Chaos, Solitons & Fractals 76 (2015) 218–230
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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Stick motions and grazing flows in an inclined impact oscillator q Xilin Fu ⇑, Yanyan Zhang 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 16 October 2014 Accepted 6 April 2015
a b s t r a c t In this paper, the dynamics of an inclined impact oscillator under periodic excitation are investigated using the flow switchability theory of the discontinuous dynamical systems. Different domains and boundaries for such system are defined according to the impact discontinuity. Based on above domains and boundaries, the analytical conditions of the stick motions and grazing motions for the inclined impact oscillator are obtained mathematically, from which it can be seen that such oscillator has more complicated and rich dynamical behaviors. The numerical simulations are given to illustrate the analytical results of complex motions, and several period-1 motions period-2 motion and chaotic motion of the ball in the inclined impact oscillator are also presented. There are more theories about such impact pair to be discussed in future. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Vibro-impact systems have been drawing considerable attentions of researchers not only because impact phenomena exist widely in practical machinery and engineering, such as impact print hammer, rattling gears, heat exchangers, train suspension, pile driving and ground moling, but also because such systems have very rich and complicated dynamical behavior derived from their impact discontinuity and strong nonlinear characteristic. Among the vibro-impact systems, periodic motions and their stability analysis of the horizontal impact systems were more intensively investigated. Masri and Caughey [1] investigated a horizontal impact damper, obtained the exact solution of periodic motion with equispaced two alternative impacts per cycle and its asymptotically stable regions. An idealized impact-pair under periodically
q This work was supported by the National Natural Science Foundation of China (11171192), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123704110001). ⇑ Corresponding author. E-mail address:
[email protected] (X. Fu).
http://dx.doi.org/10.1016/j.chaos.2015.04.005 0960-0779/Ó 2015 Elsevier Ltd. All rights reserved.
shaken excitation was investigated in Bapat et al. [2], equispaced and unequispaced periodic motions were analytically predicted using the closed form solution, and such results were also compared with experimental ones. Shaw and Holmes [3] studied periodic motions and local bifurcations, chaotic motions of a single-degree-of-freedom piecewise linear oscillator under periodic force, especially an impact oscillator – the limiting case of such oscillator – was also considered. A horizontal impact pair under generic periodic displacement was investigated in Bapat and Bapat [4], by comparison between theoretical and simulation results, the result that the effect of small higher frequency components on the periodic motion were negligible was obtained. In Hinrichs et al. [5], two oscillators under periodically external excitation – a horizontal impact oscillator and a self-sustained friction oscillator, were considered by experiments and numerical simulations, respectively. Luo [6] studied the 2n alternate impacts N cycles periodic motion in a horizontal impact pair under sinusoidally periodic excitation, the stability and perioddoubling bifurcation conditions for such motion were obtained analytically and numerically. Yue [7] studied a symmetric impact oscillator between two rigid stops,
X. Fu, Y. Zhang / Chaos, Solitons & Fractals 76 (2015) 218–230
stability and bifurcation of the symmetric period n 2 motion of such oscillator were obtained using symmetric Poincare map. Besides the single-degree-of-freedom horizontal impact systems studied, many types of two-degree-of-freedom impact system were also considered. In Kember and Babitsky [8], the dynamics of a two-degreeof-freedom impact system with periodic impulse excitation were investigated using the periodic Green function method and numerical simulation. Periodic motions with two symmetrical impacts per cycle for two harmonically excited systems with symmetrical constraints were investigated analytically by Poincare maps in Luo and Zhang [9], special attention was given to Neimark-Sacker bifurcations with different conditions by numerical simulation. Luo and Lv [10] considered dynamical behaviors of a twodegree-of-freedom plastic impact oscillator, the periodic one-impact-per-cycle motion with or without sticking were studied numerically using three-dimensional impact Poincare map, suppressing bifurcation and chaotic-impact motion were investigated by several different methods. The bouncing ball systems were also one of the studied focuses. In Holmes [11], the dynamical behavior of a ball bouncing vertically on a periodically vibrating table was studied using a difference equation derived. Peterka [12] presented the transition regions between different types of impact motions, the regions of existence and stability of periodic motion in a simple ball mechanical system under periodic excitation. Luo and Han [13] investigated the dynamics of a bouncing ball impacting on a sinusoidally excited table, stability and bifurcation of periodic motions were predicted analytically using differential equations derived and Poincare mapping. Using Poincare maps, Oknin´ski and Radziszewski [14] investigated dynamics of a bouncing ball colliding with a table moving periodically and vertically under a piecewise constant velocity. An embedded vibro-impact system derived from hammer-drilling technique was investigated in Aguiar and Weber [15], the dynamical behavior of an impact hammer was studied by experimental investigation and numerical simulations. Recently Oknin´ski and Radziszewski [16] investigated the dynamical behaviors of bouncing ball moving vertically and impacting with a vibrating table under two cases – sinusoidal excitation and four cubic polynomials excitation – by analytical and numerical investigation based on Implicit Function Theorem. Due to the influence of gravity and inclination angle, the dynamics of inclined impact oscillators were more rich and complicated. The paper about such systems was relatively less. Heiman et al. [17] investigated the periodic motions with two alternate impacts per cycle and their bifurcations in an inclined impact pair under periodic excitation using Pioncare maps, the periodic motions with one impact, two impacts or three impacts per cycle of such impact pair were also investigated in Heiman et al. [18], and digital simulations in the forms of stability plots were given to illustrate the theoretical results. The dynamical motions of an inclined impact damper with friction under sinusoidal force were studied in Bapat [19], the theoretical predictions and numerical simulation of periodic motions were agreed. The two-impacts-N-cycles periodic motions in an inclined impact pair under periodically external
219
excitation were investigated using the discrete maps theory of discontinuous dynamical systems in Zhang and Fu [20], the analytical prediction for such periodic motions were obtained. The aforementioned researches paid little attention to the motion switchability, stick and grazing motions in these discontinuous dynamical systems, so the discussions on the dynamical complexity of such systems were not enough. Luo [21] gave a general theory for discontinuous dynamical systems on accessible and connectable domains, various basic passibility of flow to ðn 1Þ-dimensional boundary planes were defined and the necessary and sufficient conditions for such passibility were presented. Using this theory, the grazing bifurcation for the generic mappings, fragmentation of strange attractors of a harmonically forced, piecewise, linear systems with impacts were investigated mathematically in Luo and Chen [22]. To study singularity in discontinuous dynamical systems, the G-functions for such systems were introduced in Luo [23], the flow switchability to the separation boundary, the switching bifurcations between non-passable and passable flows were investigated using such G-functions. Furthermore Luo [24] presented the flow switchability theory of discontinuous dynamical systems on time-varying domains, and applied such theory to different practical problems, such as the sliding motion and grazing motions in a periodically forced discontinuous dynamical system with incline line boundary, the grazing motion and stick motions for a periodically traveling belt with dry friction, the dynamics of gear transmission systems. In recent years several kinds of vibro-impact systems have been well studied by above G-functions and flow switching