity matrix

J

Generalized jacobian matrix

J

Moment of inertia

j

Jacobian terms that represent the rehonomic constraints

L

Lagrangian

l

Length

M

System mass matrix (positive and definite and symmetric)

m

Mass

n

Normal unit vector

n

Number of frictional unilateral constraints

PN

Vector of the generalized normal percussions

PT

Vector of the generalized tangential percussions

Notation

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q

Vector of the generalized coordinates

q

Vector that contains the generalized velocities, a.e.

q

Vector that contains the generalized accelerations, a.e.

R

Radius

r

Global position vector

T

Kinetic energy

t

Tangential unit vector

t

Time variable

tF

Final time simulation

u

Vector of the generalized velocities

u-

Pre-impact velocities

u+

Post-impact velocities

u

Vector of generalizes accelerations

V

Potential energy

v

Global velocity vector

wN

Generalized normal force direction

wT

Generalized tangential force direction

w

Jacobian terms that represent the rehonomic constraints

XY

2D global coordinate system

Greek symbols Δt

Time step size

ΛN

Normal impulsive force

ΛT

Normal impulsive force

εN

Normal coefficient of restitution

εT

Tangential coefficient of restitution

γ

Post-impact relative contact velocity

γN

Normal velocity

γT

Tangential velocity

γ-

Pre-impact relative contact velocity

λN

Normal contact force

λT

Tangential contact force

μ

Coefficient of friction

θ

Angular coordinate

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+

Subscripts 0

Initial conditions

A

Start point of the integration time step

E

End point of the integration time step

F

Final time simulation

M

Midpoint of the integration time step

N

Normal direction

S

Center of mass

T

Tangential direction

Operators (⋅) ⋅⋅

First derivative with respect to time

( )

Second derivative with respect to time

(∂)

Partial derivative

()

T

Δ

Matrix or vector transpose Increment

Symbols ∅

Empty set

∀

For all

∈

Inclusion

⊥

Orthogonal

\+

Set of all positive real numbers

\

Set of all real numbers

∃

There exits

:=

Definition

|

For which

Notation

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Any sufficiently advanced technology is indistinguishable from magic. Arthur Clarke

Abbreviations a.e.

Almost everywhere

ALA

Augmented Lagrangian approach

col

Column vector

diag

Diagonal matrix

DAE

Differential algebraic equation

DOF

Degrees of freedom

LCP

Linear complementarity problem

mat

Matrix of columns

max

Maximum value

MBS

Multibody system

NCP

Nonlinear complementarity problem

NSDS

Non-smooth dynamic systems

ODE

Ordinary differential equation

s.t.

Such that

Sgn

Set-valued sign function

Upr

Unilateral primitive

Abbreviations

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If you cannot convince them, confuse them. Harry Truman

Terminology Algorithm – a finite set of well defined instructions whose execution accomplish a given task on a given set of input data in finite time, with a well defined end state. Analytic mechanics – in contrast with the Newtonian formulation, analytic mechanics is entirely stated in terms of energy which is divided into kinetic and potential. These two types of energy are scalar functions which can be expressed in terms of the kinematic variables, namely, the generalized coordinates and the generalized velocities. Bilateral constraint – a general constraint relation which is satisfied as a strict equality. For such a constraint, the corresponding multiplier is generally unrestricted, unless another general constraint is imposed on this multiplier. Body frame – a rigid frame of reference attached to a moving body. Conservative systems – mechanical systems which conserve energy over time. The study of these systems is central to analytic mechanics. Constraint – any restriction on the generalized coordinates, the generalized velocities, any of the generalized forces of a mechanical system. Coulomb friction – the model of tangential force at a contact point. Equality of measures – deals with formulation of the equations of motion for multibody systems with unilateral constraints at the velocity level, allowing accounting for the atomic quantities that express the velocities jumps and impulsive forces. Euclidian distance – is the most common use of distance; Euclidian distance, or simply distance, examines the root of square differences between coordinates of a pair of points. Frictional unilateral constraint – represents a mechanical constraint that prevents the interpenetration between two bodies when the friction phenomenon is present in the tangent contact plane. Holonomic constraint – is an algebraic equation imposed to the system that is expressed as function of the displacement and, possibly, time. Impact-free motion – represents the motion without impact phases, i.e., smooth motion. Multibody system – consists of a finite number of rigid bodies subjected to geometric and physical interactions. Non holonomic constraint – is a constraint that contains inequalities or relations between velocity components that are not integrable in closed form. Non-smooth system – is a system that contains nonlinearities or discontinuities, such as those caused by impacts in clearance joints. Rehonomic constraint – represents the restrictions that are explicitly dependent on the time variable. Rigid body – is a collection of material points whose relative positions remain constant during the system’s motion. Terminology

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Scleronomic constraint – constraints in which the time does not appear explicitly in the constraint equation. Set-valued force law – includes the non-smooth representation of unilateral contacts, friction elements or other non-smooth interaction like pre-stressed springs. A set-valued force law can be interpreted as a constraint, which constraint force λi is restricted to a convex set Ci. Unilateral constraint – represents a mechanical constraint that avoids the interpenetration between two bodies when there is no friction. This refers to the one-side nature of the constraint in the measure that: if two surfaces are touching, then motion in one direction is allowed, but motion in the opposite direction is not.

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There's a way to do it better - find it! Thomas A. Edison

1. Introduction 1.1. Motivation and objectives The general motivation for this work comes from current interest in developing mathematical and computational tools for the dynamics of constrained multibody systems involving contactimpact events, in which the effects of contact, impact and friction are taken into account. It is well known that the contact-impact behavior strongly depends on the material properties of the colliding surfaces, the nature of the contact-impact problem and the level of contact-impact forces/impulses produced. Therefore, the investigation on the contact-impact field is one of the most challenging and demanding issues the engineering. In addition, contact-impact events can frequently occur in multibody systems and in many engineering applications the function of mechanical systems is based on them. Common examples can be described by the contact between tire and road in vehicles, wheel and rail in railway systems, contact in robotics and grasping machines, cam and follower mechanisms, contact in granular media, just to mention a few. The contact-impact phenomena are characterized by abrupt changes in the values of system variables, most commonly discontinuities in the system kinematics, namely: the velocities and accelerations. Other effects directly related to the impact phenomena are those of vibration propagation through the system, local elastic/plastic deformations at the contact zone and frictional energy dissipation. Impact is a prominent phenomenon in many mechanical systems such as mechanisms with intermittent motion and mechanisms with clearance joints. The impact is characterized by large forces that are applied and removed in a short time period. The knowledge of the peak forces developed in the impact process is very important for the dynamic analysis of multibody systems having consequences in the design process. The numerical description of the collision phenomenon is strongly dependent on the contact-impact force model used to represent the interaction between the system components. Therefore, in order to correctly simulate and design these types of mechanical systems adequately, appropriate contact-impact force model must be adopted. Furthermore, during impact, a multibody system presents discontinuities in geometry and some material properties can be modified by the impact itself. By and large, contact-impact analysis focuses on the resolution of three fundamental issues: (i) definition of a representative geometric description of the contacting surfaces; (ii) detection of the potential contact points; and (iii) establishment of a constitutive force model that typically depends on the bodies material properties, on the pseudo-penetration depth. 1. Introduction

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The main objective of the present work is to use present general methodologies for contactimpact analysis under the framework of non-smooth dynamics approach, contributing towards the multibody systems formulations. This desideratum is achieved by employing a solid mathematical and computational program on this field of investigation, that is, the formulations associated with non-smooth dynamics approach, the Lagrangian dynamic systems and the impact laws are taken into account. The specific objectives associated with this work can be listed as follows: (i)

To preset the fundamental notation and formulation of Lagrangian dynamic systems with constraints;

(ii)

To get insight the formulations and representations of set-valued force laws by maximal monotone graphs;

(iii) To discuss the main issues of the Moreau time-stepping approach together with impacts and impact laws; (iv) To derive analytically the equations of motion for the general multibody systems, such as the woodpecker toy system and the slider-crank system with a translation clearance joint; (v)

To apply the methodologies presented to several multibody systems involving several frictional unilateral contacts.

1.2. Literature review 1.2.1. General issues in contact-impact mechanics Over the last centuries, a great number of researchers have been investigated the contact-impact problems. For instance, in ancient Egypt people needed to transport heavy stone blocks to build the pyramids, and thus had to overcome the frictional force associated with it. Galileo Galilei (15641642) stated that the impact forces can become unlimited. Christian Huygens (1629-1695) performed studies on completely elastic collisions between two point masses. Besides conservation of momentum and kinetic energy, he recognized the fact the relative motion between two bodies has to be taken into account in order to be able to formulate the universally valid law of impact. His law that describes the relative velocities inversion during the elastic impact was extended by Isaac Newton (1643-1727) in 1687 by the restitution coefficient in order to accommodate possible losses of energy during the contact- impact process (Glocker, 2004; Flores and Claro, 2007). Since friction takes place in many engineering applications and its knowledge is of paramount importance, famous researchers in the past have investigated frictional contact problems. For instance, Leonard da Vinci (1452-1529) measured the friction force and had already considered the 2


influence of the contact area on the friction force using blocks with different contact area but same weight (Dowson, 1979). He found that the friction force is proportional to the weight of the blocks and is independent of the apparent contact area. Associated results are often attributed to Guillaume Amontons (1663-1705) neglecting the contribution of da Vinci. When putting these findings in a formula one obtains the classical equation for friction (known as Coulomb’s friction law), which every student in engineering learns during the first semesters of study, that is, λT=μλN, in which λT is the friction force, λN is the normal contact force and μ the coefficient of friction (Newton, 1687; Wriggers, 2006; Flores et al., 2008b). However, a first analysis from the mathematical point of view was carried out by Leonhard Euler (1707-1783), who assumed triangular section asperities for the representation of surface roughness. He had already concluded from the solution of the equations of motion for a mass on a slope that the dynamic coefficient of friction has to be smaller than the static coefficient of friction. Actually, it was Euler who introduced the symbol μ for the friction coefficient, which is still the common symbol used nowadays. A comprehensive experimental study of frictional phenomena was later performed by Charles-Augustin de Coulomb (1736-1806). He considered the following facts relating to friction: normal pressure, extent of surface area, materials and their surface coatings, ambient conditions (humidity, temperature and vacuum), and time dependency of friction force (Galileo, 1638; Wriggers, 2006; Flores et al., 2010a). Undeniably, the classical problem of the contact mechanics is a quite old topic in engineering applications. The pioneering work on the frictionless collision between rigid bodies was developed by Hertz (1896). Naturally, the following step was to include the friction effect on the contact analysis. In fact, this has been an intensive topic of research over the years and deserved the attention of many authors, which led to the development of relevant work and even to the publication of relevant textbooks totally devoted to this issue such as Brach (1991), Pfeiffer and Glocker (1996), Johnson (1999), Stronge (2000), among others. In general, contact modeling in multibody systems consists of two major steps, mainly, detecting the collision between moving bodies, and computing contact-impact forces. The subject of development of contact detection problem is quite challenging and actual topic (Reichardt and Wiechert, 2007; Flores and Ambrósio, 2008; Wellmann et al., 2008; Lopes et al., 2009; Machado et al., 2009a; 2009b). From the modeling methodology point of view, several different methods have been introduced. As a rough classification, they may be divided into contact force based methods and methods based on geometrical constraints (Pfeiffer and Glocker, 1996; Wriggers, 2006; Flores, 2010). In general, the motion characteristics of a multibody system are significantly affected by contact-impact phenomena. Impact is a complex physical phenomenon for which the main characteristics are a very short duration, high force levels, rapid energy dissipation and large 1. Introduction

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changes in the bodies’ velocities (Gilardi and Sharf, 2002). Inherently, contact implies a continuous process which takes place over a finite time. Contact-impact occurs in the collision of two or more bodies, which may be external or belong to a particular multibody system (Lankarani and Nikravesh, 1990; 1994). In fact, contact events can frequently happen in multibody systems and in many cases the function of mechanical systems is based on them (Pfeiffer and Glocker, 1996). In fact, the subject of contact mechanics and its applications in multibody systems had not been developed until the last couple of decades. The contact-impact study in multibody dynamics has received a great deal of attention in the past decades and still remains an active field of research and development (Gonthier et al., 2004; Flores et al., 2004; Glocker and Studer, 2005; Sharf and Zhang, 2006; Flores et al., 2006b; Flores et al., 2008b; Qiang et al., 2009; Machado et al., 2010). The analysis of contact between two bodies can be extended to the analysis of impact in a multibody mechanical system. Whenever a link in a multibody system experiences a hard stop, contact forces of complex nature act on the links and the corresponding impulse is transmitted throughout the system. For example, in the case of worn joints with clearances, the link moves freely inside the clearance zone, because it is not constrained, until it impacts onto the joint and the link experiences a hard stop (Lee and Wang, 1983; Khulief and Shabana, 1987; Bauchau et al., 2001; Bauchau and Rodriguez, 2002; Flores and Ambrósio, 2004; Flores et al., 2006a; Flores et al., 2008a; 2008b; Najafabadi, 2008). The methodology used to describe multibody systems with hard stops and joint clearances is based on the development of different contact force models that include energy dissipation, with which a continuous analysis of the system undergoing internal impact is performed. The model for continuous contact-impact force must consider the material and geometric properties of the colliding surfaces, information on relative positions and velocities, contribute to an efficient solution of the equations of motion and account for some level of energy dissipation (Ravn, 1998; Flores et al., 2010b). These characteristics are ensured with the continuous contact force model, proposed by Lankarani and Nikravesh (1990; 1994), in which the deformation and contact forces are considered as continuous functions during the complete period of contact (Qiang et al., 2009; Flores, 2009a; Ambrósio and Verissimo, 2009). Hurmuzlu and Marghitu (1994) studied the contact problem in multibody systems, where a planar rigid-body kinematic chain undergoes an external impact and an arbitrary number of internal impacts. Based on the Keller’s work (1986), they developed a differential-integral approach and used different models for coefficient of friction. Han and Gilmore (1993) proposed a similar approach, using an algebraic formulation of the equations of motion, the Poisson’s model of restitution and the Coulomb’s law to define the tangential motion. Different conditions that characterize the motion (slipping, sticking, and reverse sliding) are detected by analyzing velocities

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and accelerations at the contact points. Han and Gilmore confirmed their simulation results with experiments for two-body and three-body impacts. Based on a canonical form of the equations of motion Pereira and Nikravesh (1996) present a methodology that solves this problem in the context of multibody dynamics impact. Haug et al. (1986) solved directly the differential equations of motion by using the Lagrange multiplier technique. Newton’s model was used for impact while the Coulomb’s law was used for friction.

1.2.2. Methods to deals with contact-impact events In a broad sense, the different methods to solve the impact problem in multibody mechanical systems are continuous and discontinuous approaches (Lankarani and Nikravesh, 1990; 1994; Flores et al., 2006b; Flores et al., 2008a). Within the continuous approach, the methods commonly used are the continuous force model, which is in fact a penalty method, and the unilateral constraint methodology, based on complementarity approaches (Pfeiffer and Glocker, 1996; Glocker, 2001; Potra et al., 2006). The compliant continuous contact force models, commonly referred as penalty methods, have been gaining significant importance in the context of multibody systems with contacts thanks to their computational simplicity and efficiency. In these models, the contact force is expressed as a continuous function of pseudo-penetration between contacting bodies. However, one of the main drawbacks associated with these force models is the difficulty to choose the contact parameters such as the equivalent stiffness or the degree of nonlinearity of the penetration, especially for complex contact scenarios (Hippmann, 2004; Flores et al., 2008b). Each approach has some advantages and some disadvantages, therefore an appropriate procedure has to be chosen. In fact, both of the approaches presented have been used in the contact of multibody systems and can model contact occurring at multiple points (Malça, 2009; Flores, 2009b). The complementarity formulations associated with the Moreau time-stepping algorithm for contact modeling in multibody systems have attracted attention of many researchers (Glocker and Pfeifer, 1992; Glocker and Pfeifer, 1993; Glocker and Pfeifer, 1995; Pfeifer and Glocker, 1996; Moreau, 1999; Pfeiffer, 1999; Bauchau et al., 2001; Stewart, 2001; Pfeiffer, 2003; Leine et al., 2003; Glocker and Studer, 2005; Förg et al., 2005; 2008; Tasora et al., 2008). Assuming that the contacting bodies are truly rigid, as opposed to locally deformable or penetrable as in the penalty approaches, the complementarity formulations resolve the contact dynamics problem by using the unilateral constraints to compute contact impulses or forces to prevent penetration from occurring. Thus, at the core of the complementarity approach is an explicit formulation of the unilateral constraints between the contacting rigid bodies (Brogliato et al., 2002). In this approach, contacts are intermittent, that is, an active constraint can become passive or inactive, while an inactive constraint can become active. In the event of a passive constraint becoming active, the change is 1. Introduction

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generally accomplished by an impact which is characterized by impulsive forces. Such an event changes the behavior of the system, the number of kinematic constraints and, therefore, the number of degrees of freedom of the system is changed (Glocker, 2001a; 2001b). One of the main features of unilateral constraints is the impenetrability, which means that points candidates for contact must not cross the boundaries of antagonist bodies. This can be expressed by writing that the distance between contacting bodies or the gap distance is non-negative. Also, it is assumed that bodies are not attracting each other, that is, the reaction force is non-negative, and this reaction force vanishes when the contact is not active (Pfeiffer and Glocker, 1996; Leine and Nijmeijer, 2004). When dealing with the formulation of frictional unilateral constraints, it is possible to distinguished active and passive set-valued force laws. An active set-valued force law is always associated with a closed unilateral contact or a frictional contact, while a passive set-valued force law is related to open unilateral contacts. An active set-valued force law can be described at the velocity level by an inclusion. It should be highlighted that a system with active unilateral constraints has variable degrees of freedom, being, in general, not known which degree of freedom is removed. This problem is usually solved by looking at all possible solutions and finding the one that is physically consistent. It is obvious that the search for a physical consistent is timeconsuming. In addition, from the numerical simulation point of view, it is quite unsuitable to change the number of the minimum generalized coordinates during each time step. Thus, the LCP or ALA formulations are quite elegant ways to solve this type of problems, being constant the number of generalized coordinates at all instants of time. The number of generalized coordinates is always equal to the number of degrees of freedom of the system without unilateral constraints (Glocker and Pfeiffer, 1993; Anitescu and Potra, 1997; Brogliato, 2003). In short, the different methods to deal with contact-impact events in MBS have inherently advantages and disadvantages for each particular application. To the belief of author, none of the formulations briefly described above can a-priori be said to be superior compared to other for all applications. It is a fact that a specific multibody problem might be easier to describe by one formulation, but this does not yield a general predominance of this formulation in all situations.

