CANADIAN APPLIEDMATHEMATICS QUARTERLYVolume 17, Number 1, Spring 2009

ANALYTICAL CONDITIONS FOR MOTION

SWITCHABILITY IN A 2-DOF

FRICTION-INDUCED OSCILLATOR MOVING

ON TWO CONSTANT SPEED BELTS

ALBERT C. J. LUO AND TINGTING MAO

ABSTRACT. In this paper, the analytical conditions arepresented for the nonsliding and sliding flows to the velocityboundary in a 2-DOF friction-induced oscillator moving on twoconstant speed belts. The mapping structures are introducedand the periodic motions of the 2-DOF friction-induced oscilla-tor are presented through the corresponding mapping structure.Velocity and force responses for stick and nonstick, periodic mo-tions in the 2-DOF friction-induced system are illustrated fora better understanding of the motion complexity in multipledegrees of freedom systems.

1 Introduction The friction-induced vibration in mechanical en-gineering extensively exists. For the early study of such a problem inengineering, den Hartog and Mikina [4] investigated the steady-state vi-bration of a harmonically excited iscillator subject to dry friction. Lev-itan [8] used the shooting method to prove the existence of periodicsolution in a friction oscillator with a periodically forced base. In 1981,Pratt and Williams [14] investigated the nonlinear analysis of stick andslip motions in the friction-induced oscillator. Awrejcewicz and Delfs [2]investigated the stability of equilibrium in a 2-DOF friction-induced os-cillator. Natsiavas [12] presented the stability analysis of the periodicmotion of piecewise linear oscillators with viscous damping and dry-friction through initial condition variations. Hinrichs et al. [5] gave thenumerical and experimental investigations on the bifurcation and sta-bility of a friction oscillator with self-excited and external excitation.Andreaus and Casini [1] analyzed the response of a periodically forced,friction oscillator on a moving base. Hong and Liu [7] gave an estimatefor the steady-state responses in a Coulomb friction oscillator under har-monic force. Xia [16] presented a computational method to determine

Copyright c©Applied Mathematics Institute, University of Alberta.

201

202 ALBERT C. J. LUO AND TINGTING MAO

the stick-slip motion caused by the dry frcition in a two-dimensional os-cillator under arbitrary excitations. Thomsen and Fidlin [15] derived theapproximate emplitude of responses for the friction-induced vibrationsof a mass-on-belt system. Hoffman et al. [6] investigated the quencingmodel-coupling friction-induced instability in a 2-DOF friction inducedoscillator. Luo and Gregg [11] presented the force criteria for the stickand nonstick motions for a 1-DOF oscillator moving on the belt withdry friction. Csernak and Stepan [3] investigated analytically and nu-merically a harmonically excited dry friction oscillator.

The analytical conditions for all possible stick and nonstick motionsin a periodically forced, nonlinear friction oscillator with two degreesof freedom will be derived from the theory of discontinuous dynamicalsystem. The velocity and force responses for stick and nonstick, periodicmotions in the 2-DOF friction-induced system will be illustrated.

2 Mathematical modeling Consider a 2-DOF, friction-inducedoscillator as shown in Figure 1. The system consists of two masses(mα, α = 1, 2), which are connected with two linear springs of kα (α =1, 2), and two dampers of dα (α = 1, 2). Both of masses move on thetwo individual belts with constant velocity Vα (α = 1, 2). Two harmonicexcitations with frequency Ω and amplitude Qα (α = 1, 2) are exertedon the two masses, respectively. Since the masses are moving on the belttraveling with constant Vα (α = 1, 2), the kinetic friction exists betweenthe mass and belt. Thus, the kinetic friction force is described by

(1) F(α)f (ẋα)

= µkF(α)N , ẋα ∈ (Vα,∞)

∈ [−µkF (α)N , µkF(α)N ], ẋα = Vα

= −µkF (α)N , ẋα ∈ (−∞, Vα)

where ẋα = dxα/dt; µk and F(α)N are a friction coefficient and a normal

force to the contact surface, respectively. For this case, F(α)N = mαg

and g is the gravitational acceleration (α = 1, 2). The friction forces are

given by F(α)f = µkmαg.

The nonfriction forces on two masses in the x-direction are

(2)F (1)s = Q1 cosΩt − k1x1 − d1ẋ1 − k2(x1 − x2) − d2(ẋ1 − ẋ2),

F (2)s = Q2 cosΩt − k2(x2 − x1) − d2(ẋ2 − ẋ1).If the mass mα sticks with the traveling belt, one obtains ẋα = Vα(α = 1, 2). Before the nonfriction force overcomes the friction force

MOTION SWITCHABILITY 203

FIGURE 1: Mechanical model

FIGURE 2: Friction forces

204 ALBERT C. J. LUO AND TINGTING MAO

on the corresponding mass (i.e., |F (α)s | ≤ F (α)f and F(α)f = µkF

(α)N ),

the two masses do not have any relative motion to the belt. Becauseof Vα = consant for stick, the corresponding mass does not have anyacceleration, i.e.,

(3) ẍα = 0 for ẋα = Vα and α ∈ {1, 2}.

If |F (α)s | > F (α)f , then such a nonfriction force overcomes the frictionforce and the nonstick motion will appear. For the nonstick motion ofthe mass mα (α ∈ {1, 2}), the total forces acting on each mass are

(4)

F (1) = Q1 cosΩt − k1x1 − d1ẋ1 − k2(x1 − x2)

− d2(ẋ1 − ẋ2) − F (1)f sgn (ẋ1 − V1) for ẋ1 6= V1,

F (2) = Q2 cosΩt − k2(x2 − x1)

− d2(ẋ2 − ẋ1) − F (2)f sgn (ẋ2 − V2) for ẋ2 6= V2.

From the above discussion, there are four cases of motions.

Case I: nonstick motion (ẋα 6= Vα, α = 1, 2)The equations of nonstick motion for the two oscillators with friction

are

(5)

m1ẍ1 + d1ẋ1 + d2(ẋ1 − ẋ2) + k1x1 + k2(x1 − x2)

= Q1 cosΩt − F (1)f sgn (ẋ1 − V1),

m2ẍ2 + d2(ẋ2 − ẋ1) + k2(x2 − x1)

= Q2 cosΩt − F (2)f sgn (ẋ2 − V2).

Case II: stick motion (ẋ1 = V1 and ẋ2 6= V2)The equations of motion for the first mass sticking and the second

mass nonsticking with belts are

(6)

ẋ1 = V1,

m2ẍ2 + d2(ẋ2 − V1) + k2(x2 − x1)

= Q2 cosΩt − F (2)f sgn (ẋ2 − V2),

with

(7) Q1 cosΩt − d1V1 − d2(V1 − ẋ2) − k1x1 − k2(x1 − x2) ≤ F (1)f .

MOTION SWITCHABILITY 205

Case III: stick motion (ẋ1 6= V1 and ẋ2 = V2)The equations of motion for the first mass nonsticking and the second

mass sticking with belts are

(8)

m1ẍ1 + d1ẋ1 + d2(ẋ1 − V2) + k1x1 + k2(x1 − x2)

= Q1 cosΩt − F (1)f sgn (ẋ1 − V1),

ẋ2 = V2,

with

(9) Q2 cosΩt − d2(V2 − ẋ1) − k2(x2 − x1) ≤ F (2)f .Cases IV: double-stick motion (ẋ1 = V1 and ẋ2 = V2)

The equations of motion for two masses sticking with belts are

(10) ẋ1 = V1 and ẋ2 = V2,

with

(11)Q1 cosΩt − d1V1 − d2(V1 − V2) − k1x1 − k2(x1 − x2) ≤ F (1)f ,

Q2 cosΩt − d2(V2 − V1) − k2(x2 − x1) ≤ F (2)f .

3 Definitions of domains, edges and vector fields Since thetwo friction forces exist in the two masses of the 2-DOF system withthe two traveling belts, the space plane of the 2-DOF system is dividedinto four 4-dimensional domains and four 3-dimensional boundaries witha 2-dimensional edge. To mathematically describe the above physicalproblem, the corresponding definitions are given as follows.

Definition 1. The vectors for state variables and vector fields of the2-DOF friction-induced oscillator are defined as

(12)

x , (x1, ẋ1, x2, ẋ2)T = (x1, y1, x2, y2)

T ,

F , (y1, F1, y2, F2)T .

