Dynamics of vibro-impact mechanical systems with large dissipation

ARTICLE IN PRESS

International Journal of Mechanical Sciences 50 (2008) 214–232 www.elsevier.com/locate/ijmecsci

Dynamics of vibro-impact mechanical systems with large dissipation G.W. Luo, X.H. Lv, X.F. Zhu School of Mechatronic Engineering, Lanzhou Jiaotong University, Lanzhou 730070, PR China Received 23 November 2006; received in revised form 27 June 2007; accepted 4 July 2007 Available online 10 July 2007

Abstract An n-degree-of-freedom system having placed single stop and subjected to periodic excitation is considered. Based on the analysis of dynamics of the vibratory system with plastic impacts, we introduce a (2n1)-dimensional map with dynamical variables defined at the impact instants. The nonlinear dynamics of the vibro-impact system is analyzed by using the Poincare´ map, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of the impact mass immediately after the impact, and the singularity of the map is generated via the grazing contact of both the impact mass and the rigid stop and corresponding instability of periodic-impact motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The single-impact periodic motions and disturbed map, associated with free flight motion of the system, are derived analytically. Stability, sliding and period-doubling bifurcations of the single-impact periodic motions are analyzed by the presentation of results for a three-degree-of-freedom plastic impact oscillator. Finally two actual examples, the impact-forming machine and inertial shaker, are considered to further analyze periodicimpact motions and bifurcations of plastic impact oscillators. The free flight and sticking solutions of two impact machines are analyzed numerically, and regions of existence and stability of different periodic-impact motions are therefore presented. The influence of nonstandard bifurcations and system parameters on dynamics of the vibro-impact machines is elucidated accordingly. r 2007 Elsevier Ltd. All rights reserved. Keywords: Vibration; Impact; Periodic motion; Bifurcation

1. Introduction In a large number of diverse engineering fields, design or working conditions lead to collisions or impacts between the moving components of the system. This occurs when the vibration amplitudes of some components of systems exceed critical values (clearances or gaps). The physical process during impact is complex and highly nonlinear, but it is necessary to be able to accurately model the dynamics of an impacting system, so as to minimize adverse effects such as pitting, scoring and high noise levels. The broad interest in analyzing and understanding the performance of such systems is reflected by a still increasing amount of investigations devoted to this area. Several methods of the theoretical analysis have been developed and different Corresponding author.

E-mail addresses: [email protected], [email protected] (G.W. Luo). 0020-7403/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2007.07.001

models of impacts have been assumed in the past several years. Stability and bifurcations of different types of impact oscillators were reported in Refs. [1–8]. A special feature of impacting systems that might prove to be useful in the current study is the instability caused by low-velocity collisions, the so-called grazing effects. The first important work in this area was done by Nordmark [9], who studied analytically the occurrence of singularities in a piecewise linear system. This work has been further expanded by thorough investigations of two-dimensional maps, where some universal behavior has been found [10–20]. de Souza and Caldas [21] applied a model-based algorithm for the calculation of the spectrum of the Lyapunov exponents of attractors of mechanical systems with impacts. Chattering impact phenomenon was found to exist in an oscillator with limiting stops by Nguyen et al. [22]. Peterka et al. [23] studied chaotic motion of an intermittency type of the impact oscillator appearing near segments of saddle-node stability boundaries of subharmonic motions with two

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different impacts in motion period. Wagg [24] considered the rising phenomena which occur in sticking solutions of a impact oscillator. Quasi-periodic motions in multipledegree-of-freedom oscillators with impacts were analyzed by numerical and analytical studies [25–28]. Hu [29] presented how to control the chaos of dynamical systems with discontinuous vector field through the paradigm of a harmonically forced oscillator having a set-up elastic stop. The algorithms of position control of impact oscillator and synchronization of two impact oscillators are demonstrated by Lee and Yan [30]. de Souza et al. [31] proposed a feedback control method to suppress chaotic behavior in oscillators with limited power supply. Experimental study of base-excited symmetrically piecewise linear oscillator was performed by Wiercigroch and Sin [32]. Along with the basic research into vibro-impact dynamics, a wide range of impacting models have been applied to simulate and analyze engineering systems operating within bounded dynamic responses. For example, in wheel-rail impacts of railway coaches [33,34], impact noise analyses [35,36], inertial shakers [37,38], impact-forming machine [39], offshore structure [40], Jeffcott rotor with bearing clearance [41,42], impact dampers [43–46], gears [47,48], vibrating hammer [49], impact tools with progressive motions [50–54], etc., impacting models have proved to be useful. It is important to notice that most studies of vibro-impact dynamics have been carried out for vibratory systems with elastic impacts in the past several years (see, for example, Refs. [1–48]). Very few have considered the plastic impact oscillators. Periodic motion and bifurcations of singledegree-of-freedom oscillators with plastic impacts have only been investigated by Shaw and Holmes [55,56] and Xie [49]. However, such systems with plastic impacts have a wide range of practical applications as, for example, vibro-impact machinery with large dissipation, inertial shakers, pile drivers, compacting machines, milling and forming machines, etc. So the dynamics of plastic vibro-impact systems is of a considerable importance in practical engineering applications. To fill this gap, we give a detailed mathematical modeling and nonlinear dynamic analysis of a multi-degreeof-freedom plastic impact oscillator in this paper, and focus attention on the sliding and grazing bifurcations which occur in sticking solutions of the plastic impact oscillator. Sticking in impact systems is mathematically analogous to sliding in electrical systems [57]. The sliding orbits in these electrical systems have been shown to exhibit particular types of sliding bifurcations under parameter variation [58]. In the paper, dynamics of an n-degree-of-freedom plastic impact oscillator is analyzed by using a (2n1)-dimensional map, which describes free fight and sticking solutions of the vibro-impact system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the map. The piecewise property is caused by the transitions of free flight and sticking motions of the impact mass immediately after the impact, and the singularity of the map is generated via the grazing contact of both the

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impact mass and the rigid stop. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The single-impact periodic motions and disturbed map, associated with free flight of the system, are derived analytically by the set of periodicity and matching conditions. Local bifurcations of nonsticking single-impact periodic motions can be determined by computing and analyzing eigenvalues of Jacobian matrix of the disturbed map. Stability, sliding and period-doubling bifurcations of the single-impact periodic motions are analyzed by the presentation of results for a three-degree-of-freedom plastic impact oscillator. The extensive and systematic regularities, associated with bifurcations and transitions, are found. Finally two actual examples, the impact-forming machine and inertial shaker, are considered to further analyze periodic-impact motions and bifurcations of plastic impact oscillators. The free flight and sticking solutions of two impact machines are studied numerically, and regions of existence and stability of different periodic-impact motions are therefore presented. The bifurcation diagrams for relative before-impact velocities of these machines versus the forcing frequency, associated with the optimized parameters, are plotted, which enable the practicing engineer to select excitation frequency ranges in which stable period-1 response can be expected to occur, and to predict the larger impact velocities and shorter impact period of such a response. 2. Mechanical model A multi-degree-of-freedom system having placed single stop and subjected to periodic excitation is shown schematically in Fig. 1. Displacements of the masses M1, M2, y, Mn1 and Mn are represented by X1, X2, y, Xn1 and Xn, respectively. The masses are connected to linear springs with stiffnesses K1, K2, y, Kn1 and Kn, and linear viscous dashpots with damping constants C1, C2, y, Cn1 and Cn. Damping in the mechanical model is assumed as proportional damping. The excitations on the masses are harmonic with amplitudes P1, P2, y, Pn1 and Pn. The excitation frequency O and phase angle t are the same for these masses. The masses move only in the horizontal direction. For small forcing amplitudes the system will undergo simple oscillations and behave as a linear system. As

Fig. 1. Schematic of a multi-degree-of-freedom vibratory system contacting single rigid stop.

