Flow switchability of motions in a horizontal impact pair with dry friction

Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

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Research paper

Flow switchability of motions in a horizontal impact pair with dry friction Yanyan Zhang, Xilin Fu∗ School of Mathematical Sciences, Shandong Normal University, Ji’nan, 250014, PR China

a r t i c l e

i n f o

Article history: Received 24 January 2016 Revised 15 June 2016 Accepted 18 July 2016 Available online 30 July 2016 Keywords: Horizontal impact pair Dry friction Flow switchability theory Stick motions Grazing motions

a b s t r a c t Using the flow switchability theory of the discontinuous dynamical systems, the present paper is to develop mechanical complexity in a periodic-excited horizontal impact pair with dry friction. The impact pair studied models the motions of a single bolted connection which is vibrated in the plane perpendicular to the bolt axis. According to motion character, the phase space can be partitioned into several domains and boundaries, in which a continuous dynamical system is defined in each domain, and it possesses dynamical properties different from its adjacent subsystem, the boundaries have different properties and can fall into two kinds – displacement boundaries and velocity boundaries. In this paper, using G–functions defined on separation boundaries to study flow switching on corresponding boundaries, the analytical switching conditions on each boundary are developed: the sufficient and necessary conditions of occurrence and disappearance of sliding-stick motion and side-stick motion are obtained, the sufficient and necessary conditions of grazing motion appearing on velocity boundaries are also obtained, and the analytical conditions of appearance for grazing motion on displacement boundaries are preliminarily discussed. Thus it can be seen that dynamical behaviors of the horizontal impact pair with or without dry friction are essentially different, in particular flow switchability on displacement boundaries depend on whether the conditions of passable flows on velocity boundaries are satisfied. The numerical simulations are given to demonstrate the analytical results of two stick motions and grazing motions in such pair. More details of the motions for the object reaching the intersection point of displacement boundary and velocity boundary need to be considered further in the future. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Impact and friction are universal in mechanical engineering, and are the common and important contacts between two or more dynamical systems. Modeling of impact or friction in practical problems and research on their dynamical behaviors can provide information for using or controlling them. The dynamical systems derived from impact or friction have been extensively studied not only due to their universality, but also because of their strongly nonlinear, discontinuous and complex dynamical behaviors. Whenever the clearance or gaps exist in machinery, the impacts take place. To use or decrease the effect of impacts, the dynamical behaviors of the impact oscillators were widely studied. Using a difference equations, Holmes [1] studied ∗

Corresponding author. E-mail address: [email protected] (X. Fu).

http://dx.doi.org/10.1016/j.cnsns.2016.07.015 1007-5704/© 2016 Elsevier B.V. All rights reserved.

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Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

periodic motions and chaotic motions of a ball bouncing on a periodically excited table. In Heiman et al.[2], the steady state 2:1 motions and the corresponding stability for an inclined impact pair were studied by return map. Bapat and Bapat [3] investigated the dynamical behaviors of a horizontal impact pair under periodic displacement excitation, and obtained the stability regions of the periodic motion with two symmetrical impacts theoretically and numerically. Introducing discontinuities into discrete time Poincare maps, Foale and Bishop [4] investigated the complex motions of forced systems with impacts. A horizontal impact pair under sinusoidal displacement excitation was studied by Poincare mapping structure in Han et al. [5], period-1 motion with symmetrically alternative impacts was investigated in greater detail. Cheng and Xu [6] investigated the Hopf bifurcations of periodic motions with one impact under resonance cases in a two-degree-offreedom vibro-impact damper system. Zhang and Fu [7] developed the analytical conditions to predict periodic motions in an inclined impact pair using discrete mapping theory of discontinuous dynamical systems. Friction can exist wherever two or more moving parts contact each other, which derives a non-smooth and strongly nonlinear systems. For a mass-spring-dashpot with dry friction, Shaw [8] studied its periodic motion with or without sticking and corresponding stability using bifurcation theory. A non-smooth friction oscillator under self- and external excitation were investigated in Hinrichs et al. [9], the bifurcation behaviors were predicted using numerical simulations and experimental method. Ko et al. [10] studied the dynamics of a mass-damper-spring frictional system with or without disturbance theoretically and experimentally. The dynamical behaviors of a block-on-belt system with spring and harmonically external force were investigated in Cheng and Zu [11], the complex behaviors characterized by periodic, quasi-periodic and chaotic attractors were studied using numerical simulations. Pascal [12] studied the dynamics of a two-degree-of-freedom oscillator under dry friction, and obtained two families of periodic motions using analytical method. Using Fourier expansion and iteration perturbation method, Eigoli and Vossoughi [13] obtained an accurate analytical solution of vibrations of strongly nonlinear friction drive robots. For above systems, the effect of impact and friction were discussed independently, but impact and friction often presented at the same time in mechanical engineering. So it was not enough to independently study the dynamical behaviors of the impact oscillator or the friction-induced oscillator. Chin et al. [14] investigated several kinds of grazing bifurcations in a periodically forced impact oscillator with the addition of friction using Nordmark map. Bapat [15] studied the N-impactper-cycle periodic motions in an inclined impact damper with friction by theoretical predictions and numerical simulations. Periodic motions of an impact oscillator in the presence of dry friction were investigated in Cone and Zadoks [16], the corresponding stability and bifurcations were developed by numerical simulations. The dynamic response of a revolute joint in a four-bar mechanism with a clearance was discussed in Rhee and Akay [17], periodic motions of a pin were given using Poincare maps. Blazejczyk-Okolewska [18] investigated the bifurcation diagrams of an impact oscillator with external periodic force and dry friction by numerical analysis. For a rigid-body mechanisms with impact and friction in Burns and Piiroinen [19], the Brach impact mapping and an energetic impact mapping were investigated, and the results of two mappings were compared using a slender rod as a model example. The above results on impact and friction mainly discussed the dynamics of mentioned systems, but did not study their motions switching, so the complexity of dynamical behaviors cannot be fully investigated. In recent years, the theory of the discontinuous dynamical systems, which regards the domains and boundaries where impact or friction occur as timedependent, was initially formed. The phase space of the discontinuous dynamical systems was partitioned into several subdomains and boundaries in Luo [20], various fundamental flows passability and their decision theorems were given. Using this theory, Luo and Gegg [21] gave the sufficient and necessary conditions of grazing motions and stick motions in a periodically forced dry-friction oscillator, and illustration of special motions were carried out. Furthermore Luo [22,23] introduced G−functions on discontinuous boundaries for the discontinuous dynamical system to study motion complexity, and the flows passability and switching bifurcations were classified. A simplified brake system under periodic excitation was investigated in Luo [24], periodic motions and local stability were analytically predicted by mapping structure. Luo [25] developed the flow switchability theory of the discontinuous dynamical systems on time-varying domains, and applied such theory to different systems modeling practical problems. The analytical prediction of periodic motion in a horizontal impact pair under a periodic displacement excitation was presented in Guo and Luo [26], switching bifurcations and chaos were carried out. Luo and Huang [27] developed the analytical condition of flow switchability in a non-linear, friction-induced, periodical force oscillator. Using the theory of the discontinuous dynamical system, bouncing ball systems, horizontal impact pair and a generalized Fermi-acceleration oscillator were discussed in Luo and Guo [28], the analytical prediction of periodic motion with or without stick and corresponding stability analysis in these oscillators were developed. From above discussions, using the theory of the discontinuous dynamical systems, the dynamical behaviors of the impact oscillator or the friction-induced oscillator were sufficiently investigated, but the researches on the dynamical systems influenced by impact and friction simultaneously were relatively less, particularly few articles discussed the flow switchability. In this paper, the switching mechanism in a periodic-excited horizontal impact pair with dry friction are investigated using the flow switchability theory of the discontinuous dynamical systems. This impact pair studied models the motions of a single bolted connection which is vibrated in the plane perpendicular to the bolt axis. According to the occurring of impact and the direction of friction, the phase space can be divided into several domains and their boundaries, and these boundaries can fall into two categories due to their different properties - displacement boundaries and velocity boundaries. And based on such domains and boundaries, the behaviors of the object can be classified into four cases: slipping, impacting, slide-sticking due to dry friction and side-sticking due to limited clearance. By studying the flow switching on the boundaries of two adjacent domains, the sufficient and necessary conditions for the occurring and vanishing of two stick motions

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

d 2

P

91

d 2

e m

M

x(t ) Fig. 1. Physical model.

Fig. 2. The friction.

and the onset of grazing flows on the velocity boundaries are obtained, the preliminary results of the grazing flows on the displacement boundaries are developed. It can be seen that the side-stick motions and grazing motions on the displacement boundaries are essentially different between the motions of the horizontal impact pair with or without dry friction. The numerical simulations of sliding-stick motions, side-stick motions and grazing motions are given to provide a better understanding of complex dynamics of the horizontal impact pair with dry friction. About the motions of the object reaching the intersection point of displacement boundary and velocity boundary, they need to be considered further in the future.

2. Mechanical model A horizontal impact pair consists of a base with mass M and an object with mass m, which is shown in Fig. 1. The base, in which there is a horizontal clearance with length d, is driven under sinusoidal displacement excitation x(t ) = A sin(ωt + τ ) in the horizontal direction, where A, ω and τ are the excitation amplitude, frequency and phase angle, respectively. The object moves in such clearance with the effect of dry friction. A clamping force P¯ , perpendicular to the plane of motion, exerts on the object. The absolute coordinate system y(t) is for the object. If |x − y| = d/2, the object may impact the wall of the slot with ideal elastic restitution coefficient e. Assuming that the channel is uniform and m  M, so the continuous impacts and friction between the object and the base cannot affect the motion of the base. The origin of the absolute coordinates is set at the middle point of the clearance while the base is at the equilibrium position. Without loss of generality, it is assumed that maximum static friction equals to the kinetic friction at y˙ = x˙ , and the kinetic friction, which is shown in Fig. 2, is depicted as

 F¯f (y˙ )

= μk FN , ∈ [−μk FN , μk FN ], = −μk FN ,

y˙ > x˙ , y˙ = x˙ , y˙ < x˙ ,

(1)

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˙ = d ()/dt, and μk , FN are the friction coefficient and the normal force exerting on the object, respectively. So the where () normal force FN = P¯ + mg, where g is the gravitational acceleration. Due to limited clearance and friction, there are four motion states for the object: slipping in the slot; impacting the wall of the slot; sliding-stick motions due to friction and side-stick motions due to limited clearance. When the object moves in the slot and does not reach the wall of the slot, its behavior is only influenced by friction, such motion is called slipping motion or non-stick motion. The equation of the motion for the object is

my¨ = −μk (P¯ + mg) · sgn(y˙ − x˙ ), where

 sgn(y˙ − x˙ ) =

= 1, = −1,

if y˙ − x˙ > 0, if y˙ − x˙ < 0.