Definition 2. Four domains in phase space of the 2-DOF friction-inducedoscillator are defined as

(13)

Ω1 = {(x1, y1, x2, y2) | y1 ∈ (V1, +∞) and y2 ∈ (V2, +∞)},Ω2 = {(x1, y1, x2, y2) | y1 ∈ (V1, +∞) and y2 ∈ (−∞, V2)},Ω3 = {(x1, y1, x2, y2) | y1 ∈ (−∞, V1) and y2 ∈ (−∞, V2)},Ω4 = {(x1, y1, x2, y2) | y1 ∈ (−∞, V1) and y2 ∈ (V2, +∞)}.

206 ALBERT C. J. LUO AND TINGTING MAO

Definition 3. The 3-dimensional boundaries of four domains in phasespace of the 2-DOF friction-induced oscillator are defined as ∂Ωα1α2 =Ωα1 ∩ Ωα2 for (αi ∈ {1, 2, 3, 4}, i = 1, 2; α1 6= α2 without repeating),i.e.,

(14)

∂Ω12 = ∂Ω21 = {(x1, y1, x2, y2) | ϕ12 = y2 − V2 = 0, y1 ≥ V1},∂Ω23 = ∂Ω32 = {(x1, y1, x2, y2) | ϕ24 = y1 − V1 = 0, y2 ≤ V2},∂Ω34 = ∂Ω43 = {(x1, y1, x2, y2) | ϕ34 = y2 − V2 = 0, y1 ≤ V1},∂Ω41 = ∂Ω14 = {(x1, y1, x2, y2) | ϕ41 = y1 − V1 = 0, y2 ≥ V2},

where the boundary ∂Ωα1α2 presents the boundary between domainsΩα1 and Ωα2 .

Definition 4. The 2-dimensional edge of the 3-dimensional boundariesis defined as

(15) ∠Ωα1α2α3 = ∂Ωα1α2 ∩ ∂Ωα2α3 =3

⋂

i=1

Ωαi

for αi ∈ {1, 2, 3, 4}, i = 1, 2, 3, α1 6= α2 6= α3 without repeating, andthe union of four 2-dimensional edges is defined as

]Ω1234 = ∪∠Ωα1α2α3

=

{

(x1, y1, x2, y2)

∣

∣

∣

∣

ϕ12 = ϕ34 = y2 − V2 = 0ϕ23 = ϕ41 = y1 − V1 = 0

}

.

(16)

For illustration of the domains, the velocity plane is used for illustra-tion and the four domains, boundary and vertex are illustrated in Fig-ure 3. From the above definitions and the description in Section 2, thedynamical systems for the 2-DOF friction-induced oscillator are givenas follows.

Definition 5. Dynamical systems for the 2-DOF friction-induced oscil-lator are defined by

(17)

ẋ(α) = F(α)(x(α), t,pα) on Ωα

ẋ(α1α2) = F(α1α2)(x(α1α2), t,pα1α2) on ∂Ωα1α2

ẋ(α1α2α3) = F(α1α2α3)(x(α1α2α3), t,pα1α2α3) on ∠Ωα1α2α3

MOTION SWITCHABILITY 207

FIGURE 3: Domain partition in velocity plane of 2-DOF oscillatorswith dry friction.

where

(18)

x(α) = x(α1α2) = x(α1α2α3) = (x1, y1, x2, y2)T ,

F(α) =

(

y1,1

m1F

(α)1 , y2,

1

m2F

(α)2

)T

,

F(α1α2) =

(

y1,1

m1F

(α1α2)1 , y2,

1

m2F

(α1α2)2

)T

,

F(α1α2α3) =

(

y1,1

m1F

(α1α2α3)1 , y2,

1

m2F

(α1α2α3)2

)T

.

In the above definition, the forces for the 2-DOF friction-inducedoscillator in the domains Ωα (α = 1, 2, 3, 4) are

(19)

F(1)1 = F

(2)1 = Q1 cosΩt − F

(1)f − d1y1 − d2(y1 − y2)

−k1x1 − k2(x1 − x2),F

(3)1 = F

(4)1 = Q1 cosΩt + F

(1)f − d1y1 − d2(y1 − y2)

−k1x1 − k2(x1 − x2),F

(1)2 = F

(4)2 = Q1 cosΩt − F

(2)f − d2(y2 − y1) − k2(x2 − x1),

F(2)2 = F

(3)2 = Q1 cosΩt + F

(2)f − d2(y2 − y1) − k2(x2 − x1).

To describe the motion on the boundary, the forces for the 2-DOFfriction-induced oscillator on the boundaries ∂Ωα1α2 for αi ∈ {1, 2, 3, 4},

208 ALBERT C. J. LUO AND TINGTING MAO

i = 1, 2, α1 6= α2 without repeating, are

(20)

F(12)1 ≡ Q1 cosΩt − F

(1)f − d1y1 − d2(y1 − y2)

−k1x1 − k2(x1 − x2),F

(12)2 = 0 for stick on ∂Ω12

F(12)2 ∈ [F

(1)2 , F

(2)2 ] for nonstick on ∂Ω12;

F(23)1 = 0 for stick on ∂Ω23,

F(23)1 ∈ [F

(2)1 , F

(3)1 ] for nonstick on ∂Ω23,

F(23)2 = Q1 cosΩt + F

(2)f − d2(y2 − y1) − k2(x2 − x1);

F(34)1 = Q1 cosΩt + F

(1)f − d1y1 − d2(y1 − y2)

−k1x1 − k2(x1 − x2),F

(34)2 = 0 for stick on ∂Ω34,

F(34)2 ∈ [F

(4)2 , F

(3)2 ] for nonstick on ∂Ω34;

F(41)1 = Q1 cosΩt − F

(2)f − d2(y2 − y1) − k2(x2 − x1),

F(41)2 = 0 for stick on ∂Ω41,

F(41)2 ∈ [F

(1)2 , F

(4)2 ] for nonstick on ∂Ω41.

Similarly, to describe the motion on the boundary edges, the forces forthe 2-DOF friction-induced oscillator on the 2-D edges ∠Ωα1α2α3 forαi ∈ {1, 2, 3, 4}, i = 1, 2, α1 6= α2 6= α3 without repeating, are

(21)(F

(α1α2α3)1 , F

(α1α2α3)2 ) ∈ (F

(α1α2)1 , F

(α2α3)2 ) on ∠Ωα1α2α3 ,

(F(α1α2α3)1 , F

(α1α2α3)2 ) = (0, 0) for full stick on ∠Ωα1α2α3 .

MOTION SWITCHABILITY 209

In other words, one obtains

(22)

(F(123)1 , F

(123)2 ) ∈ (F

(12)1 , F

(23)2 ) on ∠Ω123,

(F(123)1 , F

(123)2 ) = (0, 0) for full stick on ∠Ω123;

(F(234)1 , F

(234)2 ) ∈ (F

(23)1 , F

(34)2 ) on ∠Ω234,

(F(234)1 , F

(234)2 ) = (0, 0) for full stick on ∠Ω234,

(F(341)1 , F

(341)2 ) ∈ (F

(34)1 , F

(41)2 ) on ∠Ω341,

(F(341)1 , F

(341)2 ) = (0, 0) for full stick on ∠Ω341,

(F(412)1 , F

(412)2 ) ∈ (F

(41)1 , F

(12)2 ) on ∠Ω412,

(F(412)1 , F

(412)2 ) = (0, 0) for full stick on ∠Ω412.

From the preceding definition, at the 2-dimensional edges, there arefour possible states: (i) a passable motion to the edge, (ii) two passable-sliding motions to the edge, (iii) a full stick motion to the edge.

4 Existence conditions of sliding and nonsliding flows Inthis section, the analytical conditions for sliding and nonsliding flows(or physically called stick or nonstick motions) to the boundary aredeveloped from the theory of discontinuous dynamical systems. First,the main results for those analytical conditions will be presented throughthe following theorem.

Theorem 1. For dynamical system in equation (17), the nonslidingflow exists at the boundary if and only if

(23)

F(2)2 (tm−) > 0 and F

(1)2 (tm+) > 0 from Ω2 → Ω1,

F(1)2 (tm−) < 0 and F

(2)2 (tm+) < 0 from Ω1 → Ω2,

F(3)2 (tm−) > 0 and F

(4)2 (tm+) > 0 from Ω3 → Ω4,

F(4)2 (tm−) < 0 and F

(3)2 (tm+) < 0 from Ω4 → Ω3,

F(4)1 (tm−) > 0 and F

(1)1 (tm+) > 0 from Ω4 → Ω1,

F(1)1 (tm−) < 0 and F

(4)1 (tm+) < 0 from Ω1 → Ω4,

210 ALBERT C. J. LUO AND TINGTING MAO

(23)F

(2)1 (tm−) < 0 and F

(3)1 (tm+) < 0 from Ω2 → Ω3,

F(3)1 (tm−) > 0 and F

(2)1 (tm+) > 0 from Ω3 → Ω2.

