# Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: Diversity and parameter matching of periodic-impact motions

G.W. Luo, X.H. Lv, Y.Q. Shi

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G.W. Luo, X.H. Lv, Y.Q. Shi. Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: Diversity and parameter matching of periodic-impact motions. International Journal of Non-Linear Mechanics, 2014, 65, pp.173 - 195. 10.1016/j.ijnonlinmec.2014.04.013. hal-01403576

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# Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: Diversity and parameter matching of periodic-impact motions

G.W. Luo, X.H. Lv, Y.Q. Shi

Key Laboratory of System Dynamics and Reliability of Rail Transport Equipment of Gansu Province, Lanzhou Jiaotong University, Lanzhou 730070, PR China

A two-degree-of-freedom system with a clearance and subjected to harmonic excitation is considered. The correlative relationship and matching law between dynamic performance and system parameters are studied by multi-parameter and multi-performance co-simulation analysis. Two key parameters of the system, the exciting frequency $\omega$ and clearance $\delta$, are emphasized to reveal the influence of the main factors on dynamic performance of the system. Diversity and evolution of periodic impact motions are analyzed. The fundamental group of impact motions is defined, which have the period of exciting force and differ by the numbers $p$ and $q$ of impacts occurring at the left and right constraints of the clearance. The occurrence mechanism of chattering-impact vibration of the system is studied. As the clearance $\delta$ is small or small enough, the transition from 1-p-p to 1-(p|1)-(p|1) motion (the fundamental group of motions, pZ1) basically goes through the processes as follows: pitchfork bifurcation of symmetric 1-p-p motion, period-doubling bifurcation of asymmetric 1-p-p motion, non-periodic or chaotic motions caused by a succession of period-doubling bifurcations, symmetric 1-(p|1)-(p|1) motion generated by a degeneration of chaos. As for slightly large clearance, a series of grazing bifurcations of periodic symmetrical impact motions occur with decreasing the exciting frequency so that the number $p$ of impacts of the fundamental group of motions increases two by two. As $p$ becomes big enough, the incomplete chattering-impact motion will appear which exhibits a chattering sequence in an excitation period followed by a finite sequence of impacts with successively reduced velocity and reaches the non-sticking region. Finally, the complete chattering-impact motion with sticking will occur with decreasing the exciting frequency $\omega$ up to the sliding bifurcation boundary. A series of singular points on the boundaries between existence regions of any adjacent symmetrical impact motions with fundamental period are found, i.e., two different saddle-node bifurcation boundaries of one of them, real-grazing and bare-grazing bifurcation boundaries of the other alternately and mutually cross themselves at the points of intersection and create inevitably two types of transition regions: narrow hysteresis and small tongue-shaped regions. A series of zones of regular periodic and subharmonic impact motions are found to exist in the tongue-shaped regions. Based on the sampling ranges of parameters, the influence of dynamic parameters on impact velocities, existence regions and correlative distribution of different types of periodic-impact motions of the system is emphatically analyzed.

## 1. Introduction

The clearances or constraints among mechanical components exist inevitably in mechanical systems due to the restriction on technological level in the mechanical machining process, the demand for mechanical design and assembly errors, the need for the heat-expansion and cold-contraction of some part of mechanical components or other reasons. The existence of the clearances

or motion limiting constraints is bound to bring about the vibro-impact effects which have a great influence on performance of mechanical equipments, increase noise levels and intensify fatigue and abrasion of mechanical parts. The non-smooth natures of phase trajectories caused by the impacts complicate the dynamic analysis of such systems inevitably, but they can be described theoretically and numerically by the discontinuities in good agreement with reality. It is necessary to be able accurately to model the dynamics of the mechanical systems with clearances or motion limiting constraints from the multi-parameter level, so as to utilize effectively and enlarge profitable effects by reasonable
parameter matching design, to furthest minimize the influence of
unfavorable factors such as abrasion, chattering-impact, high noise
levels, etc. The broad interest in revealing dynamical performance
of vibro-impact systems is reflected by an ever increasing amount
of investigations devoted to this aspect. Several effective methods
of qualitative, numerical and experimental analyses have been
proposed and their validity has been demonstrated by different
vibro-impact models in the past several years. Dynamics of
different types of vibratory systems with clearances or motion
limiting constraints, including detailed observations about stabi-
lity and bifurcations using numerical simulations and qualitative
analyses, were partly reported in Refs. [1–10]. Shaw and
Holmes [1] studied a single-degree-of-freedom vibro-impact sys-
tem by casting the traditional approaches used for analyzing the
periodic responses in the system in the form of a more general
Poincaré map. The notable work showed how even a very simple
impact oscillator could display the full range of typical nonlinear
behavior: saddle node and flip bifurcations, multiple coexisting
attracting solutions, chaos, etc. Periodic and chaotic behavior of a
threshold-limited two-degree-of-freedom system was analyzed by
Aidanpää and Gupta [2]. Souza and Caldas [3] put forward an
algorithm of the spectrum of Lyapunov exponents of attractors of a
vibro-impact system for interpreting bifurcations and chaotic
transitions. Peterka and Kotera [4] studied the appearance and
characteristics of intermittency chaos of an impact oscillator
occurring near saddle-node bifurcation boundaries of neighboring
fundamental motions with two different impacts in an excitation
period. Pavlovskaia and Wiercigroch [5,6] developed an efficient
semi-analytical method to study the percussive drilling principle
for the impact systems with drift and accurately predicted a range
of control parameters for which the best progression rates were
obtained. Quinn and Bairavarasu [7] considered a three-mass
collision model with compliant contacts, a two-dimensional map
is derived to describe initial conditions that lead to pairwise
sequences, and the variation in the final velocities is characterized
in terms of the initial configuration of the system. Luo [8]
determined analytically and numerically stability and bifurcation
conditions for the LR model motion in an impact oscillator, for
which the regions are developed in parameter space. Leine [9]
presented an asymptotic approximation method for the critical
restitution coefficient of a parametrically excited impact oscillator
and described its dynamics by an unilaterally constrained Hill's
equation. More studies on vibro-impact dynamics were summar-
ized in Ibrahim's monograph [10]. A special feature of vibro-
impact systems that might prove to be useful in the current study
is the instability caused by low-velocity collisions, which is the
so-called grazing effect. The pioneer work in this field was done by
Nordmark [11], who developed systematic methods for investigat-
ing grazing dynamics and their attendant bifurcations, providing
the results which laid the foundation for many subsequent studies.
Thereafter, this work has been further expanded by discontinuity-
geometry approaches of a series of impact maps, where the
increasingly thorough investigations of grazing bifurcations of
such maps have been reported in Refs. [12–25]. The chattering-
impact vibration denotes an accumulation of impact events in
finite time (or a motion period), and sticking is where the
dynamics becomes finally constrained to the impact surface as
the impact attenuation ends. Detailed studies of chattering-impact
vibration with sticking were carried out for the vibro-impact
systems in Ref. [26–35]. Rising events occurring in multi-sticking
solutions are discussed by Wagg [29,30], the study on simulation
and stability analysis of impacting systems with complete chatter-
ing has been further expanded by Nordmark and Piiriinen [31]. de
Souza and Caldas [32] numerically investigated the basins of
attraction of coexisting periodic and chaotic attractors in a
gear-rattling impact model. Luo [33,34] presented an idealized,

piecewise linear system to model non-smooth vibration of gear
transmission arising from impacts between the gear teeth and
predicted analytically occurrence of the periodic impact-sticking
motions. Hôsa and Champneysb [35] considered a mechanical
model of a pressure relief valve and analyzed the geometry of the
chattering region via the computation of several pre-images of the
grazing set. The vibro-impact systems display many of the com-
plex dynamical features near critical points of high codimension
bifurcations; in particular, different types of codimension two
bifurcations and parameter unfolding associated with the interac-
tion of saddle-node and grazing bifurcations [23], double flip
bifurcation [36], Hopf-flip bifurcation [37], Hopf-pitchfork bifurca-
tion and double Neimark-Sacker bifurcation [38], codimension-
two grazing bifurcations [39,40], grazing-sliding bifurcation [41],
etc., were studied by numerical simulations and qualitative ana-
lyses in recent years. Several methods of experimental analyses
have been developed for some impact models in Refs. [42-46].
Wiercigroch and Sin [42] performed an experimental study of base
excited symmetrically piecewise linear oscillator for observing and
proving symmetric and asymmetric periodic-impact motions and
their attendant bifurcations. Jin and Hu [43] designed an experi-
mental setup consisting of both vibro-impacting cantilever beams
with lumped mass blocks for testing various types of vibro-impact
motions such as subharmonic, superharmonic, quasi-periodic and
chaotic motions, etc. Quinn [44] considered the effect of external
forces during finite duration collisions using an incremental model
of impact. Based on the results of the experimental system and the
incremental model, the standard algebraic model of restitution is
modified to include these finite duration effects. Nguyen et al. [45]
designed an experimental scheme of vibro-impact rig to mimic a
mole penetration through the soil, which was based on electro-
mechanical interactions of a conductor with an oscillating mag-
netic field. Sitnikova and Pavlovskaia [46] studied bifurcations and
chaotic attractors of an impact oscillator with a shape memory
alloy constraint. The evolution of chaotic attractors is recorded
experimentally, and changes in the structure of the attractors are
shown. Along with the requirement of engineering application
and the increasing of basic research on non-smooth dynamics,
vibro-impact dynamics has been applied to a wide range of
practical mechanical systems for finding the correlative relation-
ship between dynamic performance and model parameters,
e.g., wheel-rail impacts of railway coaches [47,48], vibrating
hammer [49], pile driver [50,51], ground moling dynamics [52],
mill rolls [53], the fly-wheel model of the bouncing ball [54], link
mechanism [55], ultrasonic percussive drilling [56,57], the
vibro-impact capsule system [58], Jeffcott rotors with bearing
clearance [59-61], impact dampers [62,63], excited pendula with
impacts [64], high frequency vibro-impact drilling [65], pipes
conveying [66], vibro-impact interaction of ships with ice [67],
gears transmission systems [32-34,68-71], etc. The dynamical
models designed in Refs. [47-71] have proved to be useful for
revealing dynamical performance of such practical systems and
the related studies have provided much information of the funda-
mental nature that broadened the scope of knowledge on the non-
smooth motions of mechanical systems with impacts.