(2)

(3)

Simplifying Eq. (2) obtains

y¨ = −μk (P + g) · sgn(y˙ − x˙ ),

(4)

where P = P¯ /m. At the same time the equation of the motion for the base is

x¨ = −Aω2 sin(ωt + τ ).

(5)

When the object approaches the wall of the slot with nonzero relative velocity, an impact between the object and the base occurs. And then the velocity of the object will change instantaneously. This process is described as

x+ = x− , y+ = y− , |y+ − x+ | = d/2; x˙ + = x˙ − , y˙ + = [(m − Me )y˙ − + M (1 + e )x˙ − ]/(M + m ),

(6)

where ()+ and ()− denote after and before an impact between the object and the base, respectively. If the object moves together with the base for some time, such motion is called stick motion. For such motion, it can be divided into two cases: the object reaches the wall of the slot with the velocity equalling to the base’s and then tries to pass through this wall, such stick motion is called side-stick motion; another is due to friction between the object and the base, the object suspends in the base, such stick motion is called sliding-stick motion. For two stick motions, the equations of the motion for the object and the base can be shown as

x¨ = y¨ = −

M Aω2 sin(ωt + τ ). M+m

(7)

3. Domains and boundaries Because of impacts which can abruptly change the velocity of the object and friction whose direction is dependent on the direction of the relative velocity, the motions of the impact pair become discontinuous and more complicated. In order to develop analytical conditions of the flow switching, the phase space will be partitioned into different domains and boundaries in this section, and then the corresponding equations of the motions will be given on each domain through vector fields. 3.1. Domains and boundaries in absolute coordinates From the discontinuity caused by impacts and friction, the phase space in absolute coordinates can be divided into several domains and boundaries with or without side-stick motion. The phase space without side-stick motion is partitioned into two domains

1 = {(y, y˙ )| y ∈ (x − d/2, x + d/2 ), y˙ − x˙ > 0}, 2 = {(y, y˙ )| y ∈ (x − d/2, x + d/2 ), y˙ − x˙ < 0}.

(8)

The corresponding boundaries are defined as

∂ 12 = ∂ 21 = {(y, y˙ )|ϕ12 ≡ y˙ − x˙ = 0, y ∈ (x − d/2, x + d/2 )}, ∂ 1(+∞) = {(y, y˙ )| ϕ1(+∞) ≡ y − x − d/2 = 0, y˙ − x˙ > 0}, ∂ 1(−∞) = {(y, y˙ )| ϕ1(−∞) ≡ y − x + d/2 = 0, y˙ − x˙ > 0}, ∂ 2(+∞) = {(y, y˙ )| ϕ2(+∞) ≡ y − x − d/2 = 0, y˙ − x˙ < 0}, ∂ 2(−∞) = {(y, y˙ )| ϕ2(−∞) ≡ y − x + d/2 = 0, y˙ − x˙ < 0},

(9)

where equation ϕαβ = 0 determines the boundary ∂ αβ in phase space. Herein α = 1, 2 and β = ±∞ represent the infinite boundaries. The domains and boundaries without side-stick motions are pictured in Fig. 3. ∂ i(±∞ ) (i = 1, 2 ) and ∂ 12 (or ∂ 21 ) are the boundaries with different properties - the former are displacement boundaries which the object cannot pass through, and the latter are velocity boundaries on which the relative velocity can change direction. The domains

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

93

Fig. 3. Absolute domains and boundaries without side-stick motions.

Fig. 4. Absolute domains and boundaries with side-stick motions.

1 and 2 are covered with horizontal lines and vertical lines, respectively, the boundaries ∂ 1(+∞) and ∂ 2(−∞) are represented by red dashed curves, the boundaries ∂ 1(−∞ ) and ∂ 2(+∞ ) are represented by blue dashed curves, and the boundary ∂ 12 is depicted by green dashed curve. The appearance and vanishing of the side-stick motion will form new domains and boundaries in phase space. The absolute domains 1 , 2 and 3 , 4 for the object with side-stick motion are defined as

1 = {(y, y˙ )| 2 = {(y, y˙ )| 3 = {(y, y˙ )| 4 = {(y, y˙ )|

y∈ y∈ y∈ y∈

(xcr − d/2, xcr + d/2 ), y˙ − x˙ > 0}, (xcr − d/2, xcr + d/2 ), y˙ − x˙ < 0}, (−∞, xcr − d/2 ), y˙ = x˙ , y = x − d/2}, (xcr + d/2, +∞ ), y˙ = x˙ , y = x + d/2}.

(10)

The corresponding boundaries are defined as

∂ 12 = ∂ 21 = {(y, y˙ )|ϕ12 ≡ y˙ − x˙ = 0, y ∈ (xcr − d/2, xcr + d/2 )}, ∂ i3 = ∂ 3i = {(y, y˙ )| ϕi3 ≡ y − xcr + d/2 = 0, y˙ = x˙ cr }, ∂ i4 = ∂ 4i = {(y, y˙ )| ϕi4 ≡ y − xcr − d/2 = 0, y˙ = x˙ cr },

(11)

where i = 1, 2, xcr and x˙ cr represent the displacement and velocity of the base for the occurrence and disappearance of the side-stick motion. As illustrated in Fig. 4, the domains 1 and 2 are represented as the horizontal lines area and vertical lines area, respectively, the domains 3 , 4 are pictured as the mesh regions, the boundaries ∂ 14 , ∂ 23 and ∂ 24 , ∂ 13 are depicted by red and blue dashed curves, respectively, the boundary ∂ 12 is depicted as green dashed curve. From the previous definitions of domains and boundaries, the state variable vector and vector field vector in the absolute frame can be signified as

y(λ) = (y(λ) , y˙ (λ) )T ,

f(λ) = (y˙ (λ) , f (λ) )T ,

λ = 0, 1, 2, 3, 4,

(12)

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Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

where λ = 0 stands for the sliding-stick motion on boundary ∂ αβ (α = β ∈ {1, 2}), λ = 1, 2 and λ = 3, 4 stand for the slipping motions in domains 1 , 2 and the side-stick motions on the left and right wall of the slot in domains 3 , 4 , respectively. The equations of the motion for the object in the absolute coordinates are adapted in the vector form of

y˙ (λ) = f(λ) (y(λ) , t ),

λ = 0, 1, 2, 3, 4.

(13)

For the sliding-stick motion on velocity boundary ∂ αβ (α = β ∈ {1, 2}),

f (0 ) ( y (0 ) , t ) = −

M Aω2 sin(ωt + τ ), M+m

(14)

and for the non-stick motion in domain λ (λ = 1, 2 ),

f (λ) (y(λ) , t ) = (−1 )λ μk (P + g),

(15)

and for the side-stick motion in domain λ (λ = 3, 4 ),

f (λ ) ( y (λ ) , t ) = −

M Aω2 sin(ωt + τ ). M+m

(16)

Correspondingly, the equations of the motion for the base are also revised in the vector form of

x˙ (λ) = F(λ) (x(λ) , t ), λ = 0, 1, 2, 3, 4, where x(λ ) = (x(λ ) , x˙ (λ )

)T ,

F(λ ) = (x˙ (λ ) , F(λ )

(17)

)T ,

and

F(λ) (x(λ) , t ) = −Aω2 sin(ωt + τ ) (λ = 1, 2 ),

F(λ) (x(λ) , t ) = −

M Aω2 sin(ωt + τ ) (λ = 0, 3, 4 ). M+m

(18)

3.2. Domains and boundaries in relative coordinates In the absolute frame, the occurrence of impacts are dependent on the relative displacement and velocity, and the direction of friction is dependent on the direction of the relative velocity. That is, all boundaries are varying with time, so it is very difficult to develop analytical prediction for the flow switching. Therefore the relative coordinates should be introduced herein for simplicity. The displacement, velocity and acceleration of the object relative to the base are z = y − x, z˙ = y˙ − x˙ , z¨ = y¨ − x¨. The relative domains 1 , 2 and 3 , 4 for the motions are delimited as

1 = {(z, z˙ )| 2 = {(z, z˙ )| 3 = {(z, z˙ )| 4 = {(z, z˙ )|

z ∈ (−d/2, +d/2 ), z˙ > 0}, z ∈ (−d/2, +d/2 ), z˙ < 0}, z = −d/2, z˙ = 0}, z = +d/2, z˙ = 0}.