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Fig. 2. Schematic of a multi-degree-of-freedom vibratory system contacting single rigid stop (in sticking case).

the amplitude is increased, the kth mass Mk eventually begins to hit the rigid stop and the motion becomes nonlinear (the other masses are not allowed to impact the rigid stop). The periodically forced vibro-impact system, associated with large dissipation case, is considered in this paper. So the impact is assumed to be perfectly plastic, and it is assumed that the duration of impact is negligible compared to the period of the force. An impact occurs wherever Xk ¼ B. After the impact occurs, either the mass Mk remains in a contact with the rigid stop till the pressure of the mass Mk on the stop decreases to zero, or the mass Mk departs from the Xk ¼ B stop with zero velocity X_ k ¼ 0 immediately. Between any two consecutive impacts, the time T is always set to zero directly at the starting point A, and the phase angle t is used only to make a suitable choice for the origin of time in the calculation. The state of the vibroimpact system, immediately after impact, has become new initial conditions in the subsequent process of the motion. If the mass Mk remains in a contact with the rigid stop after the impact occurs, the n-degree-of-freedom vibratory system becomes two (nk) and (k1)-degree-of-freedom oscillators. The change from free motion of the system to the mass Mk sticking represents a reduction in the degree of freedom of the system from n to (nk) and (k1) (two oscillators); see Fig. 2. The presence of the sticking phase complicates the dynamic analysis of the vibroimpact system. Without loss of generality, the dynamic analysis of nonlinear systems can be considered in either a dimensional or a non-dimensional form. In the following work, we assume all parameters and variables are non-dimensional. Suppose Mk6¼0 and Kk6¼0 and let F0 ¼ |P1|+|P2|+ ? +|Pk|+ ? +|Pn|. The non-dimensional quantities are given by Pi , F0 Ci X iKk , zi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; xi ¼ F0 2 K kMk rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi z Mk Kk ; t¼T , g¼ i; o¼O ki Kk Mk BK k d¼ ; i ¼ 1; 2; . . . ; k; . . . ; n. F0

mi ¼

Mi ; Mk

ki ¼

Ki ; Kk

f i0 ¼

ð1Þ

¯ During the sticking motion the force NðtÞ holds the mass Mk against the stop xk ¼ d, which is given by ¯ NðtÞ ¼ kk ðxkþ1  dÞ þ 2zk x_ kþ1  kk1 ðd  xk1 Þ þ 2zk1 x_ k1 þ f k0 sin ðot þ tÞ.

ð2Þ

¯ During the sticking motion, NðtÞ40, xk ¼ d and x_ k ¼ 0. ¯ becomes zero The sticking motion ends when the force NðtÞ at t ¼ ts. Immediately after the impact occurs, the motions of the system are non-sticking if the mass Mk departs from the xk ¼ d stop with zero velocity x_ k ¼ 0 immediately. The motion of this system is a combination of smooth motions governed by linear differential equation interrupted by a series of non-smooth impacts. Between consecutive impacts, the smooth motions of the vibratory system are described by the non-dimensional equations: M l x€ l þ C l x_ l þ K l xl ¼ F l sin ðot þ tÞ þ Pl , M f x€ f þ C f x_ f þ K f xf ¼ F f sin ðot þ tÞ þ Pf , x_ k ¼ 0;

xk ¼ d;

0ptpts ,

M x€ þ C x_ þ Kx ¼ F sin ðot þ ots þ tÞ; ts otpts þ tf ,

ð3:1Þ ðxk odÞ, ð3:2Þ

where a dot (  ) denotes differentiation with respect to the non-dimensional time t; M, K and C are the non-dimensional mass, stiffness and damping matrixes of the vibratory system shown in Fig. 1, respectively, x ¼ (x1, x2, y, xn)T, F ¼ (f10, f20, y, fn0)T; Ml (Mf), Kl (Kf) and Cl (Cf) are the nondimensional mass, stiffness and damping matrixes of the vibratory system shown in Fig. 2a (Fig. 2b), respectively, xl ¼ (x(k+1), x(k+2), y, xn)T, xf ¼ (x1, x2, y, xk1)T, Fl ¼ (f(k+1)0, f(k+2)0, y, fn0)T, Ff ¼ (f10, f20, y, f(k1)0)T, Pl ¼ (kkd, 0, y, 0)T, Pf ¼ (0, 0, y,k(k1)d)T. In Eq. (3.2), tsotpts+tf, tf denotes the time of free flight of the vibro-impact system between two successive impacts, and ts is the time of the sticking motion. ts+tf denotes the time interval between two successive impacts. At the instant the sticking motion ends (i.e., t ¼ ts), the state of ¯ s Þ ¼ 0, xk ¼ d, x_ k ¼ 0. the mass Mk can be described by Nðt An impact occurs wherever xk ¼ d, After each impact, the velocity of the impacting mass Mk becomes zero due to the plastic impact, i.e., x_ kþ ¼ 0 where the subscript minus

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sign denotes the states just before impact and the subscript plus sign denotes the states just after impact. Let C represent the canonical modal matrix of Eq. (3.2), oi(i ¼ 1, 2, y, n) denote the eigenfrequencies of the system, modal damping ratio Zi, damping eigenfrequency pffiffiffiffiffiffiffiffiffiffiffiffiffi odi ¼ oi 1  Z2i . Eq. (3.2) is amenable to analytical treatment due to the proportional damping. Using the formal coordinate and modal matrix approach, one can obtain the general solutions of Eq. (3.2):

the Poincare´ section:

xðtÞ ¼ CxðtÞ,

where y ¼ ot, the mass XAR2n1, v is real parameter, vAR1;

(4)

xðtÞ ¼ GðtÞA þ HðtÞB þ F s sin ðot þ tÞ þ F c cos ðot þ tÞ,

s ¼ fðx1 ; x_ 1 ; x2 ; x_ 2 ; . . . ; xk ; x_ k ; . . . ; xn ; x_ n ; yÞ 2 R2n  S; xk ¼ d; x_ k ¼ x_ kþ ¼ 0g to establish Poincare´ map of the vibro-impact system. The disturbed map of N-1-I motion is represented briefly by X 0 ¼ f~ðn; X Þ,

(7)

X ¼ X  þ DX ; X 0 ¼ X  þ DX , ð5Þ

where A and B are the constant matrixes of integration, GðtÞ ¼ diag½eZi oi t sin ðodi tÞ, HðtÞ ¼ diag½eZi oi t cos ðodi tÞ, i ¼ 1, 2, y, n (The symbol ‘‘diag[ ]’’ is used to denote the diagonal matrix); F s ¼ ðf s1 ; f s2 ; . . . ; f sn ÞT and F c ¼ ðf c1 ; f c2 ; . . . ; f cn ÞT are the amplitude constant vectors which are expressed by ! 1 o þ odi o  odi f si ¼  f¯ i , 2odi ðo þ odi Þ2 þ Z2i o2i ðo  odi Þ2 þ Z2i o2i ! Zi oi 1 1  f ci ¼ f¯ i , ð6Þ 2odi ðo  odi Þ2 þ Z2i o2i ðo þ odi Þ2 þ Z2i o2i where F¯ ¼ CT F ¼ ½f¯ 1 ; f¯ 2 ; . . . ; f¯ n T .

3. Single-impact periodic motions Under suitable system parameter conditions, the system shown in Fig. 1 exhibits periodic-impact behavior. Periodic-impact motions of the system are characterized by the symbol N-p, where p denotes the number of impacts and N denotes the number of excitation periods, during one impact motion period, respectively. A definition is given as: N-1-I represents a type of motion in which only one impact exists during n excitation periods and the mass Mk does not stick to the stop, and it departs from the Xk ¼ B stop with zero velocity x_ k ¼ 0 immediately after the impact; N-1-II represents another type of motion in which one impact exists during N excitation periods and the mass Mk remains in a contact ¯ with the stop after the impact till the force NðtÞ equals zero. Periodic sticking and non-sticking motions of the system can be further characterized by the symbols N-p-I and N-p-II. Impacting systems are conveniently studied by using a map derived from the equations of motion. Each iterate of the map corresponds to the mass Mk striking the rigid stop once. As a Poincare´ section associated with the state of the vibro-impact system, just immediately after impact, is chosen, N-1-I motion and Poincare´ map can be analytically derived. Based on the analysis above mentioned, we chose

