Grazing bifurcations of a harmonically excited oscillator moving on a time-varying translation belt

Nonlinear Analysis: Real World Applications 9 (2008) 2156–2174www.elsevier.com/locate/na

Grazing bifurcations of a harmonically excited oscillator moving ona time-varying translation belt

Brandon C. Gegga, Albert C.J. Luob,∗, Steve C. Suha

aDepartment of Mechanical Engineering, Texas A&M University, College Station, TX, 77840, USAbDepartment of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805, USA

Received 25 February 2007; accepted 11 July 2007

Abstract

The criterion for grazing motions in a friction-induced oscillator with a time-varying transport belt is obtained. The three mappingsfor such a friction-induced oscillator are introduced for analytical prediction of the grazing motions. The sufficient and necessaryconditions of grazing are expressed from mappings. With system parameter variations, the initial and final switching sets of grazingmapping are illustrated. Numerical illustrations of grazing motions are carried out from analytical predictions. This investigationprovides a technique for how to determine the grazing on the time-varying boundary in discontinuous dynamical systems.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Non-smooth systems; Grazing motion; Dry-friction; Initial and grazing manifolds

1. Introduction

In 2004, Luo and Gegg [25] initially investigated the generalized, linear, mechanical model for the friction-inducedoscillator (see also [26]). The mechanism of stick and non-stick motions for such a generalized model was studiedto obtain the conditions for the onset and vanishing of stick motion in such a non-smooth dynamical system. Thecomprehensive investigation on the dynamics of this problem was given in Luo and Gegg [27]. In addition, the grazingand stick motions for such a friction-induced oscillator were investigated in Luo and Gegg [28,31]. For such a system,the belt translation speed is constant. However, once the belt travels with a time-varying speed, the prediction of thenonlinear dynamics of the friction-induced oscillator becomes more difficult. Therefore, Luo and Gegg [29,30] gavethe comprehensive investigation of periodic motions in such a friction-induced oscillator with a time-varying transportbelt. However, the grazing bifurcation is a key to determine the motions switching in discontinuous dynamical systems.Therefore, in this paper, grazing motions to the time-varying boundary for such a friction-induced oscillator will beinvestigated.

The friction-induced oscillations exist extensively in mechanical systems. This problem has attracted many attentionsin theory and application. However, the system discontinuity caused by friction force direction switching makes thisproblem difficult to solve analytically and numerically. Thus, the friction-induced oscillations have been of greatinterest for a long time. Den Hartog [8] initialized an investigation on the periodic motions of the forced linear

∗ Corresponding author. Tel.: +1 618 650 5389; fax: +1 618 650 2555.E-mail addresses: [email protected] (B.C. Gegg), [email protected] (A.C.J. Luo), [email protected] (S.C. Suh).

1468-1218/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2007.07.004

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B.C. Gegg et al. / Nonlinear Analysis: Real World Applications 9 (2008) 2156–2174 2157

oscillator with Coulomb and viscous damping in 1930. In 1960, Levitan [21] discussed a friction oscillation modelwith the periodically driven base, and the stability of the periodic motion was presented. In 1964, Filippov [12]presented a theory for differential equations with discontinuous right-hand sides, which started from the Coulombfriction oscillator. The concept of differential inclusion was introduced via set-valued analysis, and the existence anduniqueness of the solution for such a discontinuous differential equation were discussed. The comprehensive discussionof such discontinuous differential equations can be referred to Ref. [13]. In 1979, Hundal [16] discussed periodic motionsof the friction oscillator with a driven base. In 1986, Shaw [36] determined the periodic motions and the correspondingstability through the Poincare mapping. In 1994, the experimental and numerical investigations of chaos in a dry-frictionoscillator were completed in Feeny and Moon [9]. In 1997, Hinrichs et al. [14] presented the dynamics modeling ofoscillator with impact and friction (see also [15]), and the stick and non-stick motions and chaos for a nonlinear frictionmodel were observed. In 1998, Natsiavas [32] investigated the stability of piecewise linear oscillators with viscous anddry-friction damping through the perturbation of initial conditions (see also [33]). The limit cycles of the nonlinearfriction oscillator were obtained in Leine et al. [20] by the shooting method. Further, the bifurcation of periodic motionsfor such a friction-induced oscillator was investigated via the Floquet theory in [19]. In 1999, Virgin and Begley [38]investigated the grazing bifurcation and attraction basin of an impact-friction oscillator through the interpolated cellmapping method. In 2001, Ko et al. [18] gave the analytical expression for the friction-induced vibrations with andwithout external excitations. In 2001, Andreaus and Casini [1] investigated the stick–slip transition by the Poincaremapping, and the closed-form solutions of the friction-impact model with Coulomb friction without external excitationwere presented in [2]. The approximate amplitude for a free stick–slip vibration with nonlinear friction model waspresented analytically by Thomsen and Fidlin [37] in 2003. Kim and Perkins [17] used the harmonic balance/Galerkinmethod to investigate a non-smooth stick–slip oscillator. In 2004, Pilipchuk and Tan [35] investigated the friction-induced vibration of a two-degree-of-freedom mass-damper-spring system interacting with a decelerating rigid strip.Li and Feng [22] discussed the attractivity of equilibrium sets of system with dry-friction via the set-valued force laws.Van de Wouw and Leine [39] investigated the bifurcation and chaos existing in friction-induced vibration through anonlinear friction model.

In 1970, Feigin [10] used the Floquet theory of mappings to investigate the C-bifurcation in piecewise-continuoussystems, and the motion complexity was classified by the eigenvalues of mappings, which was also discussed in[11,5]. The C-bifurcation is also termed by the grazing bifurcation by many researchers. In 1991, Nordmark [34]investigated non-periodic motions in discontinuous dynamical systems through a discontinuous mapping, and suchmotions were caused by the grazing bifurcation. In 2000, Dankowicz and Nordmark [7] discussed the mechanism ofthe stick–slip oscillation through the discontinuous map. Further, the normal form mapping for such grazing phenomenawas developed (e.g., [3,4,6]). In 2005, Luo [23] developed a general theory for the local singularity of non-smoothdynamical systems on connectable domains. The necessary and sufficient conditions for grazing bifurcations werepresented. In 2006, Luo and Gegg [28] used the local singularity theory to investigate the grazing bifurcation of thefriction-induced oscillator with a constant belt speed comprehensively. The discontinuous mapping techniques will notbe used herein. However, the recently developed theory [23] will be employed to investigate grazing motions in thisfriction-induced oscillator. The grazing bifurcation to the separation boundary is very crucial to determine the motionswitching in discontinuous dynamical systems. Therefore, the grazing bifurcation of the friction-induced oscillatorwith a time-varying transport belt will be discussed.

