Stick motions and grazing ﬂows in a double belt friction oscillator

*(&+(1

Shandong Normal University

School of Mathematical Sciences

Ji’nan, 250014

PR C+,1$

-,1-81)$1?

Shandong Normal University

School of Mathematical Sciences

Ji’nan, 250014

PR C+,1$

Abstract: This paper is concerned with the dynamics of a double belt friction oscillator which is subjected to

periodic excitation, linear spring-loading, damping force and two friction forces using the ﬂow switchability theory

of the discontinuous dynamical systems. Different domains and boundaries for such system are deﬁned according

to the friction discontinuity, which exhibits multiple discontinuous boundaries in the phase space. Based on the

above domains and boundaries, the analytical conditions of the stick motions and grazing motions are obtained

mathematically. There are more theories about such friction oscillators to be discussed in future.

Key–Words: double belt friction oscillator; discontinuous dynamical system; switchability; stick motion; grazing

motion

1 Introduction

Discontinuous dynamical systems exist widely in the

real word, especially in mechanical engineering. In

mechanical engineering, most of the dynamical sys-

tems are discontinuous. This is because the dynamical

systems in mechanical engineering are constrained by

engineering requirements and limitations. The tradi-

tional theory of continuous dynamical systems can not

be applied to discontinuous dynamical systems and

only makes it more complicated and difﬁcult to be

solved. Therefore, a theory applicable to discontin-

uous dynamical systems should be built.

The early study of discontinuous dynamical sys-

tems goes back to Den Hartog [1] in 1931. Den Har-

tog considered a forced oscillator with Coulomb and

viscous damping. In 1960, Levitan [2] investigated

a friction oscillator with the periodically driven base,

and also discussed the stability of the periodic motion.

In 1966, Masri and Caughey [3] discussed a discontin-

uous impact damper, and obtained the stability of the

symmetrical period-1motion of the impact damper.

More detailed discussions on the general motion of

impact dampers were also developed in Masri [4].

In 1976, Utkin [5] ﬁrst controlled dynamical system

through the discontinuity, this method is called sliding

mode control. Utkin [6] applied the sliding mode con-

trol in variable structure systems, and more detailed

theory of this method was also developed in [7] by

Utkin. In 1986, Shaw [8] investigated the non-stick

periodic motion of a dry-friction oscillator, and dis-

cussed the stability of this motion through the Poincar-

e mapping. In 1988, Filippov [9] investigated the dy-

namic behaviors of a Coulomb friction oscillator and

developed differential equations with discontinuous

right-hand sides. The analytical conditions of sliding

motion along the discontinuous boundary were devel-

oped through differential inclusion, and the existence

and uniqueness of the solution were also discussed.

Leine etal. [10] investigated the stick-slip vibration

induced by an alternate friction models through the

shooting method in 1998. In 1999, Galvanetto and

Bishop [11] discussed dynamics of a simple dynami-

cal system subjected to an elastic restoring force, vis-

cous damping and dry friction forces and studied the

non-standard bifurcations with analytical and numer-

ical tools. Pilipchuk and Tan [12] studied the friction

induced vibration of a two-degree-of-freedom friction

oscillator in 2004. In 2005, Casini and Vestroni [13]

investigated dynamics of two double-belt friction os-

cillators by means of analytical and numerical tools.

However, the dynamical behaviors of discontinu-

ous dynamical system is stilled difﬁcult to investigate.

In 2005, Luo [14] developed a general theory to study

discontinuous dynamical systems on connectable do-

mains. Luo [15] introduced the imaginary, sink and

source ﬂows, and also developed the sufﬁcient and

necessary conditions of sink and source ﬂows. More

detailed deﬁnitions and theorems can be referred to

Luo [16]. In 2008, Luo [17] deﬁned G-functions and

developed a theory to determine the ﬂow switchability

to the discontinuous boundary through G-functions.

The detailed discussion can be referred to Luo [18].

Based on this theory, lots of discontinuous models can

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Ge Chen, Jinjun Fan

E-ISSN: 2944-9006

Volume 2, 2022

be investigated easily, for example [19 −25].