The sliding flow exists on the boundary if and only if

(24)

F(2)2 (tm−) > 0 and F

(1)2 (tm−) < 0 on ∂Ω12,

F(3)2 (tm−) > 0 and F

(4)2 (tm−) < 0 on ∂Ω34,

F(4)1 (tm−) > 0 and F

(1)1 (tm−) < 0 on ∂Ω14,

F(2)1 (tm−) < 0 and F

(3)1 (tm−) > 0 on ∂Ω23.

The onset of the sliding flow on the boundary occurs from the nonsliding

flow if and only if

(25)

F(2)2 (tm−) > 0, F

(1)2 (tm±) = 0 and DF

(1)2 (tm±) > 0

from Ω2 to Ω1 → ∂Ω12,

F(1)2 (tm−) < 0, F

(2)2 (tm±) = 0 and DF

(2)2 (tm±) < 0

from Ω1 to Ω2 → ∂Ω12,

F(3)2 (tm−) > 0, F

(4)2 (tm±) = 0 and DF

(4)2 (tm±) > 0

from Ω3 to Ω4 → ∂Ω34,

F(4)2 (tm−) < 0, F

(3)2 (tm−) = 0 and DF

(3)2 (tm−) < 0

from Ω4 to Ω3 → ∂Ω34,

F(4)1 (tm−) > 0, F

(1)1 (tm±) = 0 and DF

(1)1 (tm±) > 0

from Ω4 to Ω1 → ∂Ω14,

F(1)1 (tm±) < 0, F

(4)1 (tm−) = 0 and DF

(4)1 (tm±) < 0

from Ω1 to Ω4 → ∂Ω14,

F(3)1 (tm−) > 0, F

(2)1 (tm±) = 0 and DF

(2)1 (tm±) > 0

from Ω2 to Ω3 → ∂Ω23,

F(2)1 (tm−) > 0, F

(3)1 (tm±) = 0 and DF

(3)1 (tm±) < 0

from Ω3 to Ω2 → ∂Ω23,

MOTION SWITCHABILITY 211

where

(26)

DF(1)1 = DF

(2)1 = −Q1Ω sin Ωt − d1ẏ1

−d2(ẏ1 − ẏ2) − k1y1 − k2(y1 − y2),

DF(3)1 = DF

(4)1 = −Q1Ω sin Ωt − d1ẏ1

−d2(ẏ1 − ẏ2) − k1y1 − k2(y1 − y2),

DF(1)2 = DF

(4)2 = −Q1Ω sin Ωt − d2(ẏ2 − ẏ1) + k2(y2 − y1),

DF(2)2 = DF

(3)2 = −Q1Ω sin Ωt − d2(ẏ2 − ẏ1) + k2(y2 − y1).

The sliding flow on the boundary vanishes and enters the appropriate

domains if and only if

(27)

F(2)2 (tm−) > 0, F

(1)2 (tm∓) = 0 and DF

(1)2 (tm∓) > 0

from ∂Ω12 → Ω1,

F(1)2 (tm−) < 0, F

(2)2 (tm∓) = 0 and DF

(2)2 (tm∓) < 0

from ∂Ω12 → Ω2,

F(3)2 (tm−) > 0, F

(4)2 (tm∓) = 0 and DF

(4)2 (tm±) > 0

from ∂Ω34 → Ω4,

F(4)2 (tm−) < 0, F

(3)2 (tm∓) = 0 and DF

(3)2 (tm∓) < 0

from ∂Ω34 → Ω3,

F(4)1 (tm−) > 0, F

(1)1 (tm∓) = 0 and DF

(1)1 (tm∓) > 0

from ∂Ω14 → Ω1,

F(1)1 (tm−) < 0, F

(4)1 (tm∓) = 0 and DF

(4)1 (tm∓) < 0

from ∂Ω14 → Ω4,

F(3)1 (tm−) > 0, F

(2)1 (tm∓) = 0 and DF

(2)1 (tm∓) > 0

from ∂Ω23 → Ω2,

F(2)1 (tm−) > 0, F

(3)1 (tm∓) = 0 and DF

(3)1 (tm∓) < 0

from ∂Ω23 → Ω3.

The grazing flow in each domain to the boundary in phase space occurs

212 ALBERT C. J. LUO AND TINGTING MAO

if and only if

(28)

F(1)2 (tm±) = 0 and DF

(1)2 (tm±) > 0 on ∂Ω12 in Ω1,

F(2)2 (tm±) = 0 and DF

(2)2 (tm±) < 0 on ∂Ω12 in Ω2,

F(4)2 (tm±) = 0 and DF

(4)2 (tm±) > 0 on ∂Ω34 in Ω4,

F(3)2 (tm±) = 0 and DF

(3)2 (tm±) < 0 on ∂Ω34 in Ω3,

F(1)1 (tm±) = 0 and DF

(1)1 (tm±) > 0 on ∂Ω14 in Ω4,

F(2)1 (tm±) = 0 and DF

(2)1 (tm±) > 0 on ∂Ω23 in Ω2,

F(3)1 (tm±) = 0 and DF

(3)1 (tm±) < 0 on ∂Ω34 in Ω3.

Proof. Before proving the theorem, we state the G-functions which areintroduced by Luo [9, 10] as follows:

G(0,α1)(tm±) = nT∂Ωα1α2

• F(α1)(tm±)

G(1,α1)(tm±) = 2DnT∂Ωα1α2

• [F(α1)(tm±) − F(α1α2)(tm±)]

+ nT∂Ωαβ • [DF(α)(tm±) − DF(α1α2)(tm±)],

where

D =

2∑

i=1

∂

∂xiẋi +

∂

∂yiẏi +

∂

∂t.

If the boundary ∂Ωαβ is an n-D plane independent of time t, thenDn∂Ωα1α2 = 0. Because of n

T∂Ωα1α2

• F(α1α2) = 0,

DnT∂Ωα1α2 • F(α1α2) + nT∂Ωα1α2 • DF

(α1α2) = 0

can be obtained. Therefore, nT∂Ωα1α2• DF(α1α2) = 0. The first order

G-function becomes

G(1,α1)(tm±) = nT∂Ωα1α2

•DF(α1)(tm±).

(i) From the theory of discontinuous dynamical systems in Luo [9, 10],the nonsliding flow (or called the passable motion) to the boundary from

MOTION SWITCHABILITY 213

domain Ωα1 to domain Ωα2 requires

(29)

G(0,α1)(tm−) = nT∂Ωα1α2

• F(α1)(tm−) < 0

G(0,α2)(tm+) = nT∂Ωα1α2

• F(α2)(tm+) < 0

for n∂Ωα1α2→ Ωα1 ,

G(0,α1)(tm−) = nT∂Ωα1α2

• F(α1)(tm−) > 0

G(0,α2)(tm+) = nT∂Ωα1α2

• F(α2)(tm+) > 0

for n∂Ωα1α2→ Ωα2 ,

(αi ∈ {1, 2, 3, 4}, i = 1, 2, α1 6= α2 without repeating) with(30)

n∂Ωα1α2=

(

∂ϕα1α2∂x1

,∂ϕα1α2

∂y1,∂ϕα1α2

∂x2,∂ϕα1α2

∂y2

)T ∣∣

∣

∣

(x1m,y1m,x2m,y2m)

.

The time tm represents the time for the flow on the velocity boundary,and tm± = tm ± 0 reflects the responses in the regions rather than onthe boundary. Using equation (14), one obtains

(31)n∂Ω23 = n∂Ω14 = (0, 1, 0, 0)

T ,

n∂Ω12 = n∂Ω34 = (0, 0, 0, 1)T .

Thus,

(32)

nT∂Ω12 • F(α)(t) = F(α)2 for α = 1, 2,

nT∂Ω34 • F(α)(t) = F(α)2 for α = 3, 4,

nT∂Ω23 • F(α)(t) = F(α)1 for α = 2, 3,

nT∂Ω14 • F(α)(t) = F(α)1 for α = 1, 4.

