International Journal of Non-Linear Mechanics 77 (2015) 312–324

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International Journal of Non-Linear Mechanics

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A hybrid scheme for simulation of planar rigid bodies with impactsand friction using impact mappings

Shane J. Burns n, Petri T. PiiroinenSchool of Mathematics, Statistics & Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland

a r t i c l e i n f o

Article history:Received 3 June 2015Received in revised form3 September 2015Accepted 10 September 2015Available online 21 September 2015

Keywords:Rigid body mechanicsHybrid event driven methodCoulomb frictionImpactNon-smoothBifurcation

x.doi.org/10.1016/j.ijnonlinmec.2015.09.01162/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ353 91492332.ail addresses: shane.burns111@gmail.com (S.J.roinen@nuigalway.ie (P.T. Piiroinen).

a b s t r a c t

This paper introduces numerical techniques necessary for the implementation of impact maps derivedfrom an energetic impact law for rigid-body impacts with friction at isolated contact points. In particularthe work focuses on methodologies for long-term simulation with behaviours such as dynamic transi-tions and chatter. The methods are based on hybrid event-driven numerical solvers for ordinary differ-ential equations together with system states to deal with the transitions. A slender rod impacting aperiodically oscillating surface is used as an example to illustrate implementation and methods. Thenumerical scheme for the rod system is used to show how symmetry can play an important role in thepresence of friction for long-term dynamics. This will show that surface oscillations with low frequenciestend to lead to periodic motions of the rod that are independent of friction. For higher frequencieshowever the periodic solutions are not that common and irregular motion ensues.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Collisions or impacts in mechanical systems are very commonand in many mechanical engineering applications they can causeundesired wear and noise and thus be very problematic andexpensive. Two experimental examples of systems with suchissues include an engine cam follower [32] and a magnetic bearingsystem [22].

In many applications energy is dissipated during impactsthrough motion in both the normal and tangential direction(friction) and how this happens has wide reaching effects on boththe short-term and long-term dynamics. The understanding of theforces and impulses that occur at impact, together with an impactlaw, allows for some prediction of post-impact dynamics ofimpacting systems. By an impact law we mean a physical law thatis based on theory and experimentation and used to describe thephysics of a collision between bodies. Many impact law modelshave been developed, some of which include friction [7,8,11,21,28,40,18] and some which do not [9,10]. Impact laws can besplit into two main classes, those suitable for compliant rigidbodies and those suitable for non-compliant rigid bodies. The firstclass of impact laws allows for deformation of the contacting

Burns),

regions of the bodies [5], whereas the second class of impact lawsrequires perfect rigidity together with some rigidity constraint [8].

The impact law used in this paper will be for non-compliantrigid-body impacts with friction. Typically, the main assumptionsfor non-compliant impacts are: (i) there is no deformation of thecontact regions, (ii) an impact occurs at an isolated contact point,(iii) there is no moment impulse during impact, (iv) the contactduration is infinitesimal, (v) there is no change in generalisedcoordinates throughout the impact phase and (vi) the finite activeforces can be neglected during impact [8,19].

The dynamics of rigid-body systems with impacts and frictionis usually found by numerical integration of systems of ordinarydifferential equations (ODEs) corresponding to the mechanicalsystem under question. There are two main schemes for how thisis usually done, namely, time-stepping and event-driven schemes.How to choose one over the other depends on the class or type ofmechanical system that is being analysed, how the impact law isresolved, and the type of numerical analysis one would like toperform. Time-stepping schemes consist of a time discretisation ofthe dynamics in which each time step is advanced by solving anappropriate complementarity problem [1]. In these schemes themoment of each collision or when changes in relative velocitybetween bodies occurs is not exactly located but instead somelevel of penetration can occur. This is the price to pay for usingrigidly formulated time-stepping methods. These schemes arehowever very advantageous for the simulation of systems with alarge number of degrees of freedom with multiple contacts, for

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S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324 313

example flows in a granular material or masonry structures.Event-driven schemes are also basically time-stepping schemesbut the time for which a trajectory reaches a constraint or dis-continuity surface is located as precisely as possible to avoidpenetration. This class of schemes can in turn be divided into twoseparate categories, the complementarity methods and the hybridmethods. As the name suggests, in the complementarity method acomplementarity problem is solved as the event is located,whereafter the standard time-stepping scheme continues as dis-cussed above [1].

In the hybrid methods, which are the focus of this paper, theintegration is terminated when an event is located and a discretemap is applied to describe how the state changes at the event.When the map is applied the time-stepping scheme is restartedwith the post-event states as the initial conditions with a new setof ODEs that reflect the new circumstance. Hybrid methods havesome obvious drawbacks but also some very important advantagesthat we will use in this paper. The main complication of hybridmethods is that, since each event has to be identified and resolvedindividually, the complexity of all different combinations of eventsand ODEs grows very quickly with the number of possible events.Another complication is that for each event a mapping has to befound that reflects, in the case of this paper, what the impact lawdictates. These issues make hybrid schemes only feasible for sys-tems with relatively few different discontinuity surfaces [2,35,36].However, on the positive side it is worth raising at least four dif-ferent points. First, since the events are not included in the timestepping only ODEs with smooth dynamics have to be integratedand it is thus possible to use a suitable high-order integrator withwell-known convergence properties so that trajectories can befound with high accuracy. Second, since no events will be lostduring simulation hybrid methods are useful for the brute-forcebifurcation analysis and in particular when discontinuity-inducedbifurcations (DIBs) are involved. An example of a DIB, and some-thing that will be seen in this paper, is a grazing bifurcation. Agrazing bifurcation occurs when, under parameter variation, atrajectory of a periodic orbit makes tangential contact with thediscontinuity surface, resulting in a change in the system dynamics[30,15]. Third, hybrid methods make the stability analysis of per-iodic orbits relatively straight forward since it is possible to cal-culate saltation matrices that “glue” fundamental solution matricestogether for trajectories passing through regions between differentevents. Fourth, despite the common misconception, there aremethods born out of hybrid schemes for impacting systems thatcan deal with the accumulation of impacts, sometimes referred toas chatter or Zeno behaviour, and also calculate the correspondingsaltation matrices [31]. As mentioned above, event-drivenschemes are particularly useful for systems with relatively fewdegrees of freedom, but with multiple spatially and temporallyseparated contact points. Some examples include turbine bladedampers, friction clutch vibrations, landing gear dynamics [33],passive walkers [2,34] and braille printers [13].

The use of non-smooth system theory to predict and under-stand the kinematics of colliding rigid bodies in the presence ofimpact and friction is a useful commodity in engineering in par-ticular and research of such systems in general [6,16,22,25,33,37].It is well known that non-smooth systems can exhibit complexbehaviour that cannot be found in smooth systems. The class ofsystems with combinations of ODEs and maps, that we use herefor mechanical systems with impact and/or friction, are oftentermed as piecewise-smooth (PWS) systems. In recent years theinterest of DIBs found in PWS systems has increased dramatically,and as mentioned above the main driver of the analysis of DIBs hasbeen the hybrid system approach, where local behaviour cannumerically be pinpointed with high accuracy [3,12,15,30,31]. Inparticular DIBs in impacting systems without friction have been

studied extensively and some classification methodologies havebeen developed in [14,27,24], but also impacting systems withfriction have been studied from a DIB point of view [23,17,28]. Asalready mentioned, a type of behaviour that is very specific toimpacting systems with rigid body impacts is chatter, which is thephenomenon whereby a system goes through an infinite numberof impacts in a finite time period. Previous works on chatter haveconsidered both frictionless systems [9,12,31] and systems withimpacts and friction [23,28,29]. An interesting example of anengineering-based frictionless system is analysed in [26], wherethe problem of gear rattle (chatter) in Roots blower vacuumpumps is considered, and where the rattle is induced from thegear teeth losing and regaining contact. Similarly, in cam-followersystems for certain conditions the follower detaches from the cam,resulting in a series of unwanted impacts or chatter [3,4,32].