(19)

The relative boundaries are delimited as

∂ 1(+∞) = {(z, z˙ )| ϕ1(+∞) ≡ z − d/2 = 0, z˙ > 0}, ∂ 1(−∞) = {(z, z˙ )| ϕ1(−∞) ≡ z + d/2 = 0, z˙ > 0}, ∂ 2(+∞) = {(z, z˙ )| ϕ2(+∞) ≡ z − d/2 = 0, z˙ < 0}, ∂ 2(−∞) = {(z, z˙ )| ϕ2(−∞) ≡ z + d/2 = 0, z˙ < 0}, ∂ 12 = ∂ 21 = {(z, z˙ )| ϕ12 ≡ z˙ = 0, z ∈ (−d/2, d/2 )}, ∂ i3 = ∂ 3i = {(z, z˙ )| ϕi3 ≡ z˙ cr = 0, zcr = −d/2}, ∂ i4 = ∂ 4i = {(z, z˙ )| ϕi4 ≡ z˙ cr = 0, zcr = d/2},

(20)

where i = 1, 2, zcr , z˙ cr represent the relative displacement and velocity for appearance and vanishing of the side-stick motion. ∂ 12 is the velocity boundary; ∂ 1( ± ∞) , ∂ 2( ± ∞) represent the impact boundaries; ∂ i4 , ∂ i3 (i = 1, 2 ) represent the right and left side-stick boundaries, respectively; and ∂ 1( ± ∞) , ∂ 2( ± ∞) and ∂ i4 , ∂ i3 (i = 1, 2 ) are displacement boundaries. Fig. 5 depicts the relative domains and boundaries. It can be seen that the boundaries ∂ 1(+∞ ) , ∂ 2(−∞ ) are depicted as red straight lines and the boundaries ∂ 1(−∞ ) , ∂ 2(+∞ ) are depicted as blue straight lines; ∂ 12 , which is pictured as green line, also becomes a straight line on the z−axis; the side-stick domains 3 , 4 and corresponding side-stick boundaries ∂ i4 , ∂ i3 (i = 1, 2 ) become two points on the z−axis. In the relative frame, the state variable vector and vector field vector are stated as

z(λ) = (z(λ) , z˙ (λ) )T , g(λ) = (z˙ (λ) , g(λ) )T , λ = 0, 1, 2, 3, 4.

(21)

The equations of the motion are

z˙ (λ) = g(λ) (z(λ) , x(λ) , t )

with

x˙ (λ) = F(λ) (x(λ) , t ),

(22)

where λ = 0 indicates the sliding-stick motion on boundary ∂ αβ (α = β ∈ {1, 2}), λ = 1, 2 and λ = 3, 4 indicate the corresponding slipping motion in domains 1 , 2 and the side-stick motions on the left and right wall of the slot in domains 3 , 4 , respectively.

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

95

Fig. 5. Relative domains and boundaries.

For the sliding-stick motion on velocity boundary ∂ αβ (α = β ∈ {1, 2}), the equation of the motion is

g ( 0 ) ( z ( 0 ) , x ( 0 ) , t ) = 0,

(23)

and for the non-stick motion, the equations of the motion in the domain λ (λ = 1, 2 ) are

g(λ) (z(λ) , x(λ) , t ) = (−1 )λ μk (P + g) + Aω2 sin(ωt + τ ),

(24)

and for the side-stick motion, the equations of the motion in the domain λ (λ = 3, 4 ) are

g ( λ ) ( z ( λ ) , x ( λ ) , t ) = 0.

(25)

4. Motion mechanisms In this section, the analytical conditions of the flow switching from one domain into another will be developed using the theory of the discontinuous dynamical systems. The theory of flow switchability to a specific boundary in the discontinuous dynamical systems can be referred to [22,23] and [25]. 4.1. Basic theory Before presenting the switching conditions in the horizontal impact oscillator with dry friction, the fundamental theory on flow switchability of the discontinuous dynamical system will be introduced. The following notations are need: ∂ i j = ∂  ji = ¯ i ∩ ¯ j = {x| ϕi j (x, t, λ ) = 0, ϕi j is C r -continuous (r ≥ 1 )} ⊂ Rn−1 is the boundary of two adjacent domains i and −−→ j ; ∂ i j represents the boundary for the semi-passable flow from domain i into domain j ; ∂ i j represents the boundary for the non-passable flow of the first kind. Definition 4.1 [22]. Consider a dynamical system x˙ (α ) ≡ F(α ) (x(α ) , t, pα ) ∈ Rn in domain α (α ∈ {i, j}) which has a flow xt(α ) = (t0 , x0(α ) , pα , t ) with an initial condition (t0 , x0(α ) ), and on the boundary ∂ ij , there is a flow xt(0 ) = (t0 , x¯ 0(0 ) , λ, t ) with an initial condition (t0 , x¯ 0(0 ) ). The 0−order G−functions of the flow xt(α ) to the flow xt(0 ) on the boundary in the normal direction of the boundary ∂ ij are defined as

G∂(α) (xt(0) , t± , xt(±α ) , pα , λ ) = G∂(0,α ) (xt(0) , t± , xt(±α ) , pα , λ ) ij

ij

(26)

= Dx(0) t nT∂  · (xt(±α ) − xt(0) ) + t nT∂  · (x˙ t(±α ) − x˙ t(0) ). t

ij

ij

Definition 4.2 [23]. Consider a dynamical system x˙ (α ) ≡ F(α ) (x(α ) , t, pα ) ∈ Rn in domain α (α ∈ {i, j}) which has a flow xt(α ) = (t0 , x0(α ) , pα , t ) with an initial condition (t0 , x0(α ) ), and on the boundary ∂ ij , there is a flow xt(0 ) = (t0 , x¯ 0(0 ) , λ, t ) with an initial condition (t0 , x¯ 0(0 ) ). The G−functions of kth−order for a flow xt(α ) to a boundary flow xt(0 ) in the normal direction of the boundary ∂ ij are defined as

G∂(k,α ) (xt(0) , t± , xt(±α ) , pα , λ ) ij

=

k+1  s=0



−s t T Cks+1 Dk+1 n ∂ i j · xt(0)

ds xt(±α ) dt s

ds xt(0) − dt s



|(x(0) ,t,x(α ) ) . t

t±

(27)

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Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

In aforementioned definitions, the total derivative D

∂ ij at point x(0) (t) is obtained by t

n∂ i j (x(0) , t, λ ) =



ϕi j (x(0) , t, λ ) =

 ∂ϕ

ij

(0 )

∂ x1

,

(0 ) ( · ) xt

≡ ∂ ((·0)) x˙ t(0 ) + ∂∂(t· ) , the normal vector of the boundary surface ∂ xt

∂ϕi j ∂ϕi j T , · · · , (0 ) . (0 ) ∂ x2 ∂ xn (t,x(0) )

(28)

Assuming the flow reaches the boundary at time tm , that is xt(mα ) = xm = xt(m0 ) , and the boundary ∂ ij is linear independent of time t, the following equations can be obtained

G∂(0,α ) (xt(m0) , tm± , xt(mα±) , pα , λ ) = G∂(0,α ) (xm , tm± , pα , λ ) = t nT∂  · x˙ t(α ) |(xm ,tm± ) , ij ij ij G∂(1,α ) (xt(m0) , tm± , xt(mα±) , pα , λ ) = G∂(1,α ) (xm , tm± , pα , λ ) = t nT∂  · x¨ t(α ) |(xm ,tm± ) , ij ij ij ... G∂(2,α ) (xt(m0) , tm± , xt(mα±) , pα , λ ) = G∂(2,α ) (xm , tm± , pα , λ ) = t nT∂  · x t(α ) |(xm ,tm± ) . ij ij

(29)

ij

Here tm± = tm ± 0 is to depict the motion in domains rather than on the boundaries, and tm− , tm+ are the time before reaching and after leaving the corresponding boundary, respectively. Based on the G–functions, the decision theorems of semi-passable flow, non-passable flow, tangential flow to the separation boundary and switching bifurcations are stated in the form of lemma. Lemma 4.1 [25]. For a discontinuous dynamical system x˙ (α ) ≡ F(α ) (x(α ) , t, pα ) ∈ Rn , x(tm ) = xm ∈ ∂ i j 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 C[rt flow

x ( i) ( t)

and

x ( j) ( t)

m−ε ,tm )

and C(rt

m ,tm+ε ]

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

to the boundary ∂ ij is semi-passable from domain i to j iff

and

G∂( j) (xm , tm+ , p j , λ ) > 0

for n∂ i j →  j ,

) or G∂(i ( xm , tm− , pi , λ ) < 0 ij

and

G∂( j) (xm , tm+ , p j , λ ) < 0

for n∂ i j → i .

ij

< ∞ (α ∈ {i, j}). The



) either G∂(i ( xm , tm− , pi , λ ) > 0 ij

ij

dr+1 x(α ) dt r+1

(30)

Lemma 4.2 [22]. For a discontinuous dynamical system x˙ (α ) ≡ F(α ) (x(α ) , t, pα ) ∈ Rn , x(tm ) = xm ∈ ∂ i j at time tm . For an arbitrarily small ε > 0, there are a time interval [tm−ε , tm ). Suppose x(i ) (tm− ) = xm = x( j ) (tm− ). Both flows x(i) (t) and x(j) (t) are C[rt ,t ) continuous for time t and m −ε m non-passable of the first kind iff

dr+1 x(α ) dt r+1

< ∞ (α ∈ {i, j}, r ≥ 1). The flow x(i) (t) and x(j) (t) to the boundary ∂ ij is



) either G∂(i ( xm , tm− , pi , λ ) > 0 ij

and

G∂( j) (xm , tm− , p j , λ ) < 0

for n∂ i j →  j ,

) or G∂(i ( xm , tm− , pi , λ ) < 0 ij

and

G∂( j) (xm , tm− , p j , λ ) > 0

for n∂ i j → i .

ij

ij

(31)

Lemma 4.3 [25]. For a discontinuous dynamical system x˙ (α ) ≡ F(α ) (x(α ) , t, pα ) ∈ Rn , x(tm ) ≡ xm ∈ ∂ ij 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 C[rtα ,t ] − continuous (rα ≥ 2) for time t, and

drα +1 x(α ) dt rα +1

G∂(0,α ) (xm , tm , pα , λ ) = 0, ij

either or

m −ε m + ε

< ∞ (α ∈ {i, j}). A flow x(α ) (t) in α is tangential to the boundary ∂ ij iff

⎫ ⎪ ⎪ ⎬

and

G∂(1,α ) (xm , tm , pα , λ ) < 0 for ij

G∂(1,α ) (xm , tm , pα , λ ) ij

n ∂ i j → β ,

> 0 for n∂ i j

(32)

⎪ ⎪ → α . ⎭

Lemma 4.4 [23]. For a discontinuous dynamical system x˙ (α ) ≡ F(α ) (x(α ) , t, pα ) ∈ Rn , there is a point x(tm ) = xm ∈ [xm1 , xm2 ] ⊂ −−→ ∂ i j for 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 C[ti r

m−ε ,tm )