It is important to note, that most studies of vibro-impact
dynamics are based on single-parameter bifurcation analysis in
the past several years. Very few have considered multi-perfor-
mance, multi-process coupling and multi-parameter simulation
analysis and collaborative optimization for research on vibro-
impact dynamics. The purpose of the present study is to focus
attention on multi-performance and multi-parameter simulation
analysis for dynamics of a two-degree-of-freedom periodically-
forced system with a clearance represented by two symmetric
rigid stops. The correlative relationship between dynamic perfor-
mance and system parameters is particularly studied to find their
reasonable matching law. Two key parameters of the system, the exciting frequency $\omega$ and clearance $\delta$, are emphasized to analyze the influence of the main factors on dynamic performance of the system in Section 3, and diversity and evolution of periodic-impact motions are discussed. As for small clearance, the transition between adjacent fundamental motions basically goes through three processes with decrease in $\omega$, which are epitomized as follows: pitchfork bifurcation of symmetric 1-p-p motion; period-doubling bifurcation of asymmetric 1-p-p motion; non-periodic or chaotic motions caused by a succession of period-doubling bifurcations; symmetric 1-(p+1)-(p+1) motion generated by a degeneration of chaos. As for slightly large clearance, a series of grazing bifurcations occur with decreasing the exciting frequency so that the number p of impacts of the fundamental group of motions increases two by two. As p becomes big enough, the system will exhibit an incomplete chattering sequence in an excitation period followed by a finite sequence of impacts with successively reduced velocity. Finally, complete chattering-impact motion with sticking will occur with decreasing the exciting frequency $\omega$ up to the sliding bifurcation boundary. A series of singular points on the critical lines between adjacent fundamental motions are found, at which saddle-node bifurcation boundaries of one of them and real-grazing and bare-grazing bifurcation boundaries of the other alternately and mutually cross themselves. Consequently, two types of transition regions are created inevitably by the singular points, i.e., hysteresis and small tongue-shaped regions. A series of zones of subharmonic impact motions are found in the tongue-shaped regions. Based on the sampling ranges of parameters, the influence of system parameters on impact velocities, existence regions and correlative distribution of different types of periodic-impact motions of the system is emphatically analyzed in Sections 4 and 5.

## 2. Mechanical model

The mechanical model of a two-degree-of-freedom periodically-forced system with a clearance represented by two symmetric rigid stops is shown in Fig. 1. The vibration displacements of the mass blocks with masses $M_1$ and $M_2$ are expressed by $X_1$ and $X_2$, respectively. Two mass blocks are connected by a linear spring with stiffness $K_1$ and a linear viscous dashpot with damping constant $C_1$, and the mass block $M_2$ is attached to the supporting base by the linear spring $K_2$ and linear viscous dashpot $C_2$. The exciting forces acting on the mass blocks are harmonic with amplitudes $P_1$ and $P_2$. The exciting frequency and phase angle are the same for two mass blocks, which are respectively represented by $\Omega$ and $\tau$. The vibration amplitude of the system is restricted by a clearance. For small forcing amplitudes the system exhibits simple linear oscillation. With increase in the amplitude, the mass block $M_1$ eventually begins to hit the rigid stops of the clearance situated in certain distances B and -B from the equilibrium position of the mass block. Consequently, the contacts of the mass block with the stops result in discontinuous changes in the motion properties of the system. The kinetic energy of the system is dissipated partially on the impact. The presence of the

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic of a two-degree-of-freedom periodically-forced system with a clearance.</figcaption></figure>

clearance and impact leads to non-smooth nature of motion trajectory of the system and complicates its dynamic analysis. The impact is generally inelastic and modeled by a coefficient of restitution with $0 < R < 1$. For the simplicity of the induction and summarization, the dynamic analysis can be considered in a dimensionless form – in the following work, we assume all parameters and variables are dimensionless, which are given by

$$ 
\begin{aligned}
\mu_m &= \frac{M_2}{M_2+M_1}, & \mu_k &= \frac{K_2}{K_2+K_1}, & \mu_c &= \frac{C_2}{C_2+C_1}, \\
\zeta &= \frac{C_1}{2\sqrt{K_1M_1}}, & f_{20} &= \frac{P_2}{P_2+P_1}, & \omega &= \Omega\sqrt{\frac{M_1}{K_1}}, \\
\delta &= \frac{BK_1}{P_1+P_2}, & t &= T\sqrt{\frac{K_1}{M_1}}, & x_i &= \frac{X_iK_1}{P_1+P_2}, & i &= 1, 2.
\end{aligned} 
\quad (1) $$

Eq. (1) allows us to make a conclusion that the sampling range of partial dimensionless parameters can be easily identified as $\mu_m \in (0, 1)$, $\mu_k \in (0, 1)$, $\mu_c \in (0, 1)$, $f_{20} \in [0, 1]$ and $R \in (0, 1)$. It is important to note that the selection of the dimensionless parameters and variables is technically convenient to the induction and summarization of dynamics of the system.

Between any two adjacent impacts, dynamics of the two-degree-of-freedom periodically-forced system is smooth and linear so that its motion process can be governed by the following non-dimensional equations:

$$ 
\begin{aligned}
\begin{bmatrix} 1 & 0 \\ 0 & \frac{\mu_m}{1-\mu_m} \end{bmatrix} \begin{Bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{Bmatrix} + \begin{bmatrix} 2\zeta & -2\zeta \\ -2\zeta & \frac{2\zeta}{1-\mu_c} \end{bmatrix} \begin{Bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{Bmatrix} + \begin{bmatrix} 1 & -1 \\ -1 & \frac{1}{1-\mu_k} \end{bmatrix} \begin{Bmatrix} x_1 \\ x_2 \end{Bmatrix} \\
= \begin{Bmatrix} 1-f_{20} \\ f_{20} \end{Bmatrix} \sin(\omega t + \tau), \quad (|x_1| < \delta),
\end{aligned} 
\quad (2) $$

Immediately after the impact occurs, for $|x_1| = \delta$, a discontinuous change in the velocity of the mass block $M_1$ occurs and results highly in nonlinear and non-smooth properties, which is determined by the impact law

$$ \dot{x}_{1A+} = -R\dot{x}_{1A-}, \quad (x_1 = \delta); \quad \dot{x}_{1\bar{A}+} = -R\dot{x}_{1\bar{A}-}, \quad (x_1 = -\delta). \quad (3) $$

where the subscript signs “-” and “+” denote the states just before and after the impact, respectively; the subscripts A and $\bar{A}$ represent respectively the impacts occurring at the right and left stops of the clearance.

We can conveniently and visually describe different types of periodic-impact motions of the system by the symbol $n-p-q$, where p and q mean the number of impacts occurring at the left $\bar{A}$ and right A constraints respectively, and n represents the number of the exciting force period in the motion period $T_1 = 2n\pi/\omega$. At small forcing amplitudes a periodic steady state solution of the system will simply be the solution of Eq. (2) with no impacts at all. $p=0$ and $q=0$ symbolize the impactless motion, which exists for enough large clearance $\delta$ (or slightly larger clearances and higher exciting frequencies) and behaves as that of a linear forced vibration system. The quantity $\beta=(p+q)/n$ is of theoretic and practical significance for us to discuss and research thoroughly some dynamic performance on the parameter design of the periodically-forced system with a clearance, which expresses the average impact number during the exciting period.

More generally, a map derived from Eqs. (2) and (3) is used to study the correlative relationship and matching law between dynamics and parameters of the non-smooth system. There is a design technique of the map based on the impact number occurring at a certain stop of the clearance. Consequently, an impact map can be constructed; it is also called impact Poincaré map. This means that each iterate of the map corresponds to the mass block $M_1$ impacting the rigid stop once. As the Poincaré section $\sigma_p = \{(x_1, \dot{x}_1, x_2, \dot{x}_2, t) \in \mathbf{R}^4 \times \mathbf{S}, x_1 = \delta, \dot{x}_1 = \dot{x}_{1A+}\}$ related to the motion state of the vibro-impact system just immediately
after the impact occurring at stop A, is chosen, we can construct the Poincaré map of the periodically-forced system with a clearance

$$X^{(i+1)} = \tilde{f}(\nu, X^{(i)}), \qquad (4)$$

where $X \in \mathbb{R}^4$, $X^{(i)} = (\tau^{(i)}, x_2^{(i)}, \dot{x}_{1A+}^{(i)}, \dot{x}_2^{(i)})^T$, $X^{(i+1)} = (\tau^{(i+1)}, x_2^{(i+1)}, \dot{x}_{1A+}^{(i+1)}, \dot{x}_2^{(i+1)})^T$, $\nu$ are real parameters, $\nu \in \mathbb{R}^m$, $m=1$ or 2,... or 8.

It is important to note that the Poincaré section $\sigma_n = \{(x_1, \dot{x}_1, x_2, \dot{x}_2, t) \in \mathbb{R}^4 \times \mathbb{S}, x_1 = x_{1\min}, \mathrm{mod}(t = 2\pi/\omega)\}$ may be also considered to construct the other map. Consequently, the impact number $q$ of $n-p-q$ motion can be determined easily by the impact map corresponding to $\sigma_p$ and number $n$ of the exciting force periods in the motion period be ascertained by the map related to $\sigma_n$. This means that $n-p-q$ motion can be described totally by the two maps. An analytic derivation is made explicitly for the determination of the group of symmetric $n-1-1$ motions and the implicit expression of disturbed map such motions is analytically derived in Ref. [34]. A simplified stability analysis presumes the linear dependence between small differences of the system state from symmetric $n-1-1$ motion at two succeeding impacts occurring at the same stop. The stability and local bifurcations of the group of symmetric $n-1-1$ motions can be ascertained by computing and analyzing eigenvalues of Jacobian matrix of disturbed map. Correspondingly, saddle-node, period-doubling, pitchfork and Neimark-Sacker bifurcation boundaries of symmetric $n-1-1$ motions, in the double-parameter plane, are determined, near which a deeper understanding of the system behavior can be provided.

Here a special case that the impact mass contacts the rigid stop with almost zero before-velocity, i.e., the bare touching, is worth to be emphasized, which causes real or bare-grazing bifurcation of symmetric $1-p-p$ motion and creates two types of transition regions (hysteresis or tongue-shaped regions) in low exciting frequency range.

Let $\bar{F}$ represent a resultant force, which consists of spring restoring force, damping force and sinusoidal excitation acted on the mass block $M_1$ and is expressed by

$$\bar{F} = 2\zeta(\dot{x}_2 - \dot{x}_1) + (x_2 - x_1) + (1 - f_{20}) \sin(\omega t + \tau), \qquad (5)$$

If the mass block $M_1$ contacts the rigid stop A with almost zero before-velocity and the resultant force $\bar{F}$ is in the positive direction, in an instant immediately after the $i$-th impact contact, i.e.,

$$\bar{F} = 2\zeta(\dot{x}_2^{(i)} + (x_2^{(i)} - \delta) + (1 - f_{20}) \sin(\omega t + \tau^{(i-1)}) \geq 0, \qquad (6)$$

the mass $M_1$ is stuck to the stop A until the resultant force $\bar{F}$ changes its direction (sign). The change from the motion of mass block $M_1$ to its sticking represents a reduction in the degree of freedom of the system from 2 to 1. Accordingly, the change denotes also a reduction in the dimensions of the map from 4 to 3. The presence of the sticking phase leads to the segmentation and singularity of the map and complicates the dynamic analysis of the system. If the mass block contacts the rigid stop A with zero velocity and the resultant force $\bar{F}$ is in the negative direction immediately after the $i$-th impact contact, i.e.,

$$\bar{F} = 2\zeta(\dot{x}_2^{(i)} + (x_2^{(i)} - \delta) + (1 - f_{20}) \sin(\omega t + \tau^{(i-1)}) < 0, \qquad (7)$$

the mass block $M_1$ spends no time in sticking to the stop, the force $\bar{F}$ immediately drives the transient-static mass block $M_1$ off the stop.