With this in mind, the emphasis of this paper is two-fold. First,we will show how the impact mappings for impacts with frictionderived in [28] can be implemented for reliable simulations ofsystems with impacts and friction. Second, we want to exploit thefact that we have reliable simulation routines to analyse the long-term qualitative behaviour of an unconstrained mechanical systemwith impacts and friction. Since previous research has mainlyconsidered long-term dynamics for systems with impacts butwithout friction [3,14,31] the analysis of the unconstrained objectwill show that it is feasible to also consider long-term simulationsfor mechanical system with impact and friction. For this purposewe chose a hybrid event-driven technique as opposed to a time-stepping method in order to resolve DIBs in brute-force bifurcationanalysis as well as deal with accumulation of events (chatter). Wewill use the example of a planar slender rod impacting with anoscillating surface to show how to implement these techniques.This can be seen as a generalisation of a system where a machineelement detaches and is free to vibrate in the presence of frictionor an item that lies on a vibrating conveyor belt. We will show howrattle is affected by the presence of friction.

This paper is organised as follows. The equations of motion fora collision between two rigid bodies with an isolated contact pointare derived in Section 2.2 along with an extension of the energeticimpact law derived in [28] to allow for a two-body collision. Sec-tion 3 summarises the numerical methods necessary for theimplementation of the chosen impact law. In Section 3.1 we willintroduce the notion of system states and how this idea is used inthe simulation of impacting systems. The model example of aslender rod impacting a periodically oscillating surface is intro-duced in Section 4 and the basic setup that is used in thenumerical simulations is presented in Section 4.1. The paperconcludes with a discussion in Section 5 that provides an insightfor engineers and other researchers working with impact andfriction.

2. Planar rigid-body impacts with friction

In this section we will derive a general framework for a planarrigid-body collision between two unconstrained objects. We willpresent an extension of the energetic impact mapping derived in[28], which will be used for the model example in Section 4. Theextension derived here is more general than the mapping pre-sented in [28] in that the mapping in [28] is for the specific case ofa slender rod impacting a stationary non-compliant surface, butwhere we allow both bodies to be unconstrained.

2.1. Equations of motion

Consider two planar rigid bodies H and H0 whose configurationrelative to an inertial reference frame can be described in terms of

Fig. 1. (a) A free-body force diagram for two planar bodies in free flight. (b) Geometry of two planar rigid bodies H and H0 in contact at a point C. (c) A free-body forcediagram corresponding to the figure in (b). (d) A schematic showing the angle of rotation β between the two frames used for modelling and analysis.

S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324314

vectors of generalised coordinates. Further impose that the bodiesat any moment have a finite number of isolated contact points andcontact cannot occur at two separate points simultaneously. Forthis purpose we derive the equations of motion for two separatecases, when the bodies are in free flight, see Fig. 1(a), and whenthey are in contact at an isolated point C, see Fig. 1(b) and (c). Thecorresponding dynamics can be described using a Lagrangianformulation or using a Newtonian formulation. In Sections 2.1.1and 2.1.2 we consider a Newtonian formulation.

2.1.1. Free flightConsider the two bodies H and H0 in free flight as shown in

Fig. 1(a). The position and rotation of the centre of mass G of bodyH can be described in the X�Y plane by the coordinates qX and qYand the angle α, and similarly q0X and q

0Y are the coordinates and α

0

the rotation of the centre of mass G0 of body H0 (see Fig. 1(a)). Nextwe let

r¼ ðqX ; qY ;αÞT ; r0 ¼ ðq0X ; q0Y ;α0ÞT ;_r ¼ ð _qX ; _qY ; _αÞT ; _r 0 ¼ ð _q 0X ; _q 0Y ; _α 0ÞT ;€r ¼ ð €qX ; €qY ; €αÞT and €r 0 ¼ ð €q 0X ; €r 0Y ; €α 0ÞT :

The equations of motion for the two bodies can now be written as

M €r ¼ FT and M0 €r 0 ¼ F 0T ; ð1Þwhere M and M0 are, respectively, the mass matrices for H and H0

given by

M¼m 0 00 m 00 0 I

0B@

1CA and M0 ¼

m0 0 00 m0 00 0 I0

0B@

1CA;

and F and F 0 are, respectively, the external forces and torquesacting on H and H0 given by

F ¼ FX ; FY ;Rð Þ and F 0 ¼ F 0X ; F 0Y ;R0� �

:

Herem andm0 are the masses and I and I0 are the moments of inertiaof H and H0 , respectively. Further FX, F

0X and FY, F

0Y represent the force

components in the X�Y plane and R and R0 are the external torquesacting on H and H0, respectively, as shown in Fig. 1(a).

2.1.2. ContactNext we derive the equations of motion for the system when

the two objects are in contact, as shown in Fig. 1(b) and (c). To dothis we define a new coordinate system n1�n2, rotated by anangle β about the origin relative to the coordinate system X�Y ,where n1 is the tangential vector to the contact plane and n2 is thenormal vector to the contact plane (see Fig. 1(d)). We also define

θ¼ α�β and θ0 ¼ α0 �β and letq¼ ðq1; q2;θÞT ¼ qX cos ðβÞ�qY sin ðβÞ; qX sin ðβÞþqY cos ðβÞ;α�β

� �T;

and

q0 ¼ ðq01; q02;θ0ÞT ¼ q0X cos ðβÞ�q0Y sin ðβÞ; q0X sin ðβÞþq0Y cos ðβÞ;α0 �β� �T

:

Further, defining L as the distance from G to C and L0 as the dis-tance from G0 to C, the positions qC and q0C of the contact point Crelative to both bodies in the n1�n2 frame can be written as

qC≔ q1�L cos ðθÞ; q2�L sin ðθÞ� �T ð2Þ

and

q0C≔ q01�L0 cos ðθ0Þ; q02�L0 sin ðθ0Þ

� �T: ð3Þ

Note that we do not deal with the problem of finding the contactpoints in the general case but assume that there are well-definedcontact points on each object. Let d¼ qC�q0C be the relative dis-tance between the contact points of the two bodies. Then theunilateral constraint between the two bodies is d¼ ð0;0ÞT orequivalently when qC ¼ q0C .