∞ (rα ≥ 2, α ∈ {i, j}). The bifurcation of the passable flow

r

and C(tj x ( i) ( t)

m ,tm+ε ]

and

− continuous for time t, respectively, and

x ( j) ( t)

drα +1 x(α ) dt rα +1

<

at xm switching to the non-passable flow of the first

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

−−→ kind on the boundary ∂ i j occurs iff

G∂( j) (xm , tm± , p j , λ ) = 0 ij

either



) G∂(i ( xm , tm− , pi , λ ) > 0,

for n∂ i j →  j ,

ij

G∂(1, j ) (xm , tm± , p j , λ ) < 0, ij

and



) G∂(i ( xm , tm− , pi , λ ) < 0, ij

[20 pt]or

G∂(1, j ) (xm , tm± , p j , λ ) > 0,

for n∂ i j

97

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ → i .⎪ ⎪ ⎭

(33)

ij

Lemma 4.5 [23]. For a discontinuous dynamical system x˙ (α ) ≡ F(α ) (x(α ) , t, pα ) ∈ Rn , there is a point x(tm ) = xm ∈ [xm1 , xm2 ] ⊂ ∂ i j for time tm . The tangential bifurcation of the flow x(j) (t) at xm on the boundary ∂ i j is termed a fragmentation bifurcation of the non-passable flow of the first kind (or called a sliding fragmentation bifurcation) iff

G∂( j) (xm , tm± , p j , λ ) = 0 ij

either



) G∂(i ( xm , tm− , pi , λ ) > 0, ij

G∂(1, j ) (xm , tm± , p j , λ ) > 0, ij

[20 pt]or

) G∂(i ( xm , tm− , pi , λ ) ij ( 1, j ) G∂  ( xm , tm± , p j , λ ) ij

and for n∂ i j →  j ,



< 0, < 0,

for n∂ i j

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ → i .⎪ ⎪ ⎭

(34)

4.2. Switching conditions From the previous definitions and lemmas, the analytical switching conditions on discontinuous boundaries in the horizontal impact pair with dry friction will be developed in this subsection. For the horizontal impact pair described in Section 2, the normal vector of the relative boundaries are obtained by

n ∂ i j = t n ∂ i j = (

∂ϕi j ∂ϕi j T , ) . ∂ z ∂ z˙

(35)

In relative frame, the discontinuous boundaries become the straight line. Thus from the constraints in Eq. (20), the normal vectors are specified as

n∂ i4 = n∂ i3 = n∂ 12 = (0, 1 )T (i = 1, 2 ),

n∂ 1(±∞) = n∂ 2(±∞) = (1, 0 )T .

(36)

α) The G-functions in relative frame are simplified as G∂(k, ( z ( α ) , tm± ) ( k = 0, 1, 2 ).  ij

4.2.1. On the velocity boundaries Due to flow passability on velocity boundary ∂ 12 or ∂ 21 , the analytical conditions of passable flows appearing, slidingstick motions onset or vanishing and two cases of grazing motions occurring on such boundaries will be considered. Theorem 4.1. For the impact pair described in Section 2, when the flow of the motion in domain i (i = 1, 2 ) comes to the boundary ∂ ij at time tm , the flow will pass through such boundary into domain  j ( j = 2, 1 ) if and only if the following conditions can be obtained

⎫ ⎫⎪ ⎪ ⎪ μ (P + g ) μk ( P + g ) ⎬⎪ ⎬ either mod (ωtm + τ , 2π ) ∈ (π + arcsin k , 2 π − arcsin ) for  →  , 1 2 2 2 Aω Aω μ (P + g ) μ (P + g ) ⎭⎪ or mod (ωtm + τ , 2π ) ∈ (arcsin k , π − arcsin k ) for 2 → 1 . ⎪ ⎪ ⎪ Aω 2 Aω 2 ⎭ [20 pt]if Aω2 ≤ μk (P + g), no passable motion. if Aω2 > μk (P + g),

(37)

Proof. The object moves in the base with nonzero relative velocity before time tm , and the relative velocity becomes zero at tm , then after tm the relative velocity changes its direction. From the flow switchability theory on the discontinuous dynamical systems, the flow in domain i (i = 1, 2 ) passes through the boundary ∂ ij into domain  j ( j = 2, 1 ). It can be predicted by Lemma 4.1. Thus 0−order G−functions on such boundaries are needed. From Eq. (29), 0−order G−functions on the boundary ∂ 12 (or ∂ 21 ) are

G∂(0,i ) (z(i ) , tm± ) = nT∂ 12 · g(i ) (z(i ) , x(i ) , tm± ), 12

(38)

98

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

where i = 1, 2 indicate the motion in domain 1 and 2 , respectively. tm is the switching time of the motion for the object on the corresponding boundary. From Eqs. (24) and (38) can be computed as

G∂(0,i ) (z(i ) , tm± ) = (−1 )i μk (P + g) + Aω2 sin(ωtm + τ ), i = 1, 2.

(39)

12

By Lemma 4.1, the passable conditions of flow switching via the boundary ∂ 12 (or ∂ 21 ) are

either G∂(0,1) (z(1) , tm− ) < 0 and G∂(0,2) (z(2) , tm+ ) < 0 12 12

for

1 → 2 ,

or G∂(0,2) (z(2) , tm− ) > 0 and G∂(0,1) (z(1) , tm+ ) > 0 12 12

for

2 → 1 .

Combining Eqs. (39) and (40) obtains

either



−μk (P + g) + Aω2 sin(ωtm + τ ) < 0, μk (P + g) + Aω2 sin(ωtm + τ ) < 0,

for



−μk (P + g) + Aω2 sin(ωtm + τ ) > 0, or μk (P + g) + Aω2 sin(ωtm + τ ) > 0.

(40)

⎫ ⎪ 1 → 2 ,⎪ ⎪ ⎬ (41)

⎪ ⎪ ⎭ for 2 → 1 .⎪

Solving inequalities (41) obtains the passable conditions of the motion from 1 to 2 and from 2 to 1 , respectively.  Remark 4.1. At time tm , the relative velocity of the object is zero, and before and after time tm , the relative velocity has different direction, this means that the flow of the motion is semi-passable flow on the velocity boundary ∂ 12 (or ∂ 21 ). By Theorem 4.1, if Aω2 is less than or equals to μk (P + g), there is no passable flow on such boundaries; if Aω2 is greater μ (P+g)

μ (P+g)

than μk (P + g), the condition that switching phase mod (ωtm + τ , 2π ) is in (π + arcsin kAω2 , 2π − arcsin kAω2 ) can guarantee that the flow comes from domain 1 into domain 2 ; the condition that the switching phase mod (ωtm + τ , 2π ) is in (arcsin μkA(ωP+2 g) , π − arcsin μkA(ωP+2 g) ) can make the flow come from domain 2 into domain 1 . Theorem 4.2. For the impact pair described in Section 2, the sliding-stick motion is existing on the boundary ∂ i j ((i, j ) = (1, 2 )or(2, 1 )) if and only if the following conditions can be obtained

if Aω2 > μk (P + g),



⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬



μk ( P + g ) μ (P + g ) , arcsin k Aω 2 Aω 2   μ (P + g ) μk ( P + g ) ⎪ ∪ π − arcsin k , π + arcsin .⎪ ⎪ ⎪ Aω 2 Aω 2 ⎪ ⎭

mod (ωtm + τ , 2π ) ∈

− arcsin

(42)

if Aω2 ≤ μk (P + g), sliding − stick motion exists.

Proof. The object moves together with the base due to friction, such motion is called sliding-stick motion. From the flow switchability theory on the discontinuous dynamical systems, the sliding-stick motion is that the flow in domain i (i = 1, 2 ) reaches the boundary ∂ i j ((i, j ) = (1, 2 ) or (2, 1 )) and moves along the boundary ∂ ij , in other words, the flow in domain i is non-passable flow of the first kind to the boundary ∂ ij . Therefore it can be predicted by Lemma 4.2. From Lemma 4.2, the sufficient and necessary condition for the sliding-stick motion is

G∂(0,1) (z(1) , tm− ) · G∂(0,2) (z(2) , tm− ) < 0. 12

(43)

12

From Eq. (39), Eq. (43) can be computed as

(−μk (P + g) + Aω2 sin(ωtm + τ )) · (μk (P + g) + Aω2 sin(ωtm + τ )) < 0.

(44)

Simplifying Eq. (44) obtains

−μk (P + g) < Aω2 sin(ωtm + τ ) < μk (P + g). Solving Eq. (45) obtains the results (42).

(45)



Remark 4.2. During the object is moving in the base, the object is affected by friction and relative periodic excitation. While relative periodic excitation cannot overcome friction, the object moves together with the base, such motion is called slidingstick motion. From Theorem 4.2, the condition Aω2 ≤ μk (P + g) can guarantee that the object will move together with the base consistently; if Aω2 is greater than μk (P + g), in order for the object to move together with the base, the switching phase

mod (ωtm + τ , 2π ) must be in (− arcsin

μk (P+g) μ (P+g) , arcsin kAω2 ) or Aω2

(π − arcsin μkA(ωP+2 g) , π + arcsin μkA(ωP+2 g) ).

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

99

Theorem 4.3. For the impact pair described in Section 2, once the sliding-stick motion is formed on boundary ∂ i j ((i, j ) = (1, 2 )or(2, 1 )), such motion will vanish if and only if the following conditions can be satisfied

⎫ ⎫⎪ ⎪ ⎪ for ∂ 12 → 1 ,⎬⎪ ⎬ ⎭ for ∂ 21 → 2 . ⎪ ⎪ ⎪ ⎪ ⎭

if Aω2 > μk (P + g),

μk ( P + g ) Aω 2 μ (P + g ) or mod (ωtm + τ , 2π ) = π + arcsin k Aω 2 2 if Aω ≤ μk (P + g), no sliding-stick motion vanishing. either

mod (ωtm + τ , 2π ) = arcsin

(46)

Proof. The object and the base move together for a long time, at time tm , the relative periodic excitation is to overcome static friction, the object begins to move in the slot with non-zero relative velocity. From the flow switchability theory on the discontinuous dynamical systems, the sliding-stick motion disappears, which means that the flow on the boundary ∂ i j ((i, j ) = (1, 2 )or(2, 1 )) will go into the domain i (i = 1, 2 ). Time tm is the switching time from non-passable flow of the first kind to semi-passable flow. Thus it can be predicted by Lemma 4.5. So high-order G−functions on boundary ∂ i j ((i, j ) = (1, 2 )or(2, 1 )) are needed. From Eq. (29), 1−order G−functions on the boundary ∂ 12 (or ∂ 21 ) are

G∂(1,i ) (z(i ) , tm± ) = nT∂ 12 · Dg(i ) (z(i ) , x(i ) , tm± ), i = 1, 2.