### 3. Dynamical characteristics of the periodically-forced system with symmetric rigid stops

The main aim of this paper is the characterization of dynamics of the non-smooth vibratory system to reveal the correlative

relationship and matching law between dynamic performance and system parameters. Based on the definition of dimensionless parameters and variables in Eq. (1), dynamics of the periodically-forced impact system is governed by eight dimensionless parameters: $\mu_m, \mu_k, \mu_c, \zeta, f_{20}, R, \delta$ and $\omega$. Dynamics of the system mainly depends on the relative clearance and the exciting frequency. Two key parameters of the system, the exciting frequency $\omega$ and clearance $\delta$, are firstly chosen to analyze the influence of the main factors on its dynamic characteristics. It is important to analyze the detailed influence of each parameter on dynamics of the system to obtain parameter sensitivity and reasonable parameter matching. It is of special interest to acquire the diversity and evolution of periodic-impact motions of the system on the basis of the sampling ranges of parameters. The value range of partial dimensionless parameters are easily ascertained by Eq. (1). The sampling ranges of some parameters are identified as $\mu_m \in (0, 1)$, $\mu_k \in (0, 1)$, $\mu_c \in (0, 1)$, $f_{20} \in [0, 1]$ and $R \in [0, 1]$. Taking the dimensionless parameters: $\mu_m = 0.5$, $\mu_k = 0.5$, $\mu_c = 0.5$, $\zeta = 0.1$, $f_{20} = 0$ and $R=0.8$ as the criterion, we analyze the determinants of system parameters and the influence of them on dynamic performance. Existence regions of different types of periodic-impact motions of the system, associated with the criterion parameters, are presented in the $(\omega, \delta)$-parameter plane plotted in Fig. 2(a). Periodic-impact motions ascertained by numerical analyses are characterized and marked by the symbol $n-p-q$ and corresponding colors. The local amplification and detail of Fig. 2 are demonstrated in Fig. 3. Moreover, 1-CIS is used to denote complete chatting-impact periodic motion with sticking (i.e., $1-p \bar{p}$ motion) in Figs. 2 and 3, symmetric and asymmetric $1-1-1$ motions are respectively expressed by the symbols $1-1-1_S$ and $1-1-1_{AS}$, and PF, PB, sub and sup N-S Bif represent in proper order successively pitchfork, period doubling, supercritical and subcritical Neimark-Sacker bifurcations. As for large $\delta$, symmetric $1-1-1$ motion can be found only around the first natural frequency. At other exciting frequencies the system exhibits $1-0-0$ impactless motion. With decrease in $\delta$, the window of symmetric $1-1-1$ motion becomes wide for large clearance; becomes narrow for slight large clearance; moves towards higher exciting frequencies for small clearance. At small $\delta$, complete chattering-impact and fundamental $1-p-p$ ($p \ge 2$) motions appear in low exciting frequency range, and long-periodic or chaotic motions can be observed between the existence regions of adjacent $1-p-p$ and $1-(p+1)-(p+1)$ motions. Regions of existence and stability of the impactless $1-0-0$, partial fundamental $1-p-p$ motions and complete chattering-impact motion with sticking can be obviously observed in the parameter plane of Fig. 2, which are identified by different colors, such as the impactless $1-0-0$ motion is assigned to the red color, symmetric $1-1-1$ motion to yellow, asymmetric $1-1-1$ motion to brown and $1-2-2$ motion to green, etc. It must be emphasized that the fundamental $1-p-p$ motions exhibit the properties of symmetry and asymmetry. The fundamental group of impact motions have the excitation period ($n=1$) and differ by the number of impacts $p=0, 1, 2, \dots, m$, i.e., $1-p-p$ impact motions. The practical significance is considerable that the $1-0-0$ impactless motion is included in this group, because the motion is subjected to similar transition phenomena as other $1-p-p$ impact motions. For enough large clearance, dynamic performance becomes relatively simple, and the system exhibits mainly impactless $1-0-0$ or symmetric $1-1-1$ motion in most regions of the parameter plane except small tongue-shaped regions nearby the critical line $L_0: R_{1-0-0} \cap R_{1-1-1}$ between adjacent regions of $1-0-0$ and symmetric $1-1-1$ motions; see Fig. 2. For slightly large $\delta$, the transition from $1-p-p$ to $1-(p+1)-(p+1)$ motion arises with decreasing the exciting frequency $\omega$ up to the grazing boundary $G_{1-p-p}$; see Fig. 3(c). On the grazing boundary $G_{1-p-p}$, the mass block $M_1$ of the system, associated with $1-p-p$ motion, begins to touch the stops of the clearance. It can
be regarded as the 1-(p+1)-(p+1) motion, however the impact velocity of the additional impact degenerates to touching. This means that a new periodic motion is characterized by 1-(p+1)-(p+1), in which the velocity of one of impacts increases gradually from zero with decreasing the exciting frequency ω. Crossing the boundary G₁-ₚ-ₚ, 1-(p+1)-(p+1) motion arises and the number of impacts of the periodic motion increases by two. Thus along the grazing boundary G₁-ₚ-ₚ both neighboring motions change continuously. A series of grazing bifurcations occur with decreasing the exciting frequency ω so that the number p of impacts of the fundamental group of motions increases one by one at every rigid stop. As p becomes big enough, the 1-p-p motion will exhibit the incomplete chattering-impact characteristics (i.e., a chattering sequence in an excitation period followed by a finite sequence of impacts with successively reduced velocity). The complete chattering-impact motion with sticking will occur with decreasing the exciting frequency ω up to the sliding bifurcation boundary. The lower the exciting frequency ω is, the more the number p of impacts is, the longer the sticking time t₃ in the time interval of an exciting period is. Consequently, for slightly large δ the transition from 1-1-1 motion to complete chattering-impact motion is summarized as follows:

$$ 
\begin{aligned}
\omega \downarrow : & 1 - \bar{p} - \bar{p} \\
\mathbf{S} & \xleftarrow{\text{Bif}} \dots \xleftarrow{\text{Bif}} 1 - \tilde{p} - \tilde{p} \xleftarrow{\text{Bif}} \dots \xleftarrow{\text{GBif}} 1 - (p+1) - (p+1) \xleftarrow{\text{Bif}} 1 - p - p \xleftarrow{\text{Bif}} \dots \\
\mathbf{G} & \xleftarrow{\text{Bif}} 1 - 4 - 4 \xleftarrow{\text{Bif}} 1 - 3 - 3 \xleftarrow{\text{Bif}} 1 - 2 - 2 \xleftarrow{\text{Bif}} 1 - 1 - 1,
\end{aligned}
\quad (8) $$

where let 1- p - p denote complete chattering-impact motion and 1- p - p represent incomplete chattering- impact motion, S Bif and G Bif represent sliding and grazing bifurcations, respectively. This sticking chattering-impact vibration is periodic, which undergoes three processes in an exciting period in the following order: free flight, chattering-impact and sticking for the impacting mass. The periodically-forced impact system enters the sticking set via a chattering sequence followed by an infinite sequence of impacts with successively reduced velocity. Consequently, the impact mass of the system becomes constrained to the rigid stop in finite time (t < 2π/ω). This sequence is referred to as the complete chattering- impact motion [33]. In contrast, incomplete chattering-impact motion occurs if the trajectory of the impact mass, after a finite number of impacts, reaches the non-sticking region [33]. It is important to note that chattering-impact vibration enlarges the adverse effects such as high noise levels and wear and so on.

The regular pattern of the transition, for slightly large δ, is vividly plotted in the upper half plane of Fig. 3(c) and illustrated by grazing boundaries in Fig. 4(a). The analysis describes the system behavior on the boundaries of transition regions between 1-p-p

and 1-(p+1)-(p+1) motions. For slightly large δ, one can observe that the existence region of 1-p-p motion becomes gradually narrower and narrower with increase of the impact number of the fundamental periodic motion. This means that the existence regions of incomplete chattering-impact motions (1- p - p ) are too narrow to be observed clearly. Consequently, 1- p - p motion can only be included in the blue region of 1-p-p (p > 8) motions. However, it is important to note that the existence region of complete chattering-impact periodic motions is wider than that of 1-p-p motion (p ≥ 3) and corresponds to lower exciting frequen- cies, as seen in Fig. 3(c).

Period-doubling, saddle-node and pitchfork bifurcation bound- aries are collectively referred to as stability boundaries for the convenience of the following analysis. Transition from complete to incomplete chattering-impact motion arises with increasing the exciting frequency ω up to the rising bifurcation boundary. Transition from 1-p-p to 1-(p-1)-(p-1) motion occurs with increasing the exciting frequency ω up to the saddle-node bifurca- tion boundary SN₁-ₚ-ₚ. For slightly large δ, the transition pattern from complete chattering-impact motion with sticking to 1-1-1 motion is summarized as follows:

$$ 
\begin{aligned}
\omega \uparrow : & 1 - \bar{p} - \bar{p} \\
\mathbf{R} & \xrightarrow{\text{Bif}} 1 - \tilde{p} - \tilde{p} \xrightarrow{\text{SN Bif}} 1 - (\tilde{p} - 1) - (\tilde{p} - 1) \\
\mathbf{SN} & \xrightarrow{\text{Bif}} \dots \xrightarrow{\text{SN Bif}} 1 - p - p \xrightarrow{\text{SN Bif}} 1 - (p - 1) - (p - 1) \xrightarrow{\text{SN Bif}} \dots \\
\mathbf{SN} & \xrightarrow{\text{Bif}} 1 - 3 - 3 \xrightarrow{\text{SN Bif}} 1 - 2 - 2 \xrightarrow{\text{SN Bif}} 1 - 1 - 1.
\end{aligned}
\quad (9) $$

In Eq. (9), R Bif and SN Bif represent rising and saddle-node bifurcations, respectively.

Saddle-node bifurcation of 1-(p+1)-(p+1) motion corre- sponds to stability boundary SN₁-(p+1)-(p+1), on which two of impacts disappears and 1-p-p motion stabilizes after a jump transition. The transition between neighbor fundamental motions 1-p-p and 1-(p+1)-(p+1) is non-reversible except at the singular points of the alternate intersections of grazing and stability boundaries. Narrow hysteresis region exists between grazing bifurcation boundary G₁-ₚ-ₚ of 1-p-p motion and saddle-node bifurcation boundary SN₁-(p+1)-(p+1) of 1-(p+1)-(p+1) motion, as seen in Fig. 4(b). This means that both neighboring 1-p-p and 1-(p+1)-(p+1) motions are stable and can coexist in the depen- dence on the initial conditions of the system or on the way of the system parameters change in the hysteresis zones and they transit by a jump from one to the other on the grazing and saddle-node bifurcation boundaries of these regions. One neighboring motion switches to the other on the segments of the boundaries defining the hysteresis regions. Along the segment of grazing boundary, the

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance, associated with the criterion parameters: µₘ = 0.5, µₖ = 0.5, µₑ = 0.5, ζ = 0.1, f₂₀ = 0 and R = 0.8 in the (ω, δ) -parameter plane. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** The local amplification and details of Fig. 2: (a) existence regions of 1–1–0, 2–1–1 and 3–1–1 motions in the tongue-shaped zones nearby the boundary L: $R_{1-0-0} \cap R_{1-1-1}$; (b) existence regions of 1–2–1, 2–3–2 and 3–4–4 motions in the tongue-shaped zones nearby the boundary L: $R_{1-1-1} \cap R_{1-2-2}$; (c) existence regions of 1–p–p motions and complete chatter-impact motions; (d) existence regions of 3–(3p+1)–(3p+1), 2–(2p+1)–2p, 1–(p+1)–p motions in the tongue-shaped zones nearby the boundary L: $R_{1-p-p} \cap R_{1-(p+1)-(p+1)}$, $p \ge 2$; (e) local detail of Fig. 3(c), (P1=3–13–13, P2=3–16–16); (f) symmetric 1–1–1, asymmetric 1–1–1 motions and transition to 1–2–2 motion.</figcaption></figure>

impact number of a periodic motion increases by two. Conversely, along the other segment the impact number of the neighboring periodic motion decreases by two.