In this paper, the necessary and sufficient conditions for grazing motions in a periodically forced, linear oscillator witha time-varying transport belt will be presented. The mappings for passable and sliding motions in such an oscillator willbe introduced. Based on the mappings of two passable motions, the necessary and sufficient conditions of the grazingbifurcation will be developed. From analytical predictions, illustrations of grazing motions will be carried out.

2. Physical model

Consider a periodically forced oscillator attached to a fixed wall, which consists of a mass m, a spring of stiffnessk and a damper of viscous damping coefficient r , as shown in Fig. 1. The coordinate system (x, t) is absolute withdisplacement x and time t . The periodic driving force Q0 cos �t exerts on the mass where Q0 and � are the excitationstrength and frequency, respectively. The oscillator mass contacts the moving belt with friction. So the mass can movealong or rest on the traveling belt. The horizontal belt travels with a time-varying speed V (t):

V (t) = V0 cos(�t + �) + V1, (1)

2158 B.C. Gegg et al. / Nonlinear Analysis: Real World Applications 9 (2008) 2156–2174

V ( t )

r

km

x

Q0cosΩt

−

−

Fig. 1. Schematic mechanical model.

Ff

− �kFN

V ( t )

�k FN

t

V(t)

V1

x

−

−

−

⋅

Fig. 2. (a) Friction forces and (b) the oscillation speed of transport belt.

where � is the oscillation frequency of the traveling belt, and V0 is the oscillation amplitude of the traveling belt, andV1 is constant. � is the phase of the oscillating belt.

Without loss of generality, it is assumed that the maximum static friction force is the same as the kinetic frictionforce at ˙x = V (t), and the kinetic friction force is

Ff(x)

{ = �kFN, ˙x ∈ [V (t), ∞),

∈ [−�kFN, �kFN], ˙x = V (t),

= − �kFN, ˙x ∈ (−∞, V (t)],(2)

where ˙x�dx/dt , �k and FN are a dynamic friction coefficient and a normal force to the contact surface, respectively,as shown in Fig. 2.

The non-friction force acting on the mass in the x-direction is defined as

Fs = Q0 cos �t − rV (t) − kx for ˙x = V (t). (3)

For sliding motions (or stick motion), this non-friction force cannot overcome the friction force, i.e., |Fs|��kFN.Therefore, the mass does not have any relative motion to the time-varying belt. In other words, the relative accelerationshould be zero, i.e.,

¨x = ˙V (t) = −V0� sin(�t + �) for ˙x = V (t). (4)

If |Fs| > �kFN, the non-friction force will overcome the static friction force on the mass, and passable motions (or slipmotions) will appear. For passable motions, the total force acting on the mass is

F = Q0 cos �t − �kFN sgn( ˙x − V ) − r ˙x − kx for ˙x �= V , (5)


sgn(·) is the sign function. Therefore, without the sliding motion, the equation of motion for the friction-inducedoscillator is

¨x + 2d ˙x + cx = A0 cos �t − Ff sgn( ˙x − V ) for ˙x �= V , (6)

where A0 = Q0/m, d = r/2m, c = k/m and Ff = �kFN/m. The non-dimensional parameters are introduced as

� = �/�, t = �t , d = d/�, c = c/�2, A0 = A0/�2,

Ff = Ff/�2, V0 = V0/�, V1 = V1/�, V = V/�, x = x

}, (7)

V (t) = V0 cos(t + �i ) + V1. (8)

The phase constant �i can be used to synchronize the periodic force input and the velocity discontinuity after themodulus of time has been computed. For t = ti , X(ti) = x(ti) ≡ xi , �i = mod(�ti , 2�) − mod(ti + �i−1, 2�), and�0 = �. The integration of Eq. (8) gives

X(t) = V0[sin(t + �i ) − sin(ti + �i )] + V1 × (t − ti ) + xi for t > ti (9)

which is the displacement response of the periodically time-varying, traveling surface.Introduce the relative displacement, velocity, and acceleration as

z = x − X, z = x − V and z = x − V . (10)

The non-friction force in Eq. (3) becomes

Fs = A0 cos �t − 2d[V (t) + z(t)] − c[X(t) + z(t)] − V (t) for z = 0. (11)

Since the mass does not have any relative motion to the vibrating belt, the relative acceleration is zero, i.e.,

x = V = −V0 sin(t + �0) for x = V , (12)

z = 0 for z = 0. (13)

For passable motions, the equation of motion for this oscillator with friction becomes

x + 2dx + cx = A0 cos �t − Ff sgn(x − V ) for x �= V (14)

or

z + 2dz + cz = A0 cos �t − Ff sgn(z) − 2dV − cX − V for z �= 0. (15)

3. Analytical conditions for grazing

As in [23], since the friction force is dependent on the direction of the relative velocity, the phase plane is partitionedinto two domains in which the motion is described through the continuous dynamical systems, as shown in Fig. 3(a) and(b). Because of the discontinuity, the phase plane partition for this oscillator with friction is presented in the absoluteand relative frames. A sketch of grazing motions in domain �� (� = {1, 2}) is illustrated in Fig. 4(a) and (b). For theabsolute frame, the separation boundary is a curve varying with time. However, the discontinuous boundary in therelative frame is constant.