(47)

12

From Eq. (24), Eq. (47) can be simplified as

G∂(1,i ) (z(i ) , tm± ) = Aω3 cos(ωtm + τ ), i = 1, 2.

(48)

12

From Lemma 4.5, the criteria for disappearance of the sliding-stick motion are given by

either G∂(0,1) (z(1) , tm∓ ) = 0, G∂(1,1) (z(1) , tm∓ ) > 0, G∂(0,2) (z(2) , tm− ) > 0 for 12 12 12 or

G∂(0,2) (z(2) , tm∓ ) = 0, G∂(1,2) (z(2) , tm∓ ) < 0, G∂(0,1) (z(1) , tm− ) < 0 for 12

12

Combining Eqs. (39), (48) and (49) obtains

12



−μk (P + g) + Aω2 sin(ωtm + τ ) = 0, either Aω3 cos(ωtm + τ ) > 0, μk (P + g) + Aω2 sin(ωtm + τ ) > 0,

μk (P + g) + Aω2 sin(ωtm + τ ) = 0, 3 or Aω cos(ωtm + τ ) < 0, −μk (P + g) + Aω2 sin(ωtm + τ ) < 0,

for

∂ 12 → 1 , ∂ 21 → 2 .

(49)

⎫ ⎪ ⎪ ∂ 12 → 1 ,⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ for ∂ 21 → 2 .⎪ ⎭

Further simplification of Eq. (50) yields Eq. (46). So the vanishing conditions of the sliding-stick motion are obtained.

(50)



Remark 4.3. During the sliding-stick motion, the relative periodic excitation equals to maximum static friction at a certain time tm , and after tm the relative periodic excitation can overcome static friction, thus the velocity of the object and the base become different, which means that the object begins to slip in the slot. From Theorem 4.3, if Aω2 is less than or equals to μk (P + g), the sliding-stick motion will continue permanently; if Aω2 > μk (P + g), the condition that the switching phase μk (P+g) can guarantee that the velocity of the object will be less than the base’s, the Aω2 μ (P+g) condition that the switching phase mod (ωtm + τ , 2π ) is arcsin kAω2 can guarantee that the velocity of the object will

mod (ωtm + τ , 2π ) is π + arcsin

be greater than the base’s. In conclusion, the object cannot move together with the base in such two cases.

Theorem 4.4. For the impact pair described in Section 2, when the flow of the motion in domain i (i = 1, 2 ) comes to the boundary ∂ i j ((i, j ) = (1, 2 )or(2, 1 )) at time tm , the onset conditions of the sliding-stick motion on ∂ ij will be satisfied if and only if the following conditions can be obtained

⎫ ⎫⎪ ⎪ ⎪ μ (P + g ) either mod (ωtm + τ , 2π ) = 2π − arcsin k for 1 → ∂ 12 ,⎬⎬ 2 Aω μ (P + g ) ⎪ or mod (ωtm + τ , 2π ) = π − arcsin k for 2 → ∂ 21 .⎭⎪ ⎪ ⎭ Aω 2 if Aω2 ≤ μk (P + g), no onset of sliding-stickmotion. if Aω2 > μk (P + g),

(51)

Proof. When a flow in a domain reaches the sliding-stick boundary at time tm , and then the sliding-stick motion occurs, such time tm may be not the onset time of the sliding-stick motion on this boundary. When the conditions (40) cannot be

100

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

satisfied, the passable flow vanishes and the sliding-stick motion will begin at time tm . From the flow switchability theory on the discontinuous dynamical systems, time tm is the switching time from semi-passable flow to non-passable flow. Thus the analytical conditions of the onset of the sliding-stick motion on the boundary ∂ 12 (or ∂ 21 ) can be obtained by Lemma 4.4. From Lemma 4.4, the sufficient and necessary conditions for the onset of the sliding-stick motion are



either G∂(0,1) (z(1) , tm− ) < 0, G∂(0,2) (z(2) , tm± ) = 0, G∂(1,2) (z(2) , tm± ) > 0 for 1 → ∂ 12 , 12 12 12 G∂(0,1) (z(1) , tm± ) = 0, G∂(1,1) (z(1) , tm± ) < 0, G∂(0,2) (z(2) , tm− ) > 0 for 2 → ∂ 21 .

or

12

12

(52)

12

Using Eqs. (39) and (48), Eq. (52) can be simplified as



⎫ ⎪ ⎪ for 1 → ∂ 12 ,⎪ ⎪ ⎪ ⎪ ⎬



⎪ ⎪ ⎪ ⎪ for 2 → ∂ 21 .⎪ ⎪ ⎭

Aω2 sin(ωtm + τ ) < μk (P + g), either Aω2 sin(ωtm + τ ) = −μk (P + g), Aω3 cos(ωtm + τ ) > 0,

Aω2 sin(ωtm + τ ) = μk (P + g), or Aω2 sin(ωtm + τ ) > −μk (P + g), Aω3 cos(ωtm + τ ) < 0, Simplifying Eq. (53) can obtain the results (51).

(53)



Remark 4.4. Before time tm , the velocity of the object and the base are different, at time tm , their velocity are same, and then the relative periodic excitation cannot overcome friction, the object and the base move together. In other words, time tm is the onset time of the sliding-stick motion. From Theorem 4.4, if Aω2 > μk (P + g), the onset of sliding-stick motion on boundary ∂ 12 need condition that switching phase

μk (P+g) , the onset of sliding-stick Aω2 μ (P+g) − arcsin kAω2 ; if Aω2 is less than

mod (ωtm + τ , 2π ) is 2π − arcsin

motion on boundary ∂ 21 need condition that switching phase mod (ωtm + τ , 2π ) is π or equals to μk (P + g), the sliding-stick motion exists consistently by Theorem 4.2, so the switching from semi-passable flow to non-passable flow does not appear, there is no onset of sliding-stick motion. Theorem 4.5. For the impact pair described in Section 2, while the sliding-stick motion exists on the boundary ∂ 12 (or ∂ 21 ), the grazing motion on such boundaries will occur if and only if the following conditions can be satisfied

either or

mod (ωtm + τ , 2π ) = π /2, Aω2 = μk (P + g) on mod (ωtm + τ , 2π ) = 3π /2, Aω2 = μk (P + g) on

 ∂ 12 , ∂ 21 .

(54)

Proof. While the sliding-stick motion exists, the object moves together with the base, and static friction between them exists consistently. At time tm , the relative periodic excitation reaches maximum static friction, however the relative periodic excitation is changing periodically, and after tm , the relative periodic excitation becomes less again and cannot overcome friction, the sliding-stick motion proceeds continuously, which is called grazing motion occurs on the sliding stick boundary ∂ 12 (or ∂ 21 ) at time tm . From the flow switchability theory on the discontinuous dynamical systems, the flow of the motion is tangential to the sliding-stick boundary ∂ 12 (or ∂ 21 ) at time tm . So it can be decided by Lemma 4.3. Thus higher order G−functions on these boundaries are needed. According to Eq. (29), 2−order G−functions on the boundary ∂ 12 (or ∂ 21 ) are

G∂(2,i ) (z(i ) , tm± ) = nT∂ 12 · D2 g(i ) (z(i ) , x(i ) , tm± ), i = 1, 2.

(55)

12

From Eq. (24), Eq. (55) can be simplified as

G∂(2,i ) (z(i ) , tm± ) = −Aω4 sin(ωtm + τ ), i = 1, 2.

(56)

12

Using Lemma 4.3, the sufficient and necessary conditions of grazing motions on sliding-stick boundary ∂ 12 (or ∂ 21 ) should be obtained as

either

G∂(1,1) (z(1) , tm± ) = 0,

G∂(2,1) (z(1) , tm± ) < 0,

G∂(0,2) (z(2) , tm± ) > 0,

12

12

or



G∂(0,1) (z(1) , tm± ) = 0, G∂(0,2) (z(2) , tm± ) = 0, 12

G∂(2,2) (z(2) , tm± ) > 0, 12

12

12



G∂(1,2) (z(2) , tm± ) = 0, 12

G∂(0,1) (z(1) , tm± ) < 0, 12

⎫ ⎪ ⎪ on ∂ 12 , ⎪ ⎪ ⎬ ⎪ ⎪ on ∂ 21 .⎪ ⎪ ⎭

(57)

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

From Eqs. (39) (48) and (56), Eq. (57) can be simplified as

⎫

−μk (P + g) + Aω2 sin(ωtm + τ ) = 0,⎪ ⎬ Aω3 cos(ωtm + τ ) = 0, either on −Aω4 sin(ωtm + τ ) < 0,⎪ ⎭ 2 μk (P + g) + Aω sin(ωtm + τ ) > 0,

101

⎫ ⎪ ⎪ ⎪ ⎪ ∂ 12 ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (58)

⎪ ⎫ ⎪ ⎪ μk (P + g) + Aω2 sin(ωtm + τ ) = 0,⎪ ⎪ ⎪ ⎬ ⎪ 3 ⎪ Aω cos(ωtm + τ ) = 0, ⎪ on ∂  . 21 ⎪ 4 ⎪ −Aω sin(ωtm + τ ) > 0,⎪ ⎪ ⎭ ⎭ −μk (P + g) + Aω2 sin(ωtm + τ ) < 0,

or

Simplifying Eq. (58) obtains the results (54).