theory for discontinuous dynamical system. Luo and Guo [25] investigated impacting chatters, grazing motions and sticking motions of an extended Fermi-acceleration oscillator between a fixed wall and a piston under periodic excitation. The switching mechanisms and complex motions of a double excited Fermi-acceleration oscillator between two pistons with different excitation were also studied in Luo and Guo [26]. Guo and Luo [27,28] applied the theory of flow switchability for discontinuous dynamical systems to obtain the analytical conditions for stick and grazing motions of a horizontal impact pair under harmonic excitation. The analytical predictions for stick and grazing motions of a ball bouncing on a periodically vibrating table were developed in Guo and Luo [29], periodic motions and chaos were also studied. The detailed discussions on bouncing ball system, horizontal impact pair, generalized Fermi-acceleration oscillator can also be referred to Luo and Guo [30]. The theory of flow switchability for discontinuous dynamical system can be applied not only to the vibro-impact systems, but also to friction system, pulse phenomena of impulsive differential systems, synchronization of two differential dynamical systems. Luo and Thapa [31] investigated the singularity and switchability of periodic motions in a simplified brake system with periodic excitation. The dynamics of a train suspension system were studied in Luo and O’Conner [32], the analytical conditions for possible wall stick between the wedge and surrounding wall were obtained. Sun and Fu [33] developed the analytical conditions for synchronization of the Van
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der Pol equation with a sinusoidally forced pendulum using the theory of discontinuous dynamical systems. For a class of second-order impulsive switched systems – a certain kind of Van der Pol equations, the chatter dynamics of such system were discussed in Fu and Zheng [34], sufficient conditions to keep the pulse phenomena absent were obtained. The main goal of this paper is to study the analytical prediction conditions of stick and grazing motions in an inclined impact pair with periodically external excitation using the flow switchability theory of discontinuous dynamical systems. Different domains and boundaries for such impact system are defined due to impact discontinuity. The analytical conditions of the particular and typical motions for such system are presented mathematically based on above domains and boundaries. From the results obtained in this paper, it can be seen that the dynamical behaviors of the inclined impact pair are more complicated and rich than ones of the horizontal impact pair or the bouncing ball system. The numerical simulations are given to illustrate the analytical results of the complex motions. The period-1 motion and period-2 motion of the ball only impacting the lower hand of the inclined clearance are presented. The periodic motion and chaotic motion of the ball alternatively impacting the upper hand and lower hand of the inclined slot are also illustrated. About the inclined impact oscillator, there are more dynamical theories, such as more periodic motions with or without sticking or grazing motions, the predictions on bifurcation trees or parameter maps, to be discussed in the next, which will be reported in the future. 2. Physical model An inclined impact oscillator, which consists of a base with mass Mand a ball with mass m, is shown in Fig. 1. Inside the base, there is an rectangular clearance with inclined angle h and length d, and the ball moves freely without friction in this rectangle. The base is driven under periodically shaken displacement excitation XðtÞ in the horizontal direction. The elastic restitution coefficient for impacts between the ball and the base is e. Assume m M, so the repeated impacts of the ball can not change the motion of the base. The motion of the frame is controlled by a sinusoidal displacement as
X ¼ A sinðxt þ sÞ; X_ ¼ Ax cosðxt þ sÞ; € ¼ Ax2 sinðxt þ sÞ; X
ð1Þ
where X_ ¼ dX=dt is time derivative of X, and A; x and s are the amplitude, frequency and phase angle of the frame, respectively. The motions of the ball in the inclined impact oscillator can be divided into two cases. If the ball moves freely between the lower and upper walls of the inclined rectangle and does not move together with the base, the corresponding motion is called the non stick motion or free flight motion. If the ball arrives to lower or upper hand of the inclined slot and moves together with the base, the corresponding motion is called the stick motion. For the non-stick motion, there are two kinds of interaction between the ball and walls of the slot, which are impact and chatter. Impact means that the ball comes to one wall of the slot with some velocity, hits the wall, and then leaves this wall, chatter occurs when the ball impacts lower (or upper) wall of the slot many times continuously. The equation of the motion for the ball in such state is described as
€x ¼ g sin h;
ð2Þ
where g is the gravitational acceleration. At the same time the motion of the base remains
€ ¼ Ax2 sinðxt þ sÞ: X
ð3Þ
The following equations can be obtained from the Eq. (2) for t 2 ðt k ; tkþ1 Þ
1 x ¼ g sin hðt2 t2k Þ þ ðg sin ht k þ x_ þ ðt k ÞÞðt tk Þ þ xþ ðt k Þ; 2 x_ ¼ g sin hðt t k Þ þ x_ þ ðtk Þ; ð4Þ þ
_þ
where x ðt k Þ and x ðt k Þ are the displacement and the velocity of the ball immediately after the impact at time tk , respectively. According to m M and the conservation law of momentum, the impact process between the ball and the frame is described as
xþ ¼ x ; X þ ¼ X ; jxþ X þ cos hj ¼ d=2; X_ þ ¼ X_ ; x_ þ ¼ ½ðm MeÞx_ þ Mð1 þ eÞX_ cos h=ðM þ mÞ; ð5Þ þ
where ðÞ and ðÞ denote after and before an impact between the frame and the ball, respectively. For the stick motion, the motion of the ball can be shown as
€x ¼
M Ax2 sinðxt þ sÞ cos h Mþm
ð6Þ
and the equation of the motion for the base is
€¼ X
M Ax2 sinðxt þ sÞ: Mþm
ð7Þ
3. Domains and boundaries
Fig. 1. Physical model.
Due to impacts between the ball and the base, the motions of the ball become discontinuous and more
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complicated. In order to determine the switching complexity for the motions of the ball, different domains and boundaries in absolute and relative coordinates are defined in this section, respectively. 3.1. Domains and boundaries in absolute coordinates The origin of the absolute coordinates is set at the middle point of the inclined rectangle when the base is at the equilibrium position. The domains and boundaries with or without sticking in absolute coordinates are shown in Figs. 2 and 3, respectively. For the free-flight motion of the ball, the absolute domain X0 is defined as
d 2
_ x 2 X cos h ; X cos h þ X0 ¼ fðx; xÞj
d ; x_ 2 ð1; þ1Þg: 2 ð8Þ
The corresponding impact-chatter boundaries are defined as
n o _ u0ðþ1Þ x X cos h d=2 ¼ 0; x_ – X_ cosh ; @ X0ðþ1Þ ¼ ðx; xÞj n o _ u0ð1Þ x X cos h þ d=2 ¼ 0; x_ – X_ cosh ; @ X0ð1Þ ¼ ðx; xÞj ð9Þ
where equation uab ¼ 0 determines the boundary @ Xab in phase space. Herein a ¼ 0 and b ¼ 1 represent the permanent boundary, which means a flow of a dynamical system in a subdomain can not pass through such boundary into another subdomain without any transport law. The domain and boundaries for motions of the ball without sticking are sketched in Fig. 2. The domain X0 is represented by dotted area and the impact-chatter boundaries @ X0ðþ1Þ ; @ X0ð1Þ are depicted by the dashed curves x ¼ X cos h þ d=2; x ¼ X cos h d=2, respectively. For such system, stick motion between the ball and the base can exist under certain conditions, such appearance and vanishing of stick motion will form new domains and boundaries. For the stick motion of the ball, the absolute domains X0 and X1 ; X2 are defined as n
o
_ x 2 ðX cr cos h d=2;X cr cos h þ d=2Þ; x_ – X_ cos h ; X0 ¼ ðx; xÞj n
o
n
o
_ x 2 ð1; X cr cos h d=2Þ; x_ ¼ X_ cos h; x ¼ X cos h d=2 ; X1 ¼ ðx; xÞj _ x 2 ðX cr cos h þ d=2;þ1Þ; x_ ¼ X_ cos h; x ¼ X cos h þ d=2 : X2 ¼ ðx; xÞj ð10Þ
The corresponding stick boundaries are defined as n o _ u10 x X cr cos h þ d=2 ¼ 0; x_ ¼ X_ cr cosh ; @ X01 ¼ @ X10 ¼ ðx; xÞj n o _ u20 x X cr cos h d=2 ¼ 0; x_ ¼ X_ cr cosh ; @ X02 ¼ @ X20 ¼ ðx; xÞj ð11Þ
where X cr and X_ cr represent the displacement and velocity of the base for appearance and vanishing of the stick motion. As sketched in Fig. 3, the domain X0 is represented by the dotted area, X1 ; X2 are represented by shaded regions, the corresponding absolute boundaries @ X01 ; @ X02 are depicted by the dashed curves x ¼ X cr cos h d=2; x ¼ X cr cos h þ d=2, respectively. Based on the above domains and boundaries, the vectors for absolute motions of the ball in domains can be represented as T
xðkÞ ¼ ðxðkÞ ; x_ ðkÞ Þ ;
Fig. 2. Absolute domain and boundaries without stick.