This is a useful framework to work with, particularly whenderiving an impact mapping. It is also straightforward to translateback positions and angles to the original X�Y coordinate system.The derivation of the impact mapping introduced in [28] splitscontact forces into tangential and normal components in relationto the contact plane, i.e. using the n1�n2 coordinate system. Inorder to relate how the contact forces will affect the centre of massin terms of translations and rotations we need to consider

∂qC∂q

¼1 0 L sin ðθÞ0 1 �L cos ðθÞ

!and

∂qC∂q0

¼ 1 0 L0 sin ðθ0Þ

0 1 �L0 cos ðθ0Þ

!:

Now, the equations of motion for H and H0 during contact can,respectively, be written as

M €q ¼ F̂ T þ ∂qC∂q

� �TλT ð4Þ

and

M0 €q 0 ¼ F̂0Tþ ∂qC

∂q0

� �Tλ

0T; ð5Þ

where F̂ and F̂0are, respectively, the external forces and torques

acting on H and H0 given by

F̂ ¼ F1; F2;Rð Þ ¼ FX cos ðβÞ�FY sin ðβÞ; FX sin ðβÞþFY cos ðβÞ;R� �

;

and

F̂0 ¼ F 01; F 02;R0� �¼ F 0X cos ðβÞ�F 0Y sin ðβÞ; F 0X sin ðβÞþF 0Y cos ðβÞ;R0� �;

S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324 315

where the subscripts 1 and 2, respectively, represent the compo-nents of the external forces acting in the tangential and normaldirection, and R and R0 are as above the external torques acting onthe bodies as shown in Fig. 1(c). In a similar way we define λ and λ0

as the forces generated at impact of each body given by

λ¼ λ1; λ2� �

and λ0 ¼ �λ1; �λ2� �

:

This setup is general and does not specify the mechanism thatgenerates the tangential force λ1. For this work however weassume that any tangential force arises due to friction at thecontact point of the colliding bodies and here we use the Amon-tons–Coulomb friction law

λ1 ¼ 7μλ2 ð6Þfor some non-negative constant μ representing a coefficient offriction. The sign assigned to the tangential force λ1 is positive ðþÞwhen the relative tangential contact point velocity between thetwo bodies is negative and it is negative ð�Þ when the relativetangential contact point velocity is positive.

2.2. Energetic impact law

For collisions between rigid bodies, dissipation of energy in thedirection normal to the contact plane is modelled using a coeffi-cient of restitution. A coefficient of restitution gives a relationbetween the normal impulse applied during the restitution phaseto that applied during compression. Typically a Newtonian coef-ficient or a kinematic coefficient is used for this purpose [38],however, for situations where the direction of slip can varythroughout the impact phase both these coefficients will violateenergy conservation [39]. Stronge [38] views the impact phase asbeing composed of a compression phase followed by a restitutionphase. During compression, kinetic energy is stored as internaldeformation energy until the normal relative contact point velo-city is brought to zero. At this point the restitution phase beginsand the stored energy is released.

For this purpose Stronge defines an energetic coefficient ofrestitution en as follows:

Definition 1. The square of the coefficient of restitution is thenegative ratio of the elastic strain energy released during restitu-tion to the internal energy of deformation absorbed during com-pression,

e2n¼ �WðPf Þ�WðPcÞ

WðPcÞ;

where W ¼ R t0 Fv dt0 and for a significantly short contact durationthe force can be related to the differential of impulse, dP ¼ F dt0 sothat

e2n¼ �

R PfPc

vðPÞ dPR Pc0 vðPÞ dP

; ð7Þ

where P is the impulse, v(P) is the relative normal velocity, Pc ¼R tc0 λN dt is the normal impulse for compression, with tc the timetaken for the contact point velocity to reach zero, λN the normalcomponent of the contact force and Pf is the final impulse achievedat separation.

This energetic restitution coefficient allows for the variousstick-slip processes that can occur throughout the impact phaseand thus is a restitution coefficient that will not violate energyconservation. It is notable that the kinematic and Newtoniancoefficients do not allow for situations where the direction of slipcan vary throughout the impact phase, and the consequence ofthis is that the final impulse is not calculated correctly. Further, theenergetic restitution coefficient forms the basis for the impact

mapping derived in [28], a brief description of which will be givenbelow. When we refer to the impact phase we are consideringvelocity changes which occur as a function of normal impulse. Thisimpulse formulation is a natural framework to use given that weassume that the impact is of infinitesimal contact duration.

In order to map pre-impact velocities to post-impact ones it isnecessary to consider the terminal impulse Pf for the given colli-sion. Incorporating Amontons–Coulomb friction law (6) and theenergetic restitution coefficient (7) allows for a variety of stick-slipprocesses, each of which need to be considered and the corre-sponding Pf in each case determined. We will consider the equa-tions of motion for a planar two-body collision. It is necessary toconsider velocity changes as a function of normal impulse Pinstead of the time variable t. Consider (4) translated to the contactpoint qC so that

d _qCdt

¼ ∂qC∂q

€q ¼ ∂qC∂q

M�1∂qC∂q

� �TλT þ∂qC

∂qM�1FT ¼w�1λT þ f ðF1; F2;R; q; _qÞ

ð8Þand

d _qCdt

¼ ∂qC∂q0

€q ¼ ∂qC∂q0

ðM0Þ�1 ∂qC∂q0

� �Tλ

0T þ∂qC∂q0

ðM0Þ�1F 0T

¼ w0ð Þ�1λ0T þ f 0ðF 01; F 02;R0; q0; _q 0Þ; ð9Þwhich is the rate of change of the contact point velocities as afunction of time and where w�1 and w0ð Þ�1 are the symmetricmatrices given by

w�1 ¼ ∂qC∂q

M�1∂qC∂q

� �T¼ A B

B C

� �;

w0ð Þ�1 ¼ ∂qC∂q0

ðM0Þ�1 ∂qC∂q0

� �T¼ A

0 B0

B0 C 0

!;

where

A¼ 1mþL

2 sin 2ðθÞI

; B¼ � sin ðθÞ cos ðθÞI

; C ¼ 1mþL

2 cos 2ðθÞI

A0 ¼ 1m0

þL02 sin 2ðθ0Þ

I0; B0 ¼ � sin ðθ

0Þ cos ðθ0ÞI0

; C0 ¼ 1m0

þL02 cos 2ðθ0Þ

I0:

In this context we do not need to consider terms which do notchange throughout the impact phase and therefore we can neglectthe functions f and f 0. Further, during the impact phase we alsohave that

dPdt

¼ λ2 ð10Þ

for H and by Newton's third law of motion

dð�PÞdt

¼ �λ2 ð11Þ

for H0, since the normal impulse is a uniformly increasing scalarfunction during contact. Now, (8), (10) and (11) allow us to replacethe independent variable t with P in (10) and (11) to give

d _qCdP

¼ 1λ2w�1

λ1λ2

!; ð12Þ

which is the rate of change of contact point velocities with respectto H as a function of normal impulse. In a similar way, using (9),(10) and (11), we also have

d _q 0CdP

¼ 1λ2

w0ð Þ�1�λ1�λ2

!: ð13Þ

Subtracting (13) from (12) yields

d _~qCdP

≔d _qCdP

�d _q0C

dP¼ 1λ2

w�1þ w0ð Þ�1� � λ1

λ2

!; ð14Þ

Fig. 2. A schematic detailing the time history of a contact point p(t) in an impactingsystem.

S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324316

or in the notation of [28]

d _~qCdP

¼ 1λ2

 � B̂� B̂ Ĉ

!λ1λ2

!; ð15Þ

which is the relative change in contact point velocities as a func-tion of normal impulse. Expanding Eq. (15) and writing it in termsof tangential and normal components, respectively, give

d _~q1CdP

¼ Âλ1λ2

� B̂; d_~q2CdP

¼ � B̂λ1λ2

þ Ĉ ; ð16Þ

which will be used to define the rate constants used for the impactmappings presented below.

Stronge [38] describes four possible impact-phase processesand calculates the terminal impulse and the post-impact velocitycomponents for each phase. The four phases are:

Unidirectional slip during contact: In this case slip does not ceasethroughout the impact phase, and the tangential forcingacts in a direction opposite to the motion of the body.

Slip reversal during compression: In this situation initial sliding isbrought to rest and then reverses direction.

Slip-stick transition during compression: The case whereby initialsliding is brought to rest. The contact point sticks if thefriction coefficient μ is sufficiently large or undergoesreverse slip if it is not. It is also required that the initialsliding velocity is sufficiently small, otherwise thismotion cannot occur.