1.2.3. Multibody dynamic systems and non-smooth dynamic systems The field of multibody system dynamics (MBS) has its root in classical and analytical methods of dynamics to meet the growing demands in modeling and simulation of complex and advanced mechanical systems in industry and engineering. Multibody dynamics can be understood as the study of systems of many bodies whose interactions are modeled by forces and kinematic constraints. In other words, a multibody system can be defined as a collection of bodies that are interconnected to each other by different types of joints that constrain the relative motion of the 6


bodies in different directions. The external forces applied in the system components may include contact-impact forces, friction forces, gravitational forces, joint constraint forces, externally applied body forces and forces due to mechanical elements such as springs, dampers and actuators. Kinematic constraint types may include revolute joints, translational joints, spherical joints, and cylindrical joints, among others. The kinematic constraints may also be in the form of prescribed trajectories for given points of the system components (Nikravesh, 1988; Flores et al., 2008b). Multibody systems are ubiquitous in many fields of application, such as in aerospace, automotive systems, granular matter, circuit breakers industry, nuclear plants, bipedal locomotion, robotics, biological engineering, computer graphics, etc., (Shabana, 1989). Their numerical simulation has become a crucial step not only for the virtual prototyping process in industry, but also in academic fields like granular matter, bifurcation analysis, global behavior of complex systems, control and stability, in which it is impossible to push forward the studies without reliable simulation software packages (Leine and van de Wouw, 2008a; 2008b). Numerical simulation must in turn rely on suitable mathematical models. In particular, several points of unilateral contact usually exist in such systems, and impact phenomena and friction are an extremely important feature in most of (if not almost all) multibody systems. As a consequence, multiple impacts may occur quite frequently and become a key point for the numerical simulation of multibody systems. A multiple impact occurs when several contact points of the system undergo an impact at the same time. This is for instance the case of a biped that walks on two feet, a docking mechanism of making two aerocrafts combine together, a Newton’s cradle with several balls initially in contact, a circuit breaker, granular materials like sand piles, etc. (Glocker, 2001a; 2001b). It is well known that the actual multibody systems can present discontinuities, which can be visible in many different forms. One of the most common in mechanics is the non-smooth dynamic characteristics. Typical examples in mechanics are the noise and vibration produced in railway brakes, impact print hammers, percussion drilling machines or chattering of machine tools. These effects are due to the non-smooth characteristics such as clearances, impacts, intermittent contacts, dry friction, or a combination of these effects. In non-smooth systems the time evolution of the displacement and the velocities is not requested to be smooth. Due to the possible impacts, the velocities are even allowed to undergo jumps at certain time instances in order to fulfill the kinematical restrictions. As a consequence, the accelerations and the contact and/or friction is described by impact points. Non-smooth dynamics is that part of Mechanics that deals with mechanical systems subject to various types of non-smooth interaction laws, such as impacts, friction, or any kind of piecewise linear contact laws. The complementarity relation is one example of a non-smooth contact law that is to be coupled to the usual (smooth) dynamical equations. Coulomb’s friction is another typical and most encountered example. These two, associated with 1. Introduction

7

some impact laws, are widely used in the field of multibody dynamics, for control, stability analysis, modeling and numerical simulation. Historically, this field of Mechanics has been settled by Jean Jacques Moreau, from the University of Montpellier, in a seminal paper published in 1963. Moreau developed Non-smooth Mechanics in parallel with Convex Analysis, which is the field of analysis dealing with convex, possibly non-differentiable functions, and convex sets. Indeed, nonsmooth systems need non-smooth analysis. The reader who is aware of optimization will have notice that indeed, complementarity relations look like very much the so-called Kuhn-Tucker conditions of constrained optimization. It is a fact that non-smooth mechanics and non-smooth optimization, share a common mathematical language and both rely upon similar mathematical tools (convex analysis, non-smooth analysis, variational inequalities).

1.2.4. Penalty approach versus linear complementarity problems As an effective method in contact analysis between complex objects with multiple contacts, the surface compliance method uses a penalty formulation. It is assumed that each contact region is covered with some spring-damper elements scattered over the body surfaces. The normal force including the elastic and damping shares prevents penetration, i.e., no explicit kinematic constraint is considered (Hippmann, 2004). The magnitudes of stiffness and deflection of the spring-damper elements are computed based on the pseudo-penetration, material properties and surface geometries of the colliding bodies. In the work by Khulief and Shabana (1987) the required parameters for representing contact force laws are obtained based on the energy balance during contact. This formulation uses a force-displacement law that involves determination of material stiffness and damping coefficients. In the work by Lankarani and Nikravesh (1994) two continuous contact force models are presented for which unknown parameters are evaluated analytically. In the first model, internal damping of bodies represents the energy dissipation at low impact velocities. However, in the second model local plasticity of the surfaces in contact becomes the dominant source of energy dissipation. Dias and Pereira (1995) described the contact law using a continuous force model based on the Hertz contact law with hysteresis damping. The effect and importance of structural damping schemes in flexible bodies were also considered. A contact model with hysteresis damping is also presented by Lankarani and Nikravesh (1990). Hunt and Crossley (1975) obtained also a model for computing the stiffness coefficient from the energy balance relations. In their approach, the damping force is a linear function of the elastic penetration which is estimated from the energy dissipated during impact (Moreira et al., 2009; Machado et al., 2009b). The effect of friction in this approach is often taken into consideration by using a regularized Coulomb friction model. An overview of different models of friction together with fundamentals can be found in Oden and Martins (1985) and Feeny et al. (1998).

8


Linear Complementarity Problems (LCP) are the result of other methods which mathematically give the exact solution to the contact problem. These formulations are basically arising from the complementarity relations and unilateral contact constraints. In addition to the exact solution of contact problems, other advantages of these formulations are that they do not result in stiff equations of motion and they show a lower effort for time integration compared to applying a brute force Lagrange multipliers method. These methods are often time consuming compared to the penalty approaches and their relation to microscopic deformation is unclear. Other disadvantages that can be mentioned are the neglection of the contact patch deflection, extreme simplification of complex physical phenomena and open problems in frictional impact theory. There are extensive researches addressed in the literature about investigation of frictionless and frictional contact of rigid bodies initiated from the idea of complementarity (Pfeiffer, 2003). These approaches have been well established and frequently used in the context of dealing with rigid bodies. As a result, some algorithms were developed based on unilateral contact constraints (Kwak, 1991; Pfeiffer and Glocker, 1996; Pang and Trinkle, 1996). Initially, only planar contact kinematics was considered which led to an LCP and then, this algorithm was extended for the case of spatial contact for which a Nonlinear Complementarity Problem (NCP) has to be formulated. However, for the case of dealing with flexible bodies there is a very limited number of publications devoted to this subject which attempt to get rid of difficulties in such cases. In such treatments, the formulations of rigid contact model are extended with some modifications needed for dealing with flexible bodies (Ebrahimi and Eberhard, 2006; Ebrahimi, 2007). The basic idea of complementarity in unilateral multibody systems can be stated as for a unilateral contact either relative kinematics is zero and the corresponding constraint forces are zero, or vice versa. The product of these two groups of quantities is always zero. This leads to a complementarity problem and constitute a rule which allows the treatment of MBS with unilateral constraints (Glocker and Pfeiffer, 1993; Pang and Trinkle, 1996; Trinkle et al., 2001). One of the first published works on the complementarity problems is due to Signorini (1933), who introduced an impenetrability condition in the form of an LCP. Later, Moreau (1979) and Panagiotopoulos (1985) also applied the concept of complementarity to study non-smooth dynamic systems. Pfeiffer and Glocker (1996) extended the developments of Moreau and Panagiotopoulos to multibody dynamics with unilateral contacts, being the complementarity considered of paramount importance. Complementarity problems are a very useful way to formulating problems involving discontinuities (Brogliato, 2003). Pang et al. (1992), based on the complementarity formulations studied the problem of predicting the acceleration of three dimensional multi-rigid-body systems with contact and coulomb friction. This paper was dedicated to Professor Cottle to celebrate his sixtieth birthday, and who they consider the founder of the linear complementarity problem. 1. Introduction

9

1.2.5. Event-driven and the time-stepping methods In a broad sense there are two different approaches to deal with the numerical simulation of multibody with set-valued force laws, namely the event-driven method and the time-stepping method (Glocker, 1999; Leine and Nijmeijer, 2004; Pfeiffer et al., 2006). The event-driven integration uses a standard ODE-integrator (or DAE-integrator) in smooth phases of the system’s motion and an LCP, NCP or augmented Lagrangian approach to determine the next mode at the switching boundaries. Thus, event-driven scheme detects changes of the constraints (events), for example a stick-slip transition, and resolves the exact transition times. This method integrates the system until an event takes place, calculates the next mode and proceeds integration. The event-driven integration method therefore clearly expresses the hybrid nature of systems with friction or inclusions in general. This method is quite accurate, but the event detection can be a time-consuming task, mainly when transitions occur very often. Therefore, this method is only recommended for systems with low number of contacts. Another drawback associated with this method is the fact that the constraints are only fulfilled at the acceleration level, which results in a numerical drift effect. The event-driven method was introduced by Pfeiffer and Glocker (1996) for the dynamics of planar and spatial rigid multibody systems with impact and friction. A good detail overview on this method with applications can be found in Leine and Nijmeijer (2004). Time-stepping methods are based on using a time-discretization of generalized positions and velocities, usually with a fixed step-size. Integrals of forces over each time-step are used instead of the instantaneous values of the forces. The time-stepping method makes no distinction between impulsive forces (due to impacts) and finite forces. Only increments of the positions and velocities are computed. The acceleration u is not computed by the algorithm, as it becomes infinite for impulsive forces. The positions and velocities at the end of the time-step are found by solving an algebraic inclusion which describes the contact problem (for instance by formulating it as a nonlinear complementarity problem). Multiple events might take place during one time-step, and the algorithm computes the overall integral of the forces over this time-step, which is finite. The time-stepping method is especially useful when one is interested in the global motion of systems with many contact points, leading to a large number of events. Each individual event is for those applications not of importance but the global motion is determined by the sum of all events. The benefit of time-stepping methods over event-driven integration methods is the fact that no eventdetection and index sets are needed. This makes the algorithm less complex, more robust and will give a reduction in computation time when many contacts are involved. A second advantage of the time-stepping method is its capability to pass accumulation points of impacts. A notable disadvantage of the time-stepping method is its low-order accuracy. The time-stepping method was

10


introduced by Moreau (1988) and has been subsequently developed in Anitescu and Potra (1997), Stewart and Trinkle (1996). The time-stepping method of Moreau is very elegant due to its simplicity but suffers from the fact that it is at best a second order method. Mechanical systems can exhibit high frequency oscillations during smooth parts of the motion, which require a higher-order integration scheme (Studer, 2008).

1.3. Outline of this report This report contains nine main sections and several appendices. An introduction and an overview of the work presented within the spirit of non-smooth dynamics are provided in the First Section. In Section Two the basic set-valued elements useful to model and analyze non-smooth dynamic systems are presented, namely the linear complementarity problem, as well as the two most basic set-valued maps, that is, the unilateral primitive and the Sgn-multifunction. Section Three deals with the formulation of laws for frictional unilateral contacts in the form of set-valued force laws. The normal contact force between rigid bodies is described by a set-valued force law called Signorini’s law, while the Coulomb’s friction law is used to model the friction phenomenon as a set-valued force law. The generalized contact kinematics aspects between two planar rigid bodies that experience an oblique eccentric impact are presented in Section Four, which are formulated under the framework of multibody systems methodologies. The Fifth Section describes the formulation of the equations of motion for rigid multibody systems with frictional unilateral constraints, that is, non-smooth mechanical systems. The equations of motion are formulated at the velocity-impulse level and are expressed in the form of equality of measures. The inclusions that are necessary to solve the frictional unilateral contact events in an autonomous multibody system, based on the Newton’s impact law combined with the Coulomb’s friction law, are also stated in this section. Section Six presents the Moreau time-stepping method necessary to integrate the equations of motion. This method is based on a time-discretization of the system dynamics. In addition, a complete and general computational strategy, based on the Moreau time-stepping method, to solve the equations of motion for rigid multibody systems with frictional unilateral constraints is also presented and discussed throughout this section. In Section Seven, the linear complementarity problem formulation to solve the contact-impact problem of multibody systems with frictional unilateral contacts is presented in detail. Furthermore, 1. Introduction

11

an algorithm that summarizes the Moreau time-stepping method with a linear complementarity problem formulation is presented. This algorithm is developed under the framework of multibody systems methodologies. In Section Eight, several multibody systems which include frictional unilateral constraints are used as demonstrative examples of non-smooth mechanical systems with the intent to discuss the main assumptions and procedures adopted in this work. Section Nine summarizes the main conclusions of the present study. Finally, this report ends with a full list of references and three appendices.

12


We are what we repeatedly do. Excellence, then, is a habit. Socrates

2. Basic set-valued elements 2.1. The linear complementarity problem A linear complementarity problem (LCP) is a set of linear equations that can be written in the form (Cottle and Dantzig, 1968; Garcia, 1973; Murty, 1988; Cottle et al., 1992)

y = Ax + b

(2.1)

subjected to the inequality complementarity conditions

y ≥ 0,

x≥0,

yTx = 0

(2.2)

for which the vectors x and y have to be evaluated for given A and b. In other words, the LCP is the problem of finding solutions x ∈ \ n and y ∈ \ n of (2.1) and (2.2), where b is an n-dimensional constant column and A is a given square matrix of dimension n. In general, in contact mechanics, vector y includes contact kinematic variables, vector x contains the contact forces, being the mass matrix, friction coefficient and contact shapes included in A and b. The inequality complementarity conditions expressed by Eq. (2.2) are often written in the following form

0≤y⊥x≥0

(2.3)

where y ⊥ x denotes y T x = 0 . An LCP can have a unique solution, multiple solutions or no solution at all (Rohn, 1993; Leine et al., 2002). All existing solutions can be found by using enumerative methods, which treat the problem by a combinatorial evolution of the complementarity condition xiyi=0. From the complementarity condition it follows that when xi≥0, then yi=0, and vice versa. An LCP of dimension n provides 2n different combinations of n variables, which are allowed to be greater than zero at the same time. For large dimensions, enumerative methods become numerically expensive since 2n grows rapidly. A more efficient algorithm is the complementarity pivot algorithm, usually referred to as Lemke’s algorithm (Pang and Trinkle, 1996; Stewart and Trinkle, 1996; Leine and Nijmeijer, 2004). A drawback of Lemke’s algorithm is that it is not guaranteed to find a solution for arbitrary A (convergence is guaranteed when A is a P-matrix). The basic idea of complementarity in unilateral multibody systems can be stated as for a unilateral contact either relative kinematics is zero and the corresponding constraint forces are zero, or vice versa. The product of these two groups of quantities is always zero. This leads to a complementarity problem and constitute a rule which allows the treatment of MBS with unilateral constraints (Glocker and Pfeiffer, 1993; Pfeiffer, 1999; Trinkle et al., 2001; Pfeiffer, 2003). 2. Basic set-valued elements

13

2.2. The unilateral primitive One of the most important multifunctions (or set-valued maps) related to complementarity problems is the unilateral primitive, denoted by Upr. Unilateral primitives are used in mechanics at the displacement level and at the velocity level to model unilateral geometric and kinematic constraints, such as free plays with stops, sprag clutches among others. The unilateral primitive is a maximal monotone set-valued map on \ + defined as (Glocker, 2001a)

x>0 ⎧ {0} ⎪ Upr( x) := ⎨(−∞, 0] x = 0 ⎪ ∅ x0: Upr(ax) = Upr( x)

(2.8)

∀a >0: a ⋅ Upr( x) = Upr( x)

(2.9)

The addition of two unilateral primitives results again in a unilateral primitive

Upr( x) + Upr( x − a) = Upr( x − b)

(2.10)

with b=max(0,a). 14


2.3. The Sgn-multifunction A second maximal monotone set-valued map, frequently used in complementarity problems, is the filled-in relay function Sgn-multifunction, which is defined by (Glocker, 2001a; Leine and van de Wouw, 2008a) x>0 ⎧ {+1} ⎪ Sgn( x) := ⎨[−1, + 1] x = 0 ⎪ x0, and can only be positive when contact happens, that is, gN=0. Thus, under the assumption of impenetrability between the bodies, expressed by gN≥0, only two situations can occur, namely g N = 0 ∧ λN ≥ 0

(closed contact)

(3.1)

g N > 0 ∧ λN = 0

(open contact)

(3.2)

Equations (3.1) and (3.2) represent a complementarity behavior, being always zero the product of the relative normal gap and normal contact force, that is g N λN = 0

(3.3)

Thus, the relation between the normal gap and normal contact force can be described by gN ≥ 0 ,

λN ≥ 0 ,

g N λN = 0

(3.4)

which represents the inequality complementarity condition between gN and λN, the so-called Signorini’s condition. The inequality complementarity behavior of the normal contact law is depicted in Fig. 3.2a that shows a set-valued graph or a corner of admissible combinations between gN and λN (Leine and van de Wouw, 2008b). When two rigid bodies are contacting, the Signorini’s condition given by Eq. (3.4) needs to be complemented with an impact law, such as the well known Newton’s kinematical law that relates the pre- and post-impact velocities to the bodies’ normal coefficient of restitution, εΝ. It should be noted that the case εΝ =1 corresponds to a completely elastic contact, whereas εΝ =0 corresponds to a completely inelastic contact. 3. Set-valued force laws for frictional unilateral contacts

17

Tangent contact direction λT 1

gN

1

λN

2

λN 2

Body 1

λT

Body 1 Body 2

Body 2

(a)

(b)

Fig. 3.1 – (a) Relative normal gap; (b) Normal and tangential contact forces.