T
f ðkÞ ¼ ðx_ ðkÞ ; f ðkÞ Þ ;
k ¼ 0; 1; 2;
ð12Þ
where k ¼ 0 and k ¼ 1; 2 give the free-flight motion in domain X0 and the stick motions on the lower hand and upper hand of the rectangular clearance in domains X1 and X2 , respectively. The equations of the motion for the ball in absolute coordinates is rewritten in the vector form of
x_ ðkÞ ¼ f ðkÞ ðxðkÞ ; tÞ;
k ¼ 0; 1; 2:
ð13Þ
For the free-flight motion ðk ¼ 0Þ,
f ð0Þ ðxð0Þ ; tÞ ¼ g sin h;
ð14Þ
and for the stick motion ðk ¼ 1; 2Þ,
f ðkÞ ðxðkÞ ; tÞ ¼
M Ax2 sinðxt þ sÞ cos h: Mþm
ð15Þ
In the corresponding domains, the equations of the motion for the base in absolute coordinates are also revised in the vector forms of
X_ ðkÞ ¼ FðkÞ ðXðkÞ ; tÞ; k ¼ 0; 1; 2; Fig. 3. Absolute domains and boundaries with stick.
ð16Þ
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and 2
F ð0Þ ðXð0Þ ; tÞ ¼ Ax sinðxt þ sÞ; M Ax2 sinðxt þ sÞ; F ðkÞ ðXðkÞ ; tÞ ¼ Mþm where XðkÞ
ð17Þ
T yðkÞ ¼ ðyðkÞ ; y_ ðkÞ Þ ;
T T ¼ ðX ðkÞ ; X_ ðkÞ Þ ; FðkÞ ¼ ðX_ ðkÞ ; F ðkÞ Þ .
shows the relative domains and boundaries for the motion of the ball. The domain X0 is also represented by the dotted area, the impact-chatter boundaries @ X0ðþ1Þ ; @ X0ð1Þ are shown by the dashed lines y ¼ þd=2; y ¼ d=2, respectively, and the stick domains and boundaries become two points, which are the solid dots in line y ¼ d=2. The relative domains X0 and X1 ; X2 for the motions of the ball are defined as
_ u0ðþ1Þ y d=2 ¼ 0; y_ – 0g; @ X0ðþ1Þ ¼ fðy; yÞj _ u0ð1Þ y þ d=2 ¼ 0; y_ – 0g; @ X0ð1Þ ¼ fðy; yÞj
ð18Þ
ð21Þ
ð22Þ
where k ¼ 0 and k ¼ 1; 2 indicate the corresponding freeflight motion in domain X0 and the stick motions on the lower hand and upper hand of the inclined slot in domain X1 and X2 , respectively. For the free-flight motion ðk ¼ 0Þ,
g ð0Þ ðyð0Þ ; Xð0Þ ; tÞ ¼ g sin h þ Ax2 sinðxt þ sÞ cos h;
ð23Þ
and for the stick motion ðk ¼ 1; 2Þ,
g ðkÞ ðyðkÞ ; XðkÞ ; tÞ ¼ 0:
ð24Þ
4. Switching conditions Using the theory of flow switchability to a specific boundary in discontinuous dynamical systems in [23,24], the switching conditions of the stick motions and grazing flows of the inclined impact oscillator will be developed in this section. 4.1. Basic theory
and
ð19Þ
and the relative stick boundaries @ X0i and @ Xi0 ði ¼ 1; 2Þ for the ball, respectively, are determined by
_ u10 y_ cr ¼ 0; ycr ¼ d=2g; @ X01 ¼ @ X10 ¼ fðy; yÞj _ u20 y_ cr ¼ 0; ycr ¼ d=2g; @ X02 ¼ @ X20 ¼ fðy; yÞj
k ¼ 0; 1; 2:
y_ ðkÞ ¼ gðkÞ ðyðkÞ ; XðkÞ ; tÞ with X_ ðkÞ ¼ FðkÞ ðXðkÞ ; tÞ;
Using the absolute coordinates, it is very difficult to develop the analytical conditions for the complex motions of the ball in the inclined impact pair because the boundaries are dependent on the time, thus the relative coordinates are needed herein for simplicity. The displacement, velocity and acceleration of the ball relative to the frame € cos h. Fig. 4 €¼€ are y ¼ x X cos h; y_ ¼ x_ X_ cos h; y xX
The relative impact-chatter boundaries @ X0ðþ1Þ @ X0ð1Þ are defined as
T gðkÞ ¼ ðy_ ðkÞ ; g ðkÞ Þ ;
The equation of the relative motion for the ball is
3.2. Domains and boundaries in relative coordinates
_ y 2 ðd=2; þd=2Þ; y_ 2 ð1; þ1Þg; X0 ¼ fðy; yÞj _ y ¼ d=2; y_ ¼ 0g; X1 ¼ fðy; yÞj _ y ¼ þd=2; y_ ¼ 0g: X2 ¼ fðy; yÞj
displacement and velocity of the ball for appearance and vanishing of the stick motion. In the relative coordinates, the motions for the ball in the vector form are expressed as
ð20Þ
where @ X01 ; @ X02 represent the lower and upper stick boundaries, respectively. ycr ; y_ cr represent the relative
Before discussing the analytical switching conditions for complex motions in the inclined impact oscillator, the fundamental theory on flow switchability of discontinuous dynamical system will be presented, that is, concepts of Gfunctions and the decision theorems of semi-passable flow, tangential flow to a separation boundary are stated in the following, respectively. Definition 4.1 [23]. Consider a dynamical system x_ ðaÞ FðaÞ ðxðaÞ ; t; pa Þ 2 Rn in domain Xa ða 2 fi; jgÞ which ða Þ
has a flow xt ða Þ ðt 0 ; x0 Þ r
ðaÞ
¼ Uðt0 ; x0 ; pa ; tÞ with an initial condition
and on the boundary @ Xij ¼ fxj uij ðx; t; kÞ ¼ 0; uij ð0Þ
is C continuous ðr P 1Þg Rn1 , there is a flow xt ð0Þ Uðt0 ; x0 ; k; tÞ
¼
0 Þ. The with an initial condition ðt0 ; x ða Þ
ð0Þ
0 order Gfunctions of the flow xt to the flow xt on the boundary in the normal direction of the boundary @ Xij are defined as
ðaÞ ð0Þ ðaÞ ð0;aÞ ð0Þ ðaÞ G@ Xij xt ; t ; xt ; pa ; k ¼ G@ Xij xt ; t ; xt ; pa ; k ðaÞ ð0Þ ¼ Dtxð0Þ nT@Xij xt xt t ðaÞ ð0Þ þt nT@ Xij x_ t x_ t :
ð25Þ
Definition 4.2 [24]. Consider a dynamical system x_ ðaÞ FðaÞ ðxðaÞ ; t; pa Þ 2 Rn in domain Xa ða 2 fi; jgÞ which Fig. 4. Relative domains and boundaries.