In relative phase plane, the following vectors are introduced:

z�(z, z)T ≡ (z, w)T and F�(z, F )T = (w, F )T. (16)

The corresponding domains and boundary are

�1 = {(z, w

) ∣∣w ∈ (0, ∞)}

,

�2 = {(z, w

) ∣∣w ∈ (−∞, 0)}

,

�� = {(z, w

) ∣∣��(z, w

) = w = 0}

⎫⎪⎬⎪⎭ , (17)


x

x

V (t)

z

z

�21

�12

�21�12

Ω1

∂Ω12

Ω2

Ω1

Ω2

⋅

⋅

Fig. 3. Phase plane partition in (a) absolute and (b) relative phase planes.

or

�1 = {(x, x)|x − V > 0},�2 = {(x, x)|x − V < 0},�� = {(x, x)|��(x, x) = x − V (t) = 0}

}. (18)

The subscript (·)�� defines the boundary from �� to ��. If the absolute frame is used to develop the analytical condition,Eqs. (8) and (9) should be employed to eliminate time t . The equation of the boundary becomes very complicated. Theanalytical condition will be very difficultly derived. However, in relative frame, the boundary is a straight line on thez-axis. The analytical conditions will be relatively simple. Therefore, the equations of motion in Eqs. (13) and (15) canbe described as

z = F()

(z, t

),(, ∈ {

0, 1, 2})

, (19)

where

F(�)� (z, t) =

(w, F� (z, t)

)Tin ��(� ∈ {1, 2}),

F(�)� (z, t) =

(w, F� (z, t)

)Tin ��(� �= � ∈ {1, 2}),

F(0)0 (z, t) = (0, 0)T on �� for sliding motion,

F(0)0 (z, t) ∈

[F(�)

� (z, t) , F(�)

� (z, t)]

on �� for passable motion

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

. (20)

For the subscript and superscript ( and ) with non-zero values, they represent the two adjacent domains �� and ��

for �, � ∈ {1, 2}. F(�)� (z, t) is the true (or real) vector field in the �-domain. F(�)

� (z, t) is the fictitious (or imaginary)vector field in the �-domain, which is determined by the vector field in the �-domain. F(0)

0 (z, t) is the vector field onthe separation boundary, and the discontinuity of the vector field for the entire system is presented through such anexpression. F�(z, t) is the scalar force in the �-domain. For the system in Eq. (15), the forces in the two domains are


x

Ω2

Ω1

V (t)

F1(1) (xm+�, tm+�)

F1(1) (xm+�, tm+�)

(xm, tm)

n∂Ω12

t∂Ω12

x⋅

V (t)

(xm, tm)

F2(2) (xm−�, tm−�)

F2(2) (xm+�, tm+�)

t∂Ω12

n∂Ω12

x

x⋅

Fig. 4. Absolute vector fields of grazing motions in domains (a) �1 and (b) �2.

for � ∈ {1, 2}:F�(z, t) = A0 cos �t − b� − 2d�(V + z) − c�(X + z) − V . (21)

Note that b1 = �g, b2 = −�g, d� = d and c� = c for the model in Fig. 1.Before presenting analytical conditions, the following function is introduced as

G(�,1)(z, tm±) = 2DnT��

· [F(�)� (z, tm±) − F(0)

�� (z, tm±)]+ nT

��· [DF(�)

� (z, tm±) − DF(0)

�� (z, tm±)], (22)

where D = (�/�z)z + (�/�w)w + (�/�t). If the boundary �� is a line independent of time t , DnT��

= 0. Because

of nT��

· F(0)

�� (z, tm±) = 0, DnT��

· F(0)

�� (z, tm±) + nT��

· DF(0)

�� (z, tm±) = 0 can be obtained. Therefore, we have

nT��

· DF(0)

�� (z, tm±) = 0. Eq. (22) is reduced to

G(�,1)(zm, tm±) = nT��

· DF(�)� (zm, tm±). (23)

For a general case, Eq. (22) instead of Eq. (23) will be used. Notice that F(0)

�� (z, t) = (0, 0)T. Because the boundary inthe relative frame is a straight line, from Luo [23,24], the grazing motion is guaranteed by

nT��

· F(�)� (zm, tm±) = 0 for � = 1, 2,

G(1,1)(zm, tm±) = nT��

· DF(1)1 (zm, tm±) > 0 and

G(2,1)(zm, tm±) = nT��

· DF(2)2 (zm, tm±) < 0

⎫⎪⎬⎪⎭ , (24)


where

DF(�)� (z, t) =

(F�(z, t), ∇F�(z, t) · F(�)

� (z, t) + �F�(z, t)�t

)T

. (25)

where ∇ = �/�zi + �/�wj is the Hamilton operator. Notice that tm represents the time for the motion on the velocityboundary and tm±=tm±0 reflects the responses in the domains rather than on the boundary. From Luo [24], the passablemotion to the boundary through the real or imaginary flows is guaranteed by the product of the two correspondingnormal vector fields

L��(tm) = [nT��

· F(�)� (zm, tm−)] × [nT

��· F(�)

� (zm, tm+)] > 0 or

L��(tm) = [nT��

· F(�)� (zm, tm−)] × [nT

��· F(�)

� (zm, tm−)] > 0

}. (26)

Using the third equation of Eq. (17), the normal vector of the boundary ��12 or ��21 is

n��12 = n��21 = (0, 1)T. (27)

Therefore, we have

nT��

· F(�)� (z, t) = F�(z, t), � ∈ {1, 2},

G(�,1)(z, t) = nT��

· DF(�)� (z, t) = ∇F�(z, t) · F(�)

� (z, t) + �F�(z, t)�t

⎫⎬⎭ . (28)

From Eqs. (24) and (25), the analytical conditions for grazing motions are

F�(zm, tm±) = 0, ∇F�(zm, tm±) · F(�)� (zm, tm±) + �F�(zm, tm±)

�t

{> 0 for � = 1,

< 0 for � = 2.(29)

The grazing conditions are presented in Fig. 5(a) and (b), and the vector fields in �1 and �2 are expressed by the dashedand solid arrow-lines, respectively. The condition in Eq. (29) for the grazing motion in �� is presented through the

F1(1) (zm-�,tm-�)

F2(2) (zm+�,tm+�) F2

(2) (zm-�, tm-�)

F1(1) (zm+�, tm+�)

n∂Ω21

n∂Ω21

t∂Ω12

t∂Ω21

(zm, tm)

(zm, tm)

z

z

z

⋅

z⋅

Fig. 5. Relative vector fields of grazing motions in domains (a) �1 and (b) �2.


vector fields of F(�)� (t). In addition to F�(zm, tm±) = 0, the sufficient condition requires F1(z, t) < 0 for t ∈ [tm−�, tm)

and F1(z, t) > 0 for t ∈ (tm, tm+�] in domain �1; and F2(z, t) > 0 for t ∈ [tm−�, tm) and F2(z, t) < 0 for t ∈ (tm, tm+�]in domain �2.