Remark 4.5. While the sliding-stick motion is going, the object and the base move together due to static friction. At time tm , the relative periodic excitation equals to maximum static friction , the following two cases may appear: the relative periodic excitation overcomes static friction, sliding-stick motion vanishes as Theorem 4.3 and slipping motion will start again; another is that the relative periodic excitation becomes less than friction again, the sliding-stick motion continues, such time tm is grazing time on sliding-stick boundary. From Theorem 4.5, if Aω2 is equal to μk (P + g), the condition that switching phase mod (ωtm + τ , 2π ) is π /2 can guarantee that the grazing motion on the sliding-stick boundary ∂ 12 occurs, the occurrence of the grazing motion on the sliding-stick boundary ∂ 21 needs the condition that switching phase mod (ωtm + τ , 2π ) is 3π /2. Thus the necessary condition of grazing motion on sliding-stick boundary ∂ 12 (or ∂ 21 ) is Aω2 = μk (P + g), however it can be obtained that the sliding-stick motion exist consistently under such condition from Theorems 4.1–4.3. Thus under the conditions of Theorem 4.5, the sliding-stick motion continues permanently. Theorem 4.6. For the impact pair described in Section 2, when the flow of the motion in domain i (i = 1, 2 ) comes to the boundary ∂ 12 ( or ∂ 21 ) at time tm , the grazing motion on such boundary is to appear if and only if the following conditions can be satisfied

if Aω2 > μk (P + g),

⎫ μk ( P + g ) ⎬ on ∂  , 12 Aω 2 μ (P + g ) or mod (ωtm + τ , 2π ) = π + arcsin k on ∂ 12 .⎭ Aω 2 if Aω2 ≤ μk (P + g), no grazing motion appears. either

mod (ωtm + τ , 2π ) = arcsin

(59)

Proof. When the object is moving in the base with non-zero relative velocity, the relative velocity equals zero at time tm , and then it restores to the original relationship. That is, if the velocity of the object is greater (or less) than the base’s before time tm , the velocity of the object is greater (or less) than the base’s after tm again. From the flow switchability theory on the discontinuous dynamical systems, the flow of the motion in domain i (i = 1, 2 ) reaches the boundary ∂ i j ((i, j ) = (1, 2 ) or (2, 1 )), and then returns back into i (i = 1, 2 ). Such motion is called grazing motion on the velocity boundary ∂ 12 (or ∂ 21 ). So the analytical conditions of grazing motion on boundary ∂ 12 ( or ∂ 21 ) appearing can be obtained by Lemma 4.3. By Lemma 4.3, the sufficient and necessary conditions for a grazing motion on boundary ∂ 12 (or ∂ 21 ) are

G∂(0,i ) (z(i ) , tm± ) = 0, 12

(−1 )i · G∂(1,i12) (z(i) , tm± ) < 0 on ∂ i j .

(60)

From Eqs. (39) and (48), Eq. (60) can be simplified as

(−1 )i · μk (P + g) + Aω2 sin(ωtm + τ ) = 0, (−1 )i · Aω3 cos(ωtm + τ ) < 0 Further simplifying Eq. (61) obtains Eq. (59).

on

∂ i j .

(61)



Remark 4.6. When the object is moving in the slot with the velocity greater (or less) than the base’s, their velocity are same at time tm , the following three cases may appear: the velocity of the object becomes less (or greater) than the base’s after time tm , which can be decided by Theorem 4.1; the object and the base move together after time tm , that is, slidingstick motion appears, which can be predicted by Theorem 4.2; the velocity of the object becomes greater (or less) than the base’s again, that is the grazing motion on velocity boundary ∂ 12 ( or ∂ 21 ) occurs at time tm . From Theorem 4.6, if Aω2 is less than or equals to μk (P + g), the grazing motion on velocity boundary ∂ 12 ( or ∂ 21 ) cannot appear; if Aω2 is greater than μk (P + g), the flow in domain 1 reaches the boundary ∂ 12 at time tm , the condition that switching phase

mod (ωtm + τ , 2π ) is arcsin

μk (P+g) can guarantee the grazing motion on such boundary occurs; the flow in domain Aω2

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2 reaches the boundary ∂ 21 at time tm , the condition that switching phase mod (ωtm + τ , 2π ) is arcsin μkA(ωP+2 g) can guarantee the grazing motion on such boundary occurs.

4.2.2. On the displacement boundaries In this subsection, the analytical conditions of appearance and vanishing of side-stick motions on boundaries ∂ i j (i = 1, 2, j = 3, 4 ) are obtained, and the preliminary results of the grazing flows on the displacement boundaries ∂ i j (i = 1, 2, j = 3, 4 ) or ∂ i(±∞ ) (i = 1, 2 ) are also obtained. Theorem 4.7. For the impact pair described in Section 2, When the flow in domain i (i = 1, 2 ) approaches the side-stick boundary ∂ i j ((i, j ) = (1, 4 ) or (2, 3 )) at time tm , the side-stick motion on the corresponding boundary appears if and only if the following conditions can be satisfied

⎫ ⎪ ⎪ ⎫⎪ ⎪ μk ( P + g ) μk ( P + g ) ⎬ ⎬ either mod (ωtm + τ , 2π ) ∈ (arcsin , π − arcsin ) on ∂  , 14 Aω 2 Aω 2 ⎪ μ (P + g ) μ (P + g ) ⎪ or mod (ωtm + τ , 2π ) ∈ (π + arcsin k , 2π − arcsin k ) on ∂ 23 .⎭⎪ ⎪ 2 2 ⎭ Aω Aω 2 [20 pt]if Aω ≤ μk (P + g), no side-stick motion. if Aω2 > μk (P + g),

(62)

Proof. The object reaches the left or right wall of the slot with zero relative velocity, and then the object is to pass through such wall but cannot, so the object and the base move together. Such motion is called side-stick motion. From the flow switchability theory on the discontinuous dynamical systems, the side-stick motion occurs when the flow in domain i (i = 1, 2 ) passes through the boundary ∂ ij into the domain  j ( j = 4, 3 ). The analytical conditions of the side-stick motion appearing can be obtained using Lemma 4.1. From Eq. (29), 0−order G−functions on the side-stick boundaries ∂ i j ((i, j ) = (1, 4 ) or (2, 3 )) are

G∂(0,i ) (z(i ) , tm± ) = nT∂  · g(i ) (z(i ) , x(i ) , tm± ), i = 1, 4, 14

14

(63)

G∂(0, j ) (z( j ) , tm± ) = nT∂  · g( j ) (z( j ) , x( j ) , tm± ), j = 2, 3. 23

23

Using Eqs. (24) and (25), Eq. (63) can be computed as

G∂(0,1) (z(1) , tm± ) = −μk (P + g) + Aω2 sin(ωtm + τ ),

G∂(0,4) (z(4) , tm± ) = 0,

14

14

G∂(0,2) (z(2) , tm± ) = μk (P + g) + Aω2 sin(ωtm + τ ),

G∂(0,3) (z(3) , tm± ) = 0.

23

(64)

23

By Lemma 4.1, the sufficient and necessary conditions of the side-stick motion occurring can be obtained as



either

G∂(0,1) (z(1) , tm− ) > 0,

G∂(0,4) (z(4) , tm+ ) > 0

on

∂ 14 ,

or

G∂(0,2) (z(2) , tm− ) < 0,

G∂(0,3) (z(3) , tm+ ) < 0

on

∂ 23 .

14

23

14

23

(65)

The appearing conditions of the side-stick motions are determined from Eqs. (64) and (65) as

either − μk (P + g) + Aω2 sin(ωtm + τ ) > 0 or

on

 ∂ 14 ,

μk (P + g) + Aω2 sin(ωtm + τ ) < 0 on ∂ 23 .

Solving inequalities (66) obtains the results (62).

(66)



Remark 4.7. When the object reaches the right (or left) wall of the slot with zero relative velocity, the following three cases can be obtained: the object leaves the wall of the slot again, and slipping motion is going continuously, such motion is grazing motion, which will be discussed later; the object and the base move together due to friction, and the interaction force between the object and the wall of the slot does not exist, such motion is the sliding-stick motion on velocity boundary, which is discussed in Theorems 4.1–4.4; the object reaches the wall of the slot and wants to move rightward (or leftward), but the right (or left) wall exist, the object has to move together with the base, such motion is called side-stick motion on displacement boundary. From Theorems 4.7 and 4.3, if Aω2 is less than or equals to μk (P + g), the object and the base move together, the sliding-stick motion exist permanently, so the side-stick motion cannot appear; if Aω2 > μk (P + g), the object reaches the right wall of the slot with zero relative velocity at time tm , the condition that the switching phase μ (P+g) μ (P+g) mod (ωtm + τ , 2π ) is in (arcsin kAω2 , π − arcsin kAω2 ) can guarantee that the object moves together with the base at the right wall of the slot; the object arrives to the left wall of the slot with zero relative velocity at time tm , the condition μ (P+g) μ (P+g) that the switching phase mod (ωtm + τ , 2π ) is in (π + arcsin kAω2 , 2π − arcsin kAω2 ) can guarantee that the object moves together with the base at the left wall of the slot.

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

103

Theorem 4.8. For the impact pair described in Section 2, once the side-stick motion exists in domain i (i = 3, 4 ), such motion will vanish at time tm if and only if the following conditions can be satisfied P+g) mod (ωtm + τ , 2π ) = arcsin μkA(ω , 2

on

P+g) mod (ωtm + τ , 2π ) = π + arcsin μkA(ω , on 2

∂ 31 ,

(67)

∂ 42 .