ðaÞ
has the flow xt
ðaÞ
¼ Uðt 0 ; x0 ; pa ; tÞ with the initial condition
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X. Fu, Y. Zhang / Chaos, Solitons & Fractals 76 (2015) 218–230 ðaÞ
ðt0 ; x0 Þ and on the boundary @ Xij ¼ fxj uij ðx; t; kÞ ¼ 0;
uij is C r continuous ðr P 1Þg Rn1 , there is a flow ð0Þ
xt
ð0Þ
ð0Þ
¼ Uðt0 ; x0 ; k; tÞ with the initial condition ðt 0 ; x0 Þ. The
1 order Gfunctions for a flow
ðaÞ xt
to a boundary flow
ð0Þ
xt in the normal direction of the boundary @ Xij are defined as
t ð1;aÞ ð0Þ ðaÞ ðaÞ ð0Þ G@ Xij xt ; t ; xt ; pa ; k ¼ D2xð0Þ nT@Xij xt xt t ðaÞ ð0Þ þ 2Dtxð0Þ nT@Xij x_ t x_ t t €ðtaÞ x €tð0Þ : þt nT@ Xij x
ð0;aÞ
ð26Þ t
@ Xij at point xð0Þ ðtÞ is given by t
n@ Xij ðxð0Þ ; t; kÞ ¼ 5uij ðxð0Þ ; t; kÞ ¼
@ uij @ uij @ uij ; ð0Þ ; ; ð0Þ ð0Þ @x1 @x2 @xn
!T :
ð27Þ
ðt;xð0Þ Þ
ð0Þ
time tm , that is xtm ¼ xm ¼ xtm , and the boundary @ Xij is linear independent of time t, we obtained
ð0;aÞ ð0Þ ðaÞ ð0;aÞ G@ Xij xtm ; t m ; xtm ; pa ; k ¼ G@Xij ðxm ; t m ; pa ; kÞ ðaÞ
m
ð0Þ
ðaÞ
m
ð1;aÞ
G@ Xij ðxtm ; tm ; xtm ; pa ; kÞ ¼ G@ Xij ðxm ; tm ; pa ; kÞ ðaÞ
€t jðxð0Þ ;t ;xðaÞ Þ : ¼ t nT@ Xij x m m
ð1;aÞ
ð31Þ
More detailed theory on the flow switchability such as high-order G-functions, the definitions or the decision theorems about various flow passibility in discontinuous dynamical systems can be referred to [23,24].
m
From the aforementioned definitions and lemmas, the analytical conditions for the flows switching in the inclined impact oscillator will be developed in the following theorems in this subsection. For the inclined impact oscillator described in Section 2, the normal vector of the relative boundaries are given as
n@ Xij ¼t n@ Xij ¼
¼ t nT@ Xij x_ t jðxð0Þ ;tm ;xðaÞ Þ ; ð1;aÞ
or G@ Xij ðxm ; tm ; pa ; kÞ > 0 for n@ Xij ! Xa : ;
4.2. Main results
Considering the flow contacts with the boundary at ðaÞ
ð30Þ
9 ð1;aÞ either G@Xij ðxm ; t m ; pa ; kÞ < 0 for n@Xij ! Xb ; =
ð0Þ x_ t þ @ðÞ , the normal vector of the boundary surface @t
ð0Þ
@xt
xðaÞ ðtÞ in Xa is tangential to the boundary @ Xij iff
G@ Xij ðxm ; tm ; pa ; kÞ ¼ 0;
In above definitions, the total derivative Dxð0Þ ðÞ @ðÞ
Lemma 4.2 [23]. For a discontinuous dynamical system x_ ðaÞ FðaÞ ðxðaÞ ; t; pa Þ 2 Rn ; xðtm Þ xm 2 @ Xij at time tm . For an arbitrarily small e > 0, there is a time interval ½tme ; tmþe . Suppose xðaÞ ðt m Þ ¼ xm . The flow xðaÞ ðtÞ is C r½tame ;tmþe contin r þ1 a xðaÞ uous ðr a P 2Þ for time t, and d dtra þ1 < 1 ða 2 fi; jg). A flow
ð28Þ
Here tm ¼ t m 0 is to show the motion in different domains rather than on the boundaries, and t m ; t mþ are the time before approaching and after departing the corresponding boundary, respectively. Based on the G-functions, the decision theorems of semi-passable flow, tangential flow to the separation boundary are stated in the form of lemma.
@ uij @ uij T ; : @y @ y_
ð32Þ
Through the relative motion, the line boundaries in the relative coordinates frame take the place of the curve boundaries in the absolute coordinates frame. Thus from the line constraints in Eqs. (19) and (20), the normal vectors n@ X01 ; n@ X02 and n@ X0ðþ1Þ ; n@ X0ð1Þ are determined, respectively, as
n@ X01 ¼ n@ X02 ¼ ð0; 1ÞT ;
n@ X0ðþ1Þ ¼ n@ X0ð1Þ ¼ ð1; 0ÞT :
ð33Þ
The G-functions in relative coordinates for such impact ð0;aÞ
ð1;aÞ
oscillator are simplified as G@Xij ðyðaÞ ; tm Þ or G@ Xij ðyðaÞ ; t m Þ.