X  ¼ ðx10 ; x20 ; . . . ; xðk1Þ0 ; t0 ; xðkþ1Þ0 ; . . . ; xn0 ; x_ 10 , x_ 20 ; . . . ; x_ ðk1Þ0 ; x_ ðkþ1Þ0 ; . . . ; x_ n0 ÞT is a fixed point in the hyperplane s, of which the disturbed vectors are represented by DX ¼ ðDx1 ; Dx2 ; . . . ; Dxk1 ; Dt; Dxkþ1 ; . . . ; Dxn ; Dx_ 1 , Dx_ 2 ; . . . ; Dx_ k1 ; Dx_ kþ1 ; . . . ; Dx_ n ÞT , 0

DX ¼ ðDx01 ; Dx02 ; . . . ; Dx0k1 ; Dt0 ; Dx0kþ1 ; . . . ; Dx0n ; Dx_ 01 , Dx_ 02 ; . . . ; Dx_ 0k1 ; Dx_ 0kþ1 ; . . . ; Dx_ 0n ÞT . The single-impact periodic motion means that if the dimensionless time t is set to zero directly immediately after an impact, it becomes 2Np/o just before the next impact. After the origin of y-coordinate is displaced to an impact point o1, the determination of N-1-I motion is based on the fact that they satisfy the following set of periodicity and matching conditions " # " # " # xð2Np=oÞ xð0Þ x0 ¼D , (8) ¼ _ xð2Np=oÞ _ xð0Þ x_ 0 where D ¼ diag[di] (di ¼ 1, i ¼ 1, 2, y, k, y, n+k1, n+k+1, y, 2n; dn+k ¼ 0, x0 ¼ (x10, x20, y, x(k1)0, d, x(k+1)0, y, xn0)T, x_ 0 ¼ ðx_ 10 ; x_ 20 ; . . . ; x_ ðk1Þ0 ; x_ kþ ; x_ ðkþ1Þ0 ; . . . ; x_ n0 ÞT . Inserting t ¼ 0 into Eqs. (4) and (5), one can obtain _ matrixes A and B expressed by x(0) and xð0Þ: _ þ diag½Z oi =odi xð0Þ A ¼ diag½1=odi xð0Þ i  ðdiag½Zi oi =odi F s  diag½o=odi F c Þ sin t  ðdiag½o=odi F s þ diag½Zi oi =odi F c Þ cos t, B ¼ xð0Þ  F s sin t  F c cos t.

"

The response of N-1-I orbit is given by # " # xðtÞ xð0Þ ¼ FPðtÞF1 _ _ xðtÞ xð0Þ " # sin t þ QðtÞ ; 0ptp2Np=o, cos t

ð9Þ

ð10Þ

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where

where "

F ¼ diag½C; C; PðtÞ ¼ " QðtÞ ¼

Ps ðtÞ

Pc ðtÞ

P_ s ðtÞ

P_ c ðtÞ

#

P1 ðtÞ P2 ðtÞ P_ 1 ðtÞ P_ 2 ðtÞ

#

S~ t ¼ sinðot þ t0 þ DtÞ, C~ t ¼ cosðot þ t0 þ DtÞ, " # GðtÞ HðtÞ ~ ¼ EðtÞ , _ _ GðtÞ HðtÞ " # Fs Fc ~ , Q¼ F c F s " # I U¼ , oI

,

,

P1 ðtÞ ¼ diag½gi ðtÞZi oi =odi þ hi ðtÞ, P2 ðtÞ ¼ diag½gi ðtÞ=odi , gi ðtÞ ¼ eZi oi t sinðodi tÞ, hi ðtÞ ¼ eZi oi t cosðodi tÞ, Ps ðtÞ ¼ CðF s cos ot  F c sin ot  diag½ogi ðtÞ=odi F c þ diag½Zi gi ðtÞoi =odi  hi ðtÞF s Þ; Pc ðtÞ ¼ CðF s sin ot þ F c cos ot  diag½ogi ðtÞ=odi F s  diag½Zi gi ðtÞoi =odi þ hi ðtÞF c Þ. Substituting the formula (8) and inserting t ¼ 2Np/o to the formula (10), one obtains the following equation: " # xð0Þ ¼ ½L  DFPð2Np=oÞF1 1 _ xð0Þ " # sin t DQð2Np=oÞ , ð11Þ cos t where L is a unit matrix of degree 2n  2n. Let E ¼ ½L  DFPð2Np=oÞF1 1 DQð2Np=oÞ, then E ¼ ½eij  is a matrix of degree 2n  2. According to the periodicity and matching conditions (8), one obtains the kth component x0(0) from the formula (11), which is now xk ð0Þ ¼ d ¼ ek1 sin t0 þ ek2 cos t0 .

(12)

Solving Eq. (12), we have the phase t0 of single-impact periodic orbit, which is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi de11  e12 e211 þ e212  d2 sin t0 ¼ , e211 þ e212 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi de12  e11 e211 þ e212  d2 . ð13Þ cos t0 ¼ e211 þ e212 Substituting t0 for t in the solutions (10) and (11), we obtain the analytical expression for period N single-impact orbit. 4. Disturbed map of N-1-I motion If single-impact periodic motion is disturbed at the instant of impact by the difference DX, then one can express the difference DX0 at the instant of the next impact. Between two consecutive impacts, the disturbed solutions of N-1-I motion are written in the form: " # " # " # ~ ~ xðtÞ S~ t A ~ ¼ FEðtÞ (14) þ FU Q~ ~ , _xðtÞ ~ Ct B~

I is a unit matrix of degree n  n. For the disturbed motion of N-1-I motion, the dimensionless time is set to zero directly after an impact, it becomes (2Np/+Dy)/o just before the next impact. Here Dy ¼ Dt0 Dt. Let te ¼ (2Np/Dy)/o, the boundary conditions at two consecutive impact points are given by # " # # " " # " ~ ~ eÞ xð0Þ xðt x0 þ Dx0 x0 þ Dx ; (15) ¼ _~ e Þ ¼ x_ 0 þ Dx_ 0 . _~ xðt xð0Þ x_ 0 þ Dx_ Inserting the boundary conditions (15) into Eq. (14) for t ¼ 0, we can solve for " # " # x0 þ Dx A~ 1 ~ ¼ ½FEð0Þ x_ 0 þ Dx_ B~ " # sinðt0 þ DtÞ 1 ~  ½FEð0Þ FU Q~ . ð16Þ cosðt0 þ DtÞ For t ¼ te, Eq. (14) is written by " # x0 þ Dx0 Y~ 0 ¼ x_ 0 þ Dx_ 0 " # " # S~ Dt A~ 2 ~ eÞ ~ eÞ ¼ DFEðt þ DFU Qðt , B~ 2 C~ Dt

ð17Þ

where S~ Dt ¼ sinðt0 þ Dt þ DyÞ, C~ Dt ¼ cosðt0 þ Dt þ DyÞ. Taking t ¼ te, one obtains, from the kth term of the disturbed solution (14), the following equation: gðDX ; DyÞ ¼ x~ k ðð2Np þ DyÞ=oÞ  d ¼ 0.

(18)

Using the implicit function theorem and supposing ðqg=qDyÞjDX ¼0 a0, one can solve Eq. (18) for Dy(DX). Inserting Dy(DX) into the state vector (17), one gets finally the state vector of period N single-impact motions just immediately after the impact. It is important, to note that the kth and (n+k)th terms of the state vector Y~ 0 denote the displacement and velocity of the impact mass Mk immediately after the impact, respectively, i.e., Y~ 0ðkÞ ¼ x~ k ¼ d, Y~ 0ðnþkÞ ¼ x_~ kþ ¼ 0. Deleting the (n+k)th term of the state vector Y~ 0 , and one gets finally the disturbed map of N-1-I motions Def DX 0 ¼ D0 ðD1 Y~ 0 þ Y 1 Þ  X n ¼ f ðv; DX Þ,

(19)