Proof. During the side-stick motion, the object and the base move together, however static friction exists and is changing all the time due to their different movement trends. When the relative periodic excitation changes its direction and overcomes static friction, the side-stick motion vanishes. In other words, if the relative velocity becomes negative and the periodic non-friction force overcomes static friction, the object leaves the right wall of the slot; if the relative velocity becomes positive and the periodic non-friction force overcomes static friction, the object leaves the left wall of the slot. From the flow switchability theory on the discontinuous dynamical systems, the flow in domain i (i = 3, 4 ) passes through the boundary ∂ ij into domain  j ( j = 1, 2 ), the side-stick motion disappears. So it can be obtained by Lemma 4.1. 0−order and 1−order G−functions on the side-stick boundaries ∂ i j ((i, j ) = (4, 2 ) or (3, 1 )) are defined from Eq. (29) as

G∂(0,1) (z(1) , tm± ) = nT∂  · g(1) (z(1) , x(1) , tm± ), 13

13

G∂(0,3) (z(3) , tm± ) = nT∂  · g(3) (z(3) , x(3) , tm± ), 13

13

G∂(0,2) (z(2) , tm± ) = nT∂  · g(2) (z(2) , x(2) , tm± ), 24 24

(68)

G∂(0,4) (z(4) , tm± ) = nT∂  · g(4) (z(4) , x(4) , tm± ), 24 24 G∂(1,i ) (z(i ) , tm± ) = nT∂  · Dg(i ) (z(i ) , x(i ) , tm± ), i = 1, 3, 13

13

G∂(1, j ) (z( j ) , tm± ) = nT∂  · Dg( j ) (z( j ) , x( j ) , tm± ), j = 2, 4. 24

24

From Eqs. (24) and (25), Eq. (68) are simplified as

G∂(0,1) (z(1) , tm± ) = −μk (P + g) + Aω2 sin(ωtm + τ ), 13

G∂(0,3) (z(3) , tm± ) = 0, 13

G∂(0,2) (z(2) , tm± ) = μk (P + g) + Aω2 sin(ωtm + τ ), 24

(69)

G∂(0,4) (z(4) , tm± ) = 0, 24

G∂(1,i ) (z(i ) , tm± ) = Aω3 cos(ωtm + τ ), i = 1, 3, 13

G∂(1, j ) (z( j ) , tm± ) = Aω3 cos(ωtm + τ ), j = 2, 4. 24

By Lemma 4.1, the criteria for vanishing of the side-stick motion are given as

either

G∂(0,3) (z(3) , tm− ) = 0,

G∂(1,1) (z(1) , tm+ ) > 0,

G∂(1,3) (z(3) , tm− ) > 0,

13

13

[20 pt]or



G∂(0,1) (z(1) , tm+ ) = 0, G∂(0,2) (z(2) , tm+ ) = 0, 24

G∂(1,2) (z(2) , tm+ ) < 0, 24

13

13

on



G∂(0,4) (z(4) , tm− ) = 0, 24

G∂(1,4) (z(4) , tm− ) < 0,

on

⎫ ⎪ ⎪ ∂ 31 ,⎪ ⎪ ⎬ (70)

⎪ ⎪ ∂ 42 .⎪ ⎪ ⎭

24

From Eq. (69), Eq. (70) can be simplified as

−μk (P + g) + Aω2 sin(ωtm + τ ) = 0,

Aω3 cos(ωtm + τ ) > 0

on

∂ 31 ,

μk (P + g) + Aω2 sin(ωtm + τ ) = 0,

Aω3 cos(ωtm + τ ) < 0

on

∂ 42 .

Further simplification of Eq. (71) yields the vanishing conditions of side-stick motions.

(71) 

Remark 4.8. At the beginning of the side-stick motion, the base and the object move together with nonzero interaction force, which is because the quantity of the relative periodic excitation is greater than one of static friction. In other words, the object would continue to move forward if there is no wall of the slot. However the relative periodic excitation is changing periodically and static friction is changing all the time, and then the interaction force between the wall of the slot and the object becomes zero at some time because the relative periodic excitation is getting less. From this time, the sliding-stick motion occurs and coincides with the side-stick motion. When the relative velocity turns its direction and the relative periodic excitation overcomes friction, the object leaves the wall of the slot and begins to slip in the base, so the side-stick motion disappears. From Theorem 4.8, in order for the object to leave the left wall of the slot, the switchμ (P+g) ing phase mod (ωtm + τ , 2π ) must be arcsin kAω2 , in order for the object to leave the right wall, the switching phase

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μ (P+g)

mod (ωtm + τ , 2π ) must be π + arcsin kAω2 . Comparing with the results of Theorem 4.3, the vanishing conditions of the side-stick motion and the sliding-stick motion are same. The dynamical behaviors of the horizontal impact pair without dry friction are investigated in Luo [26], many grazing motions on the displacement boundaries occur. However under the effect of friction, the grazing motions on such displacement boundaries in the horizontal impact pair with dry friction become very complicated. When the object comes to the right (or left) wall of the slot with zero relative velocity at time tm , it means that the flow of the motion for the object reaches the intersection point of the velocity boundary and the displacement boundary. Due to existence of friction, the right (or left) displacement boundary is divided into two portions. So if the object leaves the right (or left) wall after time tm again, the relative velocity of the object must change its direction, which means that the flow of the motion for the object passes through the velocity boundary from domain i (i ∈ {1, 2}) into domain j (j = i ∈ {1, 2}). If such motion is also called grazing motion on displacement boundary, which is not the grazing motion on the single displacement boundary ∂ 1(+∞ ) (or ∂ 1(−∞ ) ) or the single displacement boundary ∂ 2(+∞ ) (or ∂ 2(−∞ ) ), but the grazing motion on right (or left) displacement boundary ∂ 1(+∞ ) (or ∂ 1(−∞ ) ) and ∂ 2(+∞ ) (or ∂ 2(−∞ ) ) at the same time. Thus from the above discussion, the occurrence of such grazing motion on right (or left) displacement boundary is dependent on whether the flow in two domains 1 and 2 can be passable, that is, in order to determine the appearance of such grazing motion, the passable conditions on velocity boundary are needed, which means that it should be obtained by Theorem 4.1. Similarly, when the object comes to the right (or left) wall of the slot with zero relative velocity and acceleration at time tm , whether the grazing motion on right (or left) side-stick boundary ∂ 14 (or ∂ 23 ) and ∂ 24 (or ∂ 13 ) occurs depends on whether the passable conditions on velocity boundary are satisfied. If the relative velocity and acceleration are zero, the following results on displacement boundary ∂ i j ((i, j ) = (1, 4 ) or (2, 3 )) are satisfied

G∂(0,i ) (z(i ) , tm± ) = (−1 )i μk (P + g) + Aω2 sin(ωtm + τ ) = 0, i = 1, 2. ij

(72)

So the results on velocity boundary ∂ i j ((i, j ) = (1, 2 ) or (2, 1 )) are obtained

⎫ ⎪ ⎪ ⎪ ⎪ ( 0,2 ) on ∂ 12 .⎪ ⎪ G∂  (z(2) , tm± ) = μk (P + g) + Aω2 sin(ωtm + τ ) ⎪ 12 ⎪ ⎪ ⎪ ⎭ ⎬ = 2 μk ( P + g ) > 0 , ⎫ ⎪ G∂(0,2) (z(2) , tm± ) = μk (P + g) + Aω2 sin(ωtm + τ ) = 0,⎪ ⎪ 12 ⎬ ⎪ ⎪ ⎪ ( 0,1 ) on ∂ 21 .⎪ ⎪ G∂  (z(1) , tm± ) = −μk (P + g) + Aω2 sin(ωtm + τ ) ⎪ 12 ⎪ ⎪ ⎭ ⎭ = −2 · μk (P + g) < 0, ⎫ ⎬

G∂(0,1) (z(1) , tm± ) = −μk (P + g) + Aω2 sin(ωtm + τ ) = 0,⎪ 12

(73)

It means that the passable conditions on velocity boundary, i.e. Eq. (40) cannot be satisfied. Thus such grazing motion might not exist. During the side-stick motion, this motion coexists with the sliding-stick motion from a certain time, and the interaction force between the object and the wall of the slot becomes zero from this time. Due to changeable friction, this interaction force between the object and the wall of the slot cannot resume. So grazing motion on the side-stick boundaries cannot occur. In summary, when the relative velocity becomes zero at time tm , that is, the flow of the motion reaches the velocity boundaries ∂ 12 (or ∂ 21 ) at time tm , this state is very complicated to be determined. If the relative displacement is in (−d/2, d/2 ), the motion state can be determined by Theorem 4.1–4.6, which is similar to the case of the systems affected only by friction. However if the relative displacement is −d/2 or d/2, that is the object reaches the wall of the slot, in other words, the flow of the motion for the object also reaches the displacement boundary, thus the flow arrives to the intersection point of the velocity boundary and the displacement boundary. The motion state of the object is different from the case of the systems affected only by impact. More details of such motions should need to be considered further in the future. 5. Numerical simulations To illustrate the analytical prediction conditions of passable flows, sliding-stick motions, side-stick motions and grazing flows, the motions of the object in the horizontal impact pair with dry friction will be demonstrated in the form of time-displacement curves, time-velocity curves, phase diagram and the force diagram. The starting points of motions are described by dark-solid circular, the switching points at which the object may change its motion state are depicted by bluesolid circular. The moving boundaries - the displacement curves or the velocity curves of the base, are depicted by dark curves, and the displacement curves or the velocity curves of the object and the corresponding trajectories in phase space are shown by red curves. In force diagram, friction are depicted by dark dashed lines, and the relative periodic excitation are shown by red curves. Consider the system parameters as A = 10, ω = π , μk = 0.3, g = 9.81, P¯ = 1, e = 0.5, τ = π /6, d = 40, M = 1, m = 0.01 to demonstrate a sliding-stick motion of the object in Fig. 6. The initial conditions are t0 = 0.40, y0 = 20.0, y˙ 0 = −40.320976. The time histories of displacement and velocity are described in Fig. 6 (a) and (b), respectively. It can be

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107

105

Fig. 6. A sliding-stick motion.

Fig. 7. A side-stick motion on right wall of the slot.

Fig. 8. A grazing motion on right wall of the slot.