Lemma 4.1 [24]. For a discontinuous dynamical system x_ ðaÞ FðaÞ ðxðaÞ ; t; pa Þ 2 Rn ; xðtm Þ ¼ xm 2 @ Xij at time tm . For an arbitrarily small e > 0, there are two time intervals ½t me ; tm Þ and ðtm ; t mþe . Suppose xðiÞ ðtm Þ ¼ xm ¼ xðjÞ ðt mþ Þ. Both flows xðiÞ ðtÞ and xðjÞ ðtÞ are C r½tme ;tm Þ and C rðtm ;tmþe
Theorem 4.1. For the inclined impact oscillator described in Section 2, (1) when the ball comes to the stick boundary @ X01 at time tm , the stick motion on the corresponding boundary appears if and only if the following conditions can be obtained:
continuous ðr P 1Þ for time t, respectively, and rþ1 d xðaÞ rþ1 < 1 ða 2 fi; jg). The flow xðiÞ ðtÞ and xðjÞ ðtÞ to the
9 S modðxtm þ s;2pÞ 2 0;arcsin gAtanh ðp arcsin gAtanh = x2 x2 ;2pÞ > if g tanh 6 Ax2 ; > ; modðxtm þ s;2pÞ 2 ð0;2pÞ if g tanh > Ax2 :
dt
boundary @ Xij is semi-passable from domain Xi to Xj iff
9 ðiÞ ðjÞ > either G@Xij ðxm ;t m ;pi ; kÞ > 0 and G@ Xij ðxm ;t mþ ; pj ;kÞ > 0 > > > > > > > = for n@Xij ! Xj ; ðiÞ
ðjÞ
or G@ Xij ðxm ; tm ; pi ;kÞ < 0 and G@Xij ðxm ;t mþ ;pj ; kÞ < 0 for n@ Xij ! Xi :
> > > > > > > > ; ð29Þ
ð34Þ (2) When the ball comes to the stick boundary @ X02 at time tm , the stick motion on the corresponding boundary appears if and only if the following conditions can be obtained:
) h g tan h modðxt m þ s; 2pÞ 2 arcsin gAtan if g tan h < Ax2 ; x2 ; p arcsin Ax2 no stick motion;
if g tanh P Ax2 : ð35Þ
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Proof. From Definition4.1, the 0 order Gfunctions for the relative stick boundaries in the inclined impact pair are ð0;iÞ
G@ X0i ðyðiÞ ; tm Þ ¼ nT@ X0i gðiÞ ðyðiÞ ; XðiÞ ; t m Þ; ð0;0Þ
G@ X0i ðyð0Þ ; t m Þ ¼ nT@ X0i gð0Þ ðyð0Þ ; Xð0Þ ; tm Þ;
ð36Þ
where i ¼ 1; 2 indicate the stick motion on the lower hand and upper hand of the slot, respectively, t m is the switching time of the motion for the ball on the corresponding boundary. From the Section 3, the Eqs. (36) can be computed as:
h g tan h must be within the ranges arcsin gAtan x2 ; p arcsin Ax2 , otherwise the ball will not stick to the upper hand of the slot. During the stick motion, the interaction force between the ball and the base is nonzero at any time interval while the force may be zero at a time point, which keeps them moves together. Theorem 4.2. For the inclined impact oscillator described in Section 2, (1) once the stick motion exists in domain X1 , such stick motion will vanish at time tm if and only if the following conditions can be satisfied: h modðxt m þ s; 2pÞ ¼ arcsin gAtan x2
ð0;iÞ
G@ X0i ðyðiÞ ; tm Þ ¼ g ðiÞ ðyðiÞ ; XðiÞ ; t m Þ ¼ 0;
no vanishing exists if
ð0;0Þ
G@ X0i ðyð0Þ ; t m Þ ¼ g ð0Þ ðyð0Þ ; Xð0Þ ; tm Þ ¼ g sin h þ Ax2 sin
g tan h 6 Ax2 ;
if
)
g tan h > Ax2 : ð41Þ
ðxt þ sÞ cos h: ð37Þ For the impact oscillator, the stick motion occurs when the flow in domain X0 passes over the boundary @ X0i into the domain Xi ði ¼ 1; 2Þ. By Lemma 4.1, the switching conditions for stick motion can be obtained as
(2) Once the stick motion exists in domain X2 , such stick motion will vanish at time tm if and only if the following conditions can be satisfied:
modðxt m þ s; 2pÞ ¼ p arcsin
g tan h Ax2
if
2
ð1Þi G@ X0i ðyð0Þ ; tm Þ > 0 and ð1Þi G@X0i ðyðiÞ ; t mþ Þ > 0: ð0;0Þ
ð0;iÞ
g tan h < Ax :
ð42Þ
ð38Þ Therefore i
i
ð1Þ g ð0Þ ðyð0Þ ; Xð0Þ ; t m Þ > 0 and ð1Þ g ðiÞ ðyðiÞ ; XðiÞ ; t mþ Þ > 0:
Proof. The stick motion vanishes, which means the flow will move back into the free-flight domain, when the flow in domain Xi ði ¼ 1; 2Þ passes through the boundary @ Xi0
ð39Þ
into domain X0 . Due to G@ X0i ðyðiÞ ; t m Þ ¼ 0, the high order
Due to the speciality of the stick domains, the onset conditions of stick motions are determined from the Eqs. (37) as
G-functions are needed. From Lemma 4.1, the criteria for vanishing of the stick motion are given for @ Xi0 by
ð0;iÞ
ð40Þ
G@X0i ðyðiÞ ; t m Þ ¼ 0;
9 ð0;0Þ G@ X0i ðyð0Þ ; tmþ Þ ¼ 0; =
Solving inequality (40) obtains the occurrence conditions for the ball to stick with the base on the lower hand or upper hand of the inclined rectangle as Eqs. (34) and (35). h
ð1Þi G@ X0i ðyðiÞ ; t m Þ < 0;
ð1Þi G@ X0i ðyð0Þ ; tmþ Þ < 0: ;
ð1Þi ðg sin h þ Ax2 sinðxt þ sÞ cos hÞ > 0:
The ball coming to the stick boundary at time tm means that the ball contacts the upper or lower wall of the rectangular clearance with zero relative velocity at time t m , that _ m Þ ¼ 0. From Eqs. (34), the ball comes is yðtm Þ ¼ d=2; yðt to the lower hand of the inclined slot with zero relative velocity, either if Ax2 is greater than or equal to g tan h and the switching phase modðxt m þ s; 2pÞ is within the h g tan h 2 ranges of ð0; arcsin gAtan x2 Þ or p arcsin Ax2 ; 2p , or if Ax is less than g tan h, the relative acceleration of the ball to the base can be less than zero, so during some time after tm , the velocity of the ball should be less than the component of one of the base in the inclined direction, but the ball can not pass through the lower wall of the inclined slot, thus the ball has to stick to the lower hand of the inclined slot and moves together with the base. Similarly, from Eqs. (35), the ball comes to the upper hand of the inclined clearance with zero relative velocity, in order for the ball to stick to this side of slot and moves together with the base, the acceleration of the ball relative to the base must be greater than zero, that is, Ax2 must be greater than g tan h, and the switching phase modðxt m þ s; 2pÞ
ð0;iÞ
ð1;iÞ
ð1;0Þ
ð43Þ From Definition 4.2, the 1 order Gfunctions for the relative stick boundaries in the inclined impact pair are ð1;iÞ
G@X0i ðyðiÞ ; t m Þ ¼ nT@ X0i DgðiÞ ðyðiÞ ; XðiÞ ; tm Þ; ð1;0Þ
G@X0i ðyð0Þ ; t m Þ ¼ nT@ X0i Dgð0Þ ðyð0Þ ; Xð0Þ ; t m Þ:
ð44Þ
From the Section 3, the above G-functions can be computed as:
d g ðy ;XðiÞ ;tm Þ ¼ Ax3 cosðxt þ sÞcosh; dt ðiÞ ðiÞ d ð1;0Þ G@X0i ðyð0Þ ;t m Þ ¼ g ð0Þ ðyð0Þ ;Xð0Þ ;t m Þ ¼ Ax3 cosðxt þ sÞcosh: dt ð45Þ ð1;iÞ
G@X0i ðyðiÞ ;t m Þ ¼
From the G-functions (37) and (45), Eqs. (43) can be simplified as 9 g ðiÞ ðyðiÞ ; XðiÞ ; t m Þ ¼ 0; > > > > = g ð0Þ ðyð0Þ ; Xð0Þ ; t mþ Þ ¼ g sinh þ Ax2 sinðxt m þ sÞcos h ¼ 0; > ð1Þi dtd g ðiÞ ðyðiÞ ;XðiÞ ; t m Þ ¼ ð1Þi ðAx3 cosðxt m þ sÞ cos hÞ < 0; > > > > > ð1Þi d g ðy ;X ; t Þ ¼ ð1Þi ðAx3 cosðxt þ sÞ cos hÞ < 0: ; dt
ð0Þ
ð0Þ
ð0Þ
mþ
m
ð46Þ
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Further simplification of the above equations yields ) 2 modðxtm þ s; 2pÞ ¼ arcsin gAtanh x2 if g tan h 6 Ax for@ X10 ; no vanishing exists if g tan h > Ax2
h modðxtm þ s; 2pÞ ¼ p arcsin gAtan x2 when g tan h <
Ax2 , no grazing motion exists if g tan h P Ax2 .