In this paper, analytical conditions for stick,

non-stick and grazing motions of the double-belt

friction oscillator will be developed using the ﬂow

switchability theory of the discontinuous dynamical

systems. Different domains and boundaries for

such system are deﬁned according to the friction

discontinuity, which exhibits multiple discontinuous

boundaries in the phase space. Based on the above

domains and boundaries, the analytical conditions of

the stick motions and grazing motions are obtained

mathematically. The switching plans and basic

mappings will be deﬁned to study grazing motions.

2 Physical Model

Consider a periodically forced oscillator, attached to

a ﬁxed wall, as shown in Fig. 1. This friction-

induced oscillator includes a mass m, a spring of s-

tiffness kand a damper of viscous damping coefﬁ-

cient c. In this conﬁguration, the mass mis contin-

uously in contact with both belts which are pushed

onto the mass with a constant forced FNand possess

the same friction characteristics. The periodic driving

force A0+B0cos Ωtexerts on the mass, where A0

,B0and Ωare the constant force, excitation strength

and frequency ratio, respectively.

Figure 1: Physical model

Since the mass contacts the moving belts with

friction, the mass can move along or rest on the belt

1or belt 2surface. Further, a kinetic friction force

shown in Fig. 2is described as

Ff( ˙x)











= (µ1+µ2)FN,˙x∈[v2,+∞),

∈[(µ1−µ2)FN,(µ1+µ2)FN],˙x=v2,

= (µ1−µ2)FN,˙x∈[v1, v2],

∈[−(µ1+µ2)FN,(µ1−µ2)FN],˙x=v1,

=−(µ1+µ2)FN,˙x∈(−∞, v1],

(1)

where ˙x:= dx/dt,FNand µk(k= 1,2) are a normal

force to the contact surface and friction coefﬁcients

between the mass mand the belt k(k= 1,2), respec-

tively. Here we assume that v2> v1and µ1≥µ2.

Figure 2: Friction force

The motions of the mass in a double-belt friction

oscillator can be divided into two cases. If the mass

moves along belt 1and belt 2, the corresponding mo-

tion is called the non-stick motion. If the mass moves

together with belt 1or belt 2, the corresponding mo-

tion is called the stick motion.

For the mass moving with the same speed of the

belt 1surface, the force acting on the mass in the x-

direction is deﬁned as

Fs1=A0+B0cos Ωt−kx −c˙x+µ2FNfor ˙x=v1.

(2)

If this force cannot overcome the friction force µ1FN

(i.e., |Fs1| ≤ µ1FN), the mass does not have any rel-

ative motion to the belt 1. The equation of the motion

for the mass in such state is described as

˙x=v1,¨x= 0.(3)

For the mass moving with the same speed of the belt 2

surface, we can also obtain the equation for the mass

as follows

˙x=v2,¨x= 0.(4)

For the non-stick motions of the friction-induced

oscillator, we can obtain the equations of the motions

as follows











m¨x=A0+B0cos Ωt−kx −c˙x+ (µ1+µ2)FN

for ˙x<v1,

m¨x=A0+B0cos Ωt−kx −c˙x−(µ1−µ2)FN

for v1<˙x<v2,

m¨x=A0+B0cos Ωt−kx −c˙x−(µ1+µ2)FN

for ˙x>v2.

(5)

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3 Domains and Boundaries

From the previous discussion, there are ﬁve motion

states including three non-stick motions in the three

regions and two stick motions on the boundaries. The

phase plane can be partitioned into three domains and

two boundaries, as shown in Fig. 3. In each domain,

the motion can be described through a continuous dy-

namical system.

Figure 3: Domains and boundaries

The three domains are expressed by Ωα(α=

1,2,3 ):

Ω1=n(x, ˙x)|x∈(−∞,+∞),˙x∈(−∞, v1)o,

Ω2=n(x, ˙x)|x∈(−∞,+∞),˙x∈(v1, v2)o,

Ω3=n(x, ˙x)|x∈(−∞,+∞),˙x∈(v2,+∞)o.

(6)

The corresponding boundaries are deﬁned as:

∂Ω12 =∂Ω21 =n(x, ˙x)|x∈(−∞,+∞),˙x=v1o,

∂Ω23 =∂Ω32 =n(x, ˙x)|x∈(−∞,+∞),˙x=v2o.