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where D1 is the matrix of degree of 2n  2n, D1 ¼ diag½d ð1Þ i , ð1Þ ð1Þ d i ¼ 1, i ¼ 1; . . . ; k  1; k þ 1; . . . ; 2n, d k ¼ 0; Y1 is a 2ndimensional vector, Y 1i ¼ 0, i ¼ 1; . . . ; k  1; k þ 1; . . . ; 2n, Y 1k ¼ t0 , t0 ¼ t0 þ DyðDX Þ þ Dt; D0 is the matrix of degree of (2n1)  2n, which is given by D0 ¼ diag½D10 ; D20 , D10 is a unit matrix of degree of (n+k1)  (n+k1), D20 is a matrix of degree of (nk)  (nk+1). The elements of the matrix D20 are denoted by d 20ðijÞ , and d 20ðijÞ ¼ 0ðjai þ 1Þ, d 20ðijÞ ¼ 1ðj ¼ i þ 1Þ. Linearizing the Poincare´ map at the fixed point X n results in the matrix

219

In this section the analysis developed in the former section is verified by the presentation of results for a threedegree-of-freedom vibratory system shown in Fig. 3. The existence and stability of N-1-I motion are analyzed explicitly. Also, local bifurcations at the points of change in stability, discussed in the previous section, are considered, thus giving some information about the existence of other types of periodic-impact motions. Let n ¼ 3 and k ¼ 2 for the system shown in Fig. 1, we obtain a three-degree-of-freedom vibratory system with a single stop. The vibro-impact system shown in Fig. 3, with non-dimensional parameters: m1 ¼ 0.2, m2 ¼ 1.0, m3 ¼ 0.6, k1 ¼ 2.0, k2 ¼ 1.0, k3 ¼ 4.0, g ¼ 0.1, f10 ¼ 0, f20 ¼ 1.0, f30 ¼ 0 and d ¼ 0 is analyzed. The forcing frequency is taken as the control parameter. The eigenvalues

of Df(o,0) associated with 1-1-I orbit can be compued by using Jacobian matrix (20). All eigenvalues of Df(o,0) are found to stay inside the unit circle for oA(2.005, 2.3379). Two critical forcing frequencies of 1-1-I orbit, oc1 ¼ 2.005 and oc2 ¼ 2.3379, are found. The real eigenvalue l1(o) escapes the unit circle of the complex plane from the (1, 0) point as o pass through o ¼ 2.005 decreasingly, and the remainder of the spectrum of Df(o,0) are strictly inside the unit circle. Consequently, a sliding bifurcation of 1-1-I orbit occurs and 1-1-II orbit stabilizes. The eigenvalue l1(o) escapes the unit circle from the (1, 0) point as o pass through o ¼ 2.3379 increasingly, period-doubling bifurcation of 1-1-I orbit correspondingly occurs. The results conform to the simulative ones below. The free flight and sticking solutions are analyzed by numerical imulation, and regions of existence and stability of different periodic-impact motions are presented. The bifurcation diagrams for the system are shown in Fig. 4, in which the quantity otf =2p and the before-impact velocity x_ 2 are plotted versus the varying forcing frequency. In Fig. 4(b), the parts of oblique lines are corresponding to N-1-II orbits while the parts of level lines to N-1-I orbits. The simulative results show that the system exhibits stable 1-1 motion in the forcing frequency interval oA[0.2, 2.3379]. A variation over the forcing frequency range from 1.0 to 4.0 sufficiently displays a large number of different types of periodic-impact motions. In the forcing frequency interval oA[0.2, 2.005), an impact occurs wherever x2 ¼ 0 during a cycle of forcing, and immediately after the impact the mass M2 is sticking to the stop till it departs once again. This means that the system exhibits stable 1-1-II motion in the frequency interval. An example of a period-1 sticking motion is given, which exists at a forcing frequency of o ¼ 1.0. A phase portrait of this periodic motion is shown in Fig. 5(a), and a time series in Fig. 5(a1). This periodic solution includes the regions of free flight and sticking for the mass M2. Stable 1-1-I motion exists in the frequency interval oA[2.005, 2.3379], as seen in Fig. 4. A phase portrait of 1-1-I motion is shown for o ¼ 2.2 in Fig. 5(b), and a time series in Fig. 5(b1). This period-1 solution only includes the region of free flight of the mass M2. The transition across the sticking boundary (o ¼ 2.005) from free flight into sticking motion is continuous and reversible

Fig. 3. Schematic of a three-degree-of-freedom vibratory system impacting a single stop

Fig. 4. Bifurcation diagrams

Df ðv; 0Þ ¼ qf ðv; DX Þ=qX jðv;DX ¼0Þ .

(20)

The stability of N-1-I motion is determined by computing and analyzing eigenvalues of Df(v, 0). Variations of the parameters of the system will cause the fixed point and its associated eigenvalues to move. If one of them passes through the unit circle of the complex plane, i.e., jli ðvc Þj ¼ 1 (vc is a bifurcation value), an instability and an associated bifurcation will occur. In general, bifurcation occurs in various ways according to the numbers of the eigenvalues on the unit circle and their position on the unit circle. In the following text, periodic-impact motions and transitions of the vibro-impact system will be analyzed. 5. Numerical analysis

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Fig. 5. Phase plane portraits and time series of the mass M2: (a) and (a1) 1-1-II motion, o ¼ 1.0; (b) and (b1) 1-1-I motion, o ¼ 2.2; (c) and (c1) 2-2-I motion, o ¼ 2.35; and (d) and (d1) 2-2-II motion, o ¼ 2.38.

Fig. 6. Phase plane portraits of the mass M2: (a) 2-2-II motion, o ¼ 2.385; (b) 2-2-II motion with grazing contact, o ¼ 2.3912; and (c) 2-1-II motion, o ¼ 2.45.

for period-1 motion. This means that sliding bifurcation of 1-1-I motion occurs and 1-1-II motion stabilizes as o passes through o ¼ 2.005 decreasingly. When o passes through o ¼ 2.3379 increasingly, period-doubling bifurcation of 1-1-I motion occurs, and 2-2-I motion stabilizes, as seen in Figs. 4, 5(c) and (c1). When o is increased to o ¼ 2.375, the system begins to exhibit stable 2-2-II motion via sliding bifurcation; see Figs. 5(d), (d1) and 6(a). The time series of period-2 double-impact sticking solution, shown in Fig. 5(d1), is composed of two parts of free flight and sticking of the mass M2. With further increase in o, the impact mass M2begins to contact the stop with zero velocity x_ 2 ¼ 0, and 2-2-II motion with grazing the stop is generated, which results in the singularity of Poincare´ map and corresponding instability of 2-2-II motion. Grazing contact instability of 2-2-II motion occurs at a forcing frequency of o ¼ 2.3912. Consequently, one impact in the motion period vanishes and the motion transits into 2-1-II motion via the grazing bifurcation, which is depicted in Fig. 6. A phase portrait of 2-2-II motion with grazing

contact is shown in Fig. 6(b). The transition across the grazing boundary (o ¼ 2.3912) from 2-2-II into 2-1-II motion is continuous and reversible for period-2 sticking motion. The system exhibits stable 2-1-II motion in the frequency interval oA(2.3912, 3.426). The transition across the sliding boundary (o ¼ 3.426) from free flight into sticking motion occurs via sliding bifurcation. An example of a period-2 sticking motion is given, which exists at a forcing frequency of o ¼ 3.0. A phase portrait and a time series of this periodic motion is shown in Fig. 7. Stable 2-1-I motion exists in the frequency interval oA[3.426, 3.8511); see Figs. 4 and 8. 2-1-I motion undergoes period-doubling bifurcation while increasing o, as a result, 4-2-I motion is generated before being transited to 4-2-II motion via sliding bifurcation. A grazing contact instability of 4-2-II motion occurs at o ¼ 3.8993, which leads to long-periodic multi-impact motion or chaos immediately. Such route to chaos via grazing contact is depicted in Figs. 9(a)–(d). Although period-doubling bifurcation of 2-1-I motion

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occurs, no period-doubling cascade emerges due to occurrence of the 4-2-II motion with grazing contact, which can be seen from Figs. 4 and 9. It is evident that the system can fall into long periodic or chaotic motion immediately via grazing bifurcation of 4-2-II motion. With the occurrence of grazing contact, 4-2-II motion undergoes a sequence of transitions that change it to 3-1-II motion with increasing o. The route from period-doubling bifurcation to chaos explicitly shows the effect of discontinuities and piecewise properties in the Poincare´ map. Grazing and sliding bifurcations result in singularity and piecewise property of the Poincare´ map. The results give great insight into the route to chaos via perioddoubling bifurcation of N-1-I orbit (nX2), sliding and grazing bifurcations. Many sets of non-dimensional para-

Fig. 7. Phase plane portrait and time series of the mass M2: 2-1-II motion, o ¼ 3.0.