Moreover, for slightly large clearance $\delta$, there exist some small tongue-shaped regions nearby the critical line $L_p: R_{1-p-p} \cap R_{1-(p+1)-(p+1)}$ between adjacent regions of 1–p–p and 1–(p+1)–(p+1) motions ($p \ge 1$). Dynamics in tongue-shaped regions is more complex and emphatically introduced in the following context. With decreasing the clearance $\delta$, the region of 1–p–p motion approaches the grazing boundary, where the

trajectory of 1–0–0 impactless motion of the mass block $M_1$ or two additional impacts of 1–p–p motion trajectory begin to touch the stops. On the contrary, with increasing the clearance $\delta$, the region of 1–p–p motion reaches the stability boundaries associated with saddle-node bifurcation, crossing one of which 1–p–p motion loses its stability and transits to 1–(p–1)–(p–1) or 1–p–(p–1) motion. There exist a series of singular points $X_i$ on the boundaries between 1–p–p and 1–(p+1)–(p+1) motions, i.e., saddle-node bifurcation boundaries $SN_{1-(p+1)-(p+1)}^{1-p-p}$ and $SN_{1-(p+1)-(p+1)}^{1-(p+1)-p}$ of 1–(p+1)–(p+1) motion, bare and real grazing
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Saddle-node, grazing bifurcations and transition phenomena of 1-p-p motions: (a) grazing bifurcation boundaries $L: R_{1-(p+1)-(p+1)} \cap R_{1-(p+1)-(p+1)}$; (b) hysteresis region $HR_p$ existing between grazing boundary $G_{1-(p+1)-(p+1)}$ and saddle-node bifurcation boundary $SN_{1-(p+1)-(p+1)}$.</figcaption></figure>

boundaries $G_{1-p-p}$ and $G_{1-(p+1)-(p+1)}^b$ of 1-p-p motion alternately and mutually cross themselves at the points $X_i$ of intersection. It needs to be emphasized that the singular points of intersection of several bifurcation boundaries have a special significance and create inevitably two types of transition regions (hysteresis and tongue-shaped regions). A special example is inserted to expound the above-mentioned analysis. A singular point X, the intersection of boundaries $SN_{1-(p+1)-(p+1)}^{1-2-2}$, $SN_{1-(p+1)-(p+1)}^{1-2-1}$, $G_{1-(p+1)-(p+1)}$ and $G_{1-(p+1)-(p+1)}^b$, is described and illustrated in Fig. 4(b). Crossing the grazing boundary $G_{1-(p+1)-(p+1)}$, the number of impacts increases by two and 1-1-1 motion transits to 1-2-2 motion. This means that the grazing boundary $G_{1-(p+1)-(p+1)}$ is real. It is important to note that the grazing boundary $G_{1-(p+1)-(p+1)}^b$ is bare. Consequently, along $G_{1-(p+1)-(p+1)}^b$, the mass block $M_1$ of the system begins to touch the stops with almost zero before-impact velocity and 1-1-1 motion develops into a 1-2-2 motion which is unstable in the whole tongue-shaped region and the system transits to other type of impact motion by the unstable 1-2-2 motion. A tongue-shaped region is surrounded by the bare-grazing bifurcation boundary $G_{1-(p+1)-(p+1)}^{1-1-1}$ of 1-1-1 motion and saddle-node bifurcation boundary $SN_{1-(p+1)-(p+1)}^{1-2-1}$ of 1-2-2 motion, and corresponding hysteresis region is limited by real-grazing boundary $G_{1-(p+1)-(p+1)}$ and saddle-node bifurcation boundary $SN_{1-(p+1)-(p+1)}^{1-1-1}$. The grazing and saddle-node boundaries between the particular regions of 1-p-p and 1-(p+1)-(p+1) motions alternately intersect at singular points and so create two kinds of transient zones. There are partly zones of hysteresis and partly zones of tongue-shaped regions including fundamental 1-(p+1)-p, subharmonic 3-(3p+1)-(3p+1) and 2-(2p+1)-2p motions. The analysis for singular points X can be extended to ascertain the limit of hysteresis and tongue-shaped regions. The hysteresis regions are limited at the top by the saddle-node bifurcation boundaries $SN_{1-(p+1)-(p+1)}$ of 1-(p+1)-(p+1) motion and at the bottom by the real-grazing boundaries $G_{1-p-p}$ of 1-p-p motion. The tongue-shaped regions are limited at the top by the bare-grazing bifurcation boundary $G_{1-p-p}^b$ of 1-p-p motion and at the bottom by the saddle-node bifurcation boundary $SN_{1-(p+1)-(p+1)}^{1-(p+1)-p}$ of 1-(p+1)-(p+1) motion. A series of zones of fundamental and subharmonic impact motions, such as 1-(p+1)-p, 2-(2p+1)-2p and 3-(3p+1)-(3p+1) impact orbits, etc., are found to occur in the tongue-shaped regions. Existence region of a periodic or subharmonic impact motion, in the parameter plane, is called parameter island of the corresponding motion. The phenomena can be observed in Fig. 3(c). Details of Fig. 3(c), associated with the tongue-shaped regions and subharmonic impact characters, are shown, respectively, in Fig. 3(b) and (d). The positions of the

corresponding parameter islands of 1-(p+1)-p, 3-(3p+1)-(3p+1) and 2-(2p+1)-2p motions arrange from the bottom up (or from left to right) in this order in the tongue-shaped region, and the area of the parameter islands becomes smaller and smaller in this order. It is important to note that no motion with the exciting force period and symmetric impacts (i.e., symmetric 1-p-p motion) is stable or can exist in the tongue-shaped regions. Consequently, 1-(p+1)-p, 2-(2p+1)-2p and 3-(3p+1)-(3p+1) motions and chaotic ones dominate in the regions. The continuous transition between neighboring 1-p-p and 1-(p+1)-(p+1) motions can exist only at the singular points X of the intersections of grazing and saddle-node boundaries $G_{1-p-p}$, $G_{1-p-p}^b$ and $SN_{1-(p+1)-(p+1)}$. The motions which the system exhibits at the singular points $X_i$ are identical. The 1-p-p motion, associated with the new-appeared periodic touching the stops of the clearance, can be regarded as that the certain two periodic impacts of the 1-(p+1)-(p+1) motion degenerates into the touching. Consequently, 1-p-p motion will end on its grazing boundary and 1-(p+1)-(p+1) motion will emerge. In contrast, the adverse transition from one neighboring fundamental motion to the other, at the singular points $X_i$, is also similar to that above-mentioned.

A tongue-shaped region nearby the critical line $L_0: R_{1-0-0} \cap R_{1-1-1}$ between adjacent regions of 1-0-0 and 1-1-1 motions are shown in Fig. 3(a). The impact motion is invariably found below the grazing boundary $G_{1-0-0}$. The regions of the kinds of impact motions ascertained are marked by different colors and corresponding symbol $n-p-q$. The tongue-shaped region is limited at the bottom by the pitchfork bifurcation boundary $PF_{1-1-1}$ of symmetric 1-1-1 motion and at the top by the bare-grazing bifurcation boundary $G_{1-0-0}^b$ of the impactless motion. Increasing the exciting frequency $\omega$ (or decreasing the clearance $\delta$) up to the boundary $L_G: R_{1-0-0} \cap R_{1-1-0}$, the mass block $M_1$ of the system, associated with 1-0-0 impactless forced vibration, begins to touch the stops with almost zero impact velocities, i.e., the bare touching. On the boundary $L_G: R_{1-0-0} \cap R_{1-1-0}$, the impactless motion develops into a 1-1-1 motion which is unstable in the whole tongue-shaped region and the system transits to other type of impact motion by the unstable 1-1-1 motion. In the tongue-shaped region $R_{1-1-0}$ no motion with the exciting force period and symmetric impacts is stable or can exist so that 1-1-0 (or 1-0-1), 2-1-0 (or 2-0-1) and 3-1-1 motions, 2-2-1 (or 2-1-2), 3-3-1 (or 3-1-3), 3-2-1 (or 3-1-2), 3-2-2 motions and chaotic motions dominate. By means of a theoretical or numerical analysis it is possible to find this zone of the existence and stability
subregions of 1-1-0, 2-1-0 and symmetric 3-1-1 motions. The zones of 1-1-0, 2-1-0 and symmetric 3-1-1 motions are limited from above by stability boundaries PD<sub>1-1-0</sub>, SN<sub>2-1-0</sub> and PF<sub>3-1-1</sub> and from below by grazing boundaries G<sub>1-1-0</sub>, G<sub>2-1-0</sub> and G<sub>3-1-1</sub> which probably correspond to stable or unstable grazing bifurcations, respectively. Partial subregions are drawn in Fig. 3(a) and surrounded by the stability and grazing boundaries. With increasing the exciting frequency ω up to crossing the boundary L<sub>PF</sub>: R<sub>1-1-1</sub>∩R<sub>T<sub>1-0-0</sub></sub>, symmetric 1-1-1 motion stabilizes. And then the symmetric 1-1-1 motion transits by jump to the impactless motion on the stability boundary L<sub>SN</sub>: R<sub>1-1-1</sub> ∩ R<sub>1-0-0</sub>, where the differences ΔX<sup>(i)</sup> = (Δτ, Δχ<sub>2</sub><sup>(i)</sup>, Δχ<sub>1+</sub><sup>(i)</sup>, Δχ<sub>2</sub><sup>(i)</sup>)<sup>T</sup> of the fixed point X* = (τ<sub>0</sub>, x<sub>20</sub>, ˙x<sub>1+</sub>, ˙x<sub>20</sub>)<sup>T</sup> of symmetric 1-1-1 motion in Poincaré section σ<sub>p</sub> grow with the same sign and the impact finally vanishes and 1-0-0 motion stabilizes. As a result, beyond the boundary L<sub>SN</sub>: R<sub>1-1-1</sub>∩R<sub>1-0-0</sub> no solution of symmetric 1-1-1 motion is real; the saddle-node bifurcation of the motion corresponds to the state along the stability boundary.