2 modðxt m þ s; 2pÞ ¼ p arcsin gAtanh x2 if g tan h < Ax for @ X20 :
ð47Þ
So the vanishing conditions of stick motions on the stick boundaries are obtained. h During the stick motion, the relative displacement of the ball to the base does not change, the relative velocity and acceleration are zero. The time t m at which the stick motion vanishes is the time that the relative jerk Ax3 cosðxt m þ sÞ cos h does not equal to zero, and after t m , the relative velocity and acceleration of the ball to the base make the ball leave the lower or upper hand of the inclined rectangle. From Eqs. (41), the conditions that g tan h is less than or equal to Ax2 and the switching phase h modðxtm þ s; 2pÞ is arcsin gAtan x2 can guarantee the stick motion on the lower hand of the slot vanishing and the ball leaves the lower hand of the slot into the free-flight domain. If g tan h is greater than Ax2 , the ball will stick to the lower hand of the slot permanently. From Eq. (42), the ball will stop the stick motion and leave the upper hand of the slot when the conditions that g tan h < Ax2 h and modðxt m þ s; 2pÞ ¼ p arcsin gAtan x2 are satisfied. Grazing phenomenon occurs in a dynamical system when the mass comes to the boundary from a subdomain and just touches the boundary with zero relative velocity or no interaction force, then moves back into this subdomain. So the analytical conditions of grazing motion can be developed from the G-functions of the flow for the impact-chatter boundaries @ X0ð1Þ and the stick boundaries @ X0i , @ Xi0 ði ¼ 1; 2Þ by using Lemma 4.2. Theorem 4.3. For the inclined impact oscillator described in Section 2, there are four cases of grazing motion on the stick boundaries in the following: (1) The flow for the motion of the ball in domain X1 reaches the lower stick boundary @ X10 at time t m without interaction force between the ball and the base, the grazing motion will occur if and only if modðxtm þ s; 2pÞ 2 ðp=2; 3p=2Þ. (2) The flow for the motion of the ball in domain X2 reaches the upper stick boundary @ X20 at time t m without interaction force between the ball and the base,, the grazing motion will occur if and only if modðxtm þ s; 2pÞ 2 ðp=2; p=2Þ. (3) The flow for the motion of the ball in domain X0 reaches the lower stick boundary @ X01 at time t m with the relative velocity and acceleration being zero, the grazing motion will occur if and only if modðxtm þ h 2 s; 2pÞ ¼ arcsin gAtan x2 when g tan h < Ax , no grazing motion exists if g tan h P Ax2 . (4) The flow for the motion of the ball reaches the upper stick boundary @ X02 the relative velocity and acceleration grazing motion will occur if
in domain X0 at time t m with being zero, the and only if
Proof. For the inclined impact oscillator described in Section 2, the grazing motion occurs when a flow in a subdomain is tangential to the boundary and then returns back to this subdomain. So from Lemma 4.2, the grazing motion conditions for the stick boundaries are given as ð0;iÞ
G@X0i ðyðiÞ ; tm Þ ¼ 0; ð0;0Þ
G@X0i ðyð0Þ ; tm Þ ¼ 0;
9 ð1;iÞ ð1Þi G@ X0i ðyðiÞ ; t m Þ > 0 for @ Xi0 ; = ð1Þi G@ X0i ðyð0Þ ; tm Þ < 0 for @ X0i : ; ð1;0Þ
ð48Þ 9 ð1Þi ðAx3 cosðxtm þ sÞ cos hÞ > 0 for @ Xi0 ; > > = ) ð1Þi ðg sin h þ Ax2 sinðxt þ sÞ cos hÞ ¼ 0; for @ X0i : > > ; ð1Þi ðAx3 cosðxt m þ sÞ cos hÞ < 0 ð49Þ Further simplication of the Eqs. (49) yields 9 modðxt m þ s;2pÞ 2 ðp=2;3p=2Þ for @ X10 ; > > > > modðxtm þ s;2pÞ 2 ðp=2; p=2Þ for @ X20 ; > > > > ) > > g tanh 2 = modðxt m þ s;2pÞ ¼ arcsin Ax2 if g tanh < Ax ; for @ X01 ; 2 > no grazing motion exists if g tanh P Ax > > ) > > 2 > > modðxtm þ s;2pÞ ¼ p arcsin gAtanh if g tanh < A x ; 2 x > for @ X02 : > > ; 2 no grazing motion exists if g tanh P Ax ð50Þ
The conditions of grazing motion on the stick boundaries are obtained. h During the stick motion, the ball moves together with the base while the nonzero interaction force between the ball and the base exists due to the acceleration of the ball itself different from the component of one of the base in the inclined direction. The interaction forces becomes zero at time t m , then there are two states may occur next, the first is that the interaction force becomes nothing, the stick motion vanishes, the second is that the interaction force rises again, the stick motion continues. For the second case, the time t m is the tangent time. Thus the flow for the motion of the ball in domain X1 reaches the stick boundary @ X10 with the state that the interaction force between the ball and the base is zero, the condition that modðxtm þ s; 2pÞ is in second or third quadrant guarantees that the interaction force comes back again, so stick motion continues, that is, the ball will graze to the stick boundary @ X10 at time tm . Similarly when the motion of the ball in domain X2 reaches the stick boundary @ X20 with the state that the interaction force between the ball and the base is zero at time t m ; modðxtm þ s; 2pÞ is in first or fourth quadrant, then the ball will graze to the upper stick boundary @ X20 . For such two cases, the stick motions continue further. The flow of the ball reaches the stick boundary from X0 at time tm with the relative velocity and acceleration of the ball to the base being zero, the conditions that Ax2 is
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h greater than g tan h and modðxt m þ s; 2pÞ is arcsin gAtan x2 can guarantee that the ball grazes to the lower stick boundary @ X01 , in order for the ball to graze to upper stick boundary @ X02 , the conditions that Ax2 is greater than g tan h and h modðxtm þ s; 2pÞ ¼ p arcsin gAtan x2 must be satisfied. And then the flow moves back to the free-flight domain and the stick motion is not formed. If the ball reaches the stick boundary from X0 with the relative velocity and acceleration of the ball to the base being zero, but Ax2 6 g tan h, no grazing motion appears on the stick boundary @ X0i ði ¼ 1; 2Þ.