221

meters have been considered and simulative results are obtained. These results from simulation show that global bifurcation of N-1-I orbit, similar to that in Fig. 4, is extensive and systematic. Consequently, N-1 orbits of the system are found to exhibit similar bifurcation regularities, which are summarized as follows: Reg.(1) N ¼ 1: 1-1-II motion’sliding bifurcation’1-1-I motion-perioddoubling bifurcation-2-2-I motionsliding bifurcation-2-2-II motion-grazing bifurcation-2-2-II motion with grazing contact-2-1-II motion; Reg.(2) NX2: N-1-II motion’sliding bifurcation’N1-I motion-perioddoubling bifurcation-2N-2-I motion-sliding bifurcation-2N-2-II motion-grazing bifurcation-2N-2-II motion with grazing contactlong periodic multi-impact motion or chaotic motion(N+1)-1-II motion. In Regs.(1) and (2), the symbol ‘-’ denotes the increasing direction of forcing frequency, ‘’’ the decreasing direction of forcing frequency. In the following text, two actual examples, the impact-forming machine and inertial shaker, are considered to further analyze periodicimpact motions and bifurcations of plastic impact oscillators. It is interesting that the bifurcation regularities are found to be similar to those of the impact-forming machine. 6. Periodic-impact motions and bifurcations of two actual impact machines 6.1. Periodic-impact motions and bifurcations of impactforming machine

Fig. 8. Phase plane portrait and time series of the mass M2: 2-1-I motion, o ¼ 3.7.

The mechanical model for a impact-forming machine with masses M1 and M2 is shown schematically in Fig. 10 [59]. Displacements of the masses M1 and M2 are represented by X1 and X2, respectively. The masses are connected to linear springs with stiffnesses K1, K2 and K3, and linear viscous dashpots with damping constants C1, C2 and C3. The excitation on the mass M2 is harmonic with amplitude P ¼ 2m0rO2. O is the excitation frequency. The

Fig. 9. Phase plane portraits of the mass M2: (a) 4-2-I motion, o ¼ 3.87; (b) 4-2-II motion, o ¼ 3.895; (c) 4-2-II motion with grazing contact, o ¼ 3.8993; and (d) chaos, o ¼ 3.8994.

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According to Eq. (22) and f1(t)/mm ¼ f2(t), one can obtain: ¯ NðtÞ ¼ ½mm sinðot þ tÞ  2zðmm mc3  mc1 Þx_ 1  ðmm mk3  mk1 Þx1 þ ðmm mk3 þ mm þ 1Þd=ð1 þ mm Þ.

¯ During the sticking motion, NðtÞ40, x1x2 ¼ d and x_ 1 ¼ x_ 2 . Immediately after the impact occurs, the motions of two masses are non-sticking if they depart with the same initial velocities immediately. Between two consecutive impacts, the motion of the vibratory system is described by the non-dimensional equations:

Fig. 10. Schematic of the impact-forming machine.

mass M1 impacts mutually with the mass M2 when the difference of their displacements equals the clearance D, i.e., X1(T)X2(T) ¼ D. The impact-forming machine, associated with large dissipation case, is considered in this paper. So the impact is assumed to be perfectly plastic. After the impact occurs, either the mass M1 moves in step with the mass M2 till the pressure of the mass M2 on M1 decreases to zero, or they depart with the same initial velocities immediately. Between any two consecutive impacts, the time T is set to zero directly at the instant the former impact is over, and the phase angle t is used only to make a suitable choice for the origin of time in the calculation. If the two masses move in step after the impact occurs, their accelerations approach to be equivalent, and the twodegree-of-freedom vibratory system becomes a singledegree-of-freedom oscillator with the mass (M1+M2). The change from free motion of both masses to their sticking represents a reduction in the degree of freedom of the system from 2 to1. The dynamic analysis can be considered in a non-dimensional form, so we assume all parameters and variables are non-dimensional, which are given by M1 Ki Ci ; mk i ¼ ; mc i ¼ , M2 K2 C2 rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi M2 K2 o¼O ; t¼T , K2 M2 C2 D  K2 , z ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; d ¼ M 2g 2 K 2M 2 X jK2 xj ¼ ; i ¼ 1; 3 and j ¼ 1; 2. P1 þ P2 mm ¼

ð21Þ

During the sticking motion, the resultant forces of two masses are respectively expressed by ¯ f 1 ðtÞ ¼  d  mk1 x1  2zmc1 x_ 1 þ NðtÞ, f 2 ðtÞ ¼ sinðot þ tÞ þ d  mk3 x2 ¯  2zmc3 x_ 2  NðtÞ,

ð23Þ

ð22Þ

where the force N(t) holds the mass M2 against the mass M1 during the sticking motion.

ð1 þ mm Þx€ 1 þ 2zðmc1 þ mc3 Þx_ 1 þ ðmk1 þ mk3 Þx1  mk3 d ¼ sinðot þ tÞ; "

mm

0

0

1 "

þ

#(

x€ 1 x€ 2

mk1 þ 1 1

"

) þ

ð0ptpts Þ,

2zð1 þ mc1 Þ

ð24Þ 2z

#(

x_ 1

)

2z 2zð1 þ mc3 Þ x_ 2 #( ) ( ) 1 x1 0 ¼ sinðot þ tÞ. ð25Þ mk3 þ 1 x2 1

In Eq. (25), tsotpts+tf, tf denotes the time of free flight of two masses between two successive impacts, and ts is the time of the sticking motion. ts+tf denotes the time interval between two successive impacts. At the instant the sticking motion ends (i.e., t ¼ ts), the state of the vibro-impact ¯ s Þ ¼ 0, x1x2 ¼ d, x_ 1 ¼ x_ 2 . system can be described by Nðt An impact occurs wherever x1x2 ¼ d. Consequently, the velocities of two masses are changed according to the impact law: mm x_ 1 þ x_ 2 ¼ ð1 þ mm Þx_ þ ;

x_ 1þ ¼ x_ 2þ ¼ x_ þ ,

(26)

where the subscript minus sign denotes the states just before impact and the subscript plus sign denotes the states just after impact. Periodic-impact motions of the system are characterized by the symbol N-p, where p denotes the number of impacts and N denotes the number of excitation periods, during one impact motion period, respectively. N-1-I represents a type of motion in which only one impact occurs during N excitation periods and two masses do not move in step, and they depart with the same initial velocities just immediately after the impact. N-1-II represents another type of motion in which one impact occurs during N excitation periods and two masses ¯ remain in a contact after the impact till the force NðtÞ equals zero. Periodic sticking and non-sticking motions of the system can be further characterized by the symbols N-p-I and N-p-II. As the state of the vibro-impact system, just immediately after impact, is chosen as Poincare´ section, N-1-I motion and Poincare´ map can be analytically derived. Based on the analysis above mentioned, we chose the Poincare´ section s ¼ fðx1 ; x_ 1 ; x2 ; x_ 2 ; tÞ 2 R4  S, x1x2 ¼ d, x_ 1 ¼ x_ 2 ¼ x_ þ g

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to construct Poincare´ map of the vibro-impact system 0

X ¼ f ðv; X Þ,

(27)

where XAR3, v is a varying parameter, vAR1, X ¼ ðx2 ; x_ 2þ ; tÞT , X 0 ¼ ðx02 ; x_ 02þ ; t0 ÞT . The vibro-impact system with the non-dimensional parameters: mm ¼ 3.0, mk1 ¼ 3.0, mk3 ¼ 3.0, mc1 ¼ 3.0, mc3 ¼ 2.0, d ¼ 0.02 and z ¼ 0.1 has been chosen for analysis. The free flight and sticking solutions are analyzed by numerical simulation, and regions of existence and stability of different periodic-impact motions are presented. The bifurcation diagrams for the system are shown in Fig. 11, in which the quantity otf =2p and the relative impact velocity x_ 2  x_ 1 are plotted versus the forcing frequency. In Fig. 11(b), the parts of oblique lines are corresponding to N-1-II orbits while the parts of level lines to N-1-I orbits. The simulative results show that the system exhibits stable 1-1 motion in the forcing frequency interval oA[0.2, 5.5597]. A variation over the forcing frequency

Fig. 11. Global bifurcation diagrams.