With (32), the conditions for the nonsliding flow to the boundary froma domain to another domain are obtained from the nonsliding flow con-

214 ALBERT C. J. LUO AND TINGTING MAO

ditions, i.e.,

F(2)2 (tm−) > 0 and F

(1)2 (tm+) > 0 from Ω2 → Ω1,

F(1)2 (tm−) < 0 and F

(2)2 (tm+) < 0 from Ω1 → Ω2,

F(3)2 (tm−) > 0 and F

(4)2 (tm+) > 0 from Ω3 → Ω4,

F(4)2 (tm−) < 0 and F

(3)2 (tm+) < 0 from Ω4 → Ω3,

F(4)1 (tm−) > 0 and F

(1)1 (tm+) > 0 from Ω4 → Ω1,

F(1)1 (tm−) < 0 and F

(4)1 (tm+) < 0 from Ω1 → Ω4,

F(2)1 (tm−) < 0 and F

(3)1 (tm+) < 0 from Ω2 → Ω3,

F(3)1 (tm−) > 0 and F

(2)1 (tm+) > 0 from Ω3 → Ω2.

(ii) Similarly, in Luo [9, 10], the sliding flow on the boundary ∂Ωα1α2(the stick motion in physics) requires

(33)

G(0,α1)(tm−) = nT∂Ωα1α2

•F(α1)(tm−) < 0

G(0,α2)(tm+) = nT∂Ωα1α2

•F(α2)(tm−) > 0

for n∂Ωα1α2→ Ωα1 ,

G(0,α1)(tm−) = nT∂Ωα1α2

•F(α1)(tm−) > 0

G(0,α2)(tm+) = nT∂Ωα1α2

•F(α2)(tm−) < 0

for n∂Ωα1α2→ Ωα2 .

With normal vectors, the preceding conditions yields

F(2)2 (tm−) > 0 and F

(1)2 (tm−) < 0 on ∂Ω12,

F(3)2 (tm−) > 0 and F

(4)2 (tm−) < 0 on ∂Ω34,

F(4)1 (tm−) > 0 and F

(1)1 (tm−) < 0 on ∂Ω14,

F(2)1 (tm−) < 0 and F

(3)1 (tm−) > 0 on ∂Ω23

for the sliding flow on the boundary. Thus, the condition in equation (24)is obtained. For stick motion with yα = Vα (α ∈ {1, 2}), one obtainsẏα = 0.

MOTION SWITCHABILITY 215

(iii) From Luo [9, 10], the onset on the sliding flow on the boundaryrequires

(34)

G(0,α1)(tm−) = nT∂Ωα1α2

• F(α1)(tm−) < 0

G(0,α2)(tm±) = nT∂Ωα1α2

• F(α2)(tm±) = 0

G(1,α2)(tm±) = nT∂Ωα1α2

• DF(α2)(tm±) < 0

for n∂Ωα1α2→ Ωα1 ,

G(0,α1)(tm−) = nT∂Ωα1α2

• F(α1)(tm−) > 0

G(0,α2)(tm±) = nT∂Ωα1α2

• F(α2)(tm±) = 0

G(1,α2)(tm±) = nT∂Ωα1α2

• DF(α2)(tm±) > 0

for n∂Ωα1α2→ Ωα2 .

The sliding flow vanishing on the boundary and entering domain Ωα2requires

(35)

G(0,α1)(tm−) = nT∂Ωα1α2

• F(α1)(tm−) < 0

G(0,α2)(tm∓) = nT∂Ωα1α2

• F(α2)(tm∓) = 0

G(1,α2)(tm∓) = nT∂Ωα1α2

• DF(α2)(tm∓) < 0

for n∂Ωα1α2→ Ωα1 ,

G(0,α1)(tm−) = nT∂Ωα1α2

• F(α1)(tm−) > 0

G(0,α2)(tm∓) = nT∂Ωα1α2

• F(α2)(tm∓) = 0

G(1,α2)(tm∓) = nT∂Ωα1α2

• DF(α2)(tm∓) > 0

for n∂Ωα1α2→ Ωα2 ;

and for vanishing of the stick motion and entering domain Ωα1

(36)

G(0,α2)(tm−) = nT∂Ωα1α2

• F(α2)(tm−) > 0

G(0,α1)(tm∓) = nT∂Ωα1α2

• F(α1)(tm∓) = 0

G(1,α1)(tm∓) = nT∂Ωα1α2

• DF(α1)(tm∓) > 0

for n∂Ωα1α2→ Ωα1 ,

G(0,α2)(tm−) = nT∂Ωα1α2

• F(α2)(tm−) < 0

G(0,α1)(tm∓) = nT∂Ωα1α2

• F(α1)(tm∓) = 0

G(1,α1)(tm∓) = nT∂Ωα1α2

• DF(α1)(tm∓) < 0

for n∂Ωα1α2→ Ωα2 .

216 ALBERT C. J. LUO AND TINGTING MAO

The conditions for grazing motion in domain Ωα1 to the boundary∂Ωα1α2 are

(37)

G(0,α1)(tm±) = nT∂Ωα1α2

• F(α1)(tm±) = 0

G(1,α1)(tm±) = nT∂Ωα1α2

• DF(α1)(tm±) > 0

for n∂Ωα1α2→ Ωα1 ,

G(0,α1)(tm±) = nT∂Ωα1α2

• F(α1)(tm±) = 0

G(1,α1)(tm±) = nT∂Ωα1α2

• DF(α1)(tm±) < 0

for n∂Ωα1α2→ Ωα2 .

From the normal vectors on the boundary in equation (31), one achieves

(38)

nT∂Ω12 • DF(α)(t) = DF(α)2 for α = 1, 2

nT∂Ω34 • DF(α)(t) = DF(α)2 for α = 3, 4

nT∂Ω23 • DF(α)(t) = DF(α)1 for α = 2, 3

nT∂Ω14 • DF(α)(t) = DF(α)1 for α = 1, 4,

where

(39)

DF(1)1 = DF

(2)1 = −Q1Ω sin Ωt − d1ẏ1

−d2(ẏ1 − ẏ2) − k1y1 − k2(y1 − y2),

DF(3)1 = DF

(4)1 = −Q1Ω sin Ωt − d1ẏ1

−d2(ẏ1 − ẏ2) − k1y1 − k2(y1 − y2),

DF(1)2 = DF

(4)2 = −Q1Ω sin Ωt − d2(ẏ2 − ẏ1) + k2(y2 − y1),

DF(2)2 = DF

(3)2 = −Q1Ω sin Ωt − d2(ẏ2 − ẏ1) + k2(y2 − y1).

Submission of equations (32) and (38) into equations (34)–(37) and ex-pansion yields all the analytical conditions in equations (25)–(28). Thiscompletes the proof of the theorem.

For the 2-dimensional edge ∠Ωα1α2α3 , for αi ∈ {1, 2, 3, 4}, i = 1, 2;α1 6= α2 6= α3 without repeating, are

(40) ∠Ωα1α2α3 = ∂Ωα1α2 ∩ ∂Ωα2α3 .

The corresponding conditions for such 2-dimensional edge can be deter-mined by the summation of the conditions of a flow to the two boundaries

MOTION SWITCHABILITY 217

∂Ωα1α2 and ∂Ωα2α3 . In other words, there are four states: (i) nonslidingflows for both of the boundaries, (ii) sliding flow on one boundary andnonsliding flow for another boundary (two cases), and (iii) sliding flowson both of the boundaries. Three critical cases with grazing flows on the2-dimensional edge include two single grazing flows to the boundary plusdouble-grazing flows to the two boundaries. For a better understandingof the afore-presented conditions, the nonsliding flow at the boundaryand the sliding flow on the boundary are sketched in Figures 4 and 5,respectively. The G-functions in equation (29) and (33) are used toshow the conditions. The vanishing conditions for the sliding flow onthe boundary are also depicted in Figure 5. Similarly, the onset condi-tions for the sliding motion on the boundary can be illustrated, and theconditions for the grazing flow to the boundary can be sketched as well.

(a)

(b)

FIGURE 4: Vector fields for non-stick motion: (a) first mass (∂Ω23),(b) second mass (∂Ω12).

218 ALBERT C. J. LUO AND TINGTING MAO

(c)

(d)

FIGURE 4: (contd.) Vector fields for non-stick motion: (c) first mass (∂ω14)

and (d) second mass (∂Ω34).

(a)

FIGURE 5: Vector fields for stick motion: (a) first mass (∂Ω23).

MOTION SWITCHABILITY 219

(b)

(c)

(d)

FIGURE 5: (contd.) Vector fields for stick motion: (b) second mass(∂Ω12), (c) first mass (∂ω14) and (d) second mass (∂Ω34).