Jam: This is the process whereby there is an increase in nor-mal acceleration at the contact point due to a largerotational acceleration. This motion occurs during aninitial period of sliding.

Nordmark et al. [28] extend this theory by describing 10 dif-ferent impact regions from which an impact law consisting ofthree mappings is derived. The impact mappings in [28] map therelative tangential and normal contact point velocities before theimpact phase _~q �C to the post-impact phase velocities

_~q þC . Using Eq.(16) and following [28] we define the rate constants kþT , k

�T , k

þN ,

k�N , kT0 and kN0. These rate constants describe how stick and

positive and negative slip can occur throughout the compressionand restitution phase. For the various stick-slip processes descri-bed above Nordmark et al. [28] define the rate constants

kþT ¼ � B̂�μÂ; kþN ¼ ĈþμB̂;k�T ¼ � B̂þμÂ; k�N ¼ Ĉ�μB̂;

k0T ¼ 0; k0N ¼ÂĈ� B̂2

Â;

from which

kT ¼kþT in positive slip;k�T in negative slip;

k0T in stick;

8>><>>: ð17Þ

and

kN ¼kþN in positive slip;k�N in negative slip;

k0N in stick;

8>><>>: ð18Þ

can be determined, where μ¼ λ1λ2

. It is worth mentioning that μ is

taken as an absolute here and the rate constants described aboveconsider all cases of positive and negative slip so it is not necessaryto assign a sign to μ. Nordmark et al. [28] also define the constantsk0T and k

0N , which are assigned one of the values of k

þT , k

�T , k

þN , k

�N ,

kT0 and kN0, and determined by the system parameters and

pre-collision conditions. For full details we refer to [28]. From thisand using Eq. (7) the following three maps, for pre-impact to post-impact contact point velocities _~q �C ↦

_~q þC can be derived:

Map I:

_~q þ1C ¼ _~q �1C� 1þenð ÞkTkN

_~q �2C ð19Þ

_~q þ2C ¼ �en _~q �2C ð20ÞMap II:

_~q þ1C ¼k0Tk0N

kNkT

_~q �1C� _~q �2Cþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�k

0N

kN

� ��kNkT

_~q �1C� _~q �2C�2

þe2n

k0NkN

�_~q �2C�2s 1A

0@

ð21Þ

_~q þ2C ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�k

0N

kN

� ��kNkT

_~q �1C� _~q �2C�2

þe2n

k0NkN

�_~q �2C

�2sð22Þ

Map III:

_~q þ1C ¼k0Tk0N

kNkT

_~q �1C� _~q �2Cþen

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1�k

0N

kN

��kNkT

_~q �1C� _~q �2C�2

þe2n

k0NkN

�_~q �2C�2s 1A

0@

ð23Þ

_~q þ2C ¼ en

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1�k

0N

kN

��kNkT

_~q �1C� _~q �2C�2

þk0N

kN

�_~q �2C�2s

ð24Þ

for kNa0 and

_~q þ1C ¼ 0 ð25Þ

_~q þ2C ¼ en

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�_~q �2C�2

þ2k0N _~q �1C_~q �2CkT

sð26Þ

for kN¼0.As mentioned above, the different combinations of segments of

stick and relative slip can be described by ten different regions,each of which corresponds to one of the three maps given above. Itis worth noting that in [40] an equivalent energetic coefficient ofrestitution is used together with the Amontons–Coulomb frictionlaw to describe the contact and impact dynamics for the case of abouncing dimer.

2.3. Dynamics

As the type of mechanical system considered for this work isunconstrained away from the discontinuity surfaces, various pos-sible modes of sustained motion can occur. To highlight this, inFig. 2 we consider a schematic of the time history of a point p(t) on

S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324 317

a rigid body that occasionally acts as a contact point duringimpacts with a non-compliant surface. We only need to definethree modes of motion, namely, unconstrained free flight, chatterand stick. For our purposes, unconstrained free flight coulddescribe, for example, the motion of a projectile that is free torotate about all axes and is entirely unconstrained. Note thatconstrained free flight could describe, for example, the motion of adouble pendulum. The system is constrained to move in a planewith two degrees of freedom due to the upper arm of the pen-dulum being constrained to a fixed point. It is also necessary todistinguish between two types of chatter, namely, complete andincomplete chatter [31]. Complete chatter is where an objectundergoes an infinite number of impacts in finite time and even-tually transitions to stick. This motion can be observed in realitywhere a bouncing a ball, for example, undergoes a large number ofimpacts and eventually comes to a stop and transitions into a stickstate. Incomplete chatter is where the system undergoes a largenumber of impacts in a short time frame but transitions to freeflight due to a change in the relative acceleration.

At an impact, after free flight, a system can evolve in a numberof different ways: it can continue in free flight motion, go into astick regime or go through a chatter sequence. A feature which isalso present in the schematic Fig. 2 is an external impact, whereanother point of the rigid-body system impacts and thus causes achange in the dynamics of the point p(t). For analysis purposes it isimportant to be able to distinguish between the different featuresand subtle changes in order to understand the mechanisms thatcause them. There are typically also a number of system-specificlong-term dynamical behaviours present. In Sections 4.1 and 4.3we discuss such examples for the case of a slender rod impactingan oscillating surface.

3. Numerical methods

In this section we will give a very brief description of the dif-ferent numerical methods employed for simulation of the rigid-body system with impacts and friction described in Section 4. Wemainly follow the methodologies described in [31,36] with someextensions and some simplifications, as we will describe below.The methodology we use here is, as mentioned in Section 1,sometimes referred to as the hybrid or event-driven approach,where continuous dynamics, described by a system of smoothODEs, is combined with discrete events, described by maps. Themaps are used when a solution trajectory reaches an event surfacedefined by the system variables and parameters. In this context weconsider the continuous dynamics as the motion between theimpacts or other transitions and the maps correspond to the actualimpacts or transitions. The transitions are typically changes fromfree-flight dynamics to stick or to chatter.

To solve a system of smooth ODEs in Section 4 we mainly usethe MATLAB's ODE solver vert ode45vertand to locate the eventsurfaces we use MATLAB's built-in event-detection routines. How-ever, the methodology described here can equally well be imple-mented in any environment that has an ODE solver and eventlocation capabilities. The ODE solver requires at least two vectorfields (one for the free-flight phase and one for stick, but seefurther Section 4 for a specific implementation), simulation times,initial conditions, error tolerances and integration step sizes. Inorder to accurately locate the event surfaces, event functions needto be described that are derived from the geometries of theimpacting rigid bodies. Finally, impact maps, like those describedin Section 2.2 have to be defined together with a process thatdetermines what impact map to use for the specific impact.

In general, when dealing with rigid bodies, it is more thanlikely that the overall system can have multiple contact points and

a large number of events can occur (impacts or vector field tran-sitions). These two factors can generally give rise to a number ofcomputational complications, making the analysis of long-termdynamics difficult. One such useful example is to define an addi-tional artificial term on the vector fields that makes the surfacelocally attractive when the system is in stick. This ensures that thecontact point does not drift away from the surface due tonumerical errors. A similar approach was used in [36] for Filippovsystems.

Following the setup of system states presented in [36,31] wecan achieve a robust numerical code capable of simulating thesystem to examine long-term dynamics. By robust we mean, usingthis method, the numerical simulator is capable of handling theevents and transitions which can occur, without breaking down.This is presented in Section 3.1, followed by a discussion in Section3.2 on brute-force bifurcation analysis in general and stabilityanalysis of symmetric period-1 solutions.