(7)

Based on the above domains and boundaries, the

vectors for motions of the mass in the domains can be

introduced as follows

x(λ)= (x(λ),˙x(λ))T,F(λ)= ( ˙x(λ), F(λ))T,(8)

where λ= 1,2,3and

F(1)(x(1), t) = −c

m˙x(1) −k

mx(1) +B0

mcos Ωt

m[A0+ (µ1+µ2)FN],

F(2)(x(2), t) = −c

m˙x(2) −k

mx(2) +B0

mcos Ωt

m[A0−(µ1−µ2)FN],

F(3)(x(3), t) = −c

m˙x(3) −k

mx(3) +B0

mcos Ωt

m[A0−(µ1+µ2)FN].

(9)

From Eq. (5), the equations of the non-stick mo-

tions for the mass are rewritten in the vector form of

x(λ)=F(λ)(x(λ), t) for λ∈ {1,2,3}.(10)

For the stick motion, the equations of the motion

for the mass are rewritten in the vector form of

x(0)

(λ)=F(0)

(λ)(x(λ), t) for λ∈ {1,2}(11)

and

F(0)

(λ)(x(0)

(λ), t)=0,(12)

where

x(0)

(λ)= (x(0)

(λ),˙x(0)

(λ))T,F(0)

(λ)= (vλ, F (0)

(λ))T.

4 Analytical Conditions

By the theory of the ﬂow switchability to a speciﬁc

boundary in discontinuous dynamical system in [17],

the switching conditions of the passability, stick mo-

tions and grazing ﬂows of the double-belt friction os-

cillator will be developed in this section.

For convenience, we ﬁrst introduce some con-

cepts and several lemmas in ﬂow switching theory.

Consider a discontinuous dynamical system

x(α)≡F(α)(x(α), t, Pα)∈Rn(13)

in domain Ωα(α=i, j)which has a ﬂow x(α)

Φ(t0,x(α)

0,Pα, t)with an initial condition (t0,x(α)

0),

and on the boundary

∂Ωij =nx|ϕij (x, t, λ)=0,

ϕij is Cr−continuous (r≥1)o⊂Rn−1,

(14)

there is a ﬂow x(0)

t=Φ(t0,x(0)

0, λ, t)with an initial

condition (t0,x(0)

0). The 0-order G-functions of the

ﬂow x(α)

tto the ﬂow x(0)

ton the boundary in the nor-

mal direction of the boundary ∂Ωij are deﬁned as

G(α)

∂Ωij (x(0)

t, t±,x(α)

t±,Pα, λ)

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≡G(0,α)

∂Ωij (x(0)

t, t±,x(α)

t±,Pα, λ)

=Dx(0)

tnT

∂Ωij ·(x(α)

t±−x(0)

+tnT

∂Ωij ·(˙

x(α)

t±−˙

x(0)

t).

(15)

The 1-order G-functions for a ﬂow x(α)

tto a boundary

ﬂow x(0)

tin the normal direction of the boundary ∂Ωij

are deﬁned as

G(1,α)

∂Ωij (x(0)

t, t(α)

±,x(α)

t±,Pα, λ)

=D2

x(0)

tnT

∂Ωij ·(x(α)

t±−x(0)

+2Dx(0)

tnT

∂Ωij ·(˙

x(α)

t±−˙

x(0)

+tnT

∂Ωij ·(¨

x(α)

t±−¨

x(0)

t),

(16)

where the total derivative

Dx(0)

(·) := ∂(·)

∂x(0)

·˙

x(0)

t+∂(·)

∂t ,

the normal vector of the boundary surface ∂Ωij at

point x(0)(t)is given by

tnT

∂Ωij (x(0), t, λ) = 5ϕij (x(0),t, λ)

= ( ∂ϕij

∂x(0)

,∂ϕij

∂x(0)

,· · · ,∂ϕij

∂x(0)

(t, x(0)),

(17)

and t±=t±0.

If the ﬂow x(α)

tcontacts with the boundary at the

time tm, that is x(α)

tm=xm=x(0)

tm, and the boundary

∂Ωij is linear, independent of time t, we have

G(0,α)

∂Ωij (xm, tm,Pα, λ)

:= G(0,α)

∂Ωij (x(0)

tm, tm±,x(α)

tm±,Pα, λ)

=tnT

∂Ωij ·˙

x(α)

t



(xm,tm±),

(18)

G(1,α)

∂Ωij (xm, tm,Pα, λ)

:= G(1,α)

∂Ωij (x(0)

tm, tm±,x(α)

tm±,Pα, λ)

=tnT

∂Ωij ·¨

x(α)

t



(xm,tm±).