Theorem 4.4. For the inclined impact oscillator described in Section 2, there are two cases of grazing motion on the impact-chatter boundaries in the following: (1) The flow for the motion of the ball in domain X0 reaches the boundary @ X0ð1Þ at time tm , the grazing motion is to occur if and only if the following conditions are obtained:
x_ ð0Þ ðt m Þ Ax cosðxtm þ sÞcos h ¼ 0; and
9 g tanh modðxt m þ s;2pÞ 2 ðarcsin gAtanh = x2 ; p arcsin Ax2 Þ > if g tanh < Ax2 ; > ; no grazing motion exists if g tanh P Ax2 :
ð51Þ
(2) The flow for the motion of the ball in domain X0 reaches the boundary @ X0ðþ1Þ at time tm , the grazing motion is to occur if and only if the following conditions are obtained:
x_ ð0Þ ðtm Þ Ax cosðxtm þ sÞcos h ¼ 0; and 9 S g tanh > modðxt m þ s;2pÞ 2 ð0; arcsin gAtanh x2 Þ ðp arcsin Ax2 ; 2pÞ > > > = if g tanh 6 Ax2 ; modðxt m þ s; 2pÞ 2 ð0; 2pÞ > > > > ; if g tanh > Ax2 : ð52Þ
Proof. From Definitions 4.1 and 4.2, the 0 order and 1 order Gfunctions for the relative impact-chatter boundaries @ X0ð1Þ are determined by ð0;0Þ
G@ X0ðþ1Þ ðyð0Þ ; t m Þ ¼ nT@X0ðþ1Þ gð0Þ ðyð0Þ ; Xð0Þ ; tm Þ; ð0;0Þ
G@ X0ð1Þ ðyð0Þ ; t m Þ ¼ nT@X0ð1Þ gð0Þ ðyð0Þ ; Xð0Þ ; tm Þ; ð1;0Þ
G@ X0ðþ1Þ ðyð0Þ ; t m Þ ¼ nT@X0ðþ1Þ Dgð0Þ ðyð0Þ ; Xð0Þ ; t m Þ;
ð53Þ
ð1;0Þ
G@ X0ð1Þ ðyð0Þ ; t m Þ ¼ nT@X0ð1Þ Dgð0Þ ðyð0Þ ; Xð0Þ ; t m Þ: From the Section 3, the above G-functions can be computed as: ð0;0Þ
G@ X0ð1Þ ðyð0Þ ; t m Þ ¼ y_ ð0Þ ¼ x_ ð0Þ Ax cosðxt m þ sÞ cos h; ð1;0Þ
G@ X0ð1Þ ðyð0Þ ; t m Þ ¼ g sin h þ Ax2 sinðxt m þ sÞ cos h: ð54Þ From Lemma 4.2, on the impact-chatter boundaries @ X0ð1Þ , the grazing motion conditions are obtained as
9 ð0;0Þ ð1;0Þ G@ X0ð1Þ ðyð0Þ ; t m Þ ¼ 0; G@X0ð1Þ ðyð0Þ ; t m Þ > 0 for @ X0ð1Þ ; = G@ X0ðþ1Þ ðyð0Þ ; t m Þ ¼ 0; G@X0ðþ1Þ ðyð0Þ ; t m Þ < 0 for @ X0ðþ1Þ : ; ð0;0Þ
ð1;0Þ
ð55Þ Combining Eqs. (54) into Eqs. (55) yields
x_ ð0Þ ðtm Þ Ax cosðxt m þ sÞ cos h ¼ 0;
and
g sin h þ Ax sinðxt m þ sÞ cos h > 0 for @ X0ð1Þ ; 2
)
g sin h þ Ax2 sinðxt m þ sÞ cos h < 0 for @ X0ðþ1Þ : ð56Þ From Eqs. (56) the results (51) and (52) can be obtained.
h
When the motion of the ball in non-stick domain comes to the impact-chatter boundary with zero relative velocity and then accelerates back towards such domain at this time, the ball is to graze to such boundary. From the Eqs. (51) and (52), when the ball comes to the boundary @ X0ð1Þ with the relative velocity of the ball to the base being zero, the conditions Ax2 being greater than g tan h and modðxtm þ s; 2pÞ h g tan h laying in ðarcsin gAtan x2 ; p arcsin Ax2 Þ are for the ball to graze on the impact-chatter boundary @ X0ð1Þ , the conditions Ax2 being less than or equal to g tan h is for no grazing motion on such boundary; the conditions Ax2 being greater than or equal to g tan h and modðxt m þ s; 2pÞ laying in h g tan h ð0; arcsin gAtan x2 Þ or ðp arcsin Ax2 ; 2pÞ are for the grazing motion to appear on the impact-chatter boundary @ X0ðþ1Þ , the conditions Ax2 is less than g tan h and modðxt m þ s; 2pÞ laying in ð0; 2pÞ are for grazing motion to occur on the corresponding boundary. 5. Numerical Simulations To illustrate the analytical conditions of stick motions and grazing flows, the motions of the ball in the inclined impact oscillator will be demonstrated through the time histories of displacement and velocity, the corresponding trajectory of the ball in phase space. The starting points of motions are represented by dark-solid circular symbols, the switching points at which the ball contacts the moving boundaries are depicted by blue-solid or red-solid circular symbols. The moving boundaries, that is, the component of displacement curves or velocity curves of the base in the inclined direction are represented by dark curves, and the displacement or the velocity of the ball, the corresponding trajectories of the motions of the ball in phase space are shown by the red curves. Consider the system parameters as A ¼ 10; x ¼ 1; h ¼ p=6; g ¼ 9:81; e ¼ 0:8; s ¼ 0; d ¼ 20; M ¼ 1; m ¼ 0:0001 to demonstrate a stick motion of the ball on the lower hand of the inclined rectangle in Fig. 5. The initial conditions are t0 ¼ 5:356754; x0 ¼ 16:923712; x_ 0 ¼ 5:202136. The time histories of displacement and velocity are depicted in Fig. 5(a) and (b), respectively. It can be seen that the stick motion appears in such initial conditions, that is, the ball reaches the lower wall of the inclined slot with the velocity equalling to the component of one of the base in the inclined direction, and then moves together with the base for some time. The red solid circular stands for the
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227
Fig. 5. Numerical simulation of a stick motion on the lower hand of the inclined rectangle: (a) displacement–time history, (b) velocity–time history, (c) phase trajectory. (A ¼ 10; x ¼ 1; h ¼ p=6; g ¼ 9:81; e ¼ 0:8; s ¼ 0; d ¼ 20; M ¼ 1; m ¼ 0:0001; t 0 ¼ 5:356754; x0 ¼ 16:923712; x_ 0 ¼ 5:202136).
Fig. 6. Numerical simulation of a grazing motion on the lower hand of the inclined rectangle: (a) displacement–time history, (b) velocity–time history, (c) phase trajectory. (A ¼ 10; x ¼ 1; h ¼ p=6; g ¼ 9:81; e ¼ 0:4; s ¼ 0; d ¼ 1; M ¼ 1; m ¼ 0:0001; t0 ¼ 0:699899; x0 ¼ 6:078421; x_ 0 ¼ 4:537097).