3.1. System states

One of the many difficulties associated with using a hybridstrategy for finding the solution to a dynamical system with dis-continuities is the accumulation of events, for instance when anincomplete or a complete chattering sequence is encountered (seeSection 2.3 and [4]). To deal with this in a systematic manner it isadvantageous to introduce the notion of system states in a similarway as was done in [36,31]. For a general system we can define ndiscrete states Sn in which one of the defined vector fields Φn isbeing used. Each vector field Φn either corresponds to free-flightor sticking motion, which has to be defined by the user. A transi-tion diagram can be used to decide if a system should transitionfrom one state to another at an event, and also provides a meansfor numerically dealing with a complete chatter sequence.

The number of free-flight and stick states depends mainly onthe geometry of the impacting rigid bodies. This point will beillustrated further in Section 4.2 for a model example showing aslender rod impacting an oscillating plane.

The mechanism for switching between the states Sn at impactinvolves evaluating relative normal contact point accelerationsand velocities, whereafter a transition diagram together with adecision tree can be used to evaluate what state the system shouldbe in after the impact. The contact point velocities are mainly usedto calculate post-impact velocities. This is in contrast to the con-tact point accelerations that are used to determine when thesystem should release from stick to free flight. The specificimplementations have to be assessed on a case-by-case basis.

3.2. Bifurcation diagrams and stability analysis

A common way in which to make an initial assessment of thelong-term dynamics of a mechanical system is to plot brute-forcebifurcation diagrams. In Section 4.3 we highlight some aspects ofhow the long-term dynamics of a mechanical system with impactsis affected by friction through bifurcation diagrams. The bifurca-tion diagrams shown in Section 4.3 are made up of, on the onehand, period-1 solutions found by a semi-analytic continuationmethod based on a shooting method and, on the other hand, otherrecurrent motions found by brute-force simulation methods. Forthe period-1 solutions the stabilities are found by calculating theeigenvalues of the fundamental solution matrix that is found bysolving the variational equations of the piecewise-smooth system.For this purpose saltation matrices are used to merge togetherfundamental solution matrices for trajectories that switch fromone state to another. These methods have successfully beenimplemented in [2,13,34] and particularly in [31] which reports

S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324318

the stability analysis for impacting systems without friction butwith complete chatter.

4. Dynamics of a slender rod impacting a periodically oscil-lating surface

In this section we will use a basic planar model of a rigidslender rod impacting a periodically oscillating surface to illustratethe techniques discussed in Section 2. The model will also be usedto describe how the numerical methods discussed in Section 3 canbe implemented and what the dynamical features presented inSection 2.3 look like for this specific case. We will also use thissetup to show how friction affects chaotic rattling behaviour ofthe rod.

4.1. The model, system states and vector fields

As discussed above we consider a planar uniform slender rigidrod and let q1, q2 be the tangential and normal position of thecentre of mass, relative to the contact plane, and let θ be the angleof rotation of the rod (see Fig. 3(a)). For this model example thetangential and normal directions correspond to the n1 and n2directions, respectively, as discussed in Section 2 and shown inFig. 1. The rod is subjected to gravity and where either of the twoisolated end points, named P1 and P2, can impact, get stuck to orslide along the periodically oscillating surface. The slender rod canessentially be in four different states: free flight (Fig. 3(a)), one ofthe two end points is stuck to the surface (Fig. 3(b) and (c)), orboth end points are stuck to the surface (Fig. 3(d)). The fourth statehere also allows for the release of the two end points at the sametime, which in effect leads to a lower-dimensional dynamicalsystem that can be treated as a simple impacting particle.

Without loss of generality and following the general setup inSection 2, we let the mass of the rod be m¼1 and the distancefrom the centre of mass to either of the two end points be L¼1.This gives the moment of inertia I¼ 13 and the radius of gyrationk2r ¼ 13. We further assume that the vertically oscillating surface isnot affected by the rod at impact and thus let D(t) represent theoscillating surface with frequency ω, amplitude A and where t istime, so that DðtÞ≔A sin ðωtÞ (see Fig. 3). This means that we onlyneed to consider one of the two impacting bodies introduced inSection 2.1.1 as the mass of the surface can be assumed to be muchgreater than that of the rod and thus only one of the two systemsof differential equations, say Eq. (4), needs to be considered. Forfuture reference we let q4 ¼ _q1; q5 ¼ _q2; q6 ¼ _θ and introduce τ asthe phase of the oscillating floor. We also let d1 be the distancebetween the end point P1 and the surface in the normal directionand let d2 be the distance between the end point P2 and the sur-face in the normal direction. The unilateral constraints for thismodel example are thus d1 ¼ 0 (see Fig. 3(b)), or d2 ¼ 0 (see Fig. 3(c)) or simultaneously d1; d2 ¼ 0 (see Fig. 3(d)). Further, usingEq. (2) we find

d1 ¼ q2� sin ðθÞ�A sin ðωtÞ;d2 ¼ q2þ sin ðθÞ�A sin ðωtÞ; ð27Þ

Fig. 3. The four possible main states of the slender rod. (a) Free-flight motion. (b) En(d) Symmetric free-flight motion (dashed) and symmetric stick motion (solid).

from which we obtain the relative velocity between the end pointP1 and the floor as

_d1 ¼ _q2� cos ðθÞ _θ�Aω cos ðωtÞ; ð28Þand the relative velocity between the end point P2 and the floor as

_d2 ¼ _q2þ cos ðθÞ _θ�Aω cos ðωtÞ;the relative acceleration between the end point P1 and the floor as

€d1 ¼ €q2þ sin ðθÞ _θ2� cos ðθÞ €θþAω2 sin ðωtÞ; ð29Þ

and the relative acceleration between the end point P2 and thefloor as

€d2 ¼ €q2� sin ðθÞ _θ2þ cos ðθÞ €θþAω2 sin ðωtÞ:

Now, we are ready to introduce the five system states, which wewill use for simulating this mathematical model of the planar rod,together with the corresponding vector fields.

State 1 – Free flight: Since we make the assumption that there isno external torque or no horizontally acting forcing present, wehave that R¼0 and F1 ¼ 0 in Eq. (4). The only external force actingin the vertical direction is due to gravity, and so F2 ¼ �g. Fol-lowing this, the equations of motion of the rod in free flight isgiven by

€q1 ¼ 0; ð30Þ

€q2 ¼ �g; ð31Þ

€θ ¼ 0; ð32Þand the corresponding dynamical system is

ð _q1; _q2; _θ ; _q4; _q5; _q6; _τÞT ¼ ðq4; q5; q6;0; �g;0;1ÞT≔Φ1ðtÞthat will be used for the numerical simulation of the free-flightmotion. Notice that we have included the phase τ in the dynamicalsystem in order to have better control of the periodic influenceof time.

State 2 – End point P1 is stuck to the floor: We will derive a newvector field for the system when P1 is stuck to the floor. First, fromEq. (4) we get that the equations of motion for the constrained barare

€q1 ¼ λ1; ð33Þ

€q2 ¼ �gþλ2; ð34Þ

€θ ¼ 3 sin ðθÞλ1�3 cos ðθÞλ2: ð35ÞNext, we need to find the forces λ1 and λ2 needed to constrain P1 tothe oscillating surface. Substituting Eqs. (34) and (35) into Eq. (29)and using the Amontons–Coulomb friction law λ1 ¼ sμλ2 gives€d1 ¼ �gþλ2þ sin ðθÞ _θ

2�3 cosθ sin ðθÞsμλ2þ3 cos 2ðθÞλ2þAω2 sin ðωtÞ; ð36Þ

where s is either þ1 or �1, depending on the relative tangentialvelocity at impact. Further, using the fact that d1 ¼ _d1 ¼ €d1 ¼ 0when the end point P1 is in contact with the surface and solvingfor λ2 gives

d point P1 constrained to the surface. (c) End point P2 constrained to the surface.