With ω increasing and crossing a tongue-shaped zone between adjacent regions of 1-0-0 and 1-1-1 motions, a most complex transition, associated with a horizontal scan for certain fixed value of clearance (or a vertical scan for certain fixed value of exciting frequency), is summarized as follows:

$$
\omega \uparrow \text{ or } \delta \downarrow: \begin{array}{l} 1-0-0 \to \text{Unstable G Bif} \to \dots \to 2-1-0 \to \text{G Bif} \to \dots \to 3-1-1_{AS} \to \\ \text{Inverse Pitchfork Bif} \to 3-1-1_{S} \to \text{G Bif} \to \dots \to 2-2-0 \to 1-1-0 \to \text{G Bif} \to \\ 1-1-1_{AS} \to \text{Inverse Pitchfork Bif} \to 1-1-1_{S}, \end{array} \tag{10}
$$

where G Bif is probably stable or unstable grazing bifurcation. The unstable grazing bifurcation causes long-periodic multi-impact or chaotic motion due to the bare touching of the mass block M₁ with the stops. Motion amplitude x₁mp(ω) and impact velocity ˙x₁A_-(ω) associated with a transition crossing a tongue-shaped zone between adjacent regions of 1-0-0 and 1-1-1 motions, along a horizontal scan for δ=5.0, are shown in Fig. 5. During a horizontal scan of change of the value of exciting frequency through regions of different types of impact motions, the diversity and evolution of the system motion can be observed by the bifurcation diagrams shown in Fig. 5(a) and (b). It is important to note that the bifurcation diagram x₁mp(ω) corresponds to the Poincaré section σₙ = {(x₁, ˙x₁, x₂, ˙x₂, t) ∈ ℝ⁴ × ℝ, x₁ = x₁ min mod(t = 2π/ω)}, which represents the minimum displacements of mass M₁ appearing in every excitation period T₁ = 2π/ω. The bifurcation diagram ˙x₁A_-(ω) is related to the impact Poincaré section σₚ = {(x₁, ˙x₁, x₂, ˙x₂, t) ∈ ℝ⁴ × ℝ, x₁ = δ, ˙x₁ = ˙x₁A_−}, which corresponds to the before-impact velocity of every impact. As a result, it is convenient to ascertain n-p-q motion according to the number p of branches

of ˙x₁A_-(ω) and the number n of branches of x₁mp(ω) of a certain impact motion. The windows of 2-1-0 (or 2-0-1), 3-1-1 and 1-1-0 (or 1-0-1) motions are easily observed by Fig. 5. Period-doubling bifurcation of 1-1-0 (or 1-0-1) motion occurs with decreasing the exciting frequency ω. However we can note that existence region of the period-doubling orbit is too small to be observed; see Fig. 6 (the detail of Fig. 5(b)). Long-periodic multi-impact or irregular impact motions arise successively via bare-grazing bifurcations of 2-1-0 (2-1-0) and asymmetric 3-1-1 motions with increasing the exciting frequency ω. A representative 3-1-1 motion is shown for ω=0.51 in Fig. 7(a). A bare-grazing contact instability of 3-1-1 motion occurs at ω =0.525659. The transition to chaos via the bare-grazing bifurcation of the motion is depicted by the phase plane portraits shown in Fig. 7(a)-(c).

The zones of tongue-shaped regions between adjacent regions of 1-p-p and 1-(p+1)-(p+1) motions are shown, respectively, in Fig. 3(b) and (d). Only the zones of the existence and stability subregions of 1-(p+1)-p, 3-(3p+1)-(3p+1) and 2-(2p+1)-2p motions are provided in the tongue-shaped regions. The zones of some impact motions caused by correlative bifurcations of the three motions are too small to be observed clearly, so these motions and other motions not ascertained in the tongue-shaped regions are all marked by gray color. Part zones of 1-(p+1)-p, 3-(3p+1)-(3p+1) and 2-(2p+1)-2p motions are drawn in Fig. 3 (c), the details of which are shown by the next enlarged subregions in Fig. 3(b) and (d). The zones of 1-(p+1)-p, 2-(2p+1)-2p and 3-(3p+1)-(3p+1) motions are limited from above in turn by period doubling bifurcation boundaries PD<sub>1-(p+1)-p</sub>, PD<sub>2-(2p+1)-2p</sub> and PD<sub>3-(3p+1)-(3p+1)</sub> and from down in turn by saddle-node bifurcation boundary SN<sub>1-(p+1)-p</sub>, grazing bifurcation boundaries G<sub>2-(2p+1)-2p</sub> and G<sub>3-(3p+1)-(3p+1)</sub>, respectively. With ω or δ decreasing and crossing a tongue-shaped zone between adjacent regions of 1-p-p and 1-(p+1)-(p+1) motions (p ≥ 1), a most complex transition, associated with a horizontal scan for certain fixed value of clearance (or a vertical scan for certain fixed value of exciting frequency), is summarized as follows:

$$
\begin{align}
\omega \downarrow \text{ or } \delta \downarrow: & \quad 1-(p+1)-(p+1)_S \leftarrow \text{SN Bif} \leftarrow 1-(p+1)-p \leftarrow 2-(2p+2)-2p \nonumber \\
& \quad \leftarrow \dots \leftarrow \text{G Bif} \leftarrow 3-(3p+1)-(3p+1) \leftarrow 6-(6p+2)-(6p+2) \nonumber \\
& \quad \leftarrow \dots \leftarrow \text{G Bif} \leftarrow 2-(2p+1)-2p \leftarrow 4-(4p+2)-4p \leftarrow \dots \leftarrow \text{unstable G Bif} \leftarrow 1-p-p_S, \tag{11}
\end{align}
$$

where G Bif is probably real or bare unstable grazing bifurcation. Motion amplitude x₁mp(δ) and impact velocity ˙x₁A_-(δ) associated with a transition crossing a tongue-shaped zone between adjacent regions of 1-1-1 and 1-2-2 motions, along a vertical scan for ω=0.26, are shown in Fig. 8. The windows of 1-2-1, 3-4-4 and 2-3-2 motions are easily found by analyzing Fig. 8(a) and (b).

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** A transition crossing a tongue-shaped zone between adjacent regions of 1-0-0 and 1-1-1 motions along a horizontal scan for δ=5.0: (a) bifurcation diagram for the lowest displacement x₁mp(ω) of the motion appearing in every period T; (b) bifurcation diagram for impact velocity ˙x₁A_-(ω).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6. Details of Fig. 5(b).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Phase plane portraits of the mass block $M_1$: (a) 3–1–1 motion, $\omega=0.51$, $\delta=5.0$; (b) 3–1–1 motion with grazing contact, $\omega=0.525659$, $\delta=5.0$; (c) chaotic impact motion, $\omega=0.5256589$, $\delta=5.0$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. A transition crossing a tongue-shaped zone between adjacent regions of 1–1–1 and 1–2–2 motions along a vertical scan for $\omega=0.26$: (a) bifurcation diagram for $x_{1mp}(\delta)$; (b) bifurcation diagram for impact velocity $x_{1A-}(\delta)$.</figcaption></figure>

Period-doubling bifurcations of 1–2–1, 3–4–4 and 2–3–2 motions occur successively with increasing the clearance $\delta$. Existence regions of 2–4–2, 6–8–8 and 4–6–4 motions are too narrow to be observed in Fig. 8. However they can be observed by Fig. 9 (details of Fig. 8(b)). Long-periodic multi-impact or chaotic motions arise successively due to the grazing bifurcation of 3–4–4 and 2–3–2 motions with decreasing the clearance $\delta$; see Fig. 8. A bare-grazing contact instability of 3–4–4 motion occurs at $\delta=2.047421$. Consequently, the unstable grazing bifurcation of the motion leads to irregular impact motion immediately, which is depicted in Fig. 10.

For very small clearance, the transition from 1–$p$–$p$ to 1–($p$+1)–($p$+1) motion ($p \ge 2$) is far more complicated than that

associated with large clearance, which basically goes through three processes. First, pitchfork bifurcation of symmetric 1–$p$–$p$ motion occurs with decreasing $\omega$. Consequently, asymmetric 1–$p$–$p$ motion stabilizes, two antisymmetric forms of which can exist in the dependence on the initial conditions of the system. Secondly, period-doubling bifurcation of asymmetric 1–$p$–$p$ motion occurs and it undergoes a succession of period-doubling bifurcations with decrease in $\omega$, which eventually result in apparently non-periodic or chaotic motions. Finally, symmetric 1–($p$+1)–($p$+1) motion is generated by a degeneration of chaos with decreasing $\omega$. Moreover, there exists the zone of 3–($p$+1)–($p$+1) motion in the transition region from 1–$p$–$p$ to 1–($p$+1)–($p$+1) motion, which are observed in Fig. 3(e) and (f). For small
<figure><img src="image_10.png" /><figcaption>Fig. 9. Details of Fig. 8(b).</figcaption></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 10. Phase plane portraits of the mass block $M_1$: (a) 3-4-4 motion, $\omega=0.26$, $\delta=2.09$; (b) 3-4-4 motion with grazing contact, $\omega=0.26$, $\delta=2.047421$; (c) chaotic impact motion, $\omega=0.26$, $\delta=2.047420$.</figcaption></figure>

clearance, another transition from asymmetric 1-1-1 to symmetric 1-2-2 motion may also occur. First, Neimark-Sacker bifurcation of asymmetric 1-1-1 motion occurs with decreasing $\omega$ so that quasi-periodic impact motions arise; secondly, the quasi-periodic impact motion transits to chaos via phase locking; finally, symmetric 1-2-2 motion is generated by a degeneration of chaos with decrease in $\omega$. The transition from 1-p-p to 1-(p+1)-(p+1) motion ($p \ge 2$) can be observed in Fig. 3(e) and Fig. 11.

For very small clearance, symmetric 1-1-1 motion first goes through pitchfork bifurcation with decreasing $\omega$. Secondly, period-doubling bifurcation of asymmetric 1-1-1 motion occurs; perhaps a succession of period-doubling bifurcations occur and eventually result in apparently non-periodic or chaotic motions with decreasing $\omega$. Finally, symmetric 1-2-2 motion is generated by a degeneration of chaos. Moreover, there exist the zones of 1-2-1, 2-3-2, 3-4-4 and 2-4-2 motions in the transition region from asymmetric 1-1-1 to symmetric 1-2-2 motion. Based on the bottom of Fig. 3(f), the whole transition can be observed.

Period doubling bifurcation of asymmetric 1-1-1 motion basically corresponds to very small clearance. However, it is possible to occur Neimark-Sacker bifurcation of asymmetric 1-1-1 motion in certain an interval associated with very small clearances, as seen in Fig. 3(f). For slightly large clearance, Neimark-Sacker bifurcation of symmetric 1-1-1 motion occurs in high exciting frequency range and quasi-periodic impact motions transit to chaotic impact motions via phase locking or torus doubling, and impactless 1-0-0 motion is finally generated by a degeneration of chaos with decrease in $\omega$; Neimark-Sacker bifurcation of asymmetric 1-1-1 motion generally corresponds to slightly low exciting frequencies, as seen in Fig. 2(a). Representative bifurcation diagrams of before-impact velocity as a function of the exciting frequency are shown for $\delta=0.1, 0.3, 0.545, 0.6, 0.8$ and

1.1 in Fig. 11(a)-(f), which illustrate bifurcation characteristics of symmetric 1-1-1 motion.