(19)

Here tm+and tm−are the time before approaching

and after departing the corresponding boundary, re-

spectively.

Lemma 1 [17] For a discontinuous dynamical sys-

tem ˙

x(α)=F(α)(x(α), t, Pα)∈Rn,x(tm) = xm∈

∂Ωij at time tm. For an arbitrarily small ε > 0, there

is a time interval [tm−ε, tm). Suppose x(i)(tm−) =

xm=x(j)(tm−). Both ﬂows x(i)(t)and x(j)(t)

are Cr

[tm−ε,tm)-continuous (r≥1) for time t, and

kdr+1x(α)/dtr+1k<∞(α∈ {i, j}). The necessary

and sufﬁcient conditions for a sliding motion on ∂Ωαβ

are

G(0,α)

∂Ωij (xm, tm−,Pα, λ)<0

G(0,β)

∂Ωij (xm, tm−,Pβ, λ)>0











for n∂Ωαβ →Ωα,

(20)

where α, β ∈ {i, j}and α6=β.

Lemma 2 [17] For a discontinuous dynamical sys-

tem ˙

x(α)=F(α)(x(α), t, Pα)∈Rn,x(tm) = xm∈

∂Ωij at time tm. For an arbitrarily small ε > 0, there

are two time intervals [tm−ε, tm)and (tm, tm+ε].

Suppose x(i)(tm−) = xm=x(j)(tm+). Both ﬂows

x(i)(t)and x(j)(t)are Cr

[tm−ε,tm)and Cr

(tm, tm+ε]-

continuous (r≥1) for time t, respectively, and

kdr+1x(α)/dtr+1k<∞(α∈ {i, j}. The ﬂow x(i)(t)

and x(j)(t)to the boundary ∂Ωij is semi-passable

from domain Ωito Ωjiff

either

G(0,i)

∂Ωij (xm, tm−,Pi, λ)>0

G(0,j)

∂Ωij (xm, tm+,Pj, λ)>0











for n∂Ωαβ →Ωj,

(21)

G(0,i)

∂Ωij (xm, tm−,Pi, λ)<0

G(0,j)

∂Ωij (xm, tm+,Pj, λ)<0











for n∂Ωαβ →Ωi.

(22)

Lemma 3 [17] For a discontinuous dynamical sys-

tem ˙

x(α)=F(α)(x(α), t, Pα)∈Rn,x(tm) = xm∈

∂Ωij at time tm. For an arbitrarily small ε > 0, there

is a time interval [tm−ε, tm+ε]. Suppose x(α)(tm±) =

xm. The ﬂow x(α)(t)is Cr

[tm−ε,tm+ε]-continuous

(rα≥2) for time t, and kdr+1x(α)/dtr+1k<

∞(α∈ {i, j}). A ﬂow x(α)(t)in Ωαis tangential

to the boundary ∂Ωij iff

G(0,α)

∂Ωij (xm, tm,Pα, λ)=0 for α∈ {i,j};(23)

either G(1,α)

∂Ωij (xm, tm,Pα, λ)<0for n∂Ωαβ →Ωβ,

or G(1,α)

∂Ωij (xm, tm,Pα, λ)>0for n∂Ωαβ →Ωα.











(24)

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Volume 2, 2022

More detailed theory on the ﬂow switchability

such as high-order G-functions, the deﬁnitions or the-

orems about various ﬂow passability in discontinuous

dynamical systems can be referred to [17] and [18].

From the aforementioned deﬁnitions and lemmas,

we give the analytical conditions for the ﬂow switch-

ing in the double-belt friction oscillator.

For the double-belt friction oscillator in Section 2,

the normal vectors of the boundaries ∂Ω12 and ∂Ω23

are given as

n∂Ω12 =n∂Ω21 = (0,1)T,n∂Ω23 =n∂Ω32 = (0,1)T.

(25)

The G-functions for such friction oscillator are sim-

pliﬁed as G(0,α)

∂Ωij (x(α), tm±) or G(1,α)

∂Ωij (x(α), tm±).