The condition for the passable motion to the boundary becomes

L��(tm) = F�(zm, tm−) × F�(zm, tm+) > 0. (30)

Select Xm = xm and Vm = xm, we have zm = 0, due to zm = 0. Therefore, the corresponding force is

F�(zm, tm±) ≡ F�(xm, Vm, tm±) = A0 cos �tm − b� − 2d�Vm − Vm − c�xm. (31)

4. Generic mappings and grazing conditions

Direct integration of Eq. (8) with initial condition (ti , zi , 0) gives the sliding displacement (i.e., Eq. (9)). For a small�-neighborhood of the sliding motion (� → 0), substitution of Eqs. (8) and (9) into (21) gives the forces in the twodomains �� (� ∈ {1, 2}). For passable motions, select the initial condition on the velocity boundary (i.e., xi = Vi andxi = Xi) in the absolute frame. Based on Eq. (14), the basic solutions of the generalized discontinuous oscillator fora certain domain in Appendix will be used for construction of mappings. In absolute phase plane, the trajectories in�� starting and ending at the velocity discontinuity (i.e., from �� to ��) are sketched in Fig. 6. The starting andending points for mappings P� in �� are (xi, Vi, ti) and (xi+1, Vi+1, ti+1), respectively. The sliding (or stick) mappingis P0.

On the boundary ��, let zi = 0, the switching planes can be defined as

0 = {(xi, �ti )|xi = Vi}, 1 = {(xi, �ti )|xi = V +

i }, 2 = {(xi, �ti )|xi = V −

i }

⎫⎬⎭ , (32)

x

x

z

⋅

z⋅

Ω1

Ω2

(xi, Ωti)

(0, Ωti)

(xi+1, Ωti+1)

(zi+1, Ωti+1)

P1

P2

P0

P2

P1

P0

V (t)

Fig. 6. Basic mappings in (a) absolute and (b) relative frames for the friction-induced oscillator with a time-varying boundary.


where V −i = lim�→0(Vi − �) and V +

i = lim�→0(Vi + �) for arbitrarily small � > 0. Therefore,

P1 : 1 → 1, P2 : 2 → 2, P0 : 0 → 0. (33)

From the foregoing two equations, we have

P0 : (xi, Vi, ti) → (xi+1, Vi+1, ti+1),

P1 : (xi, V+i , ti ) → (xi+1, V

+i+1, ti+1),

P2 : (xi, V−i , ti ) → (xi+1, V

−i+1, ti+1)

⎫⎬⎭ . (34)

From Eq. (9) and F�(zi+1, ti+1)=0, with zi+1 =zi =0 and zi+1 = zi =0, the governing equations for P0 and � ∈ {1, 2}are

xi+1 − V0[sin(ti+1 + �i ) − sin(ti + �i )] − V1 × (ti+1 − ti ) − xi = 0,

A0 cos �ti+1 − b� − 2d�Vi+1 − Vi+1 − c�xi+1 = 0

}. (35)

This paper will not use this mapping, which is presented herein for the completeness of the generic mappings. Forsliding motions, this mapping will be used, and such a discussion is arranged in another paper. For this problem, thetwo domains �� (� ∈ {1 or 2}) are unbounded. From Luo [23], it is required that the flows of the dynamical systems onthe corresponding domains should be bounded. Therefore, for any non-sliding motion, there are three possible stablemotions in the two domains �� (� ∈ {1, 2}), the governing equations of mapping P� (� ∈ {1, 2}) are obtained from thedisplacement and velocity responses for the three cases of motions in Appendix. Therefore, the governing equationsof mapping P� (� ∈ {0, 1, 2}) can be expressed by

f(�)1 (xi, �ti , xi+1, �ti+1) = 0,

f(�)2 (xi, �ti , xi+1, �ti+1) = 0

}. (36)

If the grazing for two mappings of passable motions occurs at the final state (xi+1, Vi+1, ti+1), from Eq. (29), thegrazing conditions based on mappings are obtained. With Eq. (21), the grazing condition in Eq. (29) becomes

A0 cos �ti+1 − b� − 2d�Vi+1 − c�xi+1 − Vi+1 = 0,

−2d�Vi+1 − c�Vi+1 − Vi+1 − A0� sin �ti+1

{> 0 for � = 1,

< 0 for � = 2

}. (37)

The grazing conditions for the two non-sliding mappings can be illustrated. The grazing conditions in Eq. (37) aregiven through the forces. Hence, both the initial and final switching sets of the two non-sliding mappings will vary withsystem parameters. Because the grazing characteristics of the two non-sliding mappings are different, illustrations ofgrazing conditions for the two mappings will be separated.

To ensure the initial switching sets to be passable, from Luo [24], the initial switching sets of mapping P� (� ∈ {1, 2})should satisfy the following condition as in [29,30]:

L12(tm) = F1(zm, tm∓) × F2(zm, tm±) > 0. (38)

The condition in Eq. (38) guarantees the initial switching sets of mapping P� (� ∈ {1, 2}) is passable to the discontinuousboundary (i.e., xi = Vi). The force product for the initial switching sets is also illustrated to ensure the non-slidingmapping exists. The force conditions for the final switching sets of mapping P� (� ∈ {1, 2}) are presented in Eq. (37).