Fig. 4. A transitions diagram for the planar rod system showing the five states,S1–S5 and the state transitions I–XVI. See also Table 1 for a list of event types andtransition criteria.

S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324 319

λ2 ¼g� sin ðθÞ _θ2�Aω2 sin ðωtÞ

1þ3 cos 2ðθÞ�3s cos ðθÞ sin ðθÞμ ; ð37Þ

which is the normal forcing required to ensure that the contactpoint will remain constrained to the plane.

Last, we can write the vector field for the rod with the endpoint P1 stuck to the floor as

ð _q1; _q2; _θ ; _q4; _q5; _q6; _τÞT ¼ ðq4; q5; q6;α1;α2;α3;1ÞT≔Φ2ðtÞwhere

α1 ¼sμ g� sin ðθÞ _θ2�Aω2 sin ðωtÞ� �

1þ3 cos 2ðθÞ�3sμ cos ðθÞ sin ðθÞ ;

α2 ¼ �gþg� sin ðθÞ _θ2�Aω2 sin ðωtÞ

1þ3 cos 2ðθÞ�3sμ cos ðθÞ sin ðθÞ ;

α3 ¼ð3sμ sin ðθÞ�3 cos ðθÞÞ g� sin ðθÞ _θ2�Aω2 sinωt

� �1þ3 cos 2ðθÞ�3sμ cos ðθÞ sin ðθÞ :

State 3 – End point P2 is stuck to the floor: Similarly, usingsymmetry, we can write the vector field for the rod with the endpoint P2 stuck to the floor as

ð _q1; _q2; _θ ; _q4; _q5; _q6; _τÞT ¼ ðq4; q5; q6;α4;α5;α6;1ÞT≔Φ3ðtÞwhere

α4 ¼sμ �gþ sin ðθÞ _θ2�Aω2 sin ðωtÞ� �3sμ cosθ sin ðθÞ�1�3 cos 2ðθÞ ;

α5 ¼ �gþ�gþ sin ðθÞ _θ2�Aω2 sin ðωtÞ3sμ cosθ sin ðθÞ�1�3 cos 2ðθÞ ;

α6 ¼ð3sμ sin ðθÞ�3 cos ðθÞÞ �gþ sin ðθÞ _θ2�Aω2 sin ðωtÞ

� �3sμ cosθ sin ðθÞ�1�3 cos 2ðθÞ :

State 4 – Symmetric motion: We define symmetric motion asone where θ mod π ¼ q6 ¼ 0 for all time, which means that thetwo end points will impact the floor at the same time. The dyna-mical system will be the same as in the free-flight case albeit themotion is heavily constrained, and thus the vector field Φ1 canbe used.

State 5 – Both end points P1 and P2 stuck to the floor: If both endpoints are stuck to the floor it means that the centre of mass willoscillate as DðτÞ and thus the vector field in this case is triviallyð _q1; _q2; _θ ; _q4; _q5; _q6; _τÞT ¼ ðq4; q5; q6;0; �ω2A sin ðωτÞ;0;1ÞT≔Φ4:

4.2. State transitions and impact mappings

For the planar rod model described above in Section 4.1 weintroduced five system states, two for free flight and three forstick. Our proposed scheme for dealing with chatter involvesconstraining the respective end point to the impact surface whenthat corresponding end point is going through a chatter sequence.The system then acts as a sliding hinge. For this purpose we definethe critical normal contact point relative velocity as Vtol and usethis, along with the end point accelerations, as criteria for decidingwhen the system is going through a chatter sequence. Vtol ischosen based on the system in question and what makes physicalsense. In this paper we typically let Vtol be r10�6. When animpact occurs the critical normal contact point relative velocities_d1 and _d2 together with the critical normal contact point relativeaccelerations €d1 and €d2 are evaluated. These values are then usedwith a decision tree to decide if the system should transition toanother state. It is important to note that when calculating therelative contact point accelerations when in a stick state, we use

the unconstrained values. This approach ensures that the systemwill naturally release from stick due to the change in relativeacceleration. At impact the impact law needs to be applied todetermine what type of transition should occur. For the modelexample described here the rate constants defined in Section 2.2now take the form

kþT ¼ �3 sin ðθÞ cos ðθÞ�μ 1þ3 sin 2ðθÞ� �

;

kþN ¼ 1þ3 cos 2ðθÞþ3μ sin ðθÞ cos ðθÞfor positive slip,

k�T ¼ �3 sin ðθÞ cos ðθÞþμ 1þ3 sin 2ðθÞ� �

;

k�N ¼ 1þ3 cos 2ðθÞ�3μ sin ðθÞ cos ðθÞfor negative slip, and

kT0 ¼ 0; kN0 ¼4

1þ3 sin 2ðθÞfor the stick regime. These rate constants are used in the mappingsfor the numerical implementation together with a decision tree.The process involves deciding what region the impact correspondsto depending on initial conditions and assigning the correspond-ing rate constants and impact map accordingly, see further [28].However, apart from deciding what will happen at a specificimpact the system can also switch between the different statesdescribed in Section 4.1. The transition diagram describing whattransitions are possible for the rod system is given in Fig. 4, whereS1–S5 are the states introduced in Section 4.1.

A summary of all states and transitions is given in Table 1 andbrief descriptions of the 16 transitions (I–XVI) in Fig. 4 are givenhere:

Transition I: Impact of end point P1 with j _d1 j4Vtol. The systemwill remain in free flight State 1.

Transition II: Impact of end point P2 with j _d2 j4Vtol. The systemwill remain in free flight State 1.

Transition III: If at impact j _d1 joVtol. The system will transition tostick State 2.

Transition IV: If at impact j _d2 joVtol. The system will transition tostick State 3.

Transition V: If at any moment during the constrained motion€d1 40, then the system will transition to free flightState 1.

Transition VI: If at any moment during the constrained motion €d240 then the system will transition to free flight State 1.

Transition VII: In the limiting case where the impact angle θapproaches zero with each impact and eventuallyreaches a predefined threshold in which the angle can beassumed to be zero. The system transitions to the sym-metric State 4.

Table 1Table corresponding to the transition diagram in Fig. 4. The different types of transitions are I – Impact, C – Chatter, R – Release, L – Limit, T – Two-point impact, TR – Two-point impact with Release and TC – Two-point impact with Chatter.

Transition Type Event Transition criteria Contact angle θ State

I I d1 ¼ 0 j _d1 j4Vtol a0 S1-S1II I d2 ¼ 0 j _d2 j4Vtol a0 S1-S1III C d1 ¼ 0 j _d1 joVtol a0 S1-S2IV C d2 ¼ 0 j _d2 joVtol a0 S1-S3V R €d140 a0 S2-S1VI R €d240 a0 S3-S1VII L d1 ¼ 0; d2 ¼ 0 j _d1;2 j4Vtol; θ mod πoθCrit 0 S1-S4VIII TR €d240 0 S2-S3IX TR €d1o0 0 S3-S2X T d2 ¼ 0 j _d2 j4Vtol 0 S2-S4XI T d1 ¼ 0 j _d1 j4Vtol 0 S3-S4XII TC d2 ¼ 0 j _d2 joVtol 0 S2-S5XIII TC d1 ¼ 0 j _d1 joVtol 0 S3-S5XIV R €d1;24g 0 S5-S4XV TC d1 ¼ 0; d2 ¼ 0 j _d1 joVtol; j _d2 joVtol 0 S4-S5XVI T d1 ¼ 0; d2 ¼ 0 j _d1 j4Vtol ; j _d2 j4Vtol 0 S4-S4

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Transition VIII: If at impact €d1 40 and €d2 o0, then the systemwilltransition to stick State 3.