220 ALBERT C. J. LUO AND TINGTING MAO

5 Switching sets and mapping structures From the boundary∂Ωα1α2 in equation (14), the switching sets can be defined.

Definition 6. For dynamical system in equation (17), the sets for switch-ing points of flows on the boundary are defined as

(41)

Σ+1 = {(x1(k), y1(k), x2(k), y2(k), tk) | y2(k) = V +2 , k ∈ N}

Σ01 = {(x1(k), y1(k), x2(k), y2(k), tk) | y2(k) = V2, k ∈ N}

Σ−1 = {(x1(k), y1(k), x2(k), y2(k), tk) | y2(k) = V −2 , k ∈ N}

on ∂Ω12 ∪ ∂Ω34 and

(42)

Σ+2 = {(x1(k), y1(k), x2(k), y2(k), tk) | y1(k) = V +1 , k ∈ N}

Σ02 = {(x1(k), y1(k), x2(k), y2(k), tk) | y1(k) = V1, k ∈ N}

Σ−2 = {(x1(k), y1(k), x2(k), y2(k), tk) | y1(k) = V −1 , k ∈ N}

on ∂Ω23 ∪ ∂Ω14, where

(43) V ±α = limϕ→0

(Vα ± ϕ)

and the switching set on the edge ∠Ωα1α2α3 is defined as

(44) Σ0 =

{

(x1(k), y1(k), x2(k), y2(k), tk)

∣

∣

∣

∣

y1(k) = V1 and

y2(k) = V2, k ∈ N

}

.

Definition 7. From the switching sets, mappings are defined as

(45)

P1 : Σ+1 → Σ+1 , P2 : Σ−1 → Σ−1 , P3 : Σ01 → Σ01,

P6 : Σ+2 → Σ+2 , P7 : Σ−2 → Σ−2 , P8 : Σ02 → Σ02,

P4 : Σ+2 → Σ+1 , P5 : Σ−1 → Σ−2 ,

P4 : Σ−2 → Σ−1 , P5 : Σ+1 → Σ+2 ,

P0 : Σ0 → Σ0.

The switching set Σ0 is the intersection Σ01 ∩Σ02. So the mappings P3

and P8 can be applied to Σ0, i.e.,

(46) P3 : Σ0 → Σ01, P3 : Σ01 → Σ0, P8 : Σ0 → Σ02, P8 : Σ02 → Σ0.

MOTION SWITCHABILITY 221

FIGURE 6: Switching sets and mappings.

In the total nine mappings, Pn are the local mappings for n = 0, 1, 2, 3;6, 7, 8 and the global mappings for n = 4, 5. The afore-defined nine map-pings are sketched in Figure 6. The global and local mappings are clearlyshown. Through all the mappings, the motions of the 2-DOF friction-induced oscillator can be labeled if the motion is interacted at leastwith one of two velocity boundaries. From the defined mappings, thefinal switching points can be mapped from the initial switching points.Herein, the flow not intersected with the boundary will not be interest-ing.

Theorem 2. For dynamical system in equation (17), if mapping Pn (n =0, 1, 2, . . . , 8) exists, then there is a set of nonlinear algebraic equationsas

(47) f (n)(xk, tk,xk+1, tk+1) = 0

where

(48)xk = (x1(k), y1(k), x2(k), y2(k))

T

f (n) = (f(n)1 , f

(n)2 , f

(n)3 , f

(n)4 )

T

with the constraints xk and xk+1 from the boundaries. In other words,

222 ALBERT C. J. LUO AND TINGTING MAO

one has

(49)

f(n)1 (xk, tk,xk+1, tk+1) = 0,

f(n)2 (xk, tk,xk+1, tk+1) = 0,

f(n)3 (xk, tk,xk+1, tk+1) = 0,

f(n)4 (xk, tk,xk+1, tk+1) = 0,

and

(50)

xk = (x1(k), y1(k), x2(k), V2)T

xk+1 = (x1(k+1), y1(k+1), x2(k+1), V2)T

}

for Pn (n = 1, 2, 3)

xk = (x1(k), V1, x2(k), y2(k))T

xk+1 = (x1(k+1), V1, x2(k+1), y2(k+1))T

}

for Pn (n = 6, 7, 8)

xk = (x1(k), V1, x2(k), y2(k))T

xk+1 = (x1(k+1), y1(k+1), x2(k+1), V2)T

}

for P4 and P9

xk = (x1(k), y1(k), x2(k), V2)T

xk+1 = (x1(k+1), V1, x2(k+1), y2(k+1))T

}

for P5 and P6

xk = (x1(k), V1, x2(k), V2)T

xk+1 = (x1(k+1), V1, x2(k+1), V2)T

}

for P0.

Proof. We can prove the theorem using the definitions of switchingsets and mappings in equations (41)–(46). The solutions for the linear2-DOF oscillators are listed in the Appendix. For nonstick flows, therelation for mapping can be determined by the displacement and thevelocity. From the solutions of the 2-DOF friction-induced oscillator,one obtains

(51)

f (n) = Φ(tk+1,xk, tk) − xk+1,y2(k) = y2(k+1) = V2 for n = 1, 2;

y1(k) = y1(k+1) = V1 for n = 6, 7;

y2(k) = V2, y1(k+1) = V1 for n = 4;

y1(k) = V1, y1(k+1) = V2 for n = 5.

MOTION SWITCHABILITY 223

For sliding flows, the solutions for the sliding flow should be speciallydiscussed. Consider the sliding flow relative to the mapping P3. Forthis case, the second mass is moving with the belt together and the firstmass is moving on the corresponding belt separately. The solutions inthe Appendix and sliding disappearance conditions give

(52)

f(3)1 = e

δ(II)1 (tk+1−tk){A(II)0 cos(ω

(II)1 (tk+1 − tk))

+ B(II)0 sin(ω

(II)1 (tk+1 − tk))}A3 cosΩt + B3 sin Ωt

+ A4(tk+1 − tk) + B4 − x1(k+1),

f(3)2 = e

δ(II)1 (tk+1−tk){(A(II)0 δ

(II)1 − B

(II)0 ω

(II)1 )

× cos(ω(II)1 (tk+1 − tk)) + (A(II)0 ω

(II)1 + B

(II)0 δ

(II)1 )

× cos(ω1(tk+1 − tk))} − A3Ω sin Ωt+ B3Ω cosΩt + A4 − y1(k+1),

f(3)3 = −x2(k+1) + x2(k) + V2(tk+1 − tk),

f(3)4 = Q2 cosΩtk+1 − k2(x2(k+1) − x1(k+1))

− d2(V2 − y1(k+1)) − sgn (V ±2 )m2gµk,

where

(53)

ω(II)1 =

√

4m1(k1 + k2) − (d1 + d2)22m1

,

δ(II)1 = −

d1 + d22m1

.

Similarly, if the first mass moves with the traveling belt together and thesecond mass moves with the corresponding belt separately, the governing

224 ALBERT C. J. LUO AND TINGTING MAO

equations for mapping P8 are

(54)

f(8)1 = x1(k) + V1(tk+1 − tk) − x1(k+1),

f(8)2 = Q1 cosΩtk+1 − k1x1(k+1) − d1V1 − k2(x1(k+1) − x2(k+1))

− d2(V2 − y2(k+1)) − sgn (V ±1 )m1gµk,

f(8)3 = e

δ(III)1 (tk+1−tk){A(III)0 cos(ω

(III)1 (tk+1 − tk))

+ B(III)0 sin(ω

(III)1 (tk+1 − tk))} + A5 cosΩtk+1

+ B5 sin Ωtk+1 + A6(tk+1 − tk) + B6 − x2(k+1),

f(8)4 = e

δ(III)1 (tk+1−tk){A(III)0 δ

(III)1 − B

(III)0 ω

(III)1 )

× cos(ω(III)1 (tk+1 − tk)) + (A(III)0 ω

(III)1 + B

(III)0 δ

(III)1 )

× sin(ω(III)1 (tk+1 − tk))} − A5Ω sin Ωtk+1+ B5Ω cosΩtk+1 + A6 − y2(k+1).

For the stick motion on the edge ∠Ωα1α2α3 , because the two massesmove with the two individual traveling belts together, respectively, thevelocity should be constant. The governing equations for the slidingmapping P0 is

{

f(0)1 = x1(k) + V1(tk+1 − tk) − x1(k+1),

f(0)3 = x2(k) + V2(tk+1 − tk) − x2(k+1),

f(0)2 = y1(k+1) − V1 for m2 sliding flow vanishing,

f(0)2 = Q1 cosΩtk+1 − k1x1(k+1) − d1V1

−k2(x1(k+1) − x2(k+1)) − d2(V2 − y2(k+1))−sgn (V ±1 )m1gµk for m1 sliding flow vanishing

(55)

f(0)4 = y2(k+1) − V2 for m1 sliding flow vanishing,

f(0)4 = Q2 cosΩtk+1 − k2(x2(k+1) − x1(k+1))

−d2(V2 − y1(k+1)) − sgn (V ±2 )m2gµkfor m1 sliding flow vanishing.