The grazing conditions are computed through Eqs. (36) and (37). Three equations plus an inequality with fourunknowns require one unknown be given. In all illustrations, the initial displacement or phase of mapping P� (� ∈{1, 2}) can be selected to specific values from Eq. (38). Therefore, three equations with three unknowns will givethe grazing conditions. Namely, the initial switching phase, the final switching phase and displacement of mappingP� (� ∈ {1, 2}) will be determined by Eqs. (36) and (37). From the inequality of Eq. (37), the critical value formod(�ti+1, 2�) is introduced through

�cr� = arcsin

(− ��

A0�

), (39)

where �� = c�Vi+1 + 2d�Vi+1 + Vi+1 (� = 1, 2) and the superscript “cr” represents a critical value relative to grazingand � ∈ {1, 2}.


From the second equation of Eq. (37), the final switching phase for mapping P1 has the following six cases:

mod (�ti+1, 2�) ∈ (� + |�cr1 |, 2� − |�cr

1 |) ⊂ (�, 2�) for 0 < �1 < A0�,

mod (�ti+1, 2�) ∈ (� − �cr1 , 2�] ∪ [0, �cr

1 ) for �1 < 0 and A0� > |�1|,mod (�ti+1, 2�) ∈ (�, 2�) for �1 = 0

}, (40)

and

mod(�ti+1, 2�) ∈ {∅} for �1 > 0 and A0��1,

mod(�ti+1, 2�) ∈ [0, 2�] for �1 < 0 and A0� < |�1|,mod(�ti+1, 2�) ∈ [0, 2�]/{�/2} for �1 < 0 and A0� = |�1|

}. (41)

The parameter characteristics of grazing for mapping P2 will be presented as follows. Similarly, from the secondequation of Eq. (37), the six cases of the final switching phase for mapping P2 are

mod(�ti+1, 2�) ∈ [0, � + |�cr2 |) ∪ (2� − |�cr

2 |, 2�] for 0 < �2 < A0�,

mod(�ti+1, 2�) ∈ (�cr2 , � − �cr

2 ) ⊂ (0, �) for �2 < 0 and A0� > |�2|,mod(�ti+1, 2�) ∈ (0, �) for �2 = 0

}, (42)

and

mod(�ti+1, 2�) ∈ [0, 2�] for �2 > 0 and A0� < �2,

mod(�ti+1, 2�) ∈ [0, 2�]/{3�/2} for �2 > 0 and A0� = �2,

mod(�ti+1, 2�) ∈ {∅} for �2 < 0 and A0� < |�2|

}. (43)

5. Illustrations

When the spring and damping coefficients (d1 =1, d2 =0.1, c1 =c2 =30) are fixed, the effects of external excitation,belts and friction to the grazing bifurcation will be discussed herein. The grazing vanishes when the second equationof Eq. (37) is no longer satisfied. Consider the grazing variation with excitation amplitude A0 in domain �1 withparameters (� = 1, V0 = 0.5, V1 = 1, b1 = −b2 = 0.5). Under Eqs. (37) and (38), the initial displacements xi ={−4, −3.5, . . . ,−1.5, −1.2} can be chosen. The initial switching force products and initial switching phase, and thefinal switching phase and displacement versus excitation amplitude A0 are illustrated in Fig. 7(a)–(d). For A0 = 0,only the belt provides a driving force to the oscillator. To ensure the grazing existence, the initial switching forceproduct should also satisfy the condition in Eq. (38), as shown in Fig. 7(a). With increasing excitation amplitudeA0, the grazing to the boundary ��12 will appear. For the afore given parameters, the grazing exists for A0 �36.72.However, illustrations are given only in the range of A0 ∈ [36.72, 100]. The upper bound of the grazing appearanceis determined by the zero initial switching force product. The other boundaries are determined by the condition in Eq.(37). In Fig. 7(b), the initial switching phase versus excitation amplitude is plotted. The corresponding final switchingdisplacement and phase relative to grazing points are illustrated in Fig. 7(c) and (d). Consider the grazing variation withexcitation frequency � in domain �1 with the other parameters(�=1, A0=100, V0=0.5, V1=1, b1=−b2=0.5). UnderEqs. (37) and (38), the initial switching displacements xi = {−8.9, −8, −7, . . . , 0, 0.5} can be used, and the effects ofexcitation frequency on the grazing to the boundary ��12 are presented in Fig. 8. The lower and upper boundaries ofthe initial switching displacement are about xi = −8.9 and 0.5. The initial switching force products versus excitationfrequency � are arranged in Fig. 8(a). To ensure the grazing motion of mapping P1 exists, the initial switching forceproduct should satisfy Eq. (38). The initial switching phase is plotted in Fig. 8(b) with a range of � ∈ [0.415, 3.217],and the final switching displacement and phase are presented in Fig. 8(c) and (d). The upper boundary of grazing inFig. 8(d) is determined by the critical phase in Eq. (39). The first equation of Eq. (37) produces a vanishing conditionof grazing, which gives the other boundaries of the grazing domain in parameter space. With increasing excitationamplitude, it seems the grazing to the boundary ��12 will exist always in domain �1. However, the grazing will existin a certain range of excitation frequency in domain �1.

Since the dynamical characteristics in two domains (�1 and �2) are different, the parameter effects to grazing indomain �2 will also be discussed herein. The effects of excitation forcing will be discussed first. The same dampingand spring coefficients are used. For (�=�= 1, V0 = 0.5, V1 = 1, b1 =−b2 = 10), the initial switching displacementsxi = {−0.9, 0.5, . . . , 4.5} can be used. With varying excitation amplitude, the initial switching force product, initialswitching phase, final switching displacement and phase in domain �2 are presented in Fig. 9(a)–(d), respectively.


Excitation Amplitude, A0

0 25 50 75 100

Initia

l S

witchin

g F

orc

e P

roduct, L

12(ti) (

10

2)

0.0

4.0

8.0

12.0

xi=-1.2

-4

-2

-3

-3.5

-2.5

-1.5


0 25 50 75 100In

itia

l S

witchin

g P

hase,m

od (

Ωt i,

2π)

3.2

3.6

4.0

4.4

4.8

xi=-1.2

-4-2 -3 -3.5-2.5-1.5


0 25 50 75 100

Fin

al S

witchin

g D

ispla

cem

ent, xi+

1

-3.0

-1.5

0.0

1.5

3.0

xi=-1.2

-4

-2

-3-3.5

-2.5

-1.5


0 25 50 75 100

Fin

al S

witchin

g P

hase, m

od (

Ωt i+

1,2

π)

4.0

4.4

4.8

5.2

5.6

xi=-1.2

-4-2 -3 -3.5-2.5-1.5

Fig. 7. Grazing variation with excitation amplitude A0 for mapping P1 in domain �1 with xi = {−4.0, −3.5, . . . , −1.5, −1.2}: (a) initial switchingforce product; (b) initial switching phase; (c) final switching displacement; and (d) final switching phase (� = � = 1, V0 = 0.5, V1 = 1, d1 = 1,d2 = 0.1, b1 = −b2 = 0.5, c1 = c2 = 30).