It should be highlighted that use of the Newton’s impact law in combination with Coulomb friction can, under circumstances, lead to an (unphysical) energy increase. This typically occurs when there is a wide spread in normal and tangential restitution coefficients. Therefore, alternative methods for the definition of the coefficient of restitution, such as the Poisson’s or Stronge’s definition can be considered. Sufficient conditions for energy decrease with Newton’s impact law can be found in Leine and van de Wouw (2008a). λN

λT μλN γT

gN −μλN (a)

(b)

Fig. 3.2 – (a) Signorini’s normal contact law; (b) Coulomb’s friction law.

3.2. Set-valued tangential contact law The classical Coulomb’s friction law is another typical example that can be considered as a setvalued force law (Glcoker, 2001a; Leine and Glocker, 2003). The Coulomb law states that the sliding friction is proportional to the normal force of a contact. The magnitude of the static friction force is less than or equal to the maximum static friction force, which is also proportional to the normal contact force (Beer and Johnson, 1991). Furthermore, the sliding force acts in the opposite direction of the relative velocity of the frictional contact (Jean, 1999; Pfeiffer et al., 2006). Consider again the two contacting rigid bodies depicted in Fig. 3.1, in which Coulomb friction is present at the contact points 1 and 2. The relative velocity of point 1 with respect to point 2 along their tangent plane is denoted by γT. If contact between the two bodies takes place, i.e. gN=0, then the friction phenomenon imposes a tangential force λT as it is illustrated in Fig. 3.1b. If the bodies are sliding over each other, then the friction force λT has the magnitude μλN and acts in the direction opposed to the relative tangential velocity, that is, 18


−λT = μλN Sgn ( γ T )

γT ≠ 0

(3.5)

where μ is the friction coefficient and λN is the normal contact force. If the relative tangential velocity vanishes, i.e. γT =0, then the bodies purely roll over each other without slip. Pure rolling, or no slip for locally flat objects, is denotes by stick. Thus, if the bodies stick, then the friction force must lie in the interval –μλN≤λT≤μλN. For unidirectional friction, that is, for planar contact problems, three different scenarios can occur, namely

γ T = 0 ⇒ λT ≤ μλN

(sticking)

(3.6)

γ T < 0 ⇒ λT = + μλN

(negative sliding)

(3.7)

γ T > 0 ⇒ λT = − μλN

(positive sliding)

(3.8)

These three scenarios can be summarized by a set-valued force law as −λT ∈ μλN Sgn ( γ T )

(3.9)

Figure 3.2b shows the Coulomb’s friction law as a set-valued force law (Glocker, 2001a). In short, the tangential forces are limited by a maximal friction force dependent on μλN in any direction inside the tangential plane, and where μ represents the friction coefficient. Lower forces correlate with sticking contacts and, therefore, γT=0. Forces reaching the boundary of the friction cone may indicate sliding and, therefore, γT≠0. The full description of the spatial Coulomb’s friction law as a set-valued force law can be found in Leine and Glocker (2003).

3. Set-valued force laws for frictional unilateral contacts

19

20


All generalizations are dangerous, even this one. Alexander Dumas

4. Generalized contact kinematics 4.1. General issues in contact Contact-impact occurs in the collision of two or more bodies, which can be unconstrained or may belong to a multibody system. In general, the motion characteristics of a multibody system are significantly affected by contact-impact phenomena (Flores et al., 2008a). According to Gilardi and Sharf (2002), impact can be defined as a complex physical phenomenon for which the main characteristics are a very short duration, high force levels, rapid energy dissipation and large changes in the velocities of bodies. Inherently contact implies a continuous process which takes place over a finite time. Contact events can frequently happen in multibody systems and in many cases the function of mechanical systems is based on them (Pfeiffer and Glocker, 1996).

l2 l1

θ2 2

m1,J1 Y

1

m2,J2

3

θ1

m3,J3

X

θ3

0 0

Fig. 4.1 – Slider-crank mechanism with possible contacts within the translational clearance joint.

The collision is a prominent phenomenon in many mechanical systems such as mechanisms with intermittent motion, kinematic discontinuities and clearance joints (Lee and Wang, 1983; Khulief and Shabana, 1987; Bauchau and Rodriguez, 2002; Flores et al., 2006a). Figure 4.1 shows a planar slider-crank mechanism with a translational clearance joint. As a result of an impact, the values of the system state variables change very fast, eventually looking like discontinuities in the system velocities. The impact is characterized by large forces that are applied and removed in a short time period. The knowledge of the peak forces developed in the impact process is very important for the dynamic analysis of multibody mechanical systems and it has consequences in the design process. Other effects directly related to the impact phenomena are the vibration propagation on the system components, local elastic/plastic deformations at the contact zone and energy dissipation. Thus, the selection of the most adequate contact force model plays a key role in the correct design and analysis of these types of mechanical systems (Glocker, 1999). 4. Generalized contact kinematics

21

By and large, an impact may be considered to occur in two phases: the compression or loading phase, and the restitution or unloading phase. During the compression phase, the two bodies deform in the normal direction to the impact surface, and the relative velocity of the contact points/surfaces on the two bodies in that direction is gradually reduced to zero. The end of the compression phase is referred to as the instant of maximum compression or maximum approach. The restitution phase starts at this point and ends when the two bodies separate from each other (Brach, 1991). The restitution coefficient reflects the type of collision. For a fully elastic contact the restitution coefficient is equal to the unit, while for a fully plastic contact the restitution coefficient is null. The most general and predominant type of impact is the oblique eccentric collision, which involves both relative normal velocity and relative tangential velocity (Maw et al., 1976; Zukas et al., 1982). Within the multibody system formulation, the challenging problems/issues of contact modeling and analysis are threefold: (i) geometrical definition of the contact pair of surfaces; (ii) detection of the potential contact points; (iii) computation of the contact forces. Simply stated, no matter what model of constitutive equations for contact interfaces is applied, it is firstly required to search for potential contact points in the moving bodies. Based on the geometry of the contacting surfaces and their relative positions this procedure determines whether or not bodies are in contact. For bodies with simple geometry, such as straight lines, balls and cylinders, the candidate contact points can be determined analytically (Lopes et al., 2009). For complex surfaces, as those associated with human articulations, a more complex numerical procedure to detect the contact points is required (Machado et al., 2009a, 2009b; 2009c; 2010). Contact modeling is highly dependent on the properties and geometries of contacting bodies. One of these considerations is about the rigidity or elasticity of the contacting bodies. In the nature no body is absolutely rigid but in many applications they might be considered as rigid to simplify the problem and reduce the cost of numerical computations.

4.2. Kinematic aspects of contact between rigid bodies This paragraph deals with the generalized contact kinematics between two planar rigid bodies that experience an oblique eccentric impact, as it is illustrated in Fig. 4.2, in the spirit of nonsmooth dynamics approach. From the evaluation of the kinematics of any impact problem the representation of the constraints at the displacement and velocity level must be available in order to formulate the kinematical constraints during the impact analysis. The possible motion of each body in a multibody system, which is compatible with the kinematic conditions, is restricted by constraints for normal distances and relative velocities of the potential contact points. In general, the normal distance is a function of the generalized coordinates q and time variable t. Positive values of the normal distance represents a separation, while negative values denotes penetration. Therefore, the change in sign of normal distance indicates a transition from separation to contact, or vice versa. 22


G v1

G v2

Body 1

Body 2

Fig. 4.2 – Oblique eccentric collision between two bodies.

Figure 4.3 shows two planar rigid bodies and their potential contact points denoted by numbers 1 and 2. The vector that connects these two points is represented by g, which is a gap function. The projections of the vector g onto the normal and tangential directions correspond to the normal and tangential distances, gN and gT, respectively. From the kinematic configuration of Fig. 4.3, the vector g can be expressed as g =r2 -r1

(4.1)

where r1 and r2 are the position vectors described in global coordinates with respect to the inertial reference frame XY (Nikravesh, 1988; Haug, 1989). In a simple way, when the bodies are separated gN>0. For contacting bodies gN=0. For forbidden small overlapping gN becomes negative. G t 1

G r1

gN

G n

G g gT

Body 1

2

G r2

Y

Body 2

X

Fig. 4.3 – Two planar rigid bodies and their potential contact points 1 and 2.

The scalar normal and tangential distances are given by g N =n T ( r2 -r1 )

(4.2)

gT =t T ( r2 -r1 )

(4.3)

in which vectors n and t represent the normal and tangential directions at the contact points. In a similar way, the scalar normal and tangential velocities between points 1 and 2 can be evaluated as

4. Generalized contact kinematics

g N =γ N =n T ( v 2 -v1 )

(4.4)

g T =γ T =t T ( v 2 -v1 )

(4.5)

23

where v1 and v2 are the absolute velocities of points 1 and 2, respectively. These velocity vectors are illustrated in Fig. 4.4. This way to represent the normal and tangential relative velocities is very convenient, in the measure that it is not necessary to deal with derivations of normal and tangential unit vectors because these velocity components are not directly obtained by differentiating Eqs. (4.2) and (4.3). Further, the fully-rigid body velocity kinematics can easily be applied. The absolute velocity vectors of points 1 and 2 can be written as (Pfeifer and Glocker, 1996) v1 =J1q + j1

(4.6)

v 2 =J 2q + j2

(4.7)

in which J1,2 and j 1,2 are the jacobian terms and q is the vector that contains the generalized velocities (Pfeiffer and Glocker, 1996). Substitution of Eqs. (4.6)-(4.7) into Eqs. (4.4)-(4.5) yields

γ N =n T ( J 2 -J1 ) q + n T ( j2 -j1 )

(4.8)

γ T =t T ( J 2 -J1 ) q + t T ( j2 -j1 )

(4.9)

In a compact form, Eqs. (4.8) and (4.9) can be written as

γ N =w TN q + w N

(4.10)

γ T =w TT q + w T

(4.11)

w TN =n T ( J 2 -J1 )

(4.12)

w TT =t T ( J 2 -J1 )

(4.13)

where

(

)

(4.14)

(

)

(4.15)

w N =n T j2 -j1 w T =t T j2 -j1

The terms wN and wT, that represent the generalized normal and tangential force directions, can be evaluated as (Pfeifer and Glocker, 1996) ⎛ ∂g ⎞ w =w ( q , t ) = ⎜ ⎟ ⎝ ∂q ⎠

T

(4.16)

In turn, the terms w N and w T can be evaluated by (Pfeifer and Glocker, 1996)

w =w ( q, t ) =

∂g ∂t

(4.17)

which are different from zero only for rehonomic constraints. 24


In a similar way, the normal and tangential accelerations can be written as (Pfeifer and Glocker, 1996) + wN gN =w TN q

(4.18)

+ wT gT =w TT q

(4.19)

is the vector that contains the generalized accelerations. where q

G t 1

Body 1

G v1

G n G v2

γT

2

G G v 2 − v1 γN

Body 2

Fig. 4.4 – Velocity vectors of the contact points 1 and 2.

The presented study is restricted to convex rigid bodies with a smooth surface at least in a neighborhood of the potential contact such that the contact area reduces to a single point which may move relative to the bodies’ surfaces. However, this approach can be extended to more generalized contact geometries as long as a common tangent plane of the contacting bodies is uniquely defined (Glocker, 2001b; Flores et al., 2006c).

4. Generalized contact kinematics

25

26


Never confuse movement with action. Ernest Hemingway

5. Dynamics of non-smooth rigid multibody systems 5.1. Equations of motion It is known that impacts and frictional phenomena are characterized by unilateral constraints, which usually lead to unsteady dynamical behaviors. Therefore, appropriate methodologies and procedures to deal with this class of mechanical systems are required, being the main purpose of the present section. From classical mechanics, it is well known that the Newton-Euler equations of motion of a multibody system with f degrees of freedom and with only frictionless bilateral constraints can be written as (Pfeiffer and Glocker, 1996) Mu − h = 0

(5.1)

∀t

(5.2)

q =u

where M =Μ ( q, t ) ∈ \ f × f is the positive definite and symmetric mass matrix, h =h ( q, u, t ) ∈ \ f represents the vector of all external and gyroscopic forces acting on the system (forces originating from springs and dampers are also included in vector h), q =q ( t ) ∈ \ f is the f-dimensional vector of generalized coordinates, u =u ( t ) ∈ \ f addresses the system generalized velocities and u =u ( t ) ∈ \ f is the vector that contains the system accelerations. Joint reaction forces of the bilateral constraints do not appear in the equations of motion (5.1) because the coordinates q are minimal Lagrangian coordinates with respect to bilateral constraints, i.e. the vector q represents a set of coordinates that defines uniquely the positions of all bodies in the system when all unilateral contacts are open. The choice of a different symbol for positions q and velocities u is useful for studying non-smooth system. Furthermore, the dependence of the system matrices on q, u and t has been omitted in Eq. (5.1) for brevity. The terms M and h can be derived in a straightforward manner, by taking q as a set of classical generalized system coordinates and evaluating Lagrange’s equations of second type or the associated virtual work expressions (Greenwood, 1965; Leine and Nijmeijer, 2004). It is clear that Eq. (5.1) represents a classical second order differential equation that describes the dynamic behavior of a multibody system without any contacts and contact forces. Therefore, when a system includes frictional unilateral constraints, the occurring contact forces should be taken into account in the equations of motion. In general, the magnitudes of the normal and tangential 5. Dynamics of non-smooth rigid multibody systems

27

contact forces are added to equations of motion by using the Lagrange multipliers technique (Nikravesh, 1988; Haug, 1989). Thus, adding the contact forces to Eq. (5.1), the dynamic equations of motion of a rigid multibody system with normal and tangential contact forces during an impact can be written at the acceleration level as (Pfeiffer and Glocker, 1996) Mu − h − w N λ N − w T λ T =0 q =u

a.e.

∀t

(5.3) (5.4)

where w N =w N ( q, t ) ∈ \ f and w T =w T ( q, t ) ∈ \ f represent the generalized normal and tangential force directions, respectively. The normal and tangential contact forces have magnitudes λNi and λTi for each contact point i. The dual variables to the normal contact forces λN are the variations of normal gap distances gN, while the dual variables to the generalized friction or tangential forces λT are the variations of the generalized sliding velocities γT. The remaining terms of Eq. (5.3) have the same meaning as described above. The equations of motion (5.3) have to be complemented with appropriate constitutive laws for the normal and tangential contact forces. In this work, the unilateral constraints are described by a set-valued force law of the type Signorini’s law, while the frictional contacts are characterized by a set-valued law of the type Coulomb’s law for dry friction, as it was presented in section 3. It is important to note that Eq. (5.3) requires the existence of the velocities u and accelerations u both being meaningless for the event of an impact. Therefore, it is more adequate to talk about

the left and right limit of the velocity at the impact, that is, the pre- and post-impact velocity, but never about the velocity at the impact itself, a meaningless term already from the physical point of view. When a multibody system experiences impacts, in Eq. (5.3) exactly the points of interest, i.e. the discontinuity points of the velocities, are not considered. Therefore, the substitution of Eq. (5.3) by a more suitable formulation, which also accounts for the impacts, consists of use the equalities of measures firstly introduced by Moreau (1988a) and that constitute the general framework for nonsmooth rigid multibody dynamics (Jean, 1999; Panagiotopoulos and Glocker, 2000). Moreover, motion without impulses implies that λN(t) is (locally) bounded and time-continuous. The velocities u(t) therefore exists on non-impulsive time-intervals. The friction force λT(t) is discontinuous when a slip-stick transition takes place or when the relative sliding velocity of a frictional contact reverses its sign. The acceleration u is not defined when λT(t) is discontinuous. The set of time instances for which λT(t) is discontinuous is of measure zero and Eq. (5.3), therefore, holds for almost all t, that is, q = u does not hold at single time instants at which impacts take place. Thus, due to the presence of impulsive forces, a non-smooth system can not be described 28


solely by the equations of motion (5.3). Equality of measures provide an elegant way to obtain a valid comprehensive description of a non-smooth system including the impact case. When the equations of motion for the impact case are integrated over a singleton in time yields M ( u + − u − ) − w N Λ N − w T ΛT = 0 q =u

a.e.