223

range from 2.0 to 12.5 sufficiently displays a large number of different types of periodic-impact motions; see Figs. 12–14. In these figures are included phase plane portraits and time series of relative motion of two masses. Using the phase plane portraits of relative motion, one can conveniently observe periodic-impact motions with grazing contact; one also observes clearly the process of the sticking motion by the time series of relative velocity of two masses. In the forcing frequency interval oA[0.2, 4.0968], an impact occurs during a cycle of forcing, and immediately after the impact, two masses move in step till they depart once again. This means that the system exhibits stable 1-1-II motion in the frequency interval. We give an example of a period-1 sticking motion which exists at a forcing frequency of o ¼ 2.0. A phase portrait of this periodic motion is shown in Fig. 12(a), and a time series in Fig. 12(a1). Stable 1-1-I motion exists in the frequency interval oA[4.0968, 5.5597), as seen in Fig. 11. A phase portrait of 1-1-I motion is shown for o ¼ 4.1 in Fig. 12(b), and a time series in Fig. 12(b1). The transition across the sticking boundary (o ¼ 4.0968) from free flight into sticking motion is continuous and reversible for period 1 single-impact motion. This means that sliding bifurcation of 1-1-I motion occurs and 1-1-II motion stabilizes as o passes through o ¼ 4.0968 decreasingly. When o passes through o ¼ 5.5597 increasingly, period-doubling bifurcation of 1-1-I motion occurs, and 2-2-I motion stabilizes, as seen in Figs. 11, 12(c) and (c1). When o is increased to o ¼ 5.807, the system begins to exhibit stable 2-2-II motion via sliding bifurcation. We note from Fig. 12(d) and (d1) that the time series of period 2 double-impact sticking

Fig. 12. Phase plane portraits and time series of relative motion: (a) and (a1) 1-1-II motion, o ¼ 2.0; (b) and (b1) 1-1-I motion, o ¼ 4.1; (c) and (c1) 2-2-I motion, o ¼ 5.578; and (d) and (d1) 2-2-II motion, o ¼ 5.9.

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solution is composed of two parts, which includes the regions of free flight and sticking for both of the masses. With further increase in o, two masses begin to contact with the same velocities. Consequently, 2-2-II motion with grazing contact is generated, which results in the singularity of Poincare´ map and corresponding instability of 2-2-II motion. Grazing contact instability of 2-2-II motion occurs at a forcing frequency of o ¼ 6.2181, and then one impact in the motion period vanishes and the motion transits into 2-1-II motion via the grazing bifurcation, which is depicted in Figs. 12(d) and 13. The transition across the grazing boundary (o ¼ 6.2181) from 2-2-II into 2-1-II motion is continuous and reversible for period-2 sticking motion.

Fig. 13. Phase plane portraits of relative motion: (a) 2-2-II motion with grazing contact, o ¼ 6.2181; and (b) 2-1-II motion, o ¼ 6.3.

The system exhibits stable 2-1-II motion in the frequency interval oA(6.2181, 8.8542). The transition across the sliding boundary (o ¼ 8.8542) from free flight into sticking motion occurs via sliding bifurcation. Stable 2-1-I motion exists in the frequency interval oA[8.8542, 10.3048). 2-1-I motion undergoes period-doubling bifurcation with increasing o, as a result, 4-2-I motion is generated before being transited to 4-2-II motion via sliding bifurcation. A grazing contact instability of 4-2-II motion occurs at o ¼ 11.2107, which leads to long-periodic multi-impact motion or chaos immediately. Such route to chaos via grazing contact is depicted in Figs. 14(a)–(c). Although period-doubling bifurcation of 2-1-I motion is existent, no period-doubling cascade emerges due to grazing bifurcation of 4-2-II motion, which can be seen from Figs. 11 and 14(b). It is evident that the system can fall into long periodic or chaotic motion immediately via grazing bifurcation of 4-2-II motion. With the occurrence of grazing contact, 4-2-II motion undergoes a sequence of transitions that change it to 3-2-II motion with increasing o. A representative 3-2-II motion is shown for o ¼ 11.9 in Fig. 14(d). After increasing o to 12.0278, one can find that the grazing contact of two masses occurs again to result in a grazing bifurcation that changes 3-2-II motion to 3-1-II one. Such a transition across the grazing boundary from 3-2-II to 3-1-II motion is shown in Figs. 14(d)–(f) while the phase portrait of 3-2-II orbit with grazing contact is depicted in Fig. 14(e).

Fig. 14. Phase plane portraits of relative motion: (a) 4-2-II motion, o ¼ 11.18; (b) 4-2-II motion with grazing contact, o ¼ 11.2107; (c) chaos, o ¼ 11.4; (d) 3-2-II motion, o ¼ 11.9; (e) 3-2-II motion with grazing contact, o ¼ 12.0278; and (f) 3-1-II motion, o ¼ 12.5.

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The inelastic impact system studied here involves nine non-dimensional parameters: mm, mk1, mk3, mc1, mc3, f20, d, z and o. Due to these relatively large number parameters, the detailed influence of each parameter on dynamics of the system cannot be analyzed here. However, it is of special interest to acquire an overall picture of dynamics of the system under key parameter variation. Taking the non-dimensional parameters: mm ¼ 3.0, mk1 ¼ 3.0, mk3 ¼ 3.0, mc1 ¼ 3.0, mc3 ¼ 2.0, d ¼ 0.02 and z ¼ 0.1 as the criterion, we analyze the influence of distribution of mass mm and clearance d on periodic-impact motions and bifurcations. Regions of existence and stability of different periodic-impact motions are therefore presented in (mm, o) and (d, o) planes of dimensionless mass distribution mm, clearance d and frequency o, which are plotted in Figs. 15(a) and (b), respectively. It is important to notice that global bifurcations of N-1-I motions of the impact-forming machine are found to exhibit extensive and systematic characteristics, which are summarized as follows:

(1) N ¼ 1: 1-1-II motion’sliding bifurcation’1-1-I motion-period-doubling bifurcation-2-2-I motion sliding bifurcation-2-2-II motion-grazing bifurcation-2-2-II motion with grazing contact-2-1-II motion; (2) NX2: N-1-II motion’sliding bifurcation’N-1-I motion-period-doubling bifurcation-2N-2-I motionsliding bifurcation-2N-2-II motion-grazing bifurcation-2N-2-II motion with grazing contact-long periodic multi-impact motion or chaotic motion(N+1)-2-II motion-grazing bifurcation-(N+1)-2-II motion with grazing contact-(N+1)-1-II motion.

225

In designing impact tools it is of great interest to achieve the desired periodic-impact velocities. Two components of the vibro-impact system shown in Fig. 11 collide with each other, so the relative before-impact velocity Dx_  ¼ x_ 2  x_ 1 is an important factor to working efficiency of the impact-forming machine. In Fig. 16 are plotted the bifurcation diagrams for the relative before-impact velocity Dx_  versus the forcing frequency under the conditions of different system parameters. The bifurcation diagram, with the criterion parameters, is plotted in Fig. 16(a). Only changed parameter is given in Figs. 16(b)–(p), and all the other parameters, not given in figure’s description, are the same as the criterion parameters. From these figures one observes the typical behavior of periodic windows, with one impact velocity, separated by other periodic or chaotic regions. The effects of changes in distribution of mass mm are investigated by changing its value and varying the forcing frequency. With changing the distribution of mass mm in the interval mmA[3.0, 5.0], one can find that dynamics of the system is essentially similar in comparison to that in the criterion case, no major differences can be observed, see Fig. 16(b). As the distribution of mass mm is decreased, the relative impact velocities of the system, around the period1 peak, are slightly decreased. As a result, the resonance peak becomes sharper and narrower, and the regions of N1 periodic motion are enlarged, as seen in Fig. 16(c). As the distribution of stiffness mk1 is changed in the interval [2.0, 5.0], the relative impact velocities of the mass M2 against M1 decrease slightly for low mk1, and the relative impact velocities increase slightly for high mk1; see Figs. 16(d) and (e). Low distribution of stiffness mk3 leads to large impact velocities, and the window of 1-1 motion shrinks slightly, see Fig. 16(f). For large mk3, the window of period-1 motion becomes wide and the relative impact

Fig. 15. Regions of existence and stability of different regimes of the impact motion: (a) mmA[1,5], d ¼ 0.02 and (b) dA[0.005, 0.05], mm ¼ 3.