μ (P+g)

seen that the velocity of the object equals to the base’s at t1 = 0.725008 and mod (ωt1 + τ , 2π ) = π − arcsin kAω2 , so the sliding-stick motion occurs, the object and the base move together; at t2 = 0.941658, mod (ωt2 + τ , 2π ) = π + μ (P+g) arcsin kAω2 , the sliding-stick motion vanishes, and the velocity of the object becomes less than the base’s after t2 , the object begins to move in the slot again. The corresponding trajectory and the force diagram of the sliding-stick motion are shown in Fig. 6 (c) and (d), respectively. In Fig. 6 (d), maximum static friction equals to the non-friction force at t = t1 , t2 , and the non-friction force cannot overcome friction at t1 < t < t2 , so the sliding-stick motion is in process. Based on the system parameters in Fig. 6, d = 10 is given to demonstrate a side-stick motion on the right wall of the slot in Fig. 7. The initial conditions are t0 = 0.50, y0 = 13.660254, y˙ 0 = −15.707963. It can be seen that the side-stick motion on the right wall of the clearance appears under such initial conditions. The object and the base move together from time t0 = 0.50, but this is not sliding-stick motion at beginning. Firstly, the velocity of the object and the base are identical, but μ (P+g) μ (P+g) mod (ωt0 + τ , 2π ) is not in (π − arcsin kAω2 , π + arcsin kAω2 ), so the conditions of onset or existence of the slidingstick motion cannot be satisfied. Secondly, it is shown in Fig. 7 (d) that the external periodic excitation is greater than friction from t0 = 0.50 to t = 0.725008, so the object is trying to pass through the right wall of the slot, but cannot. Thus at the beginning of such motion, it is the side-stick motion on the right wall of the clearance. In time interval (0.725008, 0.941658) the relationship of the relative periodic excitation and friction is changing, from a certain time, the relative periodic excitation is less than static friction , the interaction force between the object and the wall of the slot vanishes, the motion coincides with the sliding-stick motion; but after t1 = 0.941658, the relative periodic excitation overcomes static friction again, and the object leaves the right wall of the slot and begins to move in the slot. From Fig. 7, the impacts between the object and wall of the slot occur at t2 = 1.631372 and t3 = 2.550256, respectively. At t4 = 2.712994, the relative velocity is zero, the flow of the motion for the object reaches the velocity boundary, and then the relative velocity of the object changes from negative to positive, so the flow passes through velocity boundary, the relative velocity of the object changes its direction. It can be shown more clearly from zoomed picture in Fig. 7 (b). Based on the system parameters in Fig. 6, d = 4.393558 is given to demonstrate a grazing motion of the object on right wall of the clearance in Fig. 8. The initial conditions are t0 = 2.414413, y0 = 7.480561, y˙ 0 = −7.916003. From the time histories of the displacement and velocity in Fig. 8 (a) and (b), the velocity of the object is greater than the base’s in time interval (2.414413, 3.0), the object is moving to the right wall of the clearance; the object reaches the right wall of the

106

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Fig. 9. Another sliding-stick motion.

slot at t1 = 3.0, and at the same time the velocity of the object equals to the base’s; after t1 = 3.0, the object leaves the right wall of the slot, and the velocity of the object becomes less than the base’s. From Fig. 8 (b), the velocity of the object μ (P+g) μ (P+g) changes direction before and after t1 = 3.0. Due to t1 = 3.0 being in (2π + π + arcsin kAω2 , 2π + 2π − arcsin kAω2 ) and Fig. 8, it can be seen that the grazing motion on right wall of the slot is also the passable motion on velocity boundary, so it can be determined by the passable conditions on velocity boundary in Theorem 4.1. Based on the above system parameters in Fig. 6, μk = 0.5, d = 9.727422 are given to demonstrate another sliding-stick motion in Fig. 9. The initial conditions are t0 = 1.994955, y0 = 1.374707, y˙ 0 = 24.553138. The time histories of the displacement and velocity are depicted in Fig. 9 (a) and (b), respectively. It can be shown that the object reaches the right wall of the slot at t1 = 2.994955, and the velocity of the object and the base are identical at the same time. And then the object μ (P+g) μ (P+g) and the base move together. However in Fig. 9 (d), mod (ωt1 + τ , 2π ) is in (π − arcsin kAω2 , π + arcsin kAω2 ), that is the relative periodic excitation cannot overcome friction, so the motion is the sliding-stick motion on velocity boundary, not side-stick motion on right displacement boundary. It is also because that the relative periodic excitation is greater than friction at the beginning of the side-stick motion, so the interaction force between the object and the wall of the slot is non-zero, but during this motions, such interaction force cannot exist. From Figs. 7−9, it can be seen that the motion states of the object are very complicated when the object reaches the wall of the slot with zero relative velocity. 6. Conclusions In this paper, a periodic-excited horizontal impact pair with dry friction, which models the motions of a single bolted connection that is vibrated in the plane perpendicular to the bolt axis, is investigated using the flow switchability theory of the discontinuous dynamical systems. According to the mechanism on impact and friction, the phase space can be partitioned into several domains and boundaries. The motion in each domain is continuous, and two adjacent domains have different dynamics, that is the corresponding boundaries are discontinuous. These boundaries can be divided into velocity boundaries and displacement boundaries according to their different properties. Based on such domains and boundaries, the dynamical behaviors of the object can be classified into four cases: slipping, impacting, two sticking motions - one is due to dry friction and another is due to limited clearance. By discussing the analytical conditions of the flow switching on discontinuous boundary using the theory of the discontinuous dynamical systems, the sufficient and necessary conditions of occurrence and vanishing of sliding-stick motion and side-stick motion, appearance of the grazing motions on velocity boundaries are obtained, the preliminary results of the grazing motions on displacement boundaries are also developed. Both theoretical analysis and numerical simulations reveal the fact that essential difference between the dynamical behaviors of the horizontal impact pair with or without dry friction exists. Under the influence of friction, the second stage of the side-stick motion coincides with the sliding-stick motion; whether the grazing motions on displacement boundaries occur depends on whether the conditions of passable flows on velocity boundaries are satisfied. More details of the motions for the object reaching the displacement boundary with zero relative velocity can be considered further in the future. Acknowledgment This work was supported by the National Natural Science Foundation of China (11571208), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123704110 0 01). References [1] [2] [3] [4] [5] [6] [7] [8]

Holmes PJ. The dynamics of repeated impacts with a sinusoidally vibrating table. J Sound Vib 1982;84(2):173-189. Heiman MS, Sherman PJ, Bajaj AK. On the dynamics and stability of an inclined impact pair. J Sound Vib 1987;114(3):535-547. Bapat CN, Bapat C. Impact-pair under periodic excitation. J Sound Vib 1988;120(1):53–61. Foale S, Bishop SR. Dynamical complexities of forced impacting systems. Philos Trans R Soc London A 1992;338(1651):547-556. Han RPS, Luo ACJ, Deng W. Chaotic motion of a horizontal impact pair. J Sound Vib 1995;181(2):231-250. Cheng JL, Xu H. Nonlinear dynamic characteristics of a vibro-impact system under harmonic excitation. J Mech Mater Struct 2006;1(2):239-258. Zhang YY, Fu XL. On periodic motions of an inclined impact pair. Commun Nonlinear Sci Numer Simul 2015;20(3):1033-1042. Shaw SW. On the dynamic response of a system with dry friction. J Sound Vib 1986;108(2):305-325.

Y. Zhang, X. Fu / Commun Nonlinear Sci Numer Simulat 44 (2017) 89–107 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

107

Hinrichs N, Oestreich M, Popp K. On the modelling of friction oscillator. J Sound Vib 1998;216(3):435-459. Ko PL, Taponat MC, Pfaifer R. Friction-induced vibration - with and without external disturbance. Tribol Int 2001;34(1):7–24. Cheng G, Zu JW. A numerical study of a dry friction oscillator with parametric and external excitations. J Sound Vib 2005;287(1):329-342. Pascal M. Dynamics of coupled oscillators excited by dry friction. J Comput Nonlinear Dyn 20 08;3(3):0310 09. Eigoli AK, Vossoughi GR. A periodic solution for friction drive microrobots based on the iteration perturbation method. Scientia Iranica 2011;18(3):368-374. Chin W, Ott E, Nusse HE. Grazing bifurcations in impact oscillators. Phys Rev E 1994;50(6):4427-4444. Bapat CN. The general motion of an inclined impact damper with friction. J Sound Vib 1995;184(3):417-427. Cone KM, Zadoks RI. A numerical study of an impact oscillator with the addition of dry friction. J Sound Vib 1995;188(5):659-683. Rhee J, Akay A. Dynamic response of a revolute joint with clearance. Mech Mach Theory 1996;31(1):121-134. Blazejczyk-Okolewska B. Study of the impact oscillator with elastic coupling of masses. Chaos Solitons Fract 20 0 0;11(15):2487-2492. Burns SJ, Piiroinen PT. The complexity of a basic impact mapping for rigid bodies with impacts and friction. Regular Chaotic Dyn 2014;19(1):20–36. Luo ACJ. A theory for non-smooth dynamic systems on the connectable domains. Commun Nonlinear Sci Numer Simul 2005;10(1):1–55. Luo ACJ, Gegg BC. Grazing phenomena in a periodically forced, friction-induced, linear oscillator. Commun Nonlinear Sci Numer Simul 2006;11(7):777–802. Luo ACJ. A theory for flow switchability in discontinuous dynamical systems. Nonlinear Anal 2008;2(4):1030-1061. Luo ACJ. Global transversality, resonance and chaotic dynamics. Singapore: World Scientific; 2008. Luo ACJ, Thapa S. Periodic motions in a simplified brake system with a periodic excitation. Commun Nonlinear Sci Numer Simul 2009;14(5):2389-2414. Luo ACJ. Discontinuous dynamical systems on time-varying domains. Beijing: Higher Education Press; 2009. Guo Y, Luo ACJ. Complex motions in horizontal impact pairs with a periodic excitation. In: ASME 2011 international design engineering technical conferences and computers and information in engineering conference. American society of mechanical engineers; 2011. p. 1339-1350. Luo ACJ, Huang JZ. Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator. Nonlinear Anal 2012;13(1):241-257. Luo ACJ, Guo Y. Vibro-impact dynamics. Oxford: John Wiley & Sons; 2012.