(56)

Therefore, this theorem is proved.

MOTION SWITCHABILITY 225

Theorem 3. For the dynamical system in equation (17), if there is amapping structure of five mappings as

(57) P = P5 ◦ P1 ◦ P4 ◦ P8 ◦ P6,

then the periodic flow (xk+5, tk+5)T = P (xk, tk)

T can be determined by

(58)

f (6) = (xk , tk,xk+1, tk+1) = 0,

f (8) = (xk+1, tk+1,xk+2, tk+2) = 0,

f (4) = (xk+2, tk+2,xk+3, tk+3) = 0,

f (1) = (xk+3, tk+3,xk+4, tk+4) = 0,

f (5) = (xk+4, tk+4,xk+5, tk+5) = 0,

with the periodicity condition of the periodic flow given by

(59)

xk+5 = xk , tk+5 = tk + NT, N = 1, 2, . . . ,

T =2π

Ω.

Proof. From the mapping definition in Definitions 6–7, the correspond-ing mapping relations are

(60)

P6 : (tk, x1(k), V1,x2(k)) → (tk+1, x1(k+1), V1,x2(k+1)),P8 : (tk+1, x1(k+1), V1,x2(k+1)) → (tk+2,x1(k+2), x1(k+2), V2),P4 : (tk+2,x1(k+2), x2(k+2), V2) → (tk+3,x1(k+3), x2(k+3), V2),P1 : (tk+3,x1(k+3), x2(k+3), V2) → (tk+4, x1(k+4), V1,x2(k+4)),P5 : (tk+4, x1(k+4), V1,x2(k+4)) → (tk+5, x1(k+5), V1,x2(k+5))

where

(61)x1(j) = (x1(j), y1(j))

T

x2(j) = (x2(j), y2(j))T

}

for j = k, k + 1, . . . , k + 5.

With x = (x1,x2)T , Theorem 2 yields equation (58). With the period-

icity condition of the periodic flow in equation (59), the switching pointsof the periodic flow on the boundary is obtained. Further, the periodicflow is obtained. This theorem is proved.

226 ALBERT C. J. LUO AND TINGTING MAO

Similarly, the other mapping structures can be developed to analyt-ically predict the switching points for periodic motions in the 2-DOFfriction-induced oscillator. To intuitively show the periodic flow with amapping structure in equation (57), the simple mapping structure withfive mappings is sketched in Figure 7. For this periodic flow, the mass m1moving with the traveling belt together is for illustration of the slidingflow on the velocity boundary, and the other mapping gives the passableflows.

(a)

(b)

FIGURE 7: A mapping structure for two masses in a 2-DOF oscillatorwith dry friction: (a) phase plane of mass m1 and (b) phase plane ofmass m2.

MOTION SWITCHABILITY 227

6 Numerical results Consider a set of systems parameters fornumerical illustration

(62)m1 = 4, m2 = 1, d1 = 0.05, d2 = 0.5, k1 = 4, k2 = 1,

µk = 0.15, Q1 = −15, Q2 = 15, V1 = V2 = 2.

The bifurcation scenario for switching points versus excitation frequencyis presented in Figures 8 and 9 for the switching displacement, switchingvelocity and switching phase for the first and second masses. The exci-tation frequency for Ω ∈ (0, 8.0) are used for numerical simulation andthe dashed-dotted lines denote the grazing bifurcation (GB) and slidingbufurcation (SB). The critical values of excitation frequency (Ωcr) areabout 0.131, 0.439, 0.599, 1.457, 2.132, 2.405, 2.557 and 7.637. The pe-riodic motions for P21, P8415, P68415, P6415, P5454, P12 and the otherslie in Ω ∈ (2.557, 7.637), (2.405, 2.557), (2.132, 2.405), (1.457, 2.132),(0.599, 1.457), (0.439, 0.599) and (0.131, 0.439), respectively. It is ob-served that simple periodic flow is relative to P12. The correspondingranges are labeled in Figures 8 and 9. In Figure 8, the switching dis-placement and velocity are presented for the first and seccond masses.The switching phase versus excitation frequency is presented in Figure 9because the switching boundaries exist. The set of the switching pointsand switching phase can be used to determine periodic motions withspecific mapping structures, and the corresponding analytical predictionwill be given in sequal.

(63)Fα+ = Fα for yα > Vα

Fα− = Fα for yα < Vα

}

α = 1, 2.

Consider a periodic motion with a mapping structure of P4651. Forthe same set of system parameters, the periodic motion for Ω = 1.6 ispresented in Figures 10 and 11 with the initial condition (i.e., Ω t0 ≈5.584, x10 ≈ 3.1002, y10 ≈ −1.5826, x20 ≈ −7.1437 and y20 = 2.0).The solid and dashed curves give the real and imaginary responses. Thevelocity boundary is denoted by the straight line. The periodic responsesof the first and second masses are shown in Figures 10 and 11, respec-tively. The velocity-time history, force per unit mass-time history, phaseplane, and force versus displacement are presented. The initial point isselected on the velocity boundary of the second mass, which can be foundin Figure 11(a) and (c) with a green point. Since F2+ > 0 and F2− > 0and the initial point are shown in Figures 11(b) and (d), from Theo-rem 1, the motion flow must move to the domain relative to y2 > V2,

228 ALBERT C. J. LUO AND TINGTING MAO

which means the system switching from the domain of y2 < V2 to thedomain of y2 > V2. In such a domain, the motion flow moves to the ve-locity of the boundary of the second mass, and the F2+ < 0 and F2− < 0at such a point. From Theorem 1, the motion flow must move to thedomain relative to y2 < V2. However, in such a domain, the motion flowmoves to the velocity boundary of the first mass (i.e., y1 = V1), as shownin Figures 10(a) and (c). Because of the force condition of F1+ > 0 andF1− > 0, Theorem 1 implies that the motion flow must enter the domainof y1 > V1 from the domain of y1 < V1. The motion flow returns back tothe velocity boundary of y1 = V1. At this switching point, the motionflow moves to the domain of y1 < V1 because of F1+ < 0 and F1− < 0,shown in Figure 10(b) and (d).

A periodic motion with two sliding portions for a mapping structureof P3231 is presented in Figures 12 and 13. Choose the same systemparameters and the initial conditions Ω t0 ≈ 4.8627, x10 ≈ 1.1953, y10 ≈−0.2665, x20 ≈ 0.8086, y20 = 2.0 for Ω = 0.2. The initial condition isat the velocity boundary of y2 = V2 from Figures 13(a) and (c). Theforce conditions at this point are F2+ < 0 and F2− > 0 in Figures 13(b)and (d). Theorem 1 implies that the motion flow will slide along thisvelocity boundary of y2 = V2. Once one of two forces becomes zero andthe corresponding derivative with respect to time is less than zero, themotion flow will enter the domain of y2 < V2 from the boundary. Themotion relative to mapping P2 returns back to the velocity boundary ofy2 = V2, and the force conditions at this point are F2+ < 0 and F2− > 0.From Theorem 1, the sliding flow along such velocity boundary will beformed with mapping P3 again. Once one of the two forces becomes

(a)

FIGURE 8: Bifurcation scenario: (a) switching displacement.

MOTION SWITCHABILITY 229

(b)

(c)

(d)

FIGURE 8: (contd.) Bifurcation scenario: (b) switching velocity of thefirst mass, (c) switching displacement and (d) switching velocity of thesecond mass. (m1 = 4, m2 = 1, d1 = 0.05, d2 = 0.5, k1 = 4, k2 =1, µk = 0.15, Q1 = −15, Q2 = 15, V1 = V2 = 2.)

230 ALBERT C. J. LUO AND TINGTING MAO

FIGURE 9: Bifurcation scenario for switching phase. (m1 = 4, m2 =1, d1 = 0.05, d2 = 0.5, k1 = 4, k2 = 1, µk = 0.15, Q1 = −15, Q2 =15, V1 = V2 = 2.)

zero with an appropriate condition, the sliding flows on the boundarywill disappear. The sliding flow on the boundary disappears and entersthe domain of y2 > V2 and this point is the initial condition. Theperiodic motion will not interact with the velocity boundary of y1 = V1,and the periodic flow lies in the domain of y1 < V1, which can be foundin Figure 12.

(a)

FIGURE 10: The response of the first mass for P4651: (a) velocity-timehistory.