(5.5) (5.6)

in which u– and u+ represent the pre- and post-impact velocities, ΛN and ΛT denote the normal and tangential impulsive forces, being the remaining terms defined as previously. Note that contact forces are replaced by the impulsive forces, which are well defined in the case of an impact. Furthermore, finite forces, such as gravity or reaction forces from springs and dampers, do not contribute to the integral and, therefore, they are not considered in Eq. (5.5). The equations of motion without impacts given by Eq. (5.3) and the equations of motion for the impact case (5.5) can not be used together in the present form, because the former is written at the acceleration level and the second one is developed at the velocity level. Therefore, Eqs. (5.3) and (5.5) should be considered simultaneously. Multiplying Eqs. (5.3) and (5.5) dt and dη, yields Mu dt − hdt − w N λ N dt − wT λ T dt = 0

(5.7)

M ( u + − u − ) dη − w N Λ N dη − wT ΛT dη = 0

(5.8)

Adding now Eqs. (5.7) and (5.8) results in M ⎡⎣u dt + ( u + − u − ) dη ⎤⎦ − hdt − w N ( λ N dt + Λ N dη ) − w T ( λ T dt + ΛT dη ) = 0

(5.9)

or more briefly Mdu − hdt − w N dPN − wT dPT = 0

(5.10)

where the Lebesgue measure is represented by dt and dη represents the sum of the Dirac impulse measures at the impact times. The measure for the velocities du = u dt + (u + − u − )dη is split in Lebesgue measurable part u dt , which is continuous, and the atomic parts which occur at the discontinuity points with the left

and right limits u– and u+ and the Dirac point measure dη. For impact free motion it holds that du = u dt . Similarly, the measure for the so-called percussions corresponds to a Lagrangian

multiplier which gathers both finite contact forces λ and impulsive contact forces Λ, that is, dP=λdt+Λdη. (Glocker and Studer, 2005; Förg et al. 2005). In the case of non-impulsive motion, all measures dη vanish and a formal division by dt yields the classical Newton-Euler equations of motion given by (5.3). 5. Dynamics of non-smooth rigid multibody systems

29

In the sequel of the process to obtain the Eq. (5.10), the velocities u are assumed to be of bounded variation with differential measure du, leading to displacements that are absolutely continuous with q = u almost everywhere. The right and left limits of u(t) at t are denoted by u+(t) and u–(t), respectively, which might be different from each other in the case of an impact. The interested reader on the issues of bounded variation functions, absolute continuity and measures is referred to the works by Rudin (1981) and Moreau (1988a, 1988b). In short, as it was mentioned above, the equations of motion (5.3), which relate accelerations to forces, are not appropriate to describe motion involving impacts. Therefore, the most suitable formulation to express the Newton-Euler equations of motion for non-smooth dynamic systems is in terms of an equality of measures (5.10). In general, the motion of a non-smooth system can have its time evolution divided into piecewise smooth and non-smooth phase. The integration of Eq. (5.10) over a singleton t gives in the scalar impulsive contact forces. On the other hand, for the smooth or impact free motion, from the same equation it results in the normal and tangential scalar contact forces. In addition to non-smooth impact-free motion, this formulation covers even impulsive behavior and should be considered as the starting point of any such problem in dynamics. The basic idea of the use of equality of measures in multibody dynamics with unilateral constraints is to allow for the atomic quantities expressing velocity jumps and impulsive forces when integration of the equations of motion with respect to time is performed (Glocker and Studer, 2005).

5.2. Impact laws In this paragraph, the resolution of the equations of motion expressed in the form of equality of measures (5.10) is briefly presented and discussed. The inclusions that are necessary to solve the frictional unilateral contact events in an autonomous multibody system, based on the Newton’s impact law combined with the Coulomb’s friction law, are also stated. In addition, the force laws are related to the systems’ kinematics. The interested reader in the detail description of this formulation is referred to the works by Moreau (1988a, 1988b) and Glocker (2001a). Since the impenetrability condition between colliding bodies is required, let first consider that a rigid multibody system has a total n of frictional unilateral constraints, which can be represented by

n inequalities as g Ni ( q, t ) ≥ 0 ,

(i=1, …, n)

(5.11)

where the quantities gNi are the normal gap functions of the frictional contacts. They are formulated such that, gNi>0 indicates an open or positive contact with an Euclidian distance of the contact points given by the value of gNi, gNi=0 corresponds to a closed or active contact, and gNi0. Suppose that, for any reason, the contact does not participate in the impact, that is, that value of the normal contact impulse is zero, although the contact is closed. This situation happens normally for multiple contact scenarios. Therefore, for this case, it is allowed that the post-impact relative velocity to be higher than the value prescribed by Newton’s impact law, with the intent to express that the contact is superfluous and could be removed without changing the contact-impact process. Thus, in order to account for these possibilities, two parameters are defined as (Glocker, 2001b)

ξ Ni := γ Ni+ + ε Niγ Ni−

(5.15)

ξTi := γ Ti+ + ε Tiγ Ti−

(5.16)

where ( γ Ni± , γ Ti± ) := ( γ Ni , γ Ti )( u ± ) .

5. Dynamics of non-smooth rigid multibody systems

31

Thus, normal and tangential impact laws can be stated as two inclusions −dPNi ∈ Upr (ξ Ni )

(5.17)

−dPTi ∈ μi dPNiSgn (ξTi )

(5.18)

Finally, the complete description of the dynamics of non-smooth system, which accounts for both impact and impact-free phases, is given by Eqs. (5.10)-(5.18). This problem can be solved by using the Moreau time-stepping method, which is presented and discussed in the next section. It should be noted that the impact laws given by Eqs. (5.17) and (5.18) are actually contactimpact laws, because they hold for both impacts and smooth phases. In the first case, dPNi and dPTi have just to be replaced by the corresponding impulsive forces, ΛNi and ΛTi, when integration over t has been performed. For the second case, it is assumed that a time interval without impacts, i.e., a time interval in which the velocities are continuous u+=u–=u and the forces are non impulsive. Thus, under these assumptions, Eqs. (5.15) and (5.16) become

ξ Ni = (1 + ε Ni ) γ Ni

(5.19)

ξTi = (1 + ε Ti ) γ Ti

(5.20)

− λ Ni dt ∈ Upr ⎡⎣(1 + ε Ni ) γ Ni ⎤⎦

(5.21)

−λ Ti dt ∈ μi λ Ni dtSgn ⎡⎣(1 + ε Ti ) γ Ti ⎤⎦

(5.22)

from which Eqs. (5.17) and (5.18)

with non-negative values of the restitution coefficients, after crossing out dt, yields − λ Ni ∈ Upr ( γ Ni )

(5.23)

− λ Ti ∈ μi λ NiSgn ( γ Ti )

(5.24)

which are the force laws for impact-free motion of unilateral constraints with Coulomb’s friction.

32


The superfluous, a very necessary thing. Voltaire

6. Moreau time-stepping method 6.1. Time discretization based on the Moreau midpoint rule The time-stepping methods provide a discrete numerical scheme suitable for the simulation of non-smooth systems (Moreau, 1988a; Anitescu et al., 1999; Leine and Nijmeijer, 2004; Förg et al., 2005; Glocker and Studer, 2005; Pfeiffer et al., 2006; Studer et al., 2008). These methods are widely used due to their simplicity to implement and their robustness. In contrast to event-driven methods, the time-stepping methods enjoy convergence results. The time-stepping schemes are based on a time-discretization of the system dynamics. The whole set of discretized equations and constraints is used to compute the next state of the motion. Among the various time-stepping methods available in the literature, the Moreau midpoint method is one of the more popular and is considered in the present work (Moreau, 1988a; Jean, 1999; Studer et al., 2008). The equality of measures (5.10) together with the set-valued force for laws (5.17) and (5.18) form a measure differential inclusion which describes the time evolution of a multibody system with discontinuities in the generalized velocities, that is, a non-smooth dynamical system. A general way to solve this mathematical problem consists of applying the Moreau time-stepping method, which does not make the use of the classical equations of motion, which relate the accelerations to forces, but consider the equations of motion at the velocity level (5.10). The first step of the Moreau approach consists of the time-discretization of the measure differential equation. Thus, integrating Eq. (5.10) over a small finite time interval Δt, of which initial and end points are denoted by the indices A and E, yields the following terms

∫ Mdu = M

M

Δu = M M ( u E − u A ) ,

M M = M ( q M , tM )

(6.1)

Δt

h M = h ( q M , u A , tM )

(6.2)

dPN = w NM PN ,

w NM = w ( q M , tM )

(6.3)

dPT = wTM PT ,

w TM = w ( q M , tM )

(6.4)

∫ hdt = Δh ≈ h

M

Δt ,

Δt

∫w

N

Δt

∫w

T

Δt

where tM is the midpoint time instant of the compact time interval [tA, tE] and q M = q A + 12 u A Δt is the midpoint system’s position state. It is clear that the midpoint time instant can be evaluated as tM = t A + 12 Δt 6. Moreau time-stepping method

(6.5) 33

Finally, after the above discretization, the equations of motion expressed at the velocity level can be written as (Glocker and Studer, 2005; Flores et al., 2010a) M M ( u E − u A ) − h M Δt − w NM PN − w TM PT = 0

(6.6)

together with the set-valued contact-impact force laws − PN ∈ Upr (ξ N )

(6.7)

− PT ∈ μ PN Sgn (ξT )

(6.8)

This set of algebraic inclusions can be solved with a linear complementarity problem (LCP) formulation or by augmented Lagrangian approach (ALA) (Leine and Nijmeijer, 2004). The velocity uE, at the end of time-step tE=tA+Δt, is subsequently calculated by using Eq. (6.6). Finally, the positions at the end of the time step are calculated by q E = q M + 12 Δtu E

(6.9)

Note that (6.7) applies only to active set-valued force laws, i ∈ H (t ) , i.e. set-valued force laws that can be described at the velocity level. As friction elements are naturally defined at the velocity level, they are always active and can always be described by (6.8). Considering unilateral contacts, Moreau midpoint algorithm calculates the contact distances gNi of all unilateral contacts at the midpoint qM in order to evaluate whether these are active (gNi≤0) or not (gNi>0). Only active unilateral contacts can be modeled by inclusion (6.7). Unilateral contacts that are non-active, thus open, are disregarded because it is assumed that their contact force contribution is equal to zero. In short, the time-stepping methods are difference schemes including fully the complementarity conditions and the impact laws, by allowing a simultaneous treatment of impulsive and non impulsive forces together with all inequalities involved. The computational strategy to solve the equations of motion for non-smooth systems is presented in the next section, which is based on the Moreau midpoint rule. More advanced discretization schemes can be found in the literature, such for example the powerful Θ-method by Jean (1999), or several other well developed codes described by Stewart and Trinkle (1996) and Stewart (1998).

6.2. Computational strategy to solve the equations of motion Figure 6.1 presents the flowchart of the general computational strategy, based on the Moreau time-stepping method, to solve the equations of motion for rigid multibody systems with frictional unilateral constraints, which can be summarized by the following steps: (i)

Start the analysis by defining the initial conditions of the problem at hand, namely the initial time tA, final time of simulation tF, time step Δt, together with the given initial positions qA and velocities uA;

34


(ii)

According to the Moreau midpoint rule compute the midpoint time instant tM, the end time of the interval tE, evaluate the position’s state at the midpoint instants qM, assemble the midpoint mass matrix MM and the gyroscopic and external forces vector hM, and compute the midpoint states of the potential or candidate contact-impact points HM;

(iii) Check for contact-impact between contacting bodies. If there is no any contact-impact (open contacts) calculate the velocity at the end time uE, by using Eq. (6.6); otherwise (at least one closed contact) apply an algorithm based on the time-stepping method (for instance as an LCP or by using ALA) in order to obtain the impulsive forces PN and PT required to compute uE for the contact-impact case; (iv) Compute the position’s state at the end time qE , by solving Eq. (6.9); (v)

Increment time step. If the current time is smaller than the intended final time simulation, update the position and velocity variables and go to step (ii) to proceed with the process of a new time step; otherwise stop the simulation. START

Compute

Read

Evaluate

t A , tF

tM

Δt qA, uA

qM , M M , hM

Is there any contact or impact?

Yes

PN , PT (Apply LCP or ALA method)

HM No

Update

q A = qE

Evaluate

u A = uE

uE (Use Eq. (6.6))

No

Is tA>tF?

Increment time

t A = t A + Δt

Evaluate

qE (Use Eq. (6.9))

Yes STOP

Fig. 6.1 – Flowchart of the computational procedure for the solution of the equations of motion of constrained rigid multibody systems with frictional unilateral constraints.

6. Moreau time-stepping method

35

36


All truths are easy to understand once they are discovered; the point is to discover them. Galileo Galilei

7. Solving the contact-impact problem as an LCP 7.1. Formulation of the contact-impact problem as an LCP In this section, the LCP formulation to solve the contact-impact problem of multibody systems with frictional unilateral constraints is presented, which closely follows the work by Glocker and Studer (2005) and Flores et al. (2010a). In order to set up the LCP, let first introduce the following matrix notation WNM := mat ( w Ni ( q M , tM ) ) ∈ \ f ,i ,

i∈H

(7.1)

WTM := mat ( w Ti ( q M , tM ) ) ∈ \ f ,i ,

i∈H

(7.2)

NM := col ( w Ni ( q M , tM ) ) ∈ \ i , w

i∈H

(7.3)

TM := col ( w Ti ( q M , tM ) ) ∈ \ i , w

i∈H

(7.4)

PN := col ( PNi ) ∈ \ i ,

i∈H

(7.5)

PT := col ( PTi ) ∈ \ i ,

i∈H

(7.6)

γ NE := col ( γ NEi ) ∈ \ i ,

i∈H

(7.7)

γ TE := col ( γ TEi ) ∈ \ i ,

i∈H

(7.8)

γ NA := col ( γ NAi ) ∈ \ i ,

i∈H

(7.9)

γ TA := col ( γ TAi ) ∈ \ i ,

i∈H

(7.10)

ξ N := col (ξ Ni ) ∈ \ i ,

i∈H

(7.11)

ξT := col (ξTi ) ∈ \ i ,

i∈H

(7.12)

ε N := diag ( ε Ni ) ∈ \ i ,

i∈H

(7.13)

εT := diag ( ε Ti ) ∈ \ i ,

i∈H

(7.14)

μ := diag ( μi ) ∈ \ i ,

i∈H

(7.15)

Thus, the contact-impact problem of non-smooth systems can be summarized by the following mathematical relations M M ( u E − u A ) − h M Δt − WNM PN − WTM PT = 0 7. Solving the contact-impact problem as an LCP

(7.16) 37

T NM γ NE =WNM uE + w

(7.17)

T TM γ TE =WTM uE + w

(7.18)

T NM γ NA =WNM uA + w

(7.19)

T TM γ TA =WTM uA + w

(7.20)

ξ N = γ NE +ε N γ NA

(7.21)

ξT = γ TE +εT γ TA

(7.22)

− PN ∈ Upr ( ξ N )

(7.23)

− PT ∈ μPN Sgn ( ξ T )

(7.24)

The values of γNA and γTA can be evaluated by using Eqs. (7.19) and (7.20), respectively, since the velocities uA are known at the left endpoint of the time interval. Introducing now Eqs. (7.17) and (7.18) into (7.21) and (7.22) yields T NM +ε N γ NA ) ξ N = WNM uE + ( w

(7.25)

T TM +εT γ TA ) ξT = WTM uE + ( w

(7.26)

Now, it should be mentioned that the inclusions for the contact-impact force laws need to be formulated as complementarity conditions. Thus, the unilateral primitive of Eq. (7.23) results in − PN ∈ Upr ( ξ N ) ⇔ PN ≥ 0, ξ N ≥ 0, PNT ξ N = 0

(7.27)

In turn, the relay function (7.24) have to be decomposed into two Upr’s to achieve the desired complementarity conditions. Thus, Eq. (7.24) yields

− PT ∈ μPN Sgn ( ξT )

⎧μPN + PT ≥ 0, ξ R ≥ 0, ( μPN + PT )T ξ R = 0 ⎪⎪ T ⇔ ∃ ξ R , ξ L s.t. ⎨ μPN − PT ≥ 0, ξ L ≥ 0, ( μPN − PT ) ξ L = 0 ⎪ ξT = ξ R − ξ L ⎪⎩

(7.28)

in which the step height is [-μPN, +μPN]. In addition, to abbreviate the complementarity conditions of Eq. (7.28) the impulsive friction saturations PR and PL are defined as (Glocker, 2001a) PR := μPN + PT ,

PR ∈ \ i

(7.29)

PL := μPN − PT ,

PL ∈ \ i

(7.30)

ξ R , ξ L ∈ \i

(7.31)

together with ξT = ξ R − ξ L ,

Then, the whole set of complementarity conditions of Eq. (7.28) can be rewritten as, 38


⎛ ξ N ⎞ ⎛ PN ⎞ ⎜ ⎟ ⎜ ⎟ 0 ≤ ⎜ ξ R ⎟ ⊥ ⎜ PR ⎟ ≥ 0 ⎜P ⎟ ⎜ξ ⎟ ⎝ L⎠ ⎝ L⎠

(7.32)

The reason for this special arrangement of PL and ξL in Eq. (7.32), must be sought in optimization theory. Without this special arrangement, one is not able to be set up the LCP formulation without additional matrix inversion processes (Glocker, 2001a). Since variables ξT, PT and uE are not included in (7.32), they have to be eliminated. Thus, combining Eqs. (7.16) and (7.29), yields M M ( u E − u A ) − h M Δt − ( WNM − WTM μ ) PN − WTM PR = 0

(7.33)

Substituting now Eq. (7.31) into Eq. (7.26) results in T TM +εT γ TA ) + ξ L ξ R = WTM uE + ( w

(7.34)