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Fig. 16. Bifurcation diagrams for the relative impact velocity x_ 2  x_ 1 versus the forcing frequency: (a) mm ¼ 3.0, mk1 ¼ 3.0, mk3 ¼ 3.0, mc1 ¼ 3.0, mc3 ¼ 2.0, d ¼ 0.02, z ¼ 0.1; (b) mm ¼ 5.0; (c) mm ¼ 1.0; (d) mk1 ¼ 2.0; (e) mk1 ¼ 5.0; (f) mk3 ¼ 1.0; (g) mk3 ¼ 4.0; (h) d ¼ 0.05; (i) d ¼ 0.005; (j) z ¼ 0.05; (k) z ¼ 0.2; (l) mc1 ¼ 5.0; (m) mc1 ¼ 1.0; (n) mc3 ¼ 4.0; (o) mc3 ¼ 0.5; and (p) mc3 ¼ 0.

velocity decreases as seen in Fig. 16(g). As the clearance d is increased, the windows of 1-1, 2-2, 2-1 motion enlarge towards the region of high frequencies; see Fig. 16(h). For small value of d, dynamical behavior of the system is similar to that in the criterion case. As the damping ratio z is decreased, the resonance peak becomes sharper and narrower. Consequently, the relative impact velocities, near

the resonance peak, increase remarkably; see Fig. 16(j). Increasing damping ratio z results in lower relative impact velocities of the system (Fig. 16(k)). As the distribution of damping mc1 is increased, the system exhibits similar behavior to that in Fig. 16(a); see Fig. 16(l). Decreasing the distribution of damping mc1 leads to a higher peak of the relative impact velocity as seen in Fig. 16(m). Increase

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227

in the distribution of damping mc3 bring out the obvious decrease of the relative impact velocity Dx_  , decrease in mc3 corresponds to the contrary results; see Figs. 16(n)–(p). High impact velocity and short impact period are two important factors for raising working efficiency of the impact-forming machine. Analyses above-mentioned show that suitable system parameters generally results in both larger relative impact velocities and shorter periods for single-impact periodic motions. Low mk3, mc3 and z lead to higher peak value of relative impact velocities of period-1 motion. However, the result is contrary to low mm and high mk3, mc3 and z. It is to be noted that lower forcing frequency leads to longer period. Large impact velocity and short period are important to working efficiency of the impactforming machine, so the width of the window of 1-1 response, associated with the right side of period-1 peak, is considerable interest. When the system parameters are chosen near mm ¼ 3.0, mk1 ¼ 3.0–4.0, mk3 ¼ 2.0, mc1 ¼ 1.0–2.0, mc3 ¼ 0.0, d ¼ 0.02, z ¼ 0.05–0.1, the impact-forming machine can exhibit 1-1 motion with larger relative impact velocities. The effectiveness of the system parameters is illustrated by an example with parameters: mm ¼ 3.0, mk1 ¼ 4.0, mk3 ¼ 2.0, mc1 ¼ 2.0, mc3 ¼ 0.0, d ¼ 0.02, z ¼ 0.05 and 0.1. The bifurcation diagrams, associated with the optimized parameters, are shown in Fig. 17. It is obvious that the peak-impact velocity of relative motion of two masses, associated with 1-1 motion, increases remarkably. The simulative results shown in Fig. 17 mean that the optimized parameters bring out both high peak-impact velocity and shorter impact period for 1-1 motion.

mass M is connected to the supporting base by linear spring with stiffness K and dashpot with damping coefficient C. The excitation on the shaker is harmonic with amplitude F0. The shaker impacts mutually with the cast when they are on the same height so that the cast exhibits the bouncing motion. The impact is assumed to be perfectly plastic. After the shaker gives an impact against the cast, either the shaker M moves in step with the cast m until the pressure of the shaker on cast decreases to zero, or they depart with the same initial velocities immediately after the impact. If the shaker moves in step with the cast after the impact occurs, their accelerations are equivalent, and the two-degree-of-freedom vibro-impact system becomes a single-degree-of-freedom oscillator with the mass (M+m) and subjected to sinusoidal excitation F 0 sinðOT þ tÞ. During the sticking motion, the resultant forces of the shaker and cast are, respectively.

6.2. Periodic-impact motions and bifurcations of inertial shaker

¯ F 1 ðTÞ ¼ F 0 sinðOT þ tÞ  KX 1  C X_ 1  NðTÞ, ¯ F 2 ðTÞ ¼ mg þ NðTÞ,

The inertial shakers are the typical vibro-impact machines, which are widely used in casting industry. The mechanical model for an inertial shaker is shown schematically in Fig. 18 [60]. The masses of shaker and cast are represented by M and m, and the displacements of them are represented by X1 and X2, respectively. The shaker with

Fig. 18. Schematic of the inertial shaker.

ð28Þ

¯ where the force NðtÞ holds the shaker M against the cast m during the sticking motion. According to Eq. (28) and F1(T)/M ¼ F2(T)/m, one can obtain mðF 0 sinðOT þ tÞ  KX 1  C X_ 1 Þ þ Mmg ¯ . NðTÞ ¼ M þm

(29)

¯ During the sticking motion, NðTÞ40, X1 ¼ X2, ¯ X_ 1 ¼ X_ 2 ; the sticking motion ends when the force NðTÞ ¯ s Þ ¼ 0. becomes zero at T ¼ Ts, i.e., NðT Immediately after the impact occurs, if the shaker and cast depart with the same initial velocities immediately, their motions are non-sticking and their accelerations are not equivalent. Between consecutive impacts, the motion of the vibratory system is described by the differential equations: ðM þ mÞX€ 1 þ C X_ 1 þ KX 1 þ mg ¼ F 0 sinðOT þ dÞ; Fig. 17. Bifurcation diagrams for the relative impact velocity x_ 2  x_ 1 versus the forcing frequency (mm ¼ 3.0, mk1 ¼ 4.0, mk3 ¼ 2.0, mc1 ¼ 2.0, mc3 ¼ 0.0 and d ¼ 0.02): (a) z ¼ 0.05 and (b) z ¼ 0.1.

ð0pTpT s Þ,

M X€ 1 þ C X_ 1 þ KX 1 ¼ F 0 sinðOT þ dÞ, X€ 2 ¼ g; ðT s oTpT s þ T f Þ,

ð30Þ

ð31Þ

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228

where Tf denotes the time of free flight motion between two successive impacts, Ts is the time of the sticking motion and Ts+Tf denotes the time interval between two successive impacts. Without loss of generality, the dynamic analysis of inertial shaker can be considered in a non-dimensional form. We assume all parameters and variables are nondimensional, which are given by mm ¼

m ; M

F0 , Mg rffiffiffiffiffiffi M o¼O , K xi ¼

x_ 1 þ mm x_ 2 ¼ ð1 þ mm Þx_ þ ;

x_ 1þ ¼ x_ 2þ ¼ x_ þ .