MOTION SWITCHABILITY 231

(b)

(c)

(d)

FIGURE 10: (contd.) The response of the first mass for P4651: (b)force per unit mass-time history, (c) phase plane and (d) force versusdisplacement. (m1 = 4, m2 = 1, d1 = 0.05, d2 = 0.5, k1 = 4, k2 =1, µk = 0.15, Q1 = −15, Q2 = 15, V1 = V2 = 2, Ω = 1.6.) Theinitial condition is Ωt0 ≈ 5.5840, x10 ≈ 3.1002, y10 ≈ −1.5826, x20 ≈−7.1437, y20 = 2.0.

232 ALBERT C. J. LUO AND TINGTING MAO

(a)

(b)

(c)

FIGURE 11: The response of the second mass for P4651: (a) velocity-time history, (b) force per unit mass-time history, (c) phase plane.

MOTION SWITCHABILITY 233

(d)

FIGURE 11: (contd.) (d) force versus displacement. (m1 = 4, m2 =1, d1 = 0.05, d2 = 0.5, k1 = 4, k2 = 1, µk = 0.15, Q1 = −15, Q2 =15, V1 = V2 = 2, Ω = 1.6.) The initial condition is Ωt0 ≈ 5.5840, x10 ≈3.1002, y10 ≈ −1.5826, x20 ≈ −7.1437, y20 = 2.0.

(a)

(b)

FIGURE 12: The response of the first mass: (a) velocity-time history,(b) force per unit mass-time history.

234 ALBERT C. J. LUO AND TINGTING MAO

(c)

(d)

FIGURE 12: (contd.) The response of the first mass: (c) phase planeand (d) force versus displacement. (m1 = 4, m2 = 1, d1 = 0.05, d2 =0.5, k1 = 4, k2 = 1, µk = 0.15, Q1 = −15, Q2 = 15, V1 = V2 =2, Ω = 0.2.) The initial condition is Ωt0 ≈ 4.8627, x10 ≈ 1.1953, y10 ≈−0.2665, x20 ≈ 0.8086, y20 = 2.0.

(a)

FIGURE 13: The response of the second mass: (a) velocity-time history.

MOTION SWITCHABILITY 235

(b)

(c)

(d)

FIGURE 13: (contd.) The response of the second mass: (b) force perunit mass-time history, (c) phase plane and (d) force versus displace-ment. (m1 = 4, m2 = 1, d1 = 0.05, d2 = 0.5, k1 = 4, k2 = 1, µk =0.15, Q1 = −15, Q2 = 15, V1 = V2 = 2, Ω = 0.2.) The initial conditionis Ωt0 ≈ 4.8627, x10 ≈ 1.1953, y10 ≈ −0.2665, x20 ≈ 0.8086, y20 = 2.0.

236 ALBERT C. J. LUO AND TINGTING MAO

7 Conclusions The physical model of a 2-DOF, friction-inducedoscillators moving on the betls was presented in this paper. The analyt-ical conditions for the flow switching on the velocity boundaries for suchan oscillator were developed. The basic mappings were introduced to la-bel motions in such an oscillator. The numerical predictions of periodicmotions were carried out. The periodic flows were illustrated throughthe velocity and force responses to show the analytical conditions for themotion switching in such a dynamical system.

Appendix Case I: nonstick motion (ẋα 6= Vα, α = 1, 2)To obtain analytical solution, matrix A is given as

(A.1) A =

0 1 0 0−m−11 (k1 + k2) −m−11 (d1 + d2) m−11 k2 m−11 d2

0 0 0 1m−12 k2 m

−12 d2 −m−12 k2 −m−12 d2

Consider the matrix A possesses two pairs of eigenvalues δ(I)α ± ω(I)α i

with i =√−1 (α = 1, 2) as an example. δ(I)α and ω(I)α are positive. With

an initial condition (x1(k), y1(k), x2(k), y2(k), tk), the analytical solutionof equation (5) for this 2-DOF system is

x(t) = Φ(t,xk , tk)

= eδ(I)1 (t−tk)[C

(I)1 cosω

(I)1 (t − tk) −C

(I)2 sin ω

(I)1 (t − tk)]

+ eδ(I)2 (t−tk)[C

(I)3 cosω

(I)2 (t − tk) −C

(I)4 sinω

(I)2 (t − tk)] + xp(t).

(A.2)

Note that x = (x1, y1, x2, y2)T and Φ(t,xk , tk) = (Φ1, Φ2, Φ3, Φ4)

T . The

coefficients C(I)1 , C

(I)2 , C

(I)3 and C

(I)4 are

(A.3)

C(I)1 = [A

(I)0 , A

(I)0 δ

(I)1 − B

(I)0 ω

(I)1 , A

(I)0 M1 − B

(I)0 N1,

A(I)0 (δ

(I)1 M1 − ω

(I)1 N1) − B

(I)0 (δ

(I)1 N1 + ω

(I)1 M1)]

T ,

C(I)2 = [B

(I)0 , A

(I)0 ω

(I)1 + B

(I)0 δ

(I)1 , A

(I)0 N1 + B

(I)0 M1,

A(I)0 (δ

(I)1 N1 + ω

(I)1 M1) + B

(I)0 (δ

(I)1 M1 − ω

(I)1 N1)]

T ,

C(I)3 = [C

(I)0 , C

(I)0 δ

(I)2 − D

(I)0 ω

(I)2 , C

(I)0 M2 − D

(I)0 N2,

C(I)0 (δ

(I)2 M2 − ω

(I)2 N2) − D

(I)0 (δ

(I)2 N2 + ω

(I)2 M2)]

T ,

C(I)4 = [D

(I)0 , C

(I)0 ω

(I)2 + D

(I)0 δ

(I)2 , C

(I)0 N2 + D

(I)0 M2,

C(I)0 (δ

(I)2 N2 + ω

(I)2 M2) + D

(I)0 (δ

(I)2 M2 − ω

(I)2 N2)]

T ,

MOTION SWITCHABILITY 237

and the particular solutions are

xp(t) = (x1p, y1p, x2p, y2p)T

= (A1 cosΩt + B1 sin Ωt + C1,−A1Ω sin Ωt + B1Ω cosΩt,

A2 cosΩt + B2 sin Ωt + C2,−A2Ω sin Ωt + B2Ω cosΩt)T ,

(A.4)

where M1, N1, M2 and N2 are constants

(A.5)

M1 =

d2ω(I)1 (2m2ω

(I)1 δ

(I)1 + d2ω

(I)1 ) + (d2δ

(I)1 + k2)

×(m2(δ(I)1 )

2− m2(ω

(I)1 )

2 + d2δ(I)1 + k2)

(m2((δ(I)1 )

2− (ω

(I)1 )

2) + d2δ(I)1 + k2)

2 + (2m2δ(I)1 ω

(I)1 + d2ω

(I)1 )

2,

N1 =

−d2ω(I)1 (m2(δ

(I)1 )

2− m2(ω

(I)1 )

2 + d2δ(I)1 + k2)

+(d2δ(I)1 + k2)(2m2δ

(I)1 ω

(I)1 + d2ω

(I)1 )

(m2((δ(I)1 )

2− (ω

(I)1 )

2) + d2δ(I)1 + k2)

2 + (2m2δ(I)1 ω

(I)1 + d2ω

(I)1 )

2,

M2 =

d2ω(I)2 (2m2ω

(I)2 δ

(I)2 + d2ω

(I)2 ) + (d2δ

(I)2 + k2)

×(m2(δ(I)2 )

2− m2(ω

(I)2 )

2) + d2δ(I)2 + k2)

(m2((δ(I)2 )

2− m2(ω

(I)2 )

2) + d2δ(I)2 + k2)

2 + (2m2δ(I)2 ω

(I)2 + d2ω

(I)2 )

2,

N2 =

d2ω(I)2 (m2(δ

(I)2 )

2− m2(ω

(I)2 )

2 + d2δ(I)2 + k2)

−(d2δ(I)2 + k2)(2m2δ

(I)2 ω

(I)2 + d2ω

(I)2 )

(m2(δ(I)2 )

2− m2(ω

(I)2 )

2 + d2δ(I)2 + k2)

2 + (2m2δ(I)2 ω

(I)2 + d2ω

(I)2 )

2.

The coefficients A1, A2, B1, B2, C1 and C2 in the particular solutionare constants where

2

6

6

4

A1B1A2B2

3

7

7

5

=

2

6

6

4

k1 + k2 − m1Ω2 (d1 + d2)Ω −k2 −d2Ω

d2Ω −k2 −d2Ω k2 − m2Ω2

−k2 −d2Ω k2 − m2Ω2 d2Ω

−(d1 + d2)Ω k1 + k2 − m1Ω2 d2Ω −k2

3

7

7

5

−1

×

2

6

6

4

Q10

Q20

3

7

7

5

(A.6)

where Q1 and Q2 are amplitude of external excitation, Ω is excitationfrequency and

(A.7)

C1 = −F1 + F2

k1,

C2 = −F2k2

− F1 + F2k1

.