The lowest bound for the initial switching displacement is determined by the zero initial force product, which isabout xi ≈ 0.9. Once the initial switching displacement becomes large, the grazing range of excitation amplitudebecomes large accordingly. Compared to illustrations in domain �1, the grazing range and characteristics of excitationamplitude are distinguishing. The final switching displacement and phase distribution along excitation amplitude aremore complicated. With varying excitation amplitude, the initial switching phase in domain �2 possesses richer grazingcharacteristics than in domain �1. For excitation frequency �, the grazing variation on the boundary ��12 in domain�2 is illustrated. The initial switching displacement xi = {−0.2, 0.5, . . . , 4.5} is selectable, and the parameters by


Excitation Frequency, Ω0.0 0.9 1.8 2.7 3.6

Initia

l S

witchin

g F

orc

e P

roduct L

12(ti) (

10

3)

-10

0

10

20

30

40

-4

xi=-5

-6

-7

-8

-8.9

0.25 0.50 0.75 1.00

0.0

0.5

1.0

-3

0

-2

-1

-3

0.5


Initia

l S

witchin

g P

hase, m

od (

Ωt i,

2π)

2.4

3.2

4.0

4.8

5.6

xi=-3

-2

-4-5

-6-7

-8

-8.9

-1

0

0.5


Fin

al S

witchin

g D

ispla

cem

ent, xi+

1

-2.0

0.0

2.0

4.0

xi= -3

-2

-4

-5

-6

-7

-8

-8 .9

-1

0

0. 5


Fin

al S

witchin

g P

hase,m

od (

Ωt i+

1,2

π)

4.0

4.5

5.0

5.5

6.0

xi=-3

-2

-4

-5

-6

-7

-8

-8.9

-1

0

0.5

2π

Fig. 8. Grazing variation with excitation frequency � for mapping P1 in domain �1 with xi = {−8.9, −8, . . . , 0, 0.5}: (a) initial switching forceproduct; (b) initial switching phase; (c) final switching displacement; and (d) final switching phase (� = 1, A0 = 100, V0 = 0.5, V1 = 1, d1 = 1,d2 = 0.1, b1 = −b2 = 0.5, c1 = c2 = 30).

(�= 1, A0 = 100, V0 = 0.5, V1 = 1, b1 =−b2 = 10) are used as well. Such a grazing variation is presented through theinitial switching force product, initial switching phase, final switching displacement and phase inFig. 10(a)–(d). Thegrazing range of excitation frequency is about � ∈ [0, 1.4].

Owing to the page limitation, the other parameters effects on the grazing motion for such a discontinuous system willnot be presented herein. However, such parameter effects on the grazing motion can be discussed in a similar fashionfrom the analytical conditions in Eqs. (36)–(38).



0 25 50 75 100

InitalS

witchin

g F

orc

e P

roduct, L

12(ti) (

10

2)

-3

0

3

6

9

12

15

0.5

-0.5

0.0

1.0

xi=-0.9

1.5

2.5

2

3.0

3.5

4.0

4.5

0 25 50 75 100In

italS

witchin

g P

hase, m

od (

Ωt i,

2π)

3.0

3.5

4.0

4.5

5.0

5.5

6.0


0.5

-0.5

0.0

1.0

xi=-0.9

1.52.52

3.0 3.54.0

4.5


0 25 50 75 100

Fin

al S

witchin

g D

ispla

cem

ent, xi+

1

0.0

1.0

2.0

3.0

4.0

0.5

-0.5

0.0

1.0

xi=-0.9

1. 5

2.5

2

3.0

3.5

4.0

4.5


0 25 50 75 100

Fin

al S

witchin

g P

hase, m

od (

Ωt i+

1,2

π)

0.0

1.0

2.0

3.0

4.0

5.0

6.00.5

-0.5

0.0

1.0

xi=-0.9

1.52.52 3.0 3.5 4.0 4.5

Fig. 9. Grazing variation with excitation amplitude A0 for mapping P2 in domain �2with xi = {−0.9, −0.5, . . . , 4.5}: (a) initial switching forceproduct; (b) initial switching phase; (c) final switching displacement; and (d) final switching phase (�=1, �=1, V0 =0.5, V1 =2, d1 =1, d2 =0.1,b1 = −b2 = 10, c1 = c2 = 30).

6. Grazing motions

The parameter characteristics of grazing motion in the frictional-induced oscillator with a periodically oscillatingbelt have been investigated. From the analytical prediction of the grazing motions, illustration of grazing motions ofthe oscillator will be given through velocity responses and phase space. The grazing motions strongly depend on theforce responses in such a discontinuous dynamical system, and the force responses will be given to illustrate the forcecriteria. For illustrations, the starting and grazing points of mapping P� (� ∈ {1, 2}) are represented by the large, hollow


Excitation Frequency, Ω0.35 0.70 1.05 1.40

Initia

l S

witchin

g F

orc

e P

roduct, L

12(ti) (

10

2)

-5

0

5

10

15

xi=4.5

4.4

4.3

4.15

4

3.75

3.5

3

2.5

21.1

1

0.5

0

-0.2


Initia

l S

witchin

g P

hase, m

od(Ωt i,

2π)

0.0

0.5

1.0

1.5

2.0

4.5

xi=4

3.5 3

2.5 2

1.5

1 0.5

0

-0.2

4.3


Fin

al S

witchin

g D

ispla

cem

ent, xi+

1

-3.5

-3.0

-2.5

-2.0

-1.5

xi=4.5

4.454.4

4.3

4.154

3.75

3.5

3

2.5

2

1.5

1.1

1

0.50

-0.2


Fin

al S

witchin

g P

hase, m

od(Ωt i+

1,2

π)