Theorem 4 For the double-belt friction oscillator de-

scribed in Section 2, we have the following results:

(i) The stick motion on xm∈∂Ω12 at time tm

appears iff the following conditions can be obtained:

F(1)(xm, tm−)>0and F(2)(xm, tm−)<0.(26)

(ii) The stick motion on xm∈∂Ω23 at time tm

appears iff the following conditions can be obtained:

F(2)(xm, tm−)>0and F(3)(xm, tm−)<0.(27)

Proof: From the aforementioned deﬁnitions, the 0-

order G-functions for the stick boundaries ∂Ω12 and

∂Ω23 in the double-belt friction oscillator are

G(0,1)

∂Ω12 (xm, tm±) = nT

∂Ω12 ·F(1)(xm, tm±),

G(0,2)

∂Ω12 (xm, tm±) = nT

∂Ω12 ·F(2)(xm, tm±),

(28)

and

G(0,2)

∂Ω23 (xm, tm±) = nT

∂Ω23 ·F(2)(xm, tm±),

G(0,3)

∂Ω23 (xm, tm±) = nT

∂Ω23 ·F(3)(xm, tm±).

(29)

From (25), the Eqs. (28) and (29) can be computed as:

G(0,1)

∂Ω12 (xm, tm−) = F(1)(xm, tm−),

G(0,2)

∂Ω12 (xm, tm−) = F(2)(xm, tm−),

(30)

and

G(0,2)

∂Ω23 (xm, tm−) = F(2)(xm, tm−),

G(0,3)

∂Ω23 (xm, tm−) = F(3)(xm, tm−).

(31)

By Lemma 1, the stick motion on xm∈∂Ω12 at time

tmappears iff

G(0,1)

∂Ω12 (x(m), tm−)>0and G(0,2)

∂Ω12 (x(m), tm−)<0,

(32)

i.e.

F(1)(xm, tm−)>0and F(2)(xm, tm−)<0.(33)

Therefore, (i) holds. Similarly, (ii) holds. 2

Theorem 5 For the double-belt friction oscillator de-

scribed in Section 2, we have the following results:

(i) The non-stick motion (or called passable mo-

tion to boundary) on xm∈∂Ω12 at time tmappears

iff the following condition can be obtained:

F(1)(xm, tm±)×F(2)(xm, tm∓)>0.(34)

(ii) The non-stick motion on xm∈∂Ω23 at time

tmappears iff the following condition can be ob-

tained:

F(2)(xm, tm±)×F(3)(xm, tm∓)>0.(35)

Proof: By Lemma 2, passable motion on the boundary

xm∈∂Ω12 at time tmappears iff

G(0,1)

∂Ω12 (xm, tm±)×G(0,2)

∂Ω12 (xm, tm∓)>0.(36)

By (25), we obtain

G(0,1)

∂Ω12 (xm, tm±) = F(1)(xm, tm±),

G(0,2)

∂Ω12 (xm, tm∓) = F(2)(xm, tm∓).

(37)

The Eqs. (36) and (37) implies that (i) holds. The

proof for (ii) is similar. 2

Theorem 6 For the double-belt friction oscillator de-

scribed in Section 2, we have the following results:

(i) The grazing motion on xm∈∂Ω12 at time tm

appears iff the following conditions can be obtained:

F(α)(xm, tm±) = 0 for α∈ {1,2},(38)

∇F(1)(xm, tm±)·F(1)(xm, tm±)+∂F(1)(xm, tm±)

∂tm

<0,

(39)

∇F(2)(xm, tm±)·F(2)(xm, tm±)+∂F(2)(xm, tm±)

∂tm

>0.

(40)

(ii) The grazing motion on xm∈∂Ω23 at time tm

appears iff the following conditions can be obtained:

F(α)(xm, tm±) = 0 for α∈ {2,3},(41)

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∇F(2)(xm, tm±)·F(2)(xm, tm±)+∂F(2)(xm, tm±)

∂tm

<0,

(42)

∇F(3)(xm, tm±)·F(3)(xm, tm±)+∂F(3)(xm, tm±)

∂tm

>0.

(43)

Proof: By Lemma 3, the sufﬁcient and necessary con-

ditions for the grazing ﬂows on the boundary ∂Ω12 are

G(0,α)

∂Ω12 (xm, tm±) = 0 for α= 1,2,(44)

G(1,1)

∂Ω12 (xm, tm±)<0, G(1,2)

∂Ω12 (xm, tm±)>0.(45)

From (25), (28) and (29), we have

G(0,α)

∂Ω12 (xm, tm±) = nT

∂Ω12 ·F(α)(xm, tm±)

=F(α)(xm, tm±) for α= 1,2.