The elimination of variable PT can be done through the combination of Eqs. (7.29) and (7.30), which result can be written as PL = 2μPN − PR

(7.35)

Since the inversion of mass matrix M is always possible, then Eq. (7.33) can be solved for uE u E = u A + M −M1h M Δt + M −M1 ( WNM − WTM μ ) PN + M −M1 WTM PR

(7.36)

T T Now, Eqs. (7.19) and (7.20) are used to express WNM u A and WTM u A in terms of γ NA and γ TA ,

that is T NM WNM u A =γ NA − w

(7.37)

T TM WTM u A =γ TA − w

(7.38)

Introducing Eqs. (7.36)-(7.38) into Eqs. (7.25) and (7.34), yields T T T ξ N = WNM M −M1h M Δt +WNM M −M1 ( WNM − WTM μ ) PN + WNM M −M1 WTM PR + ( I +ε N ) γ NA

(7.39)

T T T ξ R = WTM M −M1h M Δt +WTM M −M1 ( WNM − WTM μ ) PN + WTM M −M1 WTM PR + ( I +εT ) γ TA + ξ L

(7.40)

Thus, Eqs. (7.39), (7.40) and (7.35) can be written in a matrix form as T −1 T −1 ⎛ ξ N ⎞ ⎛ WNM M M ( WNM − WTM μ ) WNM M M WTM ⎜ ⎟ ⎜ T −1 T −1 ⎜ ξ R ⎟ = ⎜ WTM M M ( WNM − WTM μ ) WTM M M WTM ⎜ ⎟ ⎜ 2μ −I ⎝ PL ⎠ ⎝

T 0 ⎞ ⎛ PN ⎞ ⎛ WNM M −M1h M Δt + ( I +ε N ) γ NA ⎞ ⎟ ⎜ ⎟ ⎜ T −1 ⎟ I ⎟ ⎜ PR ⎟ + ⎜ WTM M M h M Δt + ( I +εT ) γ TA ⎟ (7.41) ⎟ 0 ⎠⎟ ⎝⎜ ξ L ⎠⎟ ⎜⎝ 0 ⎠

Equations (7.41) together with the complementarity conditions (7.32) form the LCP for the contact-impact analysis of multibody systems with frictional unilateral constraints. The dimension 7. Solving the contact-impact problem as an LCP

39

of this LCP is 3n, where n represents the number of active contacts. The LCP (7.41) is solved in each integration time step. Then, the velocities uE and positions qE for the subsequent time steps are obtained from Eqs. (7.36) and (6.9), respectively. It should be highlighted that for Δt=0, the LCP reduces to the pure impact equations of motion and can be used, for example, for initialization of the velocities. For γNA=γTA=0, the LCP describes impact free motion at the velocity level, containing still the cases of persisting contact and stiction as well as transitions to sliding or separation and can be transformed to the acceleration level by the methods presented by Glocker (2001a).

7.2. Moreau time-stepping method with an LCP formulation Since the Moreau time-stepping method with an LCP formulation involves a good deal of mathematical manipulation, it is convenient to summarize the main steps in an appropriate algorithm. This algorithm, presented in the flowchart of Fig. 7.1, is developed under the framework of MBS formulation and can be condensed in the following steps: (i)

Specify the initial conditions of the problem at hand, tA, tF, Δt, qA and uA;

(ii)

Define the geometrical, inertial and material functions, gNi, M, h, εNi, εTi, μi, wNi, wTi, w Ni and w Ti ;

(iii) Compute the midpoint state variables: tM = t A + 12 Δt q M = q A + 12 Δtu A M M = M ( q M , tM ) h M = h ( q M , u A , tM ) g Ni = g Ni ( q M , tM )

{

}

H M = i g Ni ( q M , tM ) ≤ 0 ni = length ( H M )

(iv) For every i ∈ H M evaluate: WNM = mat ( w Ni ( q M , tM ) ) WTM = mat ( wTi ( q M , tM ) ) NM = col ( w Ni ( q M , tM ) ) w 40


TM = col ( w Ti ( q M , tM ) ) w

γ NA = col ( γ NAi ) γ TA = col ( γ TAi ) ε N = diag ( ε Ni ) εT = diag ( ε Ti ) μ = diag ( μi )

(v)

Set up the LCP in the standard form y=Ax+b; T T ⎛ WNM M −M1 ( WNM − WTM μ ) WNM M −M1 WTM ⎜ T −1 T M −M1 WTM A = ⎜ WTM M M ( WNM − WTM μ ) WTM ⎜ −I 2μ ⎝

0⎞ ⎟ I⎟ 0 ⎟⎠

T ⎛ WNM M −M1h M Δt + ( I +ε N ) γ NA ⎞ ⎜ T −1 ⎟ b = ⎜ WTM M M h M Δt + ( I +εT ) γ TA ⎟ ⎜ ⎟ 0 ⎝ ⎠

(vi) Solve the LCP using an appropriate algorithm;

( x, y ) = LCP( A, b) (vii) Split the LCP solution according to: PN = col ( xi ) ,

i=1,…,ni

PL = col ( xi ) ,

i= ni +1,…,2ni

PR = col ( y i ) ,

i=2ni +1,…,3ni

(viii) Evaluate the velocity at the end of the integration time step: u E = u A + M −M1h M Δt + M −M1 ( WNM − WTM μ ) PN + M −M1 WTM PR

(ix) Compute the positions at the end of the integration time step: q E = q M + 12 Δtu E (x)

Increment time step: t A = t A + Δt

(xi) Update system states’ variables qA=qE and uA=uE. Go to step (iii) and proceed with the process for the new time step. These steps must be performed until the final time of analysis is reached. 7. Solving the contact-impact problem as an LCP

41

Specify

t A , t F , Δt , q A , u A Define

g Ni , M, h, ε Ni , ε Ti , μi , w Ni , wTi , w Ni , w Ti Compute

tM , q M , M M , h M , g Ni , H M , ni Evaluate

NM , w TM , γ NA , γ TA , ε N , εT , μ WNM , WTM , w Set up and solve the LCP

A, b → y = Ax + b Spilt LCP solutions

PN , PL , PR Evaluate −1 M

u E = u A + M h M Δt + M

−1 M

( WNM − WTM μ ) PN + M −M1 WTM PR

Compute

q E = q M + 12 Δtu E Increment time

t A = t A + Δt Update

q A = qE , u A = uE

Fig. 7.1 – Flowchart of the Moreau time-stepping algorithm with an LCP formulation.

42


There are no facts, only interpretations. Friedrich Nietzsche

8. Results and discussion 8.1. Bouncing ball The purpose of this section is to demonstrate the application of the Moreau time-stepping method with an LCP formulation. The example considered here is the bouncing ball example, which is one of the simplest non-smooth systems. Figure 8.1 shows a rigid ball with height y, mass m, moment of inertia JS=3/5mR2 and radius R. For simplicity, the system is considered to be frictionless and the value of the coefficient of restitution is less than unit. The ball is released from the initial position, y0, under the action of gravity only, which is taken to act in the negative Y direction. Thus, the ball falls down until it collides with the table which is considered to be rigid and fixed. When the ball collides with the table, an impact takes place and the ball rebounds, producing jumps less high due to the effect of coefficient of restitution (εΝ≤1).

S

g

y0 gN Y X

Fig. 8.1 – Bouncing ball example.

The descritized equations of motion for the bouncing ball problem can be stated as M M ( u E − u A ) − h M Δt − WNM Λ N =0

(8.1)

together with the set-valued force law for impacts − Λ N ∈ Upr ( ξ N )

(8.2)

Since there is no rehonomic constraints, the relative normal contact velocities are written as T γ NA = WNM uA

(8.3)

T γ NE = WNM uE

(8.4)

Based on the definition of the parameter ξN and considering Eqs. (8.3) and (8.4) yields T T WNM u E = ξ N − ε N WNM uA

8. Results and discussion

(8.5) 43

Introducing now Eq. (8.5) into Eq. (8.1) results in T T T T ξ N = WNM M −M1 WNM Λ N + WNM u A + WNM ε N u A + WNM M −M1h M Δt

(8.6)

ξ N = AΛ N + b

(8.7)

T A = WNM M −M1 WNM

(8.8)

T T b = WNM u A (1 + ε N ) + WNM M −M1h M Δt

(8.9)

or alternatively

where

Equation (8.7), which is linear in ξN and ΛN, forms together with the complementarity condition 0 ≤ Λ N ⊥ ξ N ≥ 0 a linear complementarity problem (LCP) in the standard form, describing the contact problem at the velocity level of the bouncing ball. Since the bouncing ball has one degree of freedom, and because only the vertical motion is considered, the variables necessary to define the problem are as follows

q = ( y)

(8.10)

u = (v )

(8.11)

M = ( m)

(8.12)

h = (−mg )

(8.13)

gN = y − R

(8.14)

WN = (1)

(8.15)

Thus, introducing these values into Eqs. (8.8) and (8.9) yields, respectively A = m −1

(8.16)

b = u A (1 + ε N ) − g Δt

(8.17)

Finally, the a scalar LCP can be formulated as

ξ N = AΛ N + b

(8.18)

together with the complementarity condition 0 ≤ ΛN ⊥ ξN ≥ 0

(8.19)

which has a unique solution for A>0, that is, x = Λ N = 0 ∧ y = ξ N = b if b ≥ 0 x = ΛN = − 44

b ∧ y = ξ N = 0 if b < 0 A

(8.20) (8.21)


Ball mass – m

1.0 kg

Ball radius – R

0.1 m

Initial position – y0

1.0 m

Initial velocity – v0

0.0 m/s

Gravity acceleration – g

9.81 m/s2

Coefficient of restitution – εN

0.9

Integration time step – Δt

0.001 s

Tab. 8.1 – Parameters used in the dynamic simulation of the bouncing ball.

Thus, the LCP (8.18) is solved in each integration time step for ΛN and ξN. Then, the subsequent velocity uE and position qE are obtained from Eq. (8.1) and Eq. (6.9), respectively. Table 8.1 presents the parameters used in the simulations performed. The global results are illustrated in Fig. 8.2, namely the position and velocity of the ball, the impact force as well as the potential, kinetic and total ball energy. When the ball hits the table, an impact occurs and the ball is rebounded. The restitution coefficient is less than unit, therefore, the ball jumps less high after each impact. The smooth and impact phases can easily be observed, namely in what concerns to the velocity jumps The time lapse between the impacts tends to zero and ends up in an accumulation point, i.e.

1.0

5.0

0.9

4.0

0.8

3.0

Ball velocity [m/s]

Ball position [m]

infinitely many impacts in a finite time. After the accumulation point, the ball remains on the table.

0.7 0.6 0.5 0.4 0.3 0.2

2.0 1.0 0.0 -1.0 -2.0 -3.0

0.1

-4.0

0.0

-5.0 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

Time [s]

(a)

6

7

8

9

10

(b)

10.0

10.0

9.0

9.0

8.0

8.0

7.0

7.0

Ball energy [J]

Impulsive force [kgm/s]

5

Time [s]

6.0 5.0 4.0 3.0

6.0 5.0 4.0 3.0

2.0

2.0

1.0

1.0

0.0

Potential energy Kinetic energy Total energy

0.0 0

1

2

3

4

5

Time [s]

(c)

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

Time [s]

(d)

Fig. 8.2 – Simulation results of a ball falling on a table: (a) Ball position; (b) Ball velocity; (c) Impulsive force; (c) Potential, kinetic and total ball energy.


45

Table 8.2 presents the global results of the bouncing ball example where the first impact data can be analyzed and how it affects the problem formulation. t[s] 0.000 . . . 0.421 0.422 0.423 0.424 0.425 0.426 0.427 0.428 0.429 0.430 0.431 0.432 0.433 0.434

y[m] 0.000

v[m/s] 0.000

tM[s] qM[s] 0.0005 1.000

MM[kg] hM[N] 1.00 -9.81

gNM[m] 0.900

wNM 0.00

ΛN 0.00

ξN 0.00

0.131 0.126 0.122 0.118 0.114 0.110 0.106 0.101 0.101 0.105 0.109 0.113 0.116 0.120

-4.130 -4.140 -4.150 -4.159 -4.169 -4.179 -4.189 -4.199 3.779 3.769 3.759 3.749 3.740 3.730

0.4215 0.4225 0.4235 0.4245 0.4255 0.4265 0.4275 0.4285 0.4295 0.4305 0.4315 0.4325 0.4335 0.4345

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.029 0.024 0.020 0.016 0.012 0.008 0.004 -0.001 0.003 0.007 0.011 0.014 0.018 0.022

0.00 0.00 0.00 0.00 0.00 0.00 0.00 -7.99 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 7.99 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.129 0.124 0.120 0.116 0.112 0.108 0.104 0.099 0.103 0.107 0.111 0.114 0.118 0.122

-9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81 -9.81

Tab. 8.2 – Global results of the bouncing ball where the first impact can be observed.

1.0

5.0 Dt=0.001s

0.9

Dt=0.025s

Dt=0.025s

3.0

Ball velocity [m/s]

Ball position [m]

Dt=0.001s

4.0

0.8 0.7 0.6 0.5 0.4 0.3 0.2

2.0 1.0 0.0 -1.0 -2.0 -3.0

0.1

-4.0

0.0

-5.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

Time [s]

1.5

2.0

2.5

3.0

Time [s]

(a)

(b)

Impulsive force [kgm/s]

8.0 Dt=0.001s

7.0

Dt=0.025s

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time [s]

(c)

Fig. 8.3 – Influence of the integration time step on the ball dynamic response: (a) Ball position; (b) Ball velocity; (c) Impulsive force.

In order to understand the crucial role played by the integration time step, the bouncing ball was simulated using two different time steps, namely 0.001 and 0.025 s. The restitution coefficient used 46


is 0.7, being the system characterized by the remaining parameters listed in Tab. 8.1. The global results produced by the bouncing ball system are presented in Fig. 8.3, namely the ball position and velocity time evolutions, as well as the impulsive force produced during the impact process. From the analysis of the plots of Fig. 8.3 it can be observed that for the time step of 0.025 s, the system’s response tends to be erroneous due to the fact that the contact duration is too long, being the corresponding impulsive force a little greater than that obtained with 0.001s time step, as it can be observed in Fig. 8.3c. This phenomenon can also be observed in the deviations of the ball positions and velocities plots. Furthermore, for the time step equal to 0.025 s, after the bouncing effect, that is, after approximately 2.3 s of simulation, the ball remains in contact with the table but with a positive value for the impulsive force. Conversely, for the time step of 0.001s, when the ball stops to bounce, the impulsive force is null, as it can be observed in Fig. 8.3c. This aspect can also be seen in the ball positions and velocities diagrams. Finally, it should be mentioned that with high integration time step, the forbidden overlap between bodies is artificially too large.


47

8.2. Woodpecker toy The woodpecker toy is indubitably of the most popular and frequently used system that includes frictional unilateral constraints, that is, it combines impact and friction phenomena. Therefore, the woodpecker toy became a well known benchmark in non-smooth mechanics (Studer, 2008). The history of the woodpecker toy investigations is long and interesting. Pfeiffer (1984) was the pioneer on the study of this system and used a heuristic method to deal with the impact and friction problem. Later on, Glocker (1995), Glocker and Pfeifer (1995) and Pfeiffer and Glocker (1996) consider the woodpecker toy in the development of their work on the multiple impacts with friction in rigid multibody systems. In these studies, the woodpecker was simplified to a system with three degrees of freedom. The woodpecker system was experimentally verified by Pfeifer and Glocker (1996). The nonlinear dynamics of the woodpecker toy was performed by Leine et al. (2003). Glocker and Studer (2005) consider the woodpecker toy as an application example in the context of multibody systems with frictional constraints, using an LCP formulation to solve the contactimpact. Slavic and Boltezar (2006) also used the woodpecker toy to study and analyze nonlinearity and non-smoothness in multibody dynamics, including an extra degree of freedom in the system. Studer et al. (2008) used the woodpecker toy system as an application example to present and discuss a step size adjustment and extrapolation methods to improve the Moreau time-stepping scheme for the numerical integration of non-smooth dynamics. Y

g

lS

rO 1

hS

rM

ϕM

hM M mM, JM

S

cϕ

3

mS, JS

G

2 lM

ϕS

lG

X (a)

(b)

Fig. 8.4 – (a) Photography of the woodpecker toy; (b) Equivalent mechanical system.

The woodpecker toy consists of a pole, a sleeve with a hole that is slightly larger than the diameter of the pole, a spring and the woodpecker, as it is shown in Fig. 8.4. In operation, the woodpecker moves down the pole performing some kind of pitching motion, which is controlled by the sleeve. This mechanism of self-excitation may roughly be explained as follows. Gravitation acts 48


as an energy source. This energy is transmitted to the woodpecker and results in a vertical downward motion of the entire system. The woodpecker itself oscillates up and down. This oscillation interacts via the spring with the sleeve. It gains its energy from the downwards motion by turning the sleeve and switching on and off a frictional contact jamming of the sleeve at the pole. This mechanism ends up in a stable limit cycle with an energetic balance of the kinetic energy, gained per cycle by the falling height, and the dissipated energy due to the frictional contacts. A planar model of the woodpecker toy is shown in Fig. 8.4b, which consists of three rigid bodies, namely the woodpecker (center of mass S, mass mS, moment of inertia JS), the sleeve (center of mass M, mass mM, moment of inertia JM), and the pole that is considered to be the ground. The woodpecker and the sleeve are connected by a revolute joint with angular stiffness cϕ and are both under the influence of gravity g. The woodpecker toy is modeled as a system with 3 DOF, being the generalized coordinates vector defined as the vertical and angular displacements of the sleeve, y and

ϕM, and the angular displacement of the woodpecker, ϕS. The sleeve linear and angular velocities are denoted by v and ωM, while the angular velocity of the woodpecker is represented by ωS. The horizontal motion of the sleeve is neglected in this woodpecker model. Thus, the vectors of generalized coordinates and velocities can be written as ⎛ y ⎜ q = ⎜ ϕM ⎜ϕ ⎝ S ⎛ v ⎜ u = ⎜ ωM ⎜ω ⎝ S G lS

G 1 g1

G rO

M

(8.22)

⎞ ⎟ ⎟ , with q =u a.e. ⎟ ⎠

(8.23)

G rM gT1

G hS

gN1

G y

⎞ ⎟ ⎟ ⎟ ⎠

G lG

G lM

G rO

G hM

M

G y

G g2

G rO

gN2

G y

2 gT2

M

G g3 gN3

G rM

3 gT3

G hM

Y X

(a)

(b)

(c)

Fig. 8.5 – Closed kinematic chains for the three possible contact points.