2.0

2.5

3.0

3.5

xi=4.5

4.3

4

3.5

3

2.51.5

1

0.50

-0.2

2

Fig. 10. Grazing variation with excitation frequency � for mapping P2 in domain �2 with xi ={−0.2, 0, . . . , 4.5}: (a) initial switching force product;(b) initial switching phase; (c) final switching displacement; and (d) final switching phase (� = 1, � = 1, V0 = 0.5, V1 = 2, d1 = 1, d2 = 0.1,b1 = −b2 = 10, c1 = c2 = 30).

and dark-solid circular symbols, respectively. The oscillating separation boundaries in the relative and absolute framesare represented by the dashed lines and curves. Grazing motions are presented by thick solid curves.

In Fig. 11, relative and absolute phase trajectories, force distributions along the relative displacement and velocity,the relative and absolute velocity time histories for the grazing motion in domain �1 are illustrated. The parameters(�=1, A0 =100, V1 =1, d1 =1, d2 =0.1, b1 =−b2 =0.5, c1 =c2 =30) plus the initial conditions (xi, yi)=(−4, Vi) and�ti ≈ {3.7297, 3.5288, 3.3155} corresponding to V0 = {0, 2.5, 5} are used. If V0 = 0, this friction-induced oscillatormoves on the constant translation belt, which was discussed in Luo and Gegg [25–27]. In relative phase plane, threetrajectories are bouncing to the discontinuous boundary (z = w = 0), as shown in Fig. 11(a). The grazing points lookslike singular point. However, in Fig. 11(b), the trajectories in the absolute phase plane are tangential to the periodically


Relative Displacement, z

-1.0 0.0 1.0 2.0 3.0 4.0

Rela

tive V

elo

city, w

-2.0

0.0

2.0

4.0

6.0

8.0

0

V0=5

2.5

Displacement, x

-5.0 -4.0 -3.0 -2.0 -1.0 0.0

Velo

city, y

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

V0=0

5

2.5


-1.0 0.0 1.0 2.0 3.0 4.0

Rela

tive F

orc

e, F

1

-20

-10

0

10

20

30

40

V0=0

5

2.5

-2.0 0.0 2.0 4.0 6.0 8.0

Rela

tive F

orc

e, F

1

-20

-10

0

10

20

30

40

Relative Velocity, w

V0=52.5 0

Time, t

3.3 3.6 3.9 4.2 4.5 4.8

Rela

tive V

elo

city, w

-2

0

2

4

6

8

0

V0=5

2.5

Time,t

3.3 3.6 3.9 4.2 4.5 4.8

Velo

city, y

-4

-2

0

2

4

6

8

V0=0

5

2.5

Fig. 11. Grazing motion of mapping P1 in domain �1 for V0 = {0, 2.5, 5}: (a) relative phase trajectory; (b) absolute phase trajectory; (c) relativeforce distribution along relative displacement; (d) relative force distribution on relative velocity; (e) relative velocity time history; and (f) absolutevelocity time history (� = 1, A0 = 100, V1 = 1, d1 = 1, d2 = 0.1, b1 = −b2 = 0.5, c1 = c2 = 30). The initial conditions are (xi , yi ) = (−4, Vi )

and �ti ≈ {3.7297, 3.5288, 3.3155} with V0 = {0, 2.5, 5} accordingly.


oscillating boundary. In Fig. 11(c), the relative forces F1(t) along the relative displacement in domain �1 are presented.At grazing points, the relative forces F1(t) are zero. In vicinity of such grazing points, the relative forces F1(t) have asign change from negative to positive. In Fig. 11(d), the relative force distribution along velocity is presented and thecurves of relative force F1(t) at the grazing points are tangential to the relative velocity boundary w = 0. This indicatesthat conditions in Eq. (29) are satisfied. In Fig. 11(e), the relative velocity responses time histories are plotted, suchresponses are tangential to the boundary w = 0. However, in Fig. 11(f) the absolute velocity time history are tangentialto the discontinuous boundary V (t).

In Fig. 12, relative and absolute phase trajectories, force distributions along the relative displacement and velocity,the relative and absolute velocity time histories for the grazing motion in domain �2 are illustrated. Consider parameters(� = 1, A0 = 100, V1 = 2, d1 = 1, d2 = 0.1, b1 = −b2 = 10, c1 = c2 = 30) plus the initial conditions (xi, yi) = (4, Vi)

and �ti ≈ {0.4484, 0.3603, 0.2241} corresponding to V0 = {0, 0.75, 1.5}. In relative phase plane, the three grazingtrajectories are bouncing to the discontinuous boundary (z = w = 0) in domain �2, as shown in Fig. 12(a). In absolutephase plane, the absolute trajectories are tangential to the velocity boundary in domain �2, which are presented inFig. 12(b). In Fig. 12(c), the relative force distributions F2(t) along the relative displacement are plotted. At grazingpoints, the relative forces are zero. The relative force F2(t) has a sign change from positive to negative. The relativeforce distribution along velocity is given in Fig. 12(d). The relative forces F2(t) are tangential to the velocity boundary(z = w = 0). Again, this indicates that the conditions in Eq. (29) are satisfied. The relative and absolute velocitytime history responses are plotted in Fig. 12(e) and (f). The velocity responses in domain �2 possess the similarcharacteristics as in domain �1.

7. Conclusions

In this paper, the necessary and sufficient conditions for grazing motions in a dry-friction oscillator with a periodicallyoscillating belt are obtained in the relative frame. The three mappings from boundary to boundary in two domains andsliding on the boundary are defined. From the two non-sliding mappings in the two domains, the sufficient and necessaryconditions for the grazing motions are also expressed for grazing at the final point of mappings. To understand grazingmotions, parameter effects on grazing motions in the dry-friction oscillator are systematically investigated. Such aninvestigation provides a technique for how to determine the grazing on the time-varying boundary in discontinuousdynamical systems.