(46)

From (19), we obtain

G(1,1)

∂Ω12 (xm, tm±) = nT

∂Ω12 ·Dx(0)

F(1)(xm, tm±)

= (0,1) ·Dx(0)

tm˙

x(1), F(1)(xm, t)T



(xm,tm±)

=∇F(1)(xm, tm±)·F(1)(xm, tm±)+ ∂F(1)(xm, tm±)

∂tm

(47)

Similarly,

G(1,2)

∂Ω12 (xm, tm±)

=∇F(2)(xm, tm±)·F(2)(xm, tm±)+ ∂F(2)(xm, tm±)

∂tm

(48)

From (46),(47) and (48), (i) holds. In a similar man-

ner, (ii) holds. 2

5 Switching Plan and Mappings

The switching plans are introduced as (λ= 1,2):

Σ0

(λ)={(xi,˙xi,Ωti)|˙xi=vλ},

Σ1

(λ)={(xi,˙xi,Ωti)|˙xi=v−

λ},

Σ2

(λ)={(xi,˙xi,Ωti)|˙xi=v+

λ},

(49)

where v−

λ= limδ→0(vλ−δ)and v+

λ= limδ→0(vλ+

δ)for arbitrary small δ > 0. Therefore, eight basic

mappings will be deﬁned as:

P1: Σ0

(1) →Σ0

(1), P2: Σ1

(1) →Σ1

(1),

P3: Σ2

(1) →Σ2

(1), P4: Σ0

(2) →Σ0

(2),

P5: Σ1

(2) →Σ1

(2), P6: Σ2

(2) →Σ2

(2),

P7: Σ1

(2) →Σ2

(1), P8: Σ2

(1) →Σ1

(2).

(50)

From foregoing (49) and (50), we obtain

P1: (xi, v1,Ωti)→(xi+1, v1,Ωti+1),

P2: (xi, v−

1,Ωti)→(xi+1, v−

1,Ωti+1),

P3: (xi, v+

1,Ωti)→(xi+1, v+

1,Ωti+1),

P4: (xi, v2,Ωti)→(xi+1, v2,Ωti+1),

P5: (xi, v−

2,Ωti)→(xi+1, v−

2,Ωti+1),

P6: (xi, v+

2,Ωti)→(xi+1, v+

2,Ωti+1),

P7: (xi, v−

2,Ωti)→(xi+1, v+

1,Ωti+1),

P8: (xi, v+

1,Ωti)→(xi+1, v−

2,Ωti+1).

(51)

With (11) and (12), the governing equations for

Pλ(λ= 1,4) can be described as







xi+1 =v1(ti+1 −ti) + xi,

A0+B0cos Ωti+1 −kxi+1 −cv1+µ2FN=µ1FN,

(52)







xi+1 =v2(ti+1 −ti) + xi,

A0+B0cos Ωti+1 −kxi+1 −cv2−µ1FN=µ2FN,

(53)

respectively.

For the double-belt friction oscillator, the do-

mains Ωα(α∈ {1,2,3})are unboubded. From

the basic theorems of discontinuous dynamical sys-

tem, only three possible bounded motions exist in the

three domains, from which the governing equation-

s of mapping Pλ(λ∈ {1,2,· · · ,8})are obtained.

With (51), the governing equations of each mapping

Pλ(λ∈ {1,2,· · · ,8})can be expressed as

f(λ)

1(xi,Ωti, xi+1,Ωti+1)=0,

f(λ)

2(xi,Ωti, xi+1,Ωti+1)=0.

(54)

The grazing motion occurs when a ﬂow in a do-

main is tangential to the boundary and then returns

back to this domain. The analytical conditions for the

grazing motion in the double-belt friction oscillator

were described as Lemma 3 and Theorem 6. If the

EARTH SCIENCES AND HUMAN CONSTRUCTIONS

DOI: 10.37394/232024.2022.2.7

Ge Chen, Jinjun Fan

E-ISSN: 2944-9006

Volume 2, 2022

grazing motion occurs at (xm, tm)∈∂Ωαβ (α, β ∈

{1,2,3}), more detailed theorem on the grazing mo-

tions will be developed.