## 4. The correlative relationship between dynamic performance and system parameters

It is important to analyze the detailed influence of each parameter on dynamics of the system to reveal the correlative relationship and matching law between dynamic performance and system parameters. In the following text, we will discuss the influence of each parameter on dynamics of the system by comparing with dynamics associated with the criterion parameters. In Fig. 12 are plotted the existence regions of different types of periodic-impact motions for the non-smooth vibratory system with variation of the mass distribution $\mu_m$. The selected values of parameter $\mu_m$ are given in graphical interpretation of Fig. 12, the other parameters are the same as the remainder of the criterion parameters. For the following analyses associated with the selected parameters $\mu_k, \mu_c, \zeta, f_{20}$, and $R$, the graphical interpretations are similar to that of Fig. 12. The effects of changes in $\mu_m$ are emphatically analyzed by selecting its different values and scanning the exciting frequency $\omega$ and clearance $\delta$. As the mass distribution $\mu_m$ becomes small enough, the case corresponds to high mass concentration on the impacting mass block, the system is actually equivalent to the mechanical model that the mass block $M_1$ is connected to the supporting base by the linear springs $K_1$ and $K_2$ in series and linear viscous dashpots $C_1$ and $C_2$ in series and behaves as a single-degree-of-freedom impact oscillator. The increase of mass parameter $\mu_m$ brings great changes in the distribution and existence region of symmetric 1-1-1 motion of the system. With increasing $\mu_m$ in the interval $\mu_m \in [0.05, 0.5]$, one can find that
<figure><img src="image_15.png" /><figcaption>Fig. 11. Bifurcation diagrams of impact velocity as a function of the exciting frequency $\dot{x}_{1A-}$: (a) $\delta=0.1$; (b) $\delta=0.3$; (c) $\delta=0.545$; (d) $\delta=0.6$; (e) $\delta=0.8$; (f) $\delta=1.1$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 12. Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \delta)$-parameter plane, associated with the selected values of $\mu_m$ and the remainder of the criterion parameters: (a) $\mu_m = 0.05$; (b) $\mu_m = 0.25$; (c) $\mu_m = 0.75$; (d) $\mu_m = 0.95$. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 13. Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \delta)$-parameter plane, associated with the selected values of $\mu_k$ and the remainder of the criterion parameters: (a) $\mu_k = 0.05$; (b) $\mu_k = 0.25$; (c) $\mu_k = 0.75$; (d) $\mu_k = 0.95$.</figcaption></figure>

the existence region of symmetric 1–1–1 motion of the system shrinks remarkably, as seen in Fig. 2(a), Fig. 12(a) and (b). The existence region of symmetric 1–1–1 motion is almost divided into two parts with $\mu_m$ increasing progressively and crossing $\mu_m = 0.67$, and its left part shrinks gradually. Correspondingly, its right part expands gradually. The system exhibits complex dynamical characteristics between divided existence regions of symmetric 1–1–1 motion, impactless, subharmonic impact (2–1–1 and 3–1–1, etc.), long-periodic multi-impact, quasi-periodic impact and chaotic motions dominate. Moreover, the existence regions of $1-p-p$ motions ($p \ge 2$) shrink slightly at the outset to the lower bound and after that begin to enlarge with increase in $\mu_m$. However the existence region of complete chattering-impact motions always keep slight extension with increasing the mass distribution $\mu_m$. As $\mu_m$ becomes very large, this means that high mass concentration on the non-impacting mass block. In this case the system is actually equivalent to one in which the exciting force $P_1 \sin(\Omega T + \tau)$ directly acts on the right end of the linear springs $K_1$ and linear viscous dashpots $C_1$. The system exhibits more complex dynamical characteristics in the extreme case. Consequently, more types of periodic-impact motions arise and the existence region of complete chattering-impact motions with sticking is extended to larger clearance interval. It is possible for more complex sequences of subharmonic impact motions to occur in large $\mu_m$ case, such as 2–1–1, 3–1–1, 3–2–2 and 5–3–3, etc., which can be observed in Fig. 12(c) and (d). The regions of periodic-impact motions unascertained, quasi-periodic impact motions and chaotic impact motions are marked by blue color. We can note that these regions are very large

in the $(\omega, \delta)$-parameter plane as $\mu_m$ is large enough; see Fig. 12(d). The phenomenon can be explained by three reasons. Firstly, the existence regions of some periodic-impact motions ascertained are too small to be observed clearly; secondly, the number $n$ of excitation period or the number $p$ of impacts in the whole motion period is too big for the corresponding periodic-impact motions to be ascertained effectively; finally, the remainder are basically quasi-periodic and chaotic motions. The larger the mass distribution $\mu_m$ is, the more complex dynamics of the system is.

Existence regions of different types of periodic-impact motions for the non-smooth vibratory system with variation of the stiffness distribution $\mu_k$ are shown in Fig. 13. The effects of changes in $\mu_k$ are analyzed by comparing with the results associated with the criterion parameters. As the stiffness distribution $\mu_k$ becomes very small, the stiffness $K_1$ connecting two mass blocks is much larger than the stiffness $K_2$ connecting the mass block $M_2$ and the supporting base so that the difference between the displacements of two mass blocks is very small. The system exhibits more complex dynamical characteristics in the extreme case associated with very small $\mu_k$. Consequently, more types of periodic-impact motions arise and the existence regions of $1-p-p$ motions ($p \ge 2$) are extended to larger clearance interval. Many complex sequences of subharmonic impact motions, such as 2–1–1, 3–1–1, 3–2–2 and 5–3–3, etc., are found to exist in small $\mu_k$ case, as observed in Fig. 13(a) and (b). As $\mu_k$ is small enough, the existence region of symmetric 1–1–1 motion is almost divided into two parts, and its left part expands gradually and the right part slightly shrinks with increasing $\mu_k$ progressively. Moreover, for small $\mu_k$ the existence
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \delta)$-parameter plane, associated with the selected values of $\mu_c$ and the remainder of the criterion parameters: (a) $\mu_c = 0.05$; (b) $\mu_c = 0.25$; (c) $\mu_c = 0.75$; (d) $\mu_c = 0.95$.</figcaption></figure>

region of 1-2-2 motion expands gradually with increase in $\mu_k$, which is ascertained in comparison to the disturbance region of the motion in Fig. 13(a) and (b). As $\mu_k$ is large enough, the stiffness $K_2$ connecting the mass block $M_2$ and the supporting base is much larger than the stiffness $K_1$ connecting two mass blocks so that the amplitude of the mass block $M_2$ is much less than that of the mass $M_1$. Extremely large $\mu_k$ relates to the degenerated case of a single-degree-of-freedom system. For large $\mu_k$, the increase of stiffness parameter $\mu_k$ brings significant expansion in the existence region of symmetric 1-1-1 motion and slight shrinkage in the existence regions of 1-p-p ($p \ge 2$) and complete chattering-impact motions. The peaks of the existence regions of symmetric 1-1-1 motions of the system with large $\mu_k$ are significantly lower in the $(\omega, \delta)$-parameter plane than those with small $\mu_k$ so that the tongue-shaped region nearby the critical line $L_0$: $R_{1-0-0} \cap R_{1-1-1}$ between adjacent regions of 1-0-0 and 1-1-1 motions become larger and more obvious. The phenomenon can be observed by comparing Fig. 2(a), Fig. 13(c) and (d). It is important to note that there exist a series of tongue-shaped regions along left half part of the critical line $L_0$: $R_{1-0-0} \cap R_{1-1-1}$ and their areas become larger and larger with increase in $\delta$. Consequently, the largest tongue-shaped region nearby the critical line $L_0$: $R_{1-0-0} \cap R_{1-1-1}$ always exists near the peak of the existence region of symmetric 1-1-1 motion.

The influence of damping distribution $\mu_c$ on dynamics of the system is epitomized in Fig. 14. The correlative relationship between dynamic performance and damping distribution is analyzed to find their reasonable matching law by comparing Figs. 2 and 14. Variation of the parameter $\mu_c$ hardly affects the distribution and existence regions of 1-p-p motions and complete chattering-impact motion for

$\mu_c < 0.5$, see Fig. 2, Fig. 14(a) and (b). Some subharmonic impact motions, such as 3-2-2 and 5-3-3, etc., are found to exist in small $\mu_c$ case. As $\mu_c$ is small, the existence region of symmetric 1-1-1 motion is almost divided into two parts, but that of asymmetric 1-1-1 motion shrinks gradually with increasing $\mu_c$. For large $\mu_c$, the increase of damping parameter $\mu_c$ brings large expansion in the existence region of symmetric 1-1-1 motion in small clearance interval. The peak of the existence region of symmetric 1-1-1 motion of the system remarkably reduces with increasing $\mu_c$ so that the tongue-shaped region nearby the critical line $L_0$: $R_{1-0-0} \cap R_{1-1-1}$ between adjacent regions of 1-0-0 and 1-1-1 motions become more and more obvious, in which 1-1-0 (or 1-0-1), 2-1-1, 3-1-1, 3-2-2 and 5-3-3 motions exist. The peak of existence region of symmetric 1-1-1 motion reduces to cause the reduction of regions of other 1-p-p motions ($p \ge 2$), which exist in the zonal zones well arranged, as seen in Fig. 14(c) and (d).

The influence of damping ratio $\zeta$ on dynamics of the system is similar to that of $\mu_c$, which is epitomized in Figs. 15 and 16. It is important to note that small damping ratio $\zeta$ can bring about more complex dynamic characteristics than small $\mu_c$ does in low exciting frequency range; see Fig. 16(a). The complex distribution of existence regions of 1-p-p motions, associated with small $\zeta$, is shown for low frequency interval in Fig. 16. The serious deformation of regions of 1-p-p motions generally occurs in enough small $\zeta$ case. As a result, the existence regions of these motions mutually interweave to cause block segmentation which makes the regions of fundamental 1-p-p motions ($p \ge 2$) present irregular distribution characteristics; see Fig. 16(a). For enough small $\zeta$, the regions of long-periodic multi-impact motions unascertained
<figure><img src="image_19.png" /><figcaption>**Fig. 15.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \delta)$-parameter plane, associated with the selected values of $\zeta$ and the remainder of the criterion parameters: (a) $\zeta = 0.02$; (b) $\zeta = 0.05$; (c) $\zeta = 0.3$; (d) $\zeta = 0.5$.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 16.** Existence regions of different types of periodic-impact motions for small $\zeta$: (a) local detail of Fig.15(a), $\zeta = 0.02$; (b) local detail of Fig.15(b), $\zeta = 0.05$. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)</figcaption></figure>

or chaotic motions, marked by blue color, are much larger than that of symmetric 1-1-1 motion in high exciting frequency range; see Fig. 15(a). The reasons that some periodic-impact motions can be not marked are mainly that their existence regions are too small to be observed clearly or the number *n* of excitation period or the number *p* of impacts in the whole motion period is too big

for the corresponding periodic-impact motions to be ascertained effectively.