The woodpecker system has three potential contact points named 1, 2 and 3, as it is shown in Fig. 8.4b. Contact point 1 is between the beak of the woodpecker and the pole. This contact constraint is not necessary for the woodpecker to work, but as beak impacts have been observed in reality, it is, therefore, considered in the present model. The most important contacts are between 8. Results and discussion

49

the sleeve and the pole. The diameter of the hole in the sleeve is slightly larger than the diameter of the pole. Due to the resulting clearance, the lower or upper edge of the sleeve may come into contact with the pole. This is modeled by the unilateral constraints 2 and 3. In particular, the lower sleeve contact 2 is most essential for the jamming mechanism to be switched on and off. In order to obtain the gap function for these contact points, it is first necessary to derive the corresponding closed kinematic chains in a general position. Figure 8.5 presents the closed kinematic loops for contact points 1, 2 and 3, which are not scaled drawn. In the present analysis, two mathematical simplifications are considered, namely for small angles it is valid to write cos ϕ 1

(8.24)

sin ϕ ϕ

(8.25)

From Fig. 8.5 it is possible to write the following three vector equations

l M + l G + h S + g1 + rO + y = 0

(8.26)

rM + h M + g 2 + rO + y = 0

(8.27)

rM + h M + g 3 + rO + y = 0

(8.28)

Projecting Eqs. (8.26)-(8.28) onto the X and Y directions, taking into account the assumptions given by Eqs. (8.24) and (8.25), yields g N 1 = lM + lG − lS − rO − hSϕ S

(8.29)

gT 1 = y + lM ϕ M + lGϕ S + hS − lSϕ S

(8.30)

g N 2 = rM − rO + hM ϕ M

(8.31)

gT 2 = y + rM ϕ M − hM

(8.32)

g N 3 = rM − rO − hM ϕ M

(8.33)

gT 3 = y + rM ϕ M + hM

(8.34)

Thus, using the definition of the w vectors and of the w scalars associated with each contact point, they can be written as

50

⎛ 0 ⎞ ∂g N 1 ⎜ ⎟ =⎜ 0 ⎟ w N1 = ∂q ⎜ ⎟ ⎝ − hS ⎠

(8.35)

⎛ 1 ⎞ ∂gT 1 ⎜ ⎟ = hM ⎟ wT 1 = ∂q ⎜⎜ ⎟ ⎝ lG − lS ⎠

(8.36)


wN2

⎛ 0 ∂g N 2 ⎜ = = ⎜ hM ∂q ⎜ 0 ⎝

⎞ ⎟ ⎟ ⎟ ⎠

(8.37)

wT 2

⎛1 ∂gT 2 ⎜ = = rM ∂q ⎜⎜ ⎝0

⎞ ⎟ ⎟ ⎟ ⎠

(8.38)

wN3

⎛ 0 ∂g N 3 ⎜ = = − hM ∂q ⎜⎜ ⎝ 0

wT 3

⎛1 ∂gT 3 ⎜ = = rM ∂q ⎜⎜ ⎝0

⎞ ⎟ ⎟ ⎟ ⎠

(8.39)

⎞ ⎟ ⎟ ⎟ ⎠

(8.40)

w N 1 = w T 1 = w N 2 = w T 2 = w N 3 = w T 3 = 0

(8.41)

The system mass matrix M as well as the vector h for the dynamics of the woodpecker system can be obtained applying the equations of Lagrange of second type, that is, from the kinetic and potential system’s energy yields, respectively ⎛ mS + mM ⎜ M = ⎜ mS lM ⎜ ml S G ⎝

mS lM JM + m l

2 S M

mS lM lG

⎞ ⎟ mS lM lG ⎟ J S + mS lG2 ⎟⎠ mS lG

⎛ ⎞ − ( mS + mM ) g ⎜ ⎟ h = ⎜ −cϕ (ϕ M − ϕ S ) − mS lM g ⎟ ⎜ − c (ϕ − ϕ ) − m l g ⎟ M S G ⎝ ϕ S ⎠

(8.42)

(8.43)

The geometrical characteristics, the inertial properties, the force elements, the contact parameters and the initial conditions necessary to perform the dynamic analysis of the woodpecker toy system are listed in Tab. 8.3 (Glocker and Studer, 2005). In order to examine the effectiveness of the Moreau time-stepping method with an LCP formulation, a long time simulation of the woodpecker toy system is performed, being the total time simulation equal to 1.0 s. This amount of time corresponds about 7 periods of the system’s response. The time step of 0.0001 s is considered. The behavior of the woodpecker toy model is quantified by the values of the sleeve linear and angular positions and velocities, as well as the woodpecker angular positions and velocities. Furthermore, the phase space portraits for sleeve and woodpecker are also plotted. These global results are shown through Figs. 8.6 to 8.9. Figure 8.10 illustrated an animation sequence of the global motion produced by the woodpecker toy, where the 8. Results and discussion

51

performance of the system can easily be understood, namely in what concerns with the motion events sequence such as the sliding and locking phases. In general, the obtained results are corroborated with those published in the literature (Leine et al., 2003; Glocker and Studer, 2005). The main conclusion that can be drawn is that the woodpecker toy motion is stable with a periodic solution with a period of 0.146 s. Geometrical characteristics

rO = 0.0025 m rM = 0.0031 m hM = 0.0058 m lM = 0.0100 m lG = 0.0150 m hS = 0.0200 m lS = 0.0201 m

Inertial properties

mM = 0.0003 kg mS = 0.0045 kg JM = 5.0×10−9 kgm2 JS = 7.0×10−7 kgm2

Force elements

cϕ= 0.0056 Nm/rad g = 9.81 m/s2

Contact parameters

εN1 = 0.5 εN2 = εN3 = 0.0 εT1 = εT2 = εT3 = 0.0 μ1 = μ2 = μ 3 = 0.3

Initial conditions

y0 = 0.0 m

ϕM0 = −0.1036 rad ϕS0 = −0.2788 rad v0 = −0.3411 m/s

ωM0 = 0.0 rad/s ωS0 = −7.4583 rad/s

Tab. 8.3 – Parameters used in the dynamic simulation of the woodpecker toy model.

52


Sleeve linear position [mm]

0

-30

-60

-90

-120

-150 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.6

0.7

0.8

0.9

1.0

Time [s]

(a)

Sleeve linear velocity [mm/s]

0.1

0.0

-0.1

-0.2

-0.3

-0.4 0.0

0.1

0.2

0.3

0.4

0.5

Time [s]

(b)

Fig. 8.6 – Time history evolution (a) Sleeve linear position; (b) Sleeve linear velocity.

Sleeve angular position [rad]

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.6

0.7

0.8

0.9

1.0

Time [s]

(a)

Sleeve angular velocity [rad/s]

40

30

20

10

0

-10

-20 0.0

0.1

0.2

0.3

0.4

0.5

Time [s]

(b)

Fig. 8.7 – Time history evolution (a) Sleeve angular position; (b) Sleeve angular velocity. 8. Results and discussion

53

Woodpecker angular position [rad]

0.15

0.00

-0.15

-0.30

-0.45

-0.60 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.7

0.8

0.9

1.0

Time [s]

(a) Woodpecker angular velocity [rad/s]

20 15 10 5 0 -5 -10 -15 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Time [s]

(b)

Fig. 8.8 – Time history evolution (a) Woodpecker angular position; (b) Woodpecker angular velocity. 20

Woodpecker angular velocity [rad/s]

Sleeve angular velocity [rad/s]

40

30

20

10

0

-10

-20 -0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

15 10 5 0 -5

-10 -15 -0.60

Sleeve angular position [rad]

(a)

-0.45

-0.30

-0.15

0.00

0.15

Woodpecker angular position [rad]

(b)

Fig. 8.9 – (a) Phase space portraits for sleeve; (b) Phase space portraits for woodpecker.

Figure 8.10 presents an animation sequence of the global motion produced by the woodpecker toy model, which can be described by the following steps: (i)

t = 0.000s, initially the point 2 is the only active contact. In this phase the sleeve is sliding down, while the woodpecker is rotating downward;

(ii)

t = 0.018 s, the sleeve sticks at point 2 and the woodpecker continues to rotate downward;

(iii) t = 0.036 s, the sleeve is still sticking, while the woodpecker starts to rotate upwards; 54


(iv) t = 0.071 s, the contact point 2 becomes inactive, the sleeve is rotating upward and the woodpecker is also rotating upward; (v)

t = 0.078 s, the sleeve contact point 3 becomes active and the woodpecker continues to rotate upward;

(vi) t = 0.081 s, the contact 3 is still active, the contact point 1 becomes active, i.e., the beak impacts with the pole and the woodpecker starts to rotate downward; (vii) t = 0.085 s, the contact 3 is still active, the contact point 1 becomes inactive, the sleeve is sliding downward and the woodpecker starts to rotate downward; (viii) t = 0.090 s, the contact 3 becomes inactive, the sleeve is rotating downward while the woodpecker continues to rotate downward; (ix) t = 0.120 s, the contact 2 becomes active, the sleeve is sliding downward and the woodpecker continues to rotate downward; (x)

t = 0.146 s, the contact 2 becomes active, the sleeve sticks and the woodpecker continues to rotate downward. At this instant of time, the woodpecker toy completes a cycle and it is equivalent to instant t = 0.000 s.

Fig. 8.10 – Animation sequence of the global motion produced by the woodpecker toy model. 8. Results and discussion

55

8.3. Slider-crank mechanism with a translational clearance joint This section deals with the dynamic modeling and analysis of a planar slider-crank mechanism with a translation clearance joint. This multibody mechanical system consists of four rigid bodies, which represent the ground, the crank, the connecting rod and the slider. The body numbers and their centers of mass are shown in Fig. 8.11. The ground, the crank, the connecting rod and the slider are constrained via ideal revolute joints. The center of mass of each body is considered to be located at the mid distance of the bodies’ total length. The translational clearance joint is composed by a guide and a slider. This joint has a finite clearance, which is constant along the length of the slider. The crank rotates with a constant angular velocity of 150 rad/s. The initial configuration is taken with the crank and the connecting rod collinear, being the initial positions and velocities necessary to start the dynamic analysis obtained from kinematic simulation of the slider-crank mechanism with ideal joints only. The system is under the action of gravity force, which is taken to act in the negative Y direction.

l2 l1

θ2 2

m1,J1 Y

1

m2,J2

3

θ1

m3,J3

X

θ3

0 0

Fig. 8.11 – Slider-crank mechanism with a translational clearance joint.

Figure 8.12 shows a translational clearance joint. The clearance c is defined as the difference between the distance of the guide and the slider surfaces. The geometric characteristics of the translational clearance joint are the slider length 2a, the slider width 2b, and the distance between the guide surfaces d. In an ideal translational joint the two bodies translate with respect to each other parallel to the line of translation, so that, there is neither rotation between the bodies nor a relative translation motion in the direction perpendicular to the axis of the joint. The existence of a clearance in a translational joint introduces two extra degrees of freedom. Hence, the slider can move ‘freely’ inside the guide limits, until it reaches the upper or lower guide surfaces. The modeling of translational clearance joints is a complex task, due to the several possible contact configurations between the slider and guide (Flores et al., 2008b). Figure 8.13 illustrates four different scenarios for the slider configuration relative to guide surface, namely: (i) 56

No contact between the two elements: the slider is in free flight motion inside the guide; Application of the non-smooth dynamics approach to model and analyze contact-impact events in rigid multibody systems

(ii)

One corner of the slider is in contact with the guide surface;

(iii) Two adjacent slider corners are in contact with the guide surface, which corresponds to have a face of slider in contact with the guide surface; (iv) Two opposite slider corners are in contact with the guide surface. c Slider d

2b Guide

2a

Fig. 8.12 – Components of the translational joint with clearance, that is, the slider and guide.

The conditions for switching from one case to another depend on the system’s dynamic response as well as on the material colliding properties. 2 2 1

1 4

3

4 3

(a)

(b)

1

2

3

4

2 4 1 3

(c)

(d)

Fig. 8.13 – Different scenarios of slider and guide interaction: (a) No contact; (b) One corner in contact with the guide; (c) Two adjacent corners in contact with guide; (d) Two opposite corners in contact with guide.

Wilson and Fawcett (1974) derived the equations of motion for the different configurations of the slider motion inside the guide. Yet, in the present work, the interaction between the slider and guide is modeled considering a frictional unilateral constraint located in each slider corner. Thus, the contact kinematics of these four potential or candidate contact points is formulated in terms of gap functions and normal and tangential velocities. The constitutive laws for the normal and tangential contract forces are stated as inclusion, that is, the unilateral constraints are characterized by a set-valued force law of type Signorini’s law, in turn the frictional contacts are described by a set-valued force law of the type Coulomb’s law for dry friction (Glocker, 2001a). 8. Results and discussion

57

In order for the translational clearance joint to be simulated in the multibody system environment, is it first required that the system’s equations of motion be derived. In this work the Lagrange’s equation of second type is used and it can be written as (Greenwood, 1965) d ⎛ ∂L ⎜ dt ⎝ ∂q i

⎞ ∂L = 0, ⎟− ⎠ ∂qi

i=1,…,f

(8.44)

where L is the Lagrangian of the system, that is, the difference between kinetic and potential energies, expressed in terms of the generalized coordinates and their time derivatives. The equations represented by Eq. (8.44) are also called as Euler-Lagrange’s equations of motion, because although Lagrange was the first to formulate them specifically as the equations of motion, they were previously derived by Euler as the conditions under which a point passes from one specific place and time to another in such a way that the integral of a given function L with respect to time is stationary. Since the slider-crank mechanism represented in Fig. 8.11 has three degrees of freedom, three is also the number of generalized coordinates that uniquely represent the system’s configuration. Furthermore, the crank, the connecting rod and the slider have masses mi and moments of inertia with respect to the principal central axes perpendicular to the plane of motion Ji, where i=1, 2 and 3. Thus, the vector of generalized coordinates and velocities are defined as ⎛ θ1 ⎞ ⎜ ⎟ q = ⎜θ2 ⎟ ⎜θ ⎟ ⎝ 3⎠

(8.45)

⎛ ω1 ⎞ ⎜ ⎟ u = ⎜ ω2 ⎟ , with q =u a.e. ⎜ω ⎟ ⎝ 3⎠

(8.46)

The first step to derive the Lagrange’s equation of motion consists of express the bodies’ center of mass position in terms of the generalized coordinates. Thus, based on the information of the slider-crank mechanism presented in Fig. 8.11, the position of the center of mass of the crank, the connecting rod and the slider can be written as

58

x1 =

l1 cos θ1 2

(8.47)

y1 =

l1 sin θ1 2

(8.48)

x2 = l1 cos θ1 +

l2 cos θ 2 2

(8.49)

y2 = l1 sin θ1 +

l2 sin θ 2 2

(8.50)


x3 = l1 cos θ1 + l2 cos θ 2

(8.51)

y3 = l1 sin θ1 + l2 sin θ 2

(8.52)

The velocities of the center of mass of the crank, connecting rod and slider, expressed in terms of the generalized coordinates and their derivatives, can be obtained by simple differentiation of Eqs. (8.47)-(8.52), yielding l x1 = − 1 θ1 sin θ1 2

(8.53)

l y1 = 1 θ1 cos θ1 2

(8.54)

l x2 = −l1θ1 sin θ1 − 2 θ2 sin θ 2 2

(8.55)

l y 2 = l1θ1 cos θ1 + 2 θ2 cos θ 2 2

(8.56)

x3 = −l1θ1 sin θ1 − l2θ2 sin θ 2

(8.57)

y3 = l1θ1 cos θ1 + l2θ2 cos θ 2

(8.58)

The kinetic energy of slider-crank mechanism bodies can be expressed as Ti =

1 2 1 J iθi + mi ( xi2 + yi2 ) , 2 2

i=1, …, 3

(8.59)

Thus, using Eqs. (8.53)-(8.58) and after some mathematical manipulation yields 1 2 1 2 2 J1θ1 + m1l1 θ1 2 8

(8.60)

T2 =

⎤ 1 2 1 ⎡ 2 2 l22 2 J 2θ 2 + m2 ⎢l1 θ1 + θ 2 + l1l2θ1θ2 cos (θ 2 − θ1 ) ⎥ 2 2 ⎣ 4 ⎦

(8.61)

T3 =

1 2 1 J 3θ3 + m3 ⎡⎣l12θ12 + l22θ22 + 2l1l2θ1θ2 cos (θ 2 − θ1 ) ⎤⎦ 2 2

(8.62)