Transition IX: If at impact €d2 40 and €d1 o0, then the system willtransition to stick State 2.

Transition X: If at impact j _d2 j4Vtol, then the system transitionsto the symmetric State 4.

Transition XI: If at impact j _d1 j4Vtol, then the system transitionsto the symmetric State 4.

Transition XII: If at impact j _d2 joVtol, then the system will tran-sition to the symmetric stick State 5.

Transition XIII: If at impact j _d1 joVtol, then the system willtransition to the symmetric stick State 5.

Transition XIV: The system will remain in State 5 unless the fre-quency of oscillation exceeds a critical value ωn given byωn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig

A sin ðωτÞq

. When this value is exceeded the systemwill release from stick and transition to the symmetricState 4. For a given ω value, ωn will vary sinusoidallydepending on the phase of oscillation τ. When the flooracceleration is maximal, ωn will be minimised, and whenthe floor acceleration is minimal, ωn will be maximised.

Transition XV: If €d1;2o 0 or j _d1;2 joVtol, then the system willtransition to the symmetric stick State 5.

Transition XVI: If at impact €d1;24 0, _d14Vtol and _d24Vtol thesystem will remain in State 4.

4.3. Results

Here we will focus on the numerical analysis of some aspects ofthe long-term dynamics of the impacting rod introduced in Sec-tion 4.1. The purpose of this is twofold. First, we want to show therobustness of the numerical techniques presented in Sections 4.1and 4.2, and second, we want to display some of the behaviourthat one can expect from a rattling object where energy that isdissipated both through the impact and friction. In this context wewill focus both on steady-state dynamics and transients.

As discussed in Section 4.1 and shown in Fig. 3 the system canessentially be in five different states between events (impacts ortransitions). On top of this numerical experiments have shownthat the more energy that is removed from the system at impactthrough friction the faster the system tends from asymmetric tosymmetric motion (see Fig. 5). In terms of the model examplepresented in this work, asymmetric chaos refers to chaotic motion

when in State 1, and symmetric chaos refers to chaos when inState 4. This indicates that if the energy that is added into thesystem is acting in the normal direction relative to the impact thenthe friction, which only acts in the tangential direction, will reducethe rotational energy over time and only symmetric motion willremain. To highlight this the example in Fig. 5(a) shows a timehistory of the angle θ when the system goes through a transitionfrom asymmetric to symmetric motion. Recall that θ¼ 0 θ¼ πmeans that the rod is aligned parallel to the surface. In Fig. 5(b) wesee a close up of how the actual transition in this case happensover a very short time interval. In principal it is an accumulation ofalternative impacts between the two ends, i.e. very similar to whatwe see in complete chatter, but here we do not necessarily havestick. In Fig. 5(c) we see a similar example, where the time historyof the position of each of the two end points is shown to highlighthow the transition between transient asymmetric chaos andtransient symmetric chaos can occur. We note that up toapproximately t ¼ 302 there are two separate trajectories, one foreach end point, but suddenly the two trajectories converge and thetwo end points move in synchrony and the rotation of the rodends. The figure also indicates that the system stays chaotic butwhere all the rotational energy has dissipated due to friction. Thiswill not always be the case however, as seen in Fig. 6(a) and (b).After t ¼ 790 the system still remains in asymmetric chaos. Acomparison of Fig. 6 (b) with Fig. 5(c) shows the differencebetween asymmetric and symmetric chaos. In Fig. 6(b) the systemhas still not reached symmetric chaos (transitioned to State 4) andit is possible that it never will. In Fig. 5(c) the system has reachedsymmetric chaos (transitioned to State 4).

In Fig. 7 a schematic transition diagram for transient motionsfound in the rod system is shown. Note that this schematic isbased on observations from numerical experiments of the rodsystem and not on analytically derived conditions. In many cases,for general initial conditions, the rod system undergoes motionakin to asymmetric chaos until the rotational energy has dis-sipated and the symmetric (transient) chaos takes over. Dependingon the frequency of the external forcing the symmetric chaos maypersist or the motion turn periodic. Again, depending on the fre-quency and the value of the restitution coefficient the periodicorbits may have periods of stick. The effect of this is that, at leastfor low frequencies ω, all long term motions that we have comeacross, are symmetric and thus brute-force bifurcation diagramsonly show stable solutions, where the dynamics is symmetric. It is

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worth noting that in the limit, where the impact times betweenthe end points as well as the tangential impact velocities go tozero, Map I (see Section 2.2) is successively applied and once thetransition to symmetric motion has occurred Map I reduces to thestandard Newtonian restitution law, which is in-line with what isdiscussed in [28]. For the rod system in question this means thatthe long term behaviour can simply be approximated by a one-dimensional system of a mass impacting an oscillating surface.While the general one-dimensional system has been analysedbefore, see particularly Holmes [20], we will present some specificresults for the system analysed here in order to give us an idea on

Fig. 6. Time histories showing (a) the rod angle θ mod 2π and (b) the con

Fig. 5. Time histories showing (a) the rod angle θ mod 2π, (b) a close-up of (a) showing tend points P1ðtÞ and P2ðtÞ for the rod before and after the transition from asymmetric t

what we can predict regarding the long-term behaviour for spe-cific parameter values.

To describe how possible transitions between different types oflong-term motion in the symmetric (one-dimensional) rod systemoccur we show in Fig. 8(a) and (b) two brute-force bifurcationdiagrams, ω vs. q2, for two different values of the restitutioncoefficient, en ¼ 0:9 and en ¼ 0:8, respectively. Fig. 8(a) shows threecoexisting period-1 solutions that undergo period-doublingsequences, at three different values of ω, until the branches dis-appear in grazing bifurcations at ω� 3:5. The figures also showthat regions of chaos start at ω� 3:5, with two periodic windows

tact points P1ðtÞ and P2ðtÞ for ω¼ 4:40572; en ¼ 0:9;A¼ 1 and μ¼ 0:5.

he transition from asymmetric to symmetric motion, and (c) the surface SðtÞ and theo symmetric motion occurs. Here ω¼ 4:4050, en ¼ 0:8, A¼ 1 and μ¼ 0:05.

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(a period-1 and a period-3 orbit, see Fig. 9(a)) also existing withinthe chaos. The bifurcation diagram in Fig. 8(b), where the resti-tution coefficient is lower, we see that the onset of chaos occurs atω� 3:8 and thus the periodic orbits are sustained longer whenmore energy is taken out of the systems at impact. To show whatsome of the symmetric period-1 solutions look like in Fig. 9(a) atime series of the end points of the period-1 orbit highlighted as p1in Fig. 8(a)ðp1Þ is shown. In Fig. 9(b) a time series of the end pointsof the period-2 orbit highlighted as p2 in Fig. 8(b) is shown.Similarly in Fig. 9(c) a time series of the end points of the period-3orbit highlighted as p3 in Fig. 8(a) is shown. This shows that freelyrattling objects subject to periodic forcing have co-existing recur-rent motions, periodic and/or chaotic, as has been shown before[20]. The three co-existing period-1 orbits in Fig. 8(a) are reachedfrom different initial conditions, i.e. the rod system is initiallyimpacting the surface at different phases of the surface oscillation.The time history illustrated in Fig. 9(c) corresponds to initialconditions taken from the period-1 orbit in Fig. 8.