Fig. 7. Numerical simulation of a periodic motion of the ball impacting once with lower hand of the inclined slot: (a) displacement–time history, (b) velocity–time history, (c) phase trajectory. (A ¼ 0:7; x ¼ p; h ¼ p=6; g ¼ 9:81; e ¼ 0:5; s ¼ 0:53846; d ¼ 4; M ¼ 1; m ¼ 0:0001; t0 ¼ 0; x0 ¼ 1:68912; x_ 0 ¼ 4:90499).
vanishing points of stick motion. After vanishing of stick motion on the lower wall of the inclined rectangle, the ball moves freely and impacts this boundary again. The corresponding trajectory of the stick motion are shown in Fig. 5(c). Based on the parameters, the parameters e ¼ 0:4; d ¼ 1 are chosen to demonstrate a grazing motion of the ball on the lower hand of the inclined rectangle in Fig. 6. The initial conditions are t 0 ¼ 0:699899; x0 ¼ 6:078421; x_ 0 ¼ 4:537097. The initial point on the upper hand of the inclined slot, and then the ball contacts the lower hand of the slot as shown in Fig. 6(a). However from the Fig. 6(b), it can be
seen that the red point is the point of intersection between the black curve and the red one, in other words, the velocity of the ball equals to the component of one of the base in the inclined direction at the red point, that is, the relative velocity of the ball to the base is zero at the red point. And then the ball leaves this boundary again. So the red point is grazing point, a grazing motion on the lower hand of the slot occurs. Fig. 6(b) and (c) also show that the velocity of the ball is continuous at the red point. The following Figs. 7–10 depict the periodic motion of the ball in the inclined impact oscillator. Among them, Figs. 7–9 illustrate the periodic motion of the ball only
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Fig. 8. Numerical simulation of a periodic motion of the ball impacting twice with lower hand of the inclined slot in one period of the base excitation: (a) displacement–time history, (b) velocity–time history, (c) phase trajectory. (A ¼ 1; x ¼ p; h ¼ p=6; g ¼ 9:81; e ¼ 0:5; s ¼ 5:13028; d ¼ 4; M ¼ 1; m ¼ 0:0001; t0 ¼ 0; x0 ¼ 2:79150; x_ 0 ¼ 4:39372).
Fig. 9. Numerical simulation of a period-2 motion of the ball impacting twice on lower hand of the inclined slot in two period of the base excitation: (a) displacement–time history, (b) velocity–time history, (c) phase trajectory. (A ¼ 1:5; x ¼ p; h ¼ p=6; g ¼ 9:81; e ¼ 0:5; s ¼ 5:71953; d ¼ 8; M ¼ 1; m ¼ 0:0001; t0 ¼ 0; x0 ¼ 4:69405; x_ 0 ¼ 4:61070).
Fig. 10. Numerical simulation of a period-1 motion of the ball alternatively impacting with upper hand and lower hand of the inclined slot in one period of the base excitation: (a) displacement–time history, (b) velocity–time history, (c) phase trajectory. (A ¼ 10; x ¼ p; h ¼ p=6; g ¼ 9:81; e ¼ 0:7; s ¼ 4:70982; d ¼ 24:8041; M ¼ 1; m ¼ 0:0001; t0 ¼ 0; x0 ¼ 21:0623; x_ 0 ¼ 52:5646).
impacting with the lower hand of the inclined rectangle, which is similar to the dynamical behavior of the bouncing ball impacting on a periodically with the vibrating table. The systems parameters A ¼ 0:7; x ¼ p; h ¼ p=6; g ¼ 9:81; e ¼ 0:5; s ¼ 0:53846; d ¼ 4; M ¼ 1; m ¼ 0:0001 are chosen to illustrate a period-1 motion of the ball only impacting once the lower hand of the inclined slot in Fig. 7. The initial conditions are t 0 ¼ 0; x0 ¼ 1:68912; x_ 0 ¼ 4:90499. From the conditions making the above periodic motion appear and the eigenvalue analysis, the
following cases can take place: on the basis of the above parameters, the initial velocity of the ball which makes such periodic motion occur does not change when A is changing, when A 6 0:6, the such period-1 motion doesn’t appear, when A is within the range ð0:6; 0:83093Þ, such periodic motion is stable focus, when A is within the rang ð0:83093; 0:87619Þ, such periodic motion is stable node, when A > 0:87619, such periodic motion is the saddle of the second kind, and when A ¼ 0:87619, the period doubling bifurcation occurs and period-2 motion appears.
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Fig. 11. Numerical simulation of a chaotic motion: (a) displacement–time history, (b) velocity–time history, (c) phase trajectory. (A ¼ 10; x ¼ p; h ¼ p=6; g ¼ 9:81; e ¼ 0:4; s ¼ 0; d ¼ 20; M ¼ 1; m ¼ 0:0001; t 0 ¼ 0:318310; x0 ¼ 2:712648; x_ 0 ¼ 27:1622).
Based on the above parameters in Fig. 7, the parameters A ¼ 1; s ¼ 5:13028 are chosen to demonstrate a periodic motion of the ball impacting twice on the lower wall of the inclined clearance in one period of the base excitation in Fig. 8. The initial conditions are t0 ¼ 0; x0 ¼ 4:69405; x_ 0 ¼ 4:61070. The parameters A ¼ 1:5; s ¼ 5:71953; d ¼ 8 are given to depict a period-2 motion of the ball impacting twice on the lower wall of the inclined slot in two period of the base excitation in Fig. 9. In a similar form, for Figs. 7–9, the histories of displacement and velocity, the corresponding phase portrait of the motion of the ball are shown in corresponding Figure (a), (b) and (c), respectively. The system parameters A ¼ 10; x ¼ p; h ¼ p=6; g ¼ 9:81; e ¼ 0:7; s ¼ 4:70982; d ¼ 24:8041; M ¼ 1; m ¼ 0:0001 and the initial conditions t 0 ¼ 0; x0 ¼ 21:0623; x_ 0 ¼ 52:5646 are given to demonstrate the periodic motion of the ball alteratively impacting the upper hand and lower hand of the inclined slot in one period of the base excitation in Fig. 10. The chaotic motion is also illustrate in Fig. 11 with the parameters in Fig. 10 only changed as s ¼ 0; d ¼ 20; e ¼ 0:4. Choosing the initial conditions as t 0 ¼ 0:318310; x0 ¼ 2:712648 and x_ 0 ¼ 27:1622. In a similar pattern, the histories of displacement and velocity, the corresponding phase portrait of the motion of the ball are shown in Fig. 11(a), (b) and (c), respectively. The zoomed picture in Fig. 11(c) shows that the locations at which the ball impacts the upper wall of the inclined rectangle are different, so this motion in Fig. 11 is chaotic motion. 6. Conclusions In this paper, the analytical predictions of complex motions of an inclined impact oscillator with a harmonically external excitation were studied using the theory of flow switchability for discontinuous dynamical systems. Different domains and boundaries were defined due to impact continuity. Analytical conditions of stick motions and grazing flows corresponding to various boundaries were presented in the form of theorem to show switching complexity of such inclined oscillator. The numerical simulations of stick and grazing motions were given to provide a better understanding of complicated dynamics of such mechanical model, and several periodic motions of the ball
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