T1 =

In a similar way, the gravitational potential energy associated with each body of the slidercrank mechanism can be evaluated as Vi = mi ghi ,

i=1,…,3

(8.63)

where hi represents the height of bodies’ center of mass. Consequently, using Eqs. (8.48), (8.50) and (8.52) yields V1 =


1 m1 gl1 sin θ1 2

(8.64)

59

l ⎛ ⎞ V2 = m2 g ⎜ l1 sin θ1 + 2 sin θ 2 ⎟ 2 ⎝ ⎠

(8.65)

V3 = m3 g ( l1 sin θ1 + l2 sin θ 2 )

(8.66)

Thus, the Lagrangian L = (T1 + T2 + T )3 − (V1 + V2 + V3 ) can be written as

1 1 1 1 1 1 ⎛1 ⎞ ⎛1 ⎞ L = ⎜ J1 + m1l12 + m2l12 + m3l12 ⎟ θ12 + ⎜ J 2 + m2l22 + m3l22 ⎟ θ22 + J 3θ32 8 2 2 8 2 2 ⎝2 ⎠ ⎝2 ⎠ ⎛1 ⎞ + ⎜ m2 + m3 ⎟ l1l2θ1θ2 cos (θ 2 − θ1 ) ⎝2 ⎠

(8.67)

⎛1 ⎞ ⎛1 ⎞ − ⎜ m1 + m2 + m3 ⎟ gl1 sin θ1 − ⎜ m2 + m3 ⎟ gl2 sin θ 2 ⎝2 ⎠ ⎝2 ⎠ The derivatives necessary to obtain the Lagrangian equations of motion (8.44) can be expressed as follows ∂L ⎛ 1 ⎞ ⎛1 ⎞ = ⎜ m2 + m3 ⎟ l1l2θ1θ2 sin (θ 2 − θ1 ) − ⎜ m1 + m2 + m3 ⎟ gl1 cos θ1 ∂θ1 ⎝ 2 ⎠ ⎝2 ⎠

(8.68)

∂L ⎛1 ⎞ ⎛1 ⎞ = − ⎜ m2 + m3 ⎟ l1l2θ1θ2 sin (θ 2 − θ1 ) − ⎜ m2 + m3 ⎟ gl2 cos θ 2 ∂θ 2 ⎝2 ⎠ ⎝2 ⎠

(8.69)

∂L =0 ∂θ3

(8.70)

∂L ⎛ 1 ⎞ ⎛1 ⎞ = ⎜ J1 + m1l12 + m2l12 + m3l12 ⎟ θ1 + ⎜ m2 + m3 ⎟ l1l2 cos (θ 2 − θ1 ) θ2 ∂θ1 ⎝ 4 ⎠ ⎝2 ⎠

(8.71)

∂L ⎛ 1 ⎞ ⎛1 ⎞ = ⎜ J 2 + m2l22 + m3l22 ⎟ θ2 + ⎜ m2 + m3 ⎟ l1l2 cos (θ 2 − θ1 ) θ1 ∂θ 2 ⎝ 4 ⎠ ⎝2 ⎠

(8.72)

∂L = J 3θ3 ∂θ3

(8.73)

d ⎛ ∂L ⎞ ⎛ 1 2 ⎛1 ⎞ 2 2 ⎞ ⎜ ⎟ = ⎜ J1 + m1l1 + m2l1 + m3l1 ⎟ θ1 + ⎜ m2 + m3 ⎟ l1l2 cos (θ 2 − θ1 ) θ 2 dt ⎝ ∂θ1 ⎠ ⎝ 4 ⎠ ⎝2 ⎠ ⎛1 ⎞ − ⎜ m2 + m3 ⎟ l1l2 sin (θ 2 − θ1 ) θ2 θ2 − θ1 ⎝2 ⎠

(

)

d ⎛ ∂L ⎞ ⎛ 1 ⎛1 ⎞ 2 2 ⎞ ⎜ ⎟ = ⎜ J 2 + m2l2 + m3l2 ⎟ θ 2 + ⎜ m2 + m3 ⎟ l1l2 cos (θ1 − θ 2 ) θ1 dt ⎝ ∂θ 2 ⎠ ⎝ 4 2 ⎠ ⎝ ⎠ ⎛1 ⎞ − ⎜ m2 + m3 ⎟ l1l2 sin (θ 2 − θ1 ) θ1 θ2 − θ1 ⎝2 ⎠

(

d ⎛ ∂L ⎞ ⎜ ⎟ = J 3θ3 dt ⎝ ∂θ3 ⎠ 60

)

(8.74)

(8.75)

(8.76)


Finally, introducing Eqs. (8.68)-(8.76) into Eq. (8.44) and after mathematical treatment, the Lagrangian equations of motion can be written as ⎛ M 11 ⎜ ⎜ M 21 ⎜M ⎝ 31

M 13 ⎞ ⎛ θ1 ⎞ ⎛ h1 ⎞ ⎟⎜ ⎟ ⎜ ⎟ M 23 ⎟ ⎜ θ2 ⎟ = ⎜ h2 ⎟ M 33 ⎟⎠ ⎜⎝ θ3 ⎟⎠ ⎜⎝ h3 ⎟⎠

M 12 M 22 M 32

(8.77)

in which ⎛1 ⎞ M 11 = J1 + ⎜ m1 + m2 + m3 ⎟ l12 ⎝4 ⎠

(8.78)

⎛1 ⎞ M 12 = M 21 = ⎜ m2 + m3 ⎟ l1l2 cos (θ 2 − θ1 ) ⎝2 ⎠

(8.79)

M 13 = M 31 = M 23 = M 32 = 0

(8.80)

⎛1 ⎞ M 22 = J 2 + ⎜ m2 + m3 ⎟ l22 ⎝4 ⎠

(8.81)

M 33 = J 3

(8.82)

⎛1 ⎞ ⎛1 ⎞ h1 = ⎜ m2 + m3 ⎟ l1l2 sin (θ 2 − θ1 ) θ22 − ⎜ m1 + m2 + m3 ⎟ gl1 cos θ1 ⎝2 ⎠ ⎝2 ⎠

(8.83)

⎛1 ⎞ ⎛1 ⎞ h2 = − ⎜ m2 + m3 ⎟ l1l2 sin (θ 2 − θ1 ) θ12 − ⎜ m2 + m3 ⎟ gl2 cos θ 2 ⎝2 ⎠ ⎝2 ⎠

(8.84)

h3 = 0

(8.85)

In order to determine the gap functions let consider Fig. 8.14, where a generic position of the slider inside the guide is illustrated with the purpose to represent the closed kinematic chain of each potential contact point. G g T1 Y

G x3

2 1

G y3 3

X

G g T3

G g N2

G g T2

G g N1

G g N3

2a

2b 4

θ3

d

G g N4

G g T4

Fig. 8.14 – Generic position of the slider inside the guide where the distance between guide upper and lower surfaces is exaggerated for illustration purpose.

From analysis of Fig. 8.14 and considering Eqs. (8.51) and (8.52), the mathematical expressions of the gap functions can be written as 8. Results and discussion

61

gN1 =

d − l1 sin θ1 − l2 sin θ 2 + a sin θ3 − b cos θ3 2

gT 1 = l1 cos θ1 + l2 cos θ 2 − a cos θ 3 − b sin θ3 gN 2 =

d − l1 sin θ1 − l2 sin θ 2 − a sin θ3 − b cos θ3 2

gT 2 = l1 cos θ1 + l2 cos θ 2 + a cos θ3 − b sin θ3 gN 3 =

d + l1 sin θ1 + l2 sin θ 2 − a sin θ3 − b cos θ 3 2

gT 3 = l1 cos θ1 + l2 cos θ 2 − a cos θ 3 + b sin θ3 gN 4 =

d + l1 sin θ1 + l2 sin θ 2 + a sin θ 3 − b cos θ3 2

gT 4 = l1 cos θ1 + l2 cos θ 2 + a cos θ3 + b sin θ 3

(8.86) (8.87) (8.88) (8.89) (8.90) (8.91) (8.92) (8.93)

Then, the w vectors and the w scalars associated with each contact point can be obtained as −l1 cos θ1 ⎛ ⎞ ∂g N 1 ⎜ ⎟ =⎜ −l2 cos θ 2 w N1 = ⎟ ∂q ⎜ ⎟ ⎝ a cos θ3 + b sin θ3 ⎠

(8.94)

−l1 sin θ1 ⎛ ⎞ ∂gT 1 ⎜ ⎟ =⎜ −l2 sin θ 2 wT 1 = ⎟ ∂q ⎜ ⎟ − a sin b cos θ θ 3 3 ⎝ ⎠

(8.95)

wN2

−l1 cos θ1 ⎛ ⎞ ∂g N 2 ⎜ ⎟ = =⎜ −l2 cos θ 2 ⎟ ∂q ⎜ −a cos θ + b sin θ ⎟ 3 3⎠ ⎝

(8.96)

wT 2

−l1 sin θ1 ⎛ ⎞ ∂gT 2 ⎜ ⎟ = =⎜ −l2 sin θ 2 ⎟ ∂q ⎜ ⎟ ⎝ −a sin θ 3 − b cos θ3 ⎠

(8.97)

wN3

l1 cos θ1 ⎛ ⎞ ∂g N 3 ⎜ ⎟ = =⎜ l2 cos θ 2 ⎟ ∂q ⎜ ⎟ − + a cos b sin θ θ 3 3 ⎝ ⎠

(8.98)

−l1 sin θ1 ⎛ ⎞ ∂gT 3 ⎜ ⎟ = =⎜ −l2 sin θ 2 ⎟ ∂q ⎜ ⎟ ⎝ a sin θ 3 + b cos θ3 ⎠

(8.99)

wT 3

62


wN4

l1 cos θ1 ⎛ ⎞ ∂g N 4 ⎜ ⎟ = =⎜ l2 cos θ 2 ⎟ ∂q ⎜ a cos θ + b sin θ ⎟ 3 3⎠ ⎝

(8.100)

wT 4

−l1 sin θ1 ⎛ ⎞ ∂gT 4 ⎜ ⎟ = =⎜ −l2 sin θ 2 ⎟ ∂q ⎜ ⎟ − + a sin b cos θ θ 3 3⎠ ⎝

(8.101)

w N 1 = w T 1 = w N 2 = w T 2 = w N 3 = w T 3 = w N 4 = w T 4 = 0

(8.102)

The geometrical characteristics, the inertial properties, the force elements, the contact parameters and the initial conditions necessary to perform the dynamic analysis of the slider-crank mechanism with a translational clearance joint are listed in Tab. 8.4. Geometrical characteristics

l1 = 0.1530 m l2 = 0.3060 m a = 0.0500 m b = 0.0250 m c = 0.0010 m

Inertial properties

m1 = 0.0380 kg m2 = 0.0380 kg m3 = 0.0760 kg J1 = 7.4×10−5 kgm2 J2 = 5.9×10−4 kgm2 J3 = 2.7×10−6 kgm2

Force elements

g = 9.81 m/s2

Contact parameters

εN1 = εN2 = εN3 = εN4 = 0.4 εT1 = εT2 = εT3 = εT4 = 0.0 μ1 = μ2 = μ3 = μ4 = 0.01

Initial conditions

θ10 = 0.0 rad θ20 = 0.0 rad θ30 = 0.0 rad ω10 = 150.0 rad/s ω20 = −75.0 rad/s ω30 = 0.0 rad/s

Tab. 8.4 – Parameters used in the dynamic simulation of the slider-crank mechanism. 8. Results and discussion

63

Figure 8.15 shows the corners motion in a dimensionless form for two full crank rotations, in which the free slider motion and contact-impact events can be observed. Figure 8.16 illustrates the crank speed, the connecting-rod speed and the portraits relative to connecting-rod and slider for two complete crank rotations. The normalized slider corner motions are evaluated using the following relation yi − c , c

(i=1, 2, 3, 4)

(8.103)

1.5

1.5

1.0

1.0

0.5

0.5

Corner 2

Corner 1

where yi represents the y coordinate of the slider corners and c is the clearance size.

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

0.4

0.6

Crank revolutions

(a)

1.0

1.2

1.4

1.6

1.8

2.0

1.4

1.6

1.8

2.0

(b)

1.5

1.5

1.0

1.0

0.5

0.5

Corner 4

Corner 3

0.8

Crank revolutions

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Crank revolutions

(c)

1.4

1.6

1.8

2.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Crank revolutions

(d)

Fig. 8.15 – Dimensionless motion of the slider corners.

The dimensionless slider trajectories are shown in Fig. 8.15, where the different types of motion between the slider and guide observed are associated with the different guide-slider configurations, i.e., no contact, impact followed by rebound and permanent contact between the joint elements. The effects of impact between the slider and guide surfaces are also quite visible in the plots of Figs. 8.16(b) and 8.16(c), namely, one can observe the discontinuities in the connecting-rod speed. On the other hand, the smooth changes in the speed indicate that the slider and guide surfaces are in permanent contact for long periods, as it is illustrated in the slider portrait of Fig. 8.16(d). It should be highlighted that some numerical difficulties can arise when the clearance size is very small, which will lead to the well known drift problem. In these situations, one possible way to overcome those 64


difficulties consists of a projection technique, in which the excessive penetration between the slider and guide surfaces is eliminated in each time step in order to avoid the further interpretation of the bodies. When this scheme is implemented, special attention should be paid to the conservation of the system energy, since it can lead to overestimated total system energy associated with the contactimpact phenomena (Leine and van de Wouw, 2008a). 100

160

Ideal joint

Connecting-rod speed [rad/s]

Cank speed [rad/s]

Clearance joint 140

120

100

80

Clearance joint

80 60 40 20 0 -20 -40 -60 -80

60

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0

2.0

0.2

0.4

0.6

0.8

(a)

1.2

1.4

1.6

1.8

2.0

(b)

100

1.5

Clearance joint

80

Ideal joint 1.0

Y-Slider position

60 40 20 0 -20 -40

0.5 0.0 -0.5 -1.0

-60 -80 -0.6

1.0

Crank revolutions

Crank revolutions

Connecting-rod speed [rad/s]

Ideal joint

-0.4

-0.2

0.0

0.2

Connecting-rod angle [rad]

(c)

0.4

0.6

-1.5 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

X-Slider position

(d)

Fig. 8.16 – (a) Crank speed; (b) Connecting-rod speed; (c) Connecting-rod portrait; (d) Slider portrait.

Figure 8.17 shows the influence of the value of the restitution coefficient on the dimensionless motion of the corner 1. For this purpose four different restitution coefficient values were considered, namely, 0.1, 0.4, 0.6 and 0.9. From these plots it can be observed that the methodology is valid for different set of material properties, being the system’s response different when restitution coefficient varies. That is, for lower values, the rebounds are fewer and the slider and guide tend to have long periods of permanent or continuous contact, as it is illustrated in Fig. 8.17(a). For higher values of restitution coefficient, the free flight motion of the slider inside the guide is dominant, as Fig. 8.17(d) shows. Consequently, the slider portraits phases are also affected by the value of the coefficient of restitution, as it can be observed in the plots of Fig. 8.18.


65

1.5

1.0

1.0

0.5

0.5

Corner 1

Corner 1

1.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

0.4

0.6

Crank revolutions

0.8

(a)

1.2

1.4

1.6

1.8

2.0

1.4

1.6

1.8

2.0

(b)

1.5

1.5

1.0

1.0

0.5

0.5

Corner 1

Corner 1

1.0

Crank revolutions

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0

0.2

0.4

Crank revolutions

0.6

0.8

1.0

1.2

Crank revolutions

(c)

(d)

1.5

1.5

1.0

1.0

Y-Slider position

Y-Slider position

Fig. 8.17 – Influence of the restitution coefficient on the dimensionless motion of corner 1: (a) ε=0.1; (b) ε=0.4; (c) ε=0.6; (d) ε=0.9.

0.5 0.0 -0.5 -1.0

0.5 0.0 -0.5 -1.0

-1.5 -1.5

-1.0

-0.5

0.0

0.5

1.0

-1.5 -1.5

1.5

-1.0

X-Slider position

-0.5

(a)

0.5

1.0

1.5

1.0

1.5

(b)

1.5

1.5

1.0

1.0

Y-Slider position

Y-Slider position

0.0

X-Slider position

0.5 0.0 -0.5 -1.0

0.5 0.0 -0.5 -1.0

-1.5 -1.5

-1.0

-0.5

0.0

0.5

X-Slider position

(c)

1.0

1.5

-1.5 -1.5

-1.0

-0.5

0.0

0.5

X-Slider position

(d)

Fig. 8.18 – Influence of the restitution coefficient on the slider portraits: (a) ε=0.1; (b) ε=0.4; (c) ε=0.6; (d) ε=0.9. 66


8.4. Reciprocating cam with flat-face follower In this section, a cam follower mechanism used in an industrial machine is considered as an example of application of the non-smooth dynamics approach (Seabra et al., 2002, 2003, 2007). Figure 8.19 shows the overall view and the schematic representation of this machine-tool. The file teeth are produced by impact of the cutting beater (system composed by follower, cylinder and chisel), with a reciprocate movement. To generate this movement, the cutting bench has a wheel with six rebounds (cam) whose rotation forces the pin to move up. This will lift up the cylinder, to which the chisel is attached, which immediately falls down when the up-dead-point is reached. This falling down movement is impelled by the spring and its own weight.

(a)

(b)

Fig. 8.19 – (a) Overall view of the cutting file machine; (b) Schematic representation of the corresponding mechanical system.