A useful technique for understanding the long-term behaviourin this system is to examine how the period-1 solutions behave

Fig. 7. A schematic detailing the possible transitions from transient asymmetricchaos to transient symmetric chaos and periodic orbits that can occur in thesystem.

Fig. 8. Bifurcation diagrams showing how the steady-state of the position of the centre o(b) en ¼ 0:8;A¼ 1 and μ¼ 0:05. The period-1 orbit is labelled in (a) and the period-2 an

Fig. 9. Time histories for q2 showing (a) a period-3 solution (see label p3 in Fig. 8 (a)), (b)in Fig. 8(a)). In (a) and (c) ω¼ 3:0; en ¼ 0:9;A¼ 1 and μ¼ 0:05 and in (b) ω¼ 3:55085; en

under parameter variations. In Fig. 10(a) we plot bifurcation dia-grams using continuation methods for five different en valuesunder ω variation. The figure shows that period-1 orbits are bornat a smaller frequency ω the bigger the en value is. It seemsobvious that if less energy is taken out at impact then less energyis needed from the oscillating surface to sustain a similar periodicorbit. In the inset I we highlight how the branches of periodicorbits retract as en is increased. In Fig. 10(b) we show a magnifi-cation of the region II in Fig. 10(a) to highlight that the stablebranches born at saddle-node bifurcations (SN) undergo period-doubling bifurcations (PD) for increasing ω, which we also see inFig. 8(a). Fig. 10(b) also shows how the unstable periodic orbitsborn at SN bifurcations disappear at grazing bifurcations (G). Asmention above, these results are qualitatively in agreement withwhat is presented in [20]. The conclusion of this is that if the set ofparameters are such that they lie to the left of the correspondingsaddle-node bifurcation SN in Fig. 10(b) the long-term motion isstick and if they lie to the right of the period-doubling bifurcationPD the long-term motion is symmetric period-n ðn41Þ or sym-metric chaos. If the set of parameters are such that the system isin-between SN and PD we can expect symmetric period-1 orbits inthe long term.

5. Discussion

In this paper we introduced a framework for the numericalsimulation of rigid-body systems with impacts and friction, spe-cifically using the impact maps first derived in [28]. We present

f mass q2 varies as the frequency ω is varied. In (a) en ¼ 0:9;A¼ 1 and μ¼ 0:05 and ind period-3 orbits are labelled in (b).

a period-2 solution (see label p2 in Fig. 8(b)) and (c) a period-1 solution (see label p1¼ 0:8;A¼ 1 and μ¼ 0:05.

Fig. 10. Bifurcation diagrams showing how the period-1 solution labelled p1 in Fig. 8(a) varies with ωen ¼ 0:9;0:7;0:5;0:3 and 0:1. Solid line represents the stable limit cyclesand the dashed lines represent unstable limit cycles. In (a) all the branches are shown along with a magnification in panel I to further detail the structure. In (b) amagnification of panel II in (a) is shown. It highlights where the SN, PD and G bifurcations occur.

S.J. Burns, P.T. Piiroinen / International Journal of Non-Linear Mechanics 77 (2015) 312–324 323

the framework for a general one-point rigid-body collision andshow how the impact maps derived in [28] can be extended toallow for an impact between two unconstrained rigid bodies.Further we implemented the impact maps in a hybrid simulationenvironment in MATLAB, allowing for simulation of long-termdynamics of a planar rod where the two end points can impactan oscillating surface. For this purpose we introduced a frameworkthat includes an ODE solver, system states and a transition dia-gram. This allowed us to reliably simulate the system in free flight,through impacts, in sticking and through complete chatter. Forinstance, the approach involved switching vector fields when thesystem is transitioning between stick and free flight motion,where the vector field for stick was found by calculating the nor-mal force required to constrain the contact point to the surface.

In Table 1 together with Fig. 4 we presented a summary of what isneeded to detect an event and what state transition the system willencounter. One of the shortcomings of our approach, and an area offuture research, is that we end a complete chattering sequence whenthe contact point velocity reaches an a priori defined velocitythreshold Vtol, albeit small. This is in contrast to the method that wasderived in [31] for impacts without friction, where at the end of acomplete chatter sequence a jump in space and time is done tominimise local numerical errors. An improvement here would be toextrapolate through chatter sequences, using a similar approach as in[31], until the relative velocity is zero. Another related issue that is notfully resolved in this paper is in the limiting case whereby two impactevents accumulate, and in our case where the rod angle θ approacheszero. In this case we, again, define a tolerance θcrit a priori for whenthe transition from asymmetric to symmetric motion should occur.This was necessary to ensure robustness in the numerical scheme.

As mentioned above, we implemented this framework for themodel example of a slender rod impacting a periodically forcedsurface. This example highlighted some of the phenomena present inimpacting systems with friction, and in particular symmetric systems.The model example can, in some sense, be likened to the problem ofmachine rattle, whereby a machine element becomes detached andis free to undergo unconstrained mechanical vibration. For the rodsystem we examined the dynamics under variation of the frequencyof surface oscillation ω. We showed that for high values of the res-titution coefficient and low frequencies multiple periodic solutionscoexist, however they do not survive an increase in ω, but insteadchaotic regimes take over. Another aspect that was highlighted forlow frequencies ω is how friction removes rotational energy so thatmainly symmetric motions (impacts of both end points of the rod atthe same time) persists. One of the main limitations of the symmetricrod system is that once it is in symmetric motion it cannot get back toasymmetric motion. If an external torque was included or if the bar

was not completely uniform then such a transition could occur andmost likely make the system even more unpredictable. Althoughboth generalisations would be possible to implement, they have notbeen considered for this work since the main goal was to introduce aframework for simulating long-term dynamics for a rigid-body sys-tem with impacts and friction and to highlight some interestingfeatures that can be observed through long-term simulations.

The model example we chose to illustrate our techniques, albeit asimple geometry, was entirely unconstrained. Incorporating a morecomplex geometry, with more contact points, but with one or moreconstraints, would lead to a less complicated transition diagram thanthe one presented in Fig. 4. The advantage of the techniques presentedin this paper is that they can be easily extended to and implementedfor other mechanical systems, with relatively few contact points,where impacts and stick motions can occur. The implementation ofour method would become increasingly difficult for an unconstrainedsystem with more than two contact points, and it then may benecessary to consider a time-stepping scheme. However, it wouldthen not be possible to perform the careful stability analysis techni-ques illustrated in this paper. Further, the numerical scheme does notneed to handle large number of events at one time and it is possible todeal with complete chatter sequences. The algorithm described herecan also be exported directly to methods that locate periodic orbitsand determine their stability using a shooting method, which is theonly useful method to date that can be used for periodic orbits withincomplete or complete chatter.

This paper has opened up new research questions regardingnumerical methods, as mentioned above, but also the exact role frictionhas in dissipating rotational energy in general rattling objects. It may bepossible to find a relationship between the energy removed due tofriction and the energy introduced into the system through the peri-odic forcing. Such analysis may be a useful predictive tool for engineersworking with unconstrained impacting systems with friction.

Acknowledgements

S.B. wishes to acknowledge the economic support from theNational University of Ireland, Galway (NUI Galway) through a PhDscholarship in applied mathematics.

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A hybrid scheme for simulation of planar rigid bodies with impacts and friction using impact mappingsIntroductionPlanar rigid-body impacts with frictionEquations of motionFree flightContact

Energetic impact lawDynamics

Numerical methodsSystem statesBifurcation diagrams and stability analysis

Dynamics of a slender rod impacting a periodically oscillating surfaceThe model, system states and vector fieldsState transitions and impact mappingsResults

DiscussionAcknowledgementsReferences