Based on the criterion parameters, the correlative relationship between dynamic performance and the exciting forces acting on two mass blocks are studied. The results from simulation, associated with the influence of the exciting forces on main motions of
<figure><img src="image_21.png" /><figcaption>**Fig. 17.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \delta)$-parameter plane, associated with the selected values of $f_{20}$ and the remainder of the criterion parameters: (a) $f_{20} = 0.25$; (b) $f_{20} = 0.5$; (c) $f_{20} = 0.75$; (d) $f_{20} = 1.0$.</figcaption></figure>

the system, are shown for $f_{20} \in [0, 1]$ in Figs. 2, 17 and 25. Variation of the exciting force $f_{20}$ has an effect on the impact velocities and existence regions of $1-p-p$ and complete chattering-impact motions, but it hardly changes general forms of a series of periodic-impact motions shown in Figs. 2 and 17. No anomaly type of periodic-impact motions occurs even for two extreme cases of $f_{20}=0$ and 1. As the parameter $f_{20}$ is small enough, the exciting force acting on the mass block $M_1$ is much larger than that on the other mass, the existence regions of $1-p-p$ ($p \ge 2$) and complete chattering-impact motions enlarge towards the interval of large clearances, see Fig. 2 and Fig. 17(a). The exciting force acting on the mass block $M_1$ gradually decreases whereas that on the other mass correspondingly increases with gradual increase of $f_{20}$. The result causes a gradual reduction of the peak of the existence region of symmetric $1-1-1$ motion in the $(\omega, \delta)$-parameter plane so that the existence regions of $1-p-p$ ($p \ge 2$) and complete chattering-impact motions become correspondingly small. Because the largest tongue-shaped region nearby the critical line $L_0: R_{1-0-0} \cap R_{1-1-1}$ always exists near the peak of the existence region of symmetric $1-1-1$ motion, a following example is provided to demonstrate that increase of $f_{20}$ leads to a gradual reduction of the peak of the existence region of symmetric $1-1-1$ motion. A tongue-shaped region nearby the critical line $L_0: R_{1-0-0} \cap R_{1-1-1}$ between adjacent regions of $1-0-0$ and $1-1-1$ motions becomes larger and more obvious due to a significant reduction of the peak of existence region of symmetric $1-1-1$ motion, in which $1-1-0$ (or $1-1-0$), $3-1-1$, $2-1-0$ (or $2-0-1$),..., etc., fundamental and subharmonic impact motions are found. The positions of these tongue-shaped zones gradually lower and the shape and internal structure of one of them

become more and more observable with increase of $f_{20}$, as seen by comparing Figs. 2 and 17. It is interesting to note that change in $f_{20}$ hardly affect the distribution and existence regions of asymmetric $1-1-1$ motions in the $(\omega, \delta)$-parameter plane except very small clearance. In high exciting frequency range, the gradual increase of $f_{20}$ leads to smaller and smaller existence region of symmetric $1-1-1$ motion.

The influence of restitution coefficient $R$ of impact on dynamics of the system is epitomized in Fig. 18. Based on the existing simulation results for dynamic analysis, the correlative relationship between dynamic performance and the restitution coefficient of impact are analyzed by comparing Figs. 2 and 18. The change in restitution coefficient of impact mainly affects the impact velocities and existence regions of $1-p-p$ and complete chattering-impact motions. In the interval $R \in [0.05, 0.8]$, the total area of existence regions of $1-p-p$ motions ($p \ge 1$) and complete chattering-impact motions, especially in low exciting frequency range, is basically equal. As the restitution coefficient $R$ of impact is small enough, the existence region of complete chattering-impact motions is very large so that the regions of $1-p-p$ ($p \ge 1$) motions are badly extruded and substantially reduced. Especially the existence regions of some $1-p-p$ motions ($p \ge 4$) are too small to be shown in Fig. 18(a) and (b). It is possible that there exist no asymmetric $1-1-1$ motion for extreme small $R$. For enough small $R$, the system exhibits subharmonic $2-1-1$, $3-1-1$, $3-2-2$ motions and so on in high exciting frequency range; see Fig. 18(a). With increase in $R$, the existence region of complete chattering-impact motion reduces gradually. Consequently, the regions of $1-p-p$ motions correspondingly extends by degrees to the ranges
<figure><img src="image_22.png" /><figcaption>**Fig. 18.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \delta)$-parameter plane, associated with the selected values of $R$ and the remainder of the criterion parameters: (a) $R=0.1$; (b) $R=0.2$; (c) $R=0.4$; (d) $R=0.6$.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. 19.** Time series of complete chattering-impact motion $(\omega=0.5, \delta=0.1)$ and the criterion parameters(except for $R$): (a) $R=0.2$; (b) $R=0.5$; (c) $R=0.8$; (d) detail of Fig. 19(a); (e) detail of Fig. 19(b); (f) detail of Fig. 19(c).</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 20. Before-impact velocities and existence regions of different types of impact motions for the periodically-forced system with a clearance in the $(\omega, \mu_m)$-parameter plane related to $\delta=0.2$ and the remainder of the criterion parameters.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 21. Before-impact velocities and existence regions of different types of impact motions for the periodically-forced system with a clearance in the $(\omega, \mu_k)$-parameter plane related to $\delta=0.2$ and the remainder of the criterion parameters.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 22. Before-impact velocities and existence regions of different types of impact motions for the periodically-forced system with a clearance in the $(\omega, \mu_c)$-parameter plane related to $\delta=0.2$ and the remainder of the criterion parameters.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 23. Before-impact velocities and existence regions of different types of impact motions for the periodically-forced system with a clearance in the $(\omega, \zeta)$-parameter plane related to $\delta=0.2$ and the remainder of the criterion parameters.</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>**Fig. 24.** Before-impact velocities and existence regions of different types of impact motions for the periodically-forced system with a clearance in the $(\omega, f_{20})$-parameter plane related to $\delta=0.2$ and the remainder of the criterion parameters.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Fig. 25.** Before-impact velocities and existence regions of different types of impact motions for the periodically-forced system with a clearance in the $(\omega, R)$-parameter plane related to $\delta=0.2$ and the remainder of the criterion parameters.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>**Fig. 26.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \mu_m)$-parameter plane: (a) $\delta=0.5$; (b) $\delta=1.0$.</figcaption></figure>

observed obviously. Correspondingly, the region of symmetric 1-1-1 motion becomes wider in the exciting frequency interval and the existence region of asymmetric 1-1-1 motion extends slightly. On the other hand, the restitution coefficient of impact brings also great influences on the impact sequences and the sticking times of $1-\bar{p}-\bar{p}$ motion in a motion period. For large restitution coefficient $R$, the impact sequence of $1-\bar{p}-\bar{p}$ motion is also more obvious in the motion period and the sticking time correspondingly shorter. Conversely, as $R$ is small enough, the

impact strength is markedly attenuated, the impact sequence of $1-\bar{p}-\bar{p}$ motion decays quickly in the motion period and becomes not obvious in visual sensation and quantity (but the chatter sequence is still infinite), the sticking time is correspondingly longer. A simulation result, associated with $\omega=0.5$, $\delta=0.1$ and the remainder of the criterion parameters, is provided to observe the influence of restitution coefficient $R$ on the impact number and the sticking time of $1-\bar{p}-\bar{p}$ motion in a motion period, which is shown for $R=0.2$, 0.5 and 0.8 in Fig. 19.
<figure><img src="image_31.png" /><figcaption>**Fig. 27.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(ω, μ_k)$-parameter plane: (a) $δ=0.5$; (b) $δ=1.0$.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>**Fig. 28.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(ω, μ_c)$-parameter plane: (a) $δ=0.5$; (b) $δ=1.0$.</figcaption></figure>

**5. The influence of system parameters to before-impact velocity**

The clearances or constraints among mechanical components exist inevitably in mechanical systems due to the restriction on technological level in the mechanical machining process, the demand for mechanical design and assembly errors, the need for the heat-expansion and cold-contraction of some part of mechanical components or other reason. The existence of the clearances or constraints is bound to bring about the vibro-impact effects in service of the mechanisms. The vibroimpact effects have a great influence on performance of mechanical equipments, increase noise levels and intensify fatigue and abrasion of mechanical parts. An important scientific research subject in the field of mechanical engineering is urgent to be solved, i.e., how to reduce wear of mechanical components and improve use efficiency, service life and operation reliability of non-smooth mechanical systems. Therefore, it is a primary purpose of dynamic research on mechanical vibrating systems with clearances or constraints how both vibration and impact are effectively utilized or furthest decreased to prevent from chattering-impact or chaotic impact motions so as to restrain noise levels and reduce the abrasion between components and constraints.

The influence of clearance nonlinearity on before-import velocity of the mass block $M_1$ is epitomized in Fig. 2(b). The diagram is

useful when the distribution of impact velocity is of interest over the whole $(ω, δ)$ plane. Based on the simulation results of the system with the criterion parameters, the correlative relationship between before-impact velocity and parameters $(ω, δ)$ is analyzed to find their reasonable matching law. Impact velocities of symmetric and asymmetric 1-1-1 motions are specially marked in Fig. 2(b) by red color, the existence regions of which are shown in Fig. 2(a). With increase in the value of clearance, the impact velocity of 1-1-1 motion, associated with small $δ$, raises gradually. The resonance peak of function curve $\dot{x}_{1A-}(ω)$ of impact velocity of symmetric 1-1-1 motion, corresponding to large $δ$, rises gradually with increasing $δ$. The regularities can be observed by Fig. 2(b). In general case, small clearances lead to low impact velocity of symmetric 1-1-1 motion. The function curve $\dot{x}_{1A-}(ω)$ of impact velocity of symmetric 1-1-1 motion, associated with $δ=0.2$, is lower on the whole than that corresponding to $δ > 0.2$. As the clearance is located at the interval of approximately [0.6, 2.0], 1-p-p motions ($p \ge 2$), complete chattering-impact motions, 1-(p+1)-p, 2-(2p+1)-2p and 3-(3p+1)-(3p+1) motions ($p \ge 1$) in tongue-shaped zones play major roles in low exciting frequency range, respectively. Consequently, these motions bring about the increase of average impact number in an exciting period, which increases noise levels and intensifies wear of the impact mass and stops. The choice of small clearance and the determination of reasonable parameter matching and effective exciting
<figure><img src="image_33.png" /><figcaption>**Fig. 29.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, \zeta)$-parameter plane: (a) $\delta=0.5$; (b) $\delta=1.0$.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>**Fig. 30.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, f_{20})$-parameter plane: (a) $\delta=0.5$; (b) $\delta=1.0$.</figcaption></figure>

frequency range are conducive to obtain symmetric 1-1-1 motion, reduce its impact strength, abrasion and noise levels.

A conclusion has been obtained from Fig. 2 that the function $\dot{x}_{1A_{-}}(\omega)$ of before-impact velocity of 1-1-1 motion, associated with the criterion parameters and $\delta=0.2$, is lower on the whole than those corresponding to $\delta > 0.2$. Based on the conclusion above-mentioned, the influence of parameter variation on before-impact velocity of the mass block $M_1$ is epitomized in Figs. 20-25. The correlative relationship between impact velocity $\dot{x}_{1A_{-}}$ and parameters $(\omega, \mu_m)$ are shown for $\delta=0.2$ in Fig. 20. The change in $\mu_m$ hardly affects the impact velocity of the mass block $M_1$ for the same exciting frequency, and the range of exciting frequency of symmetric 1-1-1 motion is basically constant for small clearance and large mass distribution. As for the same mass distribution, impact velocity of symmetric 1-1-1 motion gradually raises with increase in $\omega$. As for the mass distribution interval $\mu_m \in [0.05, 0.95]$, 1-p-p fundamental motions ($p \ge 2$), complete chattering-impact motions, 1-2-1, 2-3-2 and 2-4-2 motions and chaos, etc., dominate in the exciting frequency range of $\omega < 2$, respectively. These motions bring about the increase of average impact number in a motion period. Whether large or small $\mu_m$, these motions have the negative effects on the vibrating system associated with small clearances and low exciting frequencies. With increase in $\delta$, large $\mu_m$ causes that the existence region of symmetric 1-1-1 motion becomes relatively small in the exciting

frequency range. Two examples associated with $\delta = 0.5$ and 1 are provided for analyzing the influence of parameters $\delta$ and $\mu_m$ on dynamic performance of the system. As for a little larger clearance, the existence region of symmetric 1-1-1 motion in the $(\omega, \mu_m)$-parameter plane is divided into two parts by that of 1-0-0 motion, and subharmonic 2-1-1, 3-2-2, 5-3-3, 6-4-4 and chaotic motions, etc., are found to appear in the interval of large $\mu_m$; see Fig. 26.