For the double belt friction oscillator described in

Section 2, there are four cases of grazing motions on

the boundaries: the ﬂow in domain Ω1tangential to

the boundary ∂Ω12, the ﬂow in domain Ω2tangential

to the boundary ∂Ω21, the ﬂow in domain Ω2tangen-

tial to the boundary ∂Ω23, and the ﬂow in domain Ω3

tangential to the boundary ∂Ω32, corresponding to the

mapping P2,P3,P5and P6, respectively. With (54),

we can obtain the following theorem.

Theorem 7 For the double-belt friction oscillator de-

scribed in Section 2, there are four kinds of grazing

motions:

(i) Suppose the ﬂow in domain Ω1reaches xm∈

∂Ω12 at time tm, the grazing motion on the boundary

∂Ω12 appears (i.e. the mapping P2is tangential to the

boundary ∂Ω12) iff

mod(Ωtm,2π)∈[ 0, π +|Θcr

2|)∪( 2π− |Θcr

2|,2π]

for 0< γ2<B0

mΩ;

mod(Ωtm,2π)∈[ 0,3

2π)∪(3

2π, 2π]

for 0< γ2=B0

mΩ;

mod(Ωtm,2π)∈[ 0,2π]

for 0<B0

mΩ< γ2;

mod(Ωtm,2π)∈( 0, π )

for γ2= 0;

mod(Ωtm,2π)∈( Θcr

2, π −Θcr

2)⊂( 0, π )

for γ2<0and B0

mΩ>|γ2|;

mod(Ωtm,2π)∈{Ø}

for γ2<0and B0

mΩ<|γ2|,











(55)

where

Θcr

2= arcsin(−γ2m

B0Ω),

and

γ2=c

m¨x(1)(tm) + k

m˙x(1)(tm).

(ii) Suppose the ﬂow in domain Ω2reaches xm∈

∂Ω21 at time tm, the grazing motion on the boundary

∂Ω21 appears (i.e. the mapping P3is tangential to the

boundary ∂Ω21) iff

mod(Ωtm,2π)∈(π+|Θcr

3|,2π− |Θcr

3|)⊂(π, 2π)

for 0 < γ3<B0

mΩ;

mod(Ωtm,2π)∈{Ø}

for 0 <B0

mΩ≤γ3;

mod(Ωtm,2π)∈(π, 2π)

for γ3= 0;

mod(Ωtm,2π)∈[ 0,Θcr

3)∪(π−Θcr

3,2π]

for γ3<0 and B0

mΩ>|γ3|;

mod(Ωtm,2π)∈[ 0,π

2)∪(π

2,2π]

for γ3<0 and B0

mΩ = |γ3|;

mod(Ωtm,2π)∈[ 0,2π]

for γ3<0 and B0

mΩ<|γ3|,











(56)

where

Θcr

3= arcsin(−γ3m

B0Ω),

and

γ3=c

m¨x(2)(tm) + k

m˙x(2)(tm).

(iii) Suppose the ﬂow in domain Ω2reaches xm∈

∂Ω23 at time tm, the grazing motion on the boundary

∂Ω23 appears (i.e. the mapping P5is tangential to the

boundary ∂Ω23) iff

mod(Ωtm,2π)∈[ 0, π +|Θcr

5|)∪( 2π− |Θcr

5|,2π]

for 0< γ5<B0

mΩ;

mod(Ωtm,2π)∈[ 0,3

2π)∪(3

2π, 2π]

for 0< γ5=B0

mΩ;

mod(Ωtm,2π)∈[ 0,2π]

for 0<B0

mΩ< γ5;

mod(Ωtm,2π)∈( 0, π )

for γ5= 0;

mod(Ωtm,2π)∈( Θcr

5, π −Θcr

5)⊂( 0, π )

for γ5<0and B0

mΩ>|γ5|;

mod(Ωtm,2π)∈{Ø}

for γ5<0and B0

mΩ<|γ5|,











(57)

where

Θcr

5= arcsin(−γ5m

B0Ω),

and

γ5=c

m¨x(2)(tm) + k

m˙x(2)(tm).