(32)

Let y ¼ ot, the Poincare´ section s ¼ fðx1 ; x_ 1 ; x2 ; x_ 2 ; yÞ 2 R4  S, x1 ¼ x2, x_ 1 ¼ x_ 2 ¼ x_ þ g is chosen, and dynamics of the inertial shaker, in inelastic impact case, is represented by a three-dimensional map: X 0 ¼ f~ðv; X Þ,

b0 ¼

C g ¼ pffiffiffiffiffiffiffiffiffi ; 2 KM rffiffiffiffiffiffi K 1 t¼T ; g1 ¼ ; M b0

An impact occurs wherever x1 ¼ x2. After each impact, the velocities of shaker and cast are changed according to the impact law:

(33) 3

KX i . F0

Fig. 19. Bifurcation diagrams (b ¼ 3.0, z ¼ 0.1 and m ¼ 0.3).

where XAR , v is a varying parameter, vAR1; X ¼ ðx1 ; x_ 1þ ; tÞT , X 0 ¼ ðx01 ; x_ 01þ ; t0 ÞT . Periodic-impact motions of the system can be characterized by the symbols: NpI

and

NpII,

(34)

where the symbols I and II represent free flight and sticking motion of the shaker and cast, respectively, p is the number of impacts and n the number of excitation force. The existence and transition of 1-1-II (or I) motions were analyzed explicitly in this section. A shaker with non-dimensional parameters m ¼ 0.3, b ¼ 3.0 and z ¼ 0.1 has been chosen for analyses. A variation over the forcing frequency range from 1.32 to 6.0 is sufficient to display a large number of different types of periodicimpact motions; see Figs. 19–21. In the forcing frequency interval oA(1.7122, 8), the shaker gives an impact against the cast during a cycle of forcing, and after the impact the shaker and cast move in step until they

Fig. 20. Phase plane portraits and time series: (a) and (a1) 1-1-II motion, o ¼ 6.0; (b) and (b1) 1-1-I motion, o ¼ 1.7; (c) and (c1) 2-2-I motion, o ¼ 1.52; and (d) and (d1) 4-4-I motion, o ¼ 1.5.

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Fig. 21. Phase plane portraits of relative motion: (a) 4-4-I motion with grazing contact, o ¼ 1.475; (b) 8-6-II motion, o ¼ 1.36; (c) 16-12-II motion, o ¼ 1.342; and (d) chaos; o ¼ 1.32.

Fig. 22. Bifurcation diagrams: (a) m ¼ 0.1; (b) m ¼ 0.5; (c) z ¼ 0.2; (d) z ¼ 0.05; (e) b ¼ 1.5; and (f) b ¼ 5.0.

depart once again. When one cycle is completed, another begins, which means that the system exhibits stable 1-1-II motion in the forcing frequency interval. Phase plane portrait and time series of stable 1-1-II motion are plotted for o ¼ 6.0 in Fig. 20(a) and (a1), respectively. Stable 1-1-I motion exists in the frequency interval oA(1.54653, 1.7122]. A phase portrait of 1-1-I motion is shown for o ¼ 1.7 in Fig. 20(b), and a time series in Fig. 20(b1). The transition across the sticking boundary (o ¼ 1.7122) from free flight into sticking motion is continuous and reversible for single-impact periodic motion. This means that sliding bifurcation of 1-1-I motion occurs and 1-1-II motion stabilizes as o passes through o ¼ 1.7122 increasingly.

When o passes through 1.54653 decreasingly, 1-1-I orbit has changed its stability, and period-doubling bifurcation of 1-1-I motion occurs so that 2-2-I motion stabilizes. The time and phase trajectories of 2-2-I motion are shown for o ¼ 1.52 in Fig. 20(c) and (c1). With decreasing the forcing frequency o, 2-2-I motion loses its stability so that 4-4-I motion is generated, see Fig. 20(d) and (d1). With further decrease in o, the vibro-bench begins to contact the cast with the same velocities ðx_ 1 ¼ x_ 2 Þ when x1 ¼ x2, which causes the 4-4-I motion with ‘‘grazing contact of vibrobench and cast’’. As the forcing frequency o is decreased to 1.475, a grazing contact instability of 4-4-I motion occurs, one impact in the motion period vanishes and the motion transits into 4-3-I motion. Such a case of 4-4-I orbit with

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grazing contact is shown for o ¼ 1.475 in Fig. 21(a). The transition is caused by discontinuity of impact (grazing contact of shaker and cast). With decrease in the forcing frequency o, 4-3-I motion transits to 4-3-II motion via sliding bifurcation, and then the system falls into chaotic motion via period- doubling cascades of 4-3-II motion; see Figs. 21(b)–(d). The route from discontinuous perioddoubling cascades to chaos explicitly shows the effect of discontinuities in the Poincare´ map. Grazing and sliding bifurcations result in singularity and piecewise property of the Poincare´ map. The dynamical behavior of shaker, near the nondimensional parameters: mm ¼ 0.3, b ¼ 3.0 and z ¼ 0.1, are further analyzed by changing one of the system parameters. The bifurcation diagrams are plotted in Figs. 22(a)–(f). The global bifurcation diagrams for the relative before-impact velocity of the system versus the forcing frequency enable one to select forcing frequency ranges in which stable single-impact periodic responses can be expected to occur, and to predict larger relative impact velocities of such responses. When the system parameters are chosen near mm ¼ 0.3, b ¼ 1.5, and z ¼ 0.1, the shaker can give larger impact velocities against the cast in the forcing frequency range oA[0.8, 1.5], and the system exhibits stable 1-1-I motion; see Fig. 22(e). 7. Conclusions In this paper we have analyzed sliding and grazing touch phenomena which occur in sticking solutions of plastic impact oscillators. An n-degree-of-freedom vibratory system with plastic impacts is considered. Dynamics of the vibro-impact system is described by a (2n1)-dimensional map with dynamical variables defined at the impact instants, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of the impact mass immediately after the impact, and the singularity of map is generated via the grazing contact of both the impact mass and the rigid stop and corresponding instability of periodic-impact motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. Finally two actual examples, the impact-forming machine and inertial shaker, are considered to further analyze periodic-impact motions and bifurcations of plastic impact oscillators. The free flight and sticking solutions of such systems are analyzed by numerical simulation, bifurcation and transition of N-1-I motions are therefore presented. For mass-spring-damping type of plastic impact oscillators, N-1-I orbits of such systems are found to exhibit extensive and systematic bifurcation characteristics, which are summarized as follows: (1) N ¼ 1: 1-1-II motion’sliding bifurcation’1-1-I motion-period-doubling bifurcation-2-2-I motion-

sliding bifurcation-2-2-II motion-grazing bifurcation-2-1-II motion; (2) NX2: N-1-II motion’sliding bifurcation’N-1-I motion-period-doubling bifurcation-2N-2-I motionsliding bifurcation-2N-2-II motion-grazing bifurcation-long periodic multi-impact motion or chaotic motion-(N+1)-1-II motion (or (N+1)-2-II motiongrazing bifurcation-(N+1)-1-II motion). It is important to notice that the bifurcation regularities above-mentioned do not hold for the bouncing oscillators with inelastic impacts, e.g., inertial shaker. Such systems can only exhibit 1-1-I and 1-1-II motions, no N-1-I and N-1-II motions (NX2) are generated under parameter variation. Sliding and period-doubling bifurcations of 1-1-I motion occur with increase and decrease in forcing frequency, respectively. In designing impact tools, it is of great interest to achieve the desired periodic-impact velocities. In order to facilitate such design, bifurcation diagrams of the before-impact (or relative before-impact) velocity may be useful. The global bifurcation diagrams for the before-impact velocity of such systems versus the forcing frequency enable the practicing engineer to select excitation frequency ranges in which stable single-impact periodic responses can be expected to occur, and to predict the peak velocities of such responses. In designing and remaking inelastic vibro-impact machines, the other parameters and forcing frequency can be optimized by analyses of stability and bifurcations of periodic-impact motions if some system parameters have been given or limited. The method of dynamical analyses of the mechanical model shown in Fig. 1 can be applied to some practical vibratory machines and equipment with large dissipation, e.g., hammer impact tools, pile driver, compacting machines, milling and forming machines, and so on, by which the optimal system parameters of the vibro-impact systems may be obtained. Acknowledgments The authors gratefully acknowledge the support by National Natural Science Foundation (10572055, 50475109), Natural Science Foundation of Gansu Province Government of China (3ZS061-A25-043 (key item)) and ‘Qing Lan’ Talent Engineering by Lanzhou Jiaotong University. References [1] Shaw SW, Holmes PJ. A periodically forced piecewise linear oscillator. Journal of Sound and Vibration 1983;90(1):129–55. [2] Cone KM, Zadoks RI. A numerical study of an impact oscillator with the addition of dry friction. Journal of Sound and Vibration 1995;188(5):659–83. [3] Aidanpa¨a¨ JO, Gupta BR. Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system. Journal of Sound and Vibration 1993;165(2):305–27.

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