The impact energy of the chisel depends on the relationship between the spring force and the maximum distance between the chisel and the file (adjusted by a presser foot). The chisel describes a reciprocating motion that always reaches the same up-dead-point (maximum distance between the chisel and the file), while the pin, rigidly attached to the cylinder that moves the chisel, which always passes by the tops of cam. On the other hand, the down dead point of the chisel varies, and depends on the impact energy absorbed by the file body. The chisel impact energy depends on the relation of the regulation of the spring pre-load and on the maximum distance between the chisel and file. In order to obtain a tooth with the appropriate geometry (depth of the penetration), it is required that the impact energy should be adequately adjusted, but also the maximum distance between the chisel and the file resulting from the regulation of the presser foot should have a value that allows the chisel to pass above the last produced tooth. When the machine operates correctly, the kinetic energy produced during the descending chisel movement is totally absorbed by the base body of the file. For that purpose, the presser foot must be adjusted in order to prevent impacts between cam and pin during the movement descendent of the 8. Results and discussion

67

chisel. It means that the pin should never collide with cam. When this does not happen, as consequence of incorrect positioning of the presser foot, it can be observed that the cutting operation produces a hard and increasing noise. The noise is due to the impact of the pin on the cam, and strongly depends on the spring force. This clash is undesirable for two main reasons; firstly, because it accelerates the cam and follower wear, and secondly, because it decreases the kinetic energy available to the cutting operation, since part of the energy is absorbed by that impact. Consequently, the file quality is significantly penalized. The multibody system of the cutting file machine is made of three rigid bodies (cam – the driver, follower – the driven element, and the ground), one revolute joint, and one translational joint. Figure 8.20a depicts the kinematic configuration of the cam follower mechanism. It is known that for nb rigid body system with nc independent constraint equations, the mobility or degrees of freedom (DOF) is given by (Nikravesh, 1988) DOF = 6 × nb − nc

(8.104)

This mathematical expression, usually called as Grüebler equation, can be used to determine the mobility of multibody system. Thus, from Eq. (8.104), the DOF of the cam-follower mechanism is equal to 1, implying one, and only one, motion generator. Since the follower can not rotate about its own axis, and the follower curvature radius is very large when compared to its own dimensions, the follower can be considered to be of flat-face type. Thus, to keep the analysis simple, the present study is performed for a disk cam flat follower type mechanism. The flat-face follower has the advantage of having a zero pressure angle, which is an important feature, since most of the camfollower mechanisms are designed with pressure angles as small as possible (Chen, 1982). 1.90

A

A

Follower displacement [mm]

Pre-loaded spring

Follower Cam

1.55

1.20 B D 0.85 C 0.50 Fall θ

(a)

Rise Cam angle rotation [º]

θ+60

(b)

Fig. 8.20 – (a) Kinematic configuration of the cam follower mechanism; (b) Follower displacement. 68


Figure 8.20b schematically illustrates the experimental data relative to the follower displacement diagram, corresponding to a sixth part of the cam angle rotation, since the cam has six rebounds and the cam-follower motion repeats itself six times in each complete cam rotation (Seabra et al., 2007). In Fig. 8.20b, point A represents the maximum follower displacement, point B defines the instant of impact between the follower and file body, point C corresponds to the minimum follower displacement, that is, the maximum penetration/deformation of the body file, and, finally, point D represents the re-contact between the cam and follower after the rebound effect. Observing Fig. 8.20b, it is clear that the follower motion can be divided into two main phases, namely, the fall and the rise movements. In turn, these two phases can be analyzed into two different parts. Starting from maximum follower elevation, point A, the follower motion can be described and summarized by the following steps: (i)

Fall #1 – from point A to point B: during the fall phase, the follower motion is influenced by three main factors, namely, the gravity effect, the spring action and friction phenomenon that exists between the follower and guide. At point A, the follower is pushed down by preloaded spring and gravity action. When compared with other two effects, the friction effect between the follower and guide can be neglected;

(ii)

Fall #2 – from point B to point C: this phase corresponds to the cutting file edge process. Point B represents the initial instant of impact between the follower (chisel) and the file body. The maximum penetration depth, which corresponds to the edge height, is represented by the distance between points B and C. Point C corresponds to the end of follower fall motion;

(iii) Rise #1 – from point C to point D: this phase represents the rebound effect caused by the accumulated energy during the contact-impact process (corresponding to the penetration) between the follower and file body. In this process, there is no contact between the follower and cam due to rebound effect and cam speed; (iv) Rise #2 – from point D to point A: during this phase, the follower is in permanent contact with the cam surface, consequently, the follower is rising and the spring is preloading in this process. In order to keep the analysis simple, the follower motion is considered to be of sinusoidal type, being the displacement expression given by ⎛θ 1 2πθ ⎞ s = h⎜ − sin β ⎟⎠ ⎝ β 2π


(8.105)

69

where h is maximum stroke of the follower, θ represents the angle of the cam rotation corresponding to displacement of the follower y, and β is the angle of cam rotation necessary to reach the stroke h. Since the cam follower has one degree of freedom the variables necessary to define the problem are as follows q = ( y)

(8.106)

u = ( y )

(8.107)

M = ( m)

(8.108)

h = (− mg − K s s )

(8.109)

g N = y − Rb − s

(8.110)

WN = (1)

(8.111)

N = (− s) w

(8.112)

The simulation parameters of the cam follower system are listed in Tab. 8.5. The system is considered to be frictionless. Follower mass – m

1.0 kg

Cam speed – Rb

20.94 rad/s

Cam base radius – Rb

0.003 m

Maximum follower stroke – s

0.017 m

Pre-load spring – K

0.0 N

Spring stiffness – Ks

240 N/mm

Initial position – y0

0.017 m

Initial velocity – v0

0.0 m/s

Gravity acceleration – g

9.81 m/s2

Coefficient of restitution – εN

0.4

Integration time step – Δt

0.0001 s

Tab. 8.5 – Parameters used in the dynamic simulation of the cam follower system.

Figure 8.21 shows the behavior of the follower for the data presented above and for a full cam rotation, quantified by the plots of the follower displacement and the follower velocity. From these two plots, the different contact scenarios between the cam and follower are well visible, namely the continuous or permanent contact and the impact followed be rebounds due to the impacts that take place. Figure 8.22 shows an animation sequence of the simulation of the cam follower movement during the first instants after the follower reaches the up-dead-point.

70


Follower displacement [mm]

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.000

0.010

0.020

0.030

0.040

0.050

0.040

0.050

0.060

Time [s]

(a)

Follower velocity [m/s]

2.00 1.50 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 0.000

0.010

0.020

0.030

0.060

Time [s]

(b)

Fig. 8.21 – (a) Follower displacement; (b) Follower velocity.

Fig. 8.22 – Animation sequence of the virtual simulation of the cam follower movement during the first instants after the follower reach the up-dead-point.


71

72


In the end, everything is a gag. Charlie Chaplin

9. Concluding remarks A comprehensive investigation on contact-impact analysis in multibody systems based on the non-smooth dynamics approach was presented in this study. The main conclusions of this research work have been presented throughout this report, being the most important highlighted here. Contact-impact events occur in the collision of two or more bodies that can be unconstrained or may belong to a multibody system. In many cases the function of the mechanical systems is based on the contact-impact behavior. The collision is an outstanding phenomenon in many multibody systems such as mechanisms with intermittent motion, kinematic discontinuities and joints with clearance. As a result of an impact, the values of the system state variables change very fast, eventually looking like discontinuities in the system velocities. The knowledge of the peak forces developed in the impact process is very important for the dynamic analysis of multibody systems and has consequences in the design process. Thus, the selection of the most adequate contact-impact method used to describe the process correctly plays a key role in the accurate design and analysis of the mechanical systems. In a broad sense, there are two methods to solve the contact-impact problem in multibody systems, that is, the continuous contact force model and the methodology based on complementarity approaches. The compliant continuous contact force models have been gaining significant importance in the context of multibody systems with contacts thanks to their computational simplicity and efficiency. One of the main drawbacks associated with this model is the difficulty to choose the contact parameters such as the equivalent stiffness or the degree of nonlinearity of the penetration, especially for complex contact scenarios. The complementary formulations for contact modeling assume that the contacting bodies are truly rigid, as opposed to locally deformable/penetrable as in the penalty approaches. The complementarity formulations resolve the contact dynamics problem by using the unilateral constraints to compute contact impulses or forces to prevent penetration from occurring. In order to carry out the present investigation, the dynamic modeling and analysis of planar multibody systems that experience contact-impact events was presented and discussed. The methodology was based on the non-smooth dynamics approach, in which the interaction of the colliding bodies is modeled with multiple frictional unilateral constraints. The dynamics of rigid multibody systems were stated as an equality of measures, which were formulated at the velocity-

9. Concluding remarks

73

impulse level. The equations of motion were complemented with constitutive laws for the forces and impulses the normal and tangential directions. The formulation of the generalized contact-impact kinematics in the normal and tangential directions was performed by obtaining a geometric relation for the gaps of the potential contact points. The gaps were expressed as functions of the generalized coordinates. The potential contact points were modeled as hard contacts, being the normal and tangential contact laws formulated as set-valued force laws for frictional unilateral constraints. In this study, the unilateral constraints were described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts were characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem was formulated and solved as a linear complementarity problem, which was embedded in the Moreau time-stepping method. Finally, elementary multibody systems were used to discuss the mains assumptions and procedures adopted throughout this work. The main results obtained from this research work showed that the effect of the contact-impact phenomena can have a predictable nonlinear behavior. This nonlinearity aspect is more evident when the system includes friction phenomenon. With the knowledge of nonlinearities in multibody systems, chaotic behavior may be eliminated with suitable design and/or parameter changes of a mechanical system. This feature plays a crucial role in the dynamics, design and control of general multibody systems of common application. Briefly, the design tools presented herein can be employed in an industrial setting, to predict the dynamic responses of general mechanical systems with contact-impact events. Further experimental work is needed, in order to evaluate the predictive capabilities of the methodologies presented in this report, when applied to more demanding systems with multiple contact-impact points and flexible bodies.

74


The next best thing to knowing something is knowing where to find it. Samuel Johnson

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Transactions:

Stewart DE, Trinkle JC, (1996) An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. International Journal Numerical Methods Engineering, 39(15), 2673-2691. Stronge WJ, (2000) Impact Mechanics. Cambridge University Press, Cambridge. Studer C, (2008) Augmented time-stepping integration of non-smooth dynamical systems. Ph.D. Dissertation, ETH Zurich, ETH E-Collection, Zurich, Switzerland. Studer C, Glocker C, (2007) Solving Normal Cone Inclusion Problems in Contact Mechanics by Iterative Methods. Journal of System Design and Dynamics 1(3) 458-467. Studer C, Leine RI, Glocker C, (2008) Step size adjustment and extrapolation for time-stepping schemes in non-smooth dynamics. International Journal for Numerical Methods in Engineering 76(11) 1747-1781. Tasora A, Negrut D, Anitescu A, (2008) Large-scale parallel multi-body dynamics with frictional contact on the graphical processing unit. Proceedings of the Institution of Mechanical Engineers, Part-K Journal of Multi-body Dynamics, 222, 315-326. Trinkle JC, Tzitzouris JA, Pang JS, (2001) Dynamic multi-rigid-body systems with concurrent distributed contacts. Philosophical Transactions: Mathematical, Physical and Engineering Sciences 359(1789), 2575-2593. Wellmann C, Lillie C, Wriggers P, (2008) A contact detection algorithm for superellipsoids based on the common-normal concept. Engineering Computations 25(5), 432-442. Wilson R, Fawcett JN, (1974) Dynamics of the slider-crank mechanism with clearance in the sliding bearing. Mechanism and Machine Theory 9(1), 61-80. Wriggers P, (2006) Computational Contact Mechanics, Second Edition. Springer-Verlag Berlin. Zukas JA, Nicholas T, Greszczuk LB, Curran DR, (1982) Impact Dynamics, John Wiley and Sons, New York.

References

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Once the game is over, the King and the pawn go back in the same box. Italian Proverb

Appendices Appendix I – The linear complementarity problem I.1. Introduction

One of the first published works on the complementarity problems is due to Signorini (1933), who introduced an impenetrability condition in the form of an LCP. Later, Moreau (1979) and Panagiotopoulos (1985) also applied the concept of complementarity to study non-smooth dynamic systems. Pfeiffer and Glocker (1996) extended the developments of Moreau and Panagiotopoulos to multibody dynamics with unilateral contacts, in which the complementarity is considered of paramount importance. Indeed, complementarity problems are a very useful way to formulate problems involving discontinuities (Brogliato, 2003). Prior to present and discuss in detail some of the fundamental aspects of the LCP, a particle falling down on a table is considered as a simple and quite elucidative example of an LCP. Thus, let consider a particle at a height y above a table, which can exert a normal contact force λN. Since the particle can not penetrate the table, the value of y is greater or equal to zero. Moreover, the normal contact force can only prevent penetration and can not act as glue, therefore, its magnitude is also greater or equal to zero. These two variables are related by the condition that if y>0, then the particle does not contact the table and, consequently, λN=0. Conversely, if λN>0, then contact exists and, therefore, y=0. These two scenarios are illustrated in Fig. I.1, and can be summarized as y ≥ 0,

λN ≥ 0 ,

y T λN = 0

(I.1)

λN y

y>0

y=0

λN=0

λN>0 y λN

(a)

(b)

(c)

Fig. I.1 – Particle falling down on a table: (a) Open contact; (b) Closed contact; (c) Set-valued force law.

I.2. Mathematical description

A linear complementarity problem is a set of linear equations that can be written as (Cottle and Dantzig, 1968; Garcia, 1973; Murty, 1988; Cottle et al., 1992) Appendices

83

y = Ax + b

(I.2)

subjected to the complementarity conditions y ≥ 0,

yTx = 0

x≥0,

(I.3)

for which the vectors x and y have to be evaluated for given A and b. In other words, the LCP is the problem of finding solutions x and y of (I.2) and (I.3), where b is an n-dimensional constant column, and A is a given square matrix of dimension n. In general, in contact mechanics, vector y includes contact kinematic variables, vector x contains the contact forces, being the mass matrix, friction coefficient and contact shapes included in A and b. The complementarity conditions expressed by Eq. (I.3) are often written in the form 0≤y⊥x≥0

(I.4)

where y ⊥ x denotes y T x = 0 . An LCP can have a unique solution, multiple solutions or no solution at all (Rohn, 1993; Leine et al., 2002). Let consider a scalar LCP y=Ax+b with complementarity conditions y≥0, x≥0 and yx=0. These complementarity conditions express that if y>0, then it must hold that x=0, and vice versa. The graph of the complementarity conditions forms a corner in the x-y plane, consisting of the positive x-axis, positive y-axis and the origin. A solution of the LCP is an intersection of the straight line y=Ax+b with the corner defined by the complementarity conditions. The scalar LCP has a unique solution for A>0 and b>0, two solutions for A0, no solution for A0

y

A>0, b0

b 0

(I.16)

Equations (I.6)-(I.8) and (I.14)-(I.16) provides 6 equations for the 5 unknowns accelerations x, y, ϕ, gN , gT ) and 2 forces (λN, λT). Thus, one more condition is needed to solve the system. ( Substituting the generalized accelerations ( x, y , ϕ ) from Eqs. (I.6), (I.7) and (I.8) into (I.14) and (I.15), and express with (I.16) all occurring tangential forces λT by their normal forces λN yields gN = m1 ⎡⎣1 + 3cos ϕ ( cos ϕ − μ sin ϕ ) λN + ( sϕ 2 sin ϕ − g ) ⎤⎦

(I.17)

gT = m1 ⎡⎣ − μ + 3sin ϕ ( cos ϕ − μ sin ϕ ) λN − sϕ 2 cos ϕ ⎤⎦

(I.18)

Let now take Eq. (I.17) which is a linear equation with two unknowns, gN and λN. Again, it is assumed that the rod maintains contact, which is only possible if the normal force acts with a compressive magnitude, λN≥0, and the acceleration in the normal direction is equal to zero, gN = 0 . 86


But is only one of two permitted situations. The second one is given by the take-off transition where the rod loses contact. In that case, the contact force must be equal to zero, λN=0 and the separation process can only state with values gN ≥ 0 . Thus, both situations can be stated by two inequalities and one complementarity condition as gN ≥ 0 ,

λN ≥ 0 ,

gN λN = 0

(I.19)

where the term gN λN = 0 switches between the two admissible states by demeaning at least one of both factors to be zero. Thus, Eq. (I.19) is the missing condition for determining the unknowns of Eq. (I.17). In order to simplify the analysis, let rewrite Eqs. (I.17) and (I.19) by using the following abbreviations a = m1 ⎡⎣1 + 3cos ϕ ( cos ϕ − μ sin ϕ ) ⎤⎦

(I.20)

⎛ sϕ 2 ⎞ b = g⎜ sin ϕ − 1⎟ ⎝ g ⎠

(I.21)

and note that both terms may have positive or negative values depending on the parameters ϕ, μ and sϕ 2

g

. Hence, the resulting condition gN = aλN + b ,

gN ≥ 0 ,

λN ≥ 0 ,

gN λN = 0

(I.22)

forms a one dimensional LCP which is here used to discuss the existence and uniqueness of the solutions. Hereby, the a>0 and a0 - Let b>0; in this case the solution is: gN = b > 0, λN = 0 , because the second possibility ( gN = 0, λN = − b a < 0 ) would contradict λN ≥ 0 . Here the rod leaves the surface. - Let b 0 . In short, for a>0 either separation or continual sliding takes place, depending on the sign of b. CASE II: a0; as in the first solution of case I, the solution of the LCP is: gN = b > 0, λN = 0 . But now a is negative and thus a second solution can be found as gN = 0, λN = − b a > 0 . Continual sliding and separation are possible solutions for the rod. So, there is no longer unique solution as in case I. - Let b

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