Appendix A

In the absolute frame, equation of motion for discontinuous dynamical system in domain �j (j ∈ {1, 2}) is

x(j) + 2dj x(j) + cj x

(j) = bj + A0 cos �t . (A.1)

Solution to Eq. (A.1) in two domain �j (j ∈ {1, 2}) are for Case I (i.e., d2j > cj ):

x(j)(t) = C(j)

1 (xi, xi , ti )e(j)

1 (t−ti ) + C(j)


2 (t−ti ) + A(j) cos �t + B(j) sin �t + C(j), (A.2)

x(j)(t) = (j)

1 C(j)


1 (t−ti ) + (j)

2 C(j)


2 (t−ti ) − A(j)� sin �t + B(j)� cos �t (A.3)

(j)

1,2 = −dj ±√

d2j − cj , �

(j)d =

√d2j − cj ,

C(j)

1 (xi, xi , ti ) = 1

2�(j)d

{−[B(j)� + (dj + �(j)d )A(j)] cos �ti

+[A(j)� − (dj + �(j)d )B(j)] sin �ti + xi

−(dj + �(j)d )(C(j) − xi)},

C(j)

2 (xi, xi , ti ) = 1

2�(j)d

{[B(j)� − (�(j)d − dj )A

(j)] cos �ti − [(�(j)d − dj )B

(j) + A(j)�] sin �ti

−xi + (�(j)d − dj )(xi − C(j))}.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(A.4)

A(j) = A0(cj − �2)

(cj − �2)2 + (2dj�)2, B(j) = 2dj�A0

(cj − �2)2 + (2dj�)2, C(j) = −bj

cj

. (A.5)



-16.0 -12.0 -8.0 -4.0 0.0 4.0

Rela

tive V

elo

city, w

-12.0

-4.0

0.0

4.0

V0=1.5 0.75 0.0

Displacement, x

-4.0 -2.0 0.0 2.0 4.0 6.0

Velo

cit, y

-9.0

-6.0

-3.0

0.0

3.0

V0=0 1.5

0.75


-16.0 -12.0 -8.0 -4.0 0.0 4.0

Rela

tive F

orc

e, F

2

-30

-15

0

15

30

0

V0=1.5 0.75

-12.0 -8.0 -4.0 0.0 4.0

Rela

tive F

orc

e, F

2

-30

-15

0

15

30

Relative Velocity, w

0

V0=1.5

0.75

Time, t

0.0 1.0 2.0 3.0

Rela

tive V

elo

city, w

-12.0

-8.0

-4.0

0.0

4.0

0

V0=1.50.75

Time, t

0.0 1.0 2.0 3.0

Velo

city, y

-9.0

-6.0

-3.0

0.0

3.0

0.0

V0=1.5

0.75

-8.0

Fig. 12. Grazing motion of mapping P2 in domain �2 for V0 ={0.0, 0.75, 1.5}: (a) relative phase trajectory; (b) absolute phase trajectory; (c) relativeforce distribution along relative displacement; (d) relative forces distribution on relative velocity; (e) relative velocity time history; and (f) absolutevelocity time history (�=�= 1, A0 = 100, V1 = 2, d1 = 1, d2 = 0.1, b1 = −b2 = 10, c1 = c2 = 30). The initial conditions are (xi , yi ) = (4, V (ti ))

and �ti ≈ {0.4484, 0.3603, 0.2241} with V0 = {0, 0.75, 1.5}, respectively.


For Case II (i.e., d2j < cj )

x(j)(t) = e−dj (t−ti )[C(j)

1 (xi, xi , ti ) cos �(j)d (t − ti ) + C

(j)

1 (xi, xi , ti ) sin �d(t − ti )]+ A(j) cos �t + B(j) sin �t + C(j), (A.6)

x(j)(t) = {[�dC(j)

2 (xi, xi , ti ) − djC(j)

1 (xi, xi , ti )] cos �d(t − ti )

− [�dC(j)

1 (xi, ti) + djC(j)

2 (xi, xi , ti )] sin �d(t − ti )}e−dj (t−ti )

− A(j)� sin �t + B(j)� cos �t , (A.7)

�(j)d =

√cj − d2

j ,

C(j)

1 (xi, xi , ti ) = xi − A(j) cos �ti − B(j) sin �ti − C(j),

C(j)

2 (xi, xi , ti ) = 1

�(j)d

[xi − (djA + B�) cos �ti − (djB(j) − A(j)�) sin �ti + dj (xi − C(j))]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

. (A.8)

The solution for Case III (i.e., d2j = cj ) is

x(j)(t) = [C(j)

1 (xi, xi , ti ) + C(j)

2 (xi, xi , ti ) × (t − ti )]e(j)1 (t−ti ) + A(j) cos �t + B(j) sin �t + C(j), (A.9)

x(j)(t) = {(j)

1 [C(j)

1 (xi, xi , ti ) + (j)

1 C(j)

2 (xi, xi , ti ) × (t − ti )] + C(j)

2 (xi, xi , ti )}e(j)1 (t−ti )

− A(j)� sin �t + B(j)� cos �t , (A.10)

(j)

1 = −2dj ,

C(j)

1 (xi, xi , ti ) = xi − A(j) cos �ti − B(j) sin �ti − C(j),

C(j)

2 (xi, xi , ti ) = xi + (A(j)� − djB(j)) sin �ti − (djA

(j) + B(j)�) cos �ti − dj (C(j) − xi)

⎫⎬⎭ . (A.11)

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37 (2002) 117–133.[3] M. di Bernaedo, C.J. Budd, A.R. Champney, Grazing and border-collision in piecewise-smooth systems: a unified analytical framework, Phys.

Rev. Lett. 86 (2001) 2553–2556.[4] M. di Bernardo, C.J. Budd, A.R. Champneys, Normal form maps for grazing bifurcation in n-dimensional piecewise-smooth dynamical systems,

Physica D 160 (2001) 222–254.[5] M. di Bernardo, M.I. Feigin, S.J. Hogan, M.E. Homer, Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems,

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Grazing bifurcations of a harmonically excited oscillator moving on a time-varying translation belt

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