(iv) Suppose the ﬂow in domain Ω3reaches xm∈

∂Ω32 at time tm, the grazing motion on the boundary

∂Ω32 appears (i.e. the mapping P6is tangential to the

EARTH SCIENCES AND HUMAN CONSTRUCTIONS

DOI: 10.37394/232024.2022.2.7

Ge Chen, Jinjun Fan

E-ISSN: 2944-9006

Volume 2, 2022

boundary ∂Ω32) iff

mod(Ωtm,2π)∈(π+|Θcr

6|,2π− |Θcr

6|)⊂(π, 2π)

for 0 < γ6<B0

mΩ;

mod(Ωtm,2π)∈{Ø}

for 0 <B0

mΩ≤γ6;

mod(Ωtm,2π)∈(π, 2π)

for γ6= 0;

mod(Ωtm,2π)∈[ 0,Θcr

6)∪(π−Θcr

6,2π]

for γ6<0 and B0

mΩ>|γ6|;

mod(Ωtm,2π)∈[ 0,π

2)∪(π

2,2π]

for γ6<0 and B0

mΩ = |γ6|;

mod(Ωtm,2π)∈[ 0,2π]

for γ6<0 and B0

mΩ<|γ6|,











(58)

where

Θcr

6= arcsin(−γ6m

B0Ω),

and

γ6=c

m¨x(3)(tm) + k

m˙x(3)(tm).

Proof: For the double-belt friction oscillator de-

scribed in Section 2, by Theorem 6, the grazing mo-

tion conditions for the ﬂow x(1)(t)in domain Ω1on

the boundary ∂Ω12 at time tmare given as

F(1)(xm, tm±)=0,(59)

∇F(1)(xm, tm±)·F(1)(xm, tm±)+∂F(1)(xm, tm±)

∂tm

<0.

(60)

With (9), the Eqs. (59) and (60) can be computed as

−c

m˙x(1)(tm)−k

mx(1)(tm) + B0

mcos Ωtm

m[A0+ (µ1+µ2)FN] = 0,

(61)

−c

m¨x(1)(tm)−k

m˙x(1)(tm)−B0Ω

msin Ωtm<0.

(62)

The grazing conditions are computed through (54),

(61) and (62). Three equations and an inequality have

four unknowns, then one unknown must be given.

From (62), the critical value for mod(Ωtm,2π)is

introduced through

Θcr

2= arcsin(−γ2m

B0Ω),

where γ2=c

m¨x(1)(tm) + k

m˙x(1)(tm), and the super-

script ”cr” represents a critical value relative to graz-

ing.

If 0< γ2<B0

mΩ, then −1<−γ2m

B0Ω<0, we

have

mod(Ωtm,2π)∈[ 0, π +|Θcr

2|)∪( 2π− |Θcr

2|,2π].

If 0< γ2=B0

mΩ, then −γ2m

B0Ω=−1, we have

mod(Ωtm,2π)∈[ 0,3

2π)∪(3

2π, 2π].

If 0<B0

mΩ< γ2, then −γ2m

B0Ω<−1, we have

mod(Ωtm,2π)∈[ 0,2π].

If γ2= 0, then −γ2m

B0Ω= 0, we have

mod(Ωtm,2π)∈( 0, π ).

If γ2<0 and B0

mΩ>|γ2|, then 0<−γ2m

B0Ω<1,

we have

mod(Ωtm,2π)∈( Θcr

2, π −Θcr

2)⊂( 0, π ).

If γ2<0 and B0

mΩ<|γ2|, then −γ2m

B0Ω>1, we

have

mod(Ωtm,2π)∈ {Ø}.

Therefore (i) holds. Similarly we can prove that (ii),

(iii) and (iv) hold. 2

6 Conclusion

In this paper, analytical results of complex motions of

a double belt friction oscillator which was subjected

to periodic excitation, linear spring-loading, damping

force and two friction forces were investigated using

the ﬂow switchability theory of the discontinuous dy-

namical systems. Different domains and boundaries

for such system were deﬁned according to the friction

discontinuity, which exhibited multiple discontinuous

boundaries in the phase space. Analytical condition-

s of the stick motions and grazing motions of such

system were obtained in the form of theorem mathe-

matically. More theories about the double belt friction

oscillator need to be investigated in the next.

Acknowledgements: This research was supported

by the National Natural Science Foundation of Chi-

na(No.11471196) and Natural Science Foundation of

Shandong Province(No. ZR2013AM005).

?Corresponding author: Jinjun Fan.

E-mail: fjj18@126.com(J.Fan).

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DOI: 10.37394/232024.2022.2.7

Ge Chen, Jinjun Fan

E-ISSN: 2944-9006

Volume 2, 2022

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DOI: 10.37394/232024.2022.2.7

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