238 ALBERT C. J. LUO AND TINGTING MAO

Here, F1 and F2 are dynamical frictional forces

(A.8)F1 = sgn (y1 − V1)m1 gµk,F2 = sgn (y2 − V2)m2 gµk.

A(I)0 , B

(I)0 , C

(I)0 and D

(I)0 are coefficients that depend on the initial con-

dition

2

6

6

6

6

6

6

6

4

A(I)0

B(I)0

C(I)0

D(I)0

3

7

7

7

7

7

7

7

5

=

2

6

6

6

6

6

6

6

6

4

1 0 1 0

M1 −N1 M2 −N2

δ(I)1 −ω

(I)1 δ

(I)2 −ω

(I)2

(M1δ(I)1

−N1ω(I)1 )

(−M1ω(I)1

−N1δ(I)1 )

(M2δ(I)2

−N2ω(I)2 )

(−M2ω(I)2

−N2δ(I)2 )

3

7

7

7

7

7

7

7

7

5

×

2

6

6

6

6

6

4

x1(k) − (A1 cos Ωtk + B1 sinΩtk + C1)

x2(k) − (A2 cos Ωtk + B2 sinΩtk + C2)

y1(k) − (−A1Ω sinΩtk + B1Ωcos Ωtk)

y2(k) − (−A2Ω sinΩtk + B2Ωcos Ωtk)

3

7

7

7

7

7

5

.

(A.9)

Case II: stick motion (ẋ1 6= V1 and ẋ2 = V2)The solution for the mass m2 sticking with traveling belts in the 2-

DOF system is for d1 + d2 <√

4m1(k1 + k2)

x1 = eδ(II)(t−tk )

1 [C(II)1 cos(ω

(II)1 (t − tk))

−C(II)2 sin(ω(II)1 (t − tk)) + x1p(t)

(A.10)

with x1 = [x1, y1]T and

C(II)1 = [A

(II)0 , A

(II)0 δ

(II)1 − B

(II)0 ω

(II)1 ]

T ,

C(II)2 = [B

(II)0 , A

(II)0 ω

(II)1 + B

(II)0 δ

(II)1 ]

T ,

x1p(t) = [A3 cosΩt + B3 sin Ωt + A4(t − tk) + B4;B3Ω cosΩt − A3Ω sinΩt + A4]T .

(A.11)

δ(II)1 = −

d1 + d22m1

,

ω(II)1 =

√

4m1(k1 + k2) − (d1 + d2)22m1

,

(A.12)

MOTION SWITCHABILITY 239

where the coefficients A3, B3, A4 and B4 in the particular solution areconstant and

(A.13)

A3 =[(k1 + k2) − Ω2m1]Q1

(Ω(d1 + d2))2 + [(k1 + k2) − Ω2m1]2

B3 =[Ω(d1 + d2)]Q1

[Ω(d1 + d2)]2 + [(k1 + k2) − Ω2m1]2

A4 =k2V2

k1 + k2

B4 = −(d2 + d2)A4

k1 + k2+

d2V2 + k2x2(k) − sgn (y1 − V1)m1gµkk1 + k2

.

A(II)0 and B

(II)0 are coefficients that depend on the initial condition

(A.14)

A(II)0 = x1(k) − (A3 cosΩtk + B3 sin Ωtk + B4)

B(II)0 =

−y1(k) + A(II)0 δ(II)1 + B3 cosΩtk − A3Ω sin Ωtk + A4

ω(II)1

.

Case III: stick motion (ẋ1 = V1 and ẋ2 6= V2)The solution for the mass m1 sticking on the traveling belt in the

2-DOF system is for d2 <√

4m2k2

x2 = eδ(III)1 (t−tk)[C

(III)1 cos[ω

(III)1 (t − tk)]

−C(III)2 sin[ω(III)1 (t − tk)] + x2p(t)

(A.15)

where x2 = [x2, y2]T and

C(III)1 = [A

(III)0 , A

(III)0 δ

(III)1 − B

(III)0 ω

(III)1 ]

T ,

C(III)2 = [B

(III)0 , A

(III)0 ω

(III)1 + B

(III)0 δ

(III)1 ]

T ,

x1p(t) = [A5 cosΩt + B5 sin Ωt + A6(t − tk) + B6;B5Ω cosΩt − A5Ω sinΩt + A6]T ,

(A.16)

δ(III)1 = −

d22m2

ω(II)1 =

√

4m2k2 − d222m2

,

(A.17)

240 ALBERT C. J. LUO AND TINGTING MAO

where the coefficients A5, B5, A6 and B6 in the particular solution areconstants such that

(A.18)

A5 =(k2 − Ω2m2)Q2

(Ωd2)2 + (k2 − Ω2m2)2,

B5 =(Ωd2)Q2

(Ωd2)2 + (k2 − Ω2m2)2,

A6 = V1,

B6 = −d2V1k2

+ d2V1 + k2x1(k) − sgn (y2 − V2)m2gµk.

A(III)0 and B

(III)0 are coefficients that depend on the initial condition

(A.19)

A(III)0 = x2(k) − (A5 cos Ωtk + B5 sin Ωtk + B6),

B(III)0 =

−y2(k) + A(III)0 δ

(III)1 + B5Ω cos Ωtk − A5Ω sin Ωtk + A6

ω(III)1

.

Case IV: double-stick motion (ẋ1 = V1 and ẋ2 = V2)If both of masses stick on the corresponding belts, the corresponding

solutions are

(A.20)x1 = x1(k) + V1(t − tk), y1 = V1;x2 = x1(k) + V2(t − tk), y2 = V2.

REFERENCES

1. U. Andreaus and P. Casini, Dynamics of friction oscillators excited by a mov-ing base and driving force, J. Sound Vibration 245 (2000), 685–699.

2. J. Awrejcewicz and J. Delfs, Dynamics of a self-excited stick-slip oscillatorwith two degrees of freedom, Part I. Investigation of equilibrium, Eur. J. Mech.A/Solids 9(4) (1990), 269–282.

3. G. Csernak and G. Stepan, On the periodic response of a harmonically exciteddry friction oscillator, J. Sound Vibration 295 (2006), 649–658.

4. J. P. den Hartog and S. J. Mikina, Forced vibrations with nonlinear springconstants, ASME J. Appl. Mech. 58 (1932), 157–164.

5. N. Hinrichs, M. Oestreich and K. Popp, On the modeling of friction oscillators,J. Sound Vibration 216(3) (1998), 435–459.

6. N. Hoffmann, N. Wagner and L. Gaul, Quenching model-coupling friction-induced instability using high-frequency dither, J. Sound Vibration 279 (2005),471–480.

MOTION SWITCHABILITY 241

7. H. K. Hong and C. S. Liu, Non-sticking oscillation formulae for coulomb fric-tion under harmonic loading, J. Sound Vibration 244(5) (2000), 883–898.

8. E. S. Levitan, Forced oscillation of a spring-mass system having combinedCoulomb and viscous damping, J. Acoustical Soc. Amer. 32 (1960), 1265–1269.

9. A. C. J. Luo, Global Transversality, Resonance and Chaotic Dynamics, WorldSci., Singapore, 2008.

10. A. C. J. Luo, Discontinuous Dynamical Systems on Time-Varying Domains,Higher Education Press, Springer, Beijing, 2009.

11. A. C. J. Luo and B. C. Gregg, On the mechanism of stick and nonstick, periodicmotions in a forced linear oscillator with dry friction, ASME J. Vib. Acoustics128 (2006), 97–105.

12. S. Natsiavas, Stability of piecewise linear oscillators with viscous and dry fric-tion damping, J. Sound Vibration 217 (1998), 507–522.

13. M. Oestreich, N. Hinrichs and K. Popp, Bifurcation and stability analysis fora non-smooth friction oscillator, Arch. Appl. Mech. 66 (1996), 301–314.

14. T. K. Pratt and R. Williams, Nonlinear analysis for stick/slip motion, J. SoundVibration 74 (1981), 531–542.

15. J. J. Thomsen and A. Fidlin, Analytical approximation for stick-slip vibrationamplitudes, Int. J. Nonlinear Mech. 38 (2003), 389–403.

16. F. Xia, Modeling of a two-dimensional Coulomb friction oscillator, J. SoundVibration 265 (2003), 1063–1074.

Department of Mechanical and Industrial Engineering,

Southern Illinois University Edwardsville,

Edwardsvill, IL 62062-1805 U.S.A.

E-mail address: aluo@siue.edu