The correlative relationship between before-impact velocity $\dot{x}_{1A_{-}}$ and parameters $(\omega, \mu_k)$ are shown for $\delta=0.2$ in Fig. 21. It is interesting to note that the change in $\mu_k$ hardly affects the exciting frequency range of existence regions of 1-p-p fundamental motions ($p > 3$) and complete chattering-impact motions associated with small $\delta$. The existence region of symmetric 1-1-1 motion is almost invariant in a certain range of exciting frequency as $\mu_k$ is not large enough, and the impact velocity of the motion is basically constant whether $\mu_k$ increases or decreases for the same exciting frequency. The range of exciting frequency of asymmetric 1-1-1 motion slightly increases as $\mu_k$ is large enough. As for a little larger clearance $\delta$, small $\mu_k$ causes that the existence region of symmetric 1-1-1 motion becomes narrow in the exciting frequency range, and subharmonic 2-1-1 motion is found to exist in the range of $\mu_k \in (0.05, 0.2377)$; the existence region of symmetric 1-1-1 motion, associated with large $\mu_k$, is still wide in the exciting frequency range, as seen in Fig. 27. Dynamic performance of the
<figure><img src="image_35.png" /><figcaption>**Fig. 31.** Existence regions of different types of periodic-impact motions for the periodically-forced system with a clearance in the $(\omega, R)$-parameter plane: (a) $\delta=0.5$; (b) $\delta=1.0$.</figcaption></figure>

system can be improved significantly by selecting slightly larger stiffness distribution $\mu_c$.

The correlative relationship between before-impact velocity $\dot{x}_{1A_{-}}$ and parameters $(\omega, \mu_c)$ are shown for $\delta=0.2$ in Fig. 22. As the damping distribution $\mu_c$ is not large enough, the existence regions of $1-p-p$ motions ($p \ge 3$) and complete chattering-impact motions are almost invariant in $\omega$ interval. The range of exciting frequency of symmetric $1-1-1$ motion is almost invariant for $\mu_c$, and impact velocity of the motion is basically constant for the same exciting frequency with change in $\mu_c$. The range of exciting frequency of asymmetric $1-1-1$ motion slightly increases as $\mu_c$ is large enough. As for small clearance, small damping distribution $\mu_c$ hardly affects dynamics of the system, as seen in Fig. 22. As for a little larger clearance $\delta$, small $\mu_c$ causes that the existence region of symmetric $1-1-1$ motion becomes narrow in the exciting frequency range and the system exhibits subharmonic 3-2-2 and 5-3-3 motions in a little higher exciting frequency interval; see Fig. 28.

The correlative relationship between impact velocity $\dot{x}_{1A_{-}}$ and parameters $(\omega, \zeta)$ is shown for $\delta=0.2$ in Fig. 23. With increase in $\zeta$, impact velocity of $1-1-1$ motion, corresponding to the same and relatively small exciting frequency, slightly attenuates at the outset to the lower bound and after that begins to slightly increase. As the exciting frequency keeps invariant and is higher than some value, impact velocity of $1-1-1$ motion attenuates gradually with increasing $\zeta$. It is important to note that the function curve $\dot{x}_{1A_{-}}(\omega)$ of symmetric $1-1-1$ motion is divided into two sections in the parameter range associated with large $\zeta$ and higher $\omega$. The section of the function curve $\dot{x}_{1A_{-}}(\omega)$ of symmetric $1-1-1$ motion, in low exciting frequency range, is higher than the other in high exciting frequency interval, both are mainly separated by the windows of quasi-periodic, chaotic and impactless motions. There exists Neimark-Sacker bifurcation of symmetric $1-1-1$ motion as $\zeta$ is very small or large enough. As for large $\zeta$, impact velocity of symmetric $1-1-1$ motion is very small in certain a range of high exciting frequency; see Fig. 23. The correlative relationship between dynamic performance and parameters $(\omega, \zeta)$ are shown for $\delta=0.5$ and $1$ in Fig. 29. Small $\zeta$ badly affects dynamics of the system, slightly large $\zeta$ leads to more energy consumption of the system to reduce its use efficiency. Based on comprehensive consideration of large existence region and low impact velocity of $1-1-1$ motion, the damping ratio range $\zeta \in [0.1, 0.2]$ is feasible and effective to the operation reliability of the system.

The correlative relationship between before-impact velocity $\dot{x}_{1A_{-}}$ and parameters $(\omega, f_{20})$ are shown for $\delta=0.2$ in Fig. 24. The change in $f_{20}$ mainly affects the existence regions of $1-1-1$

motions associated with small $\delta$ in high exciting frequency range. As $f_{20}$ is large enough, the existence region of $1-1-1$ motion is very narrow for small $\delta$ in the $(\omega, f_{20})$-parameter plane, as seen in Figs. 17 and 24 and Fig. 30(a). With increase in parameter $f_{20}$, impact velocity of symmetric $1-1-1$ motion, corresponding to the same exciting frequency, attenuates gradually. As for invariant exciting frequency, the impact strength of complete chattering-impact motions can enhance gradually with increasing $f_{20}$. As for small clearances, large $f_{20}$ causes that the existence region of symmetric $1-1-1$ motion becomes narrower in the exciting frequency range; see Fig. 17 and Fig. 30(a).

The correlative relationship between before-impact velocity $\dot{x}_{1A_{-}}$ and parameters $(\omega, R)$ is shown for $\delta=0.2$ in Fig. 25. As the restitution coefficient $R$ of impact is not large enough, the system exhibits complete chattering-impact motions $(1-p-p)$ in a large exciting frequency interval due to fast attenuation in impact velocities, though the first impact velocities of these motions in an exciting period are very high. Furthermore, the existence regions of $1-p-p$ motions ($p \ge 3$), associated with small $R$, are too narrow in $\omega$ interval to be observed. It is important to note that the highest branches of function curves $\dot{x}_{1A_{-}}(\omega)$ of $1-p-p$ motions, in certain an interval of $\omega$, are possibly higher than the whole curve $\dot{x}_{1A_{-}}(\omega)$ of $1-1-1$ motion for small $R$. With increase in the restitution coefficient $R$, the existence region of complete chattering-impact motions is gradually diminished in the $\omega$ interval and those of $1-p-p$ motions accordingly expand, and impact velocity of $1-1-1$ motion, corresponding to the same and relatively small exciting frequency, slightly increases at the outset to the upper bound and after that begins to slightly attenuate. However, as the exciting frequency is higher than some value and keeps invariant, impact velocity of $1-1-1$ motion increases gradually with increasing $R$. As for small $R$, the function curves $\dot{x}_{1A_{-}}(\omega)$ of impact velocity of symmetric $1-1-1$ motion attenuate gradually with increasing the exciting frequency $\omega$. The case is contrary after $R$ is larger than some value, the function curves $\dot{x}_{1A_{-}}(\omega)$ of symmetric $1-1-1$ motion increases progressively with increasing $\omega$. As $R$ is not large enough, some narrow windows of $1-1-1$ motions exist in high exciting frequency range, which are isolated by other periodic-impact or chaotic motions. As for a little larger clearance $\delta$, small $R$ leads to narrower existence region of symmetric $1-1-1$ motion in the exciting frequency range; see Fig. 31. Some characteristics of complete chattering-impact motions must be emphasized. As $R$ is small enough, the impact strength is markedly attenuated, the impact sequence of $1-p-p$ motion decays quickly in the motion period and becomes not obvious in visual sensation and
quantity (but the chatter sequence is still infinite), and the sticking time is correspondingly longer. Conversely, for large restitution coefficient $R$, the impact sequence of $1-p$ and $p$ motion is also more obvious in the motion period and the sticking time becomes correspondingly shorter due to slow attenuation in impacts. The effect, i.e., large impact velocities and increase of average impact number in a motion period, badly intensifies abrasion and pitting fatigue between the restrained mass and clearances and increase noise levels. It is valuable to avoid the existence regions of complete and incomplete chattering-impact motions and $1-p$-$p$ motions ($p \ge 2$) in parameter design as much as possible to improve service life and operation reliability of vibrating systems with clearances or constraints.

## 6. Conclusions

In this paper we consider a two-degree-of-freedom periodically-forced system with a clearance represented by two symmetric rigid stops. The correlative relationship between dynamic performance and system parameters is analyzed to find their reasonable matching law. Two key parameters of the system, the dimensionless excitation frequency $\omega$ and clearance $\delta$, are emphasized to analyze the influence of the main factors on dynamics of the system.

The system usually exhibits $1-p$-$p$ motions ($p \ge 1$) and complete chattering-impact motions in low exciting frequency or small clearance range. As the clearance $\delta$ is small or small enough, the transition between adjacent fundamental motions basically goes through three processes with decrease in $\omega$, which are epitomized as follows: pitchfork bifurcation of symmetric $1-p$-$p$ motion; period-doubling bifurcation of asymmetric $1-p$-$p$ motion; non-periodic or chaotic motions caused by a succession of period-doubling bifurcations; symmetric $1-(p+1)-(p+1)$ motion generated by a degeneration of chaos. Both $1-p$-$p$ and $1-(p+1)-(p+1)$ motions are separated by a region including other periodic-impact and chaotic motions, in which only the zones of $3-(p+1)-(p+1)$ and $1-(p+1)-p$ motions are obvious enough to be observed.

For slightly large clearance $\delta$, the transition from $1-p$ to $1-(p+1)-(p+1)$ motion arises with decreasing the exciting frequency $\omega$ up to the grazing bifurcation boundary $G_{1-(p+1)-(p+1)}$ of $1-p$-$p$ motion. A series of grazing bifurcations occur with decreasing $\omega$ so that the number $p$ of impacts of the fundamental group of motions increases two by two. As $p$ becomes big enough, $1-p$-$p$ motion will exhibit incomplete chattering-impact characteristics, which is the so-called an incomplete chattering sequence in an excitation period followed by a finite sequence of impacts with successively reduced velocity. Complete chattering-impact motion with sticking will occur with decreasing $\omega$ up to the sliding bifurcation boundary. Transition from $1-(p+1)-(p+1)$ to $1-p$-$p$ motion occurs with increasing $\omega$ up to the saddle-node bifurcation boundary $SN_{1-(p+1)-(p+1)}$ of $1-(p+1)-(p+1)$ motion. The transition between neighbor fundamental motions $1-p$-$p$ and $1-(p+1)-(p+1)$ is generally non-reversible. There exist a series of singular points on the boundaries between the existence regions of $1-p$-$p$ and $1-(p+1)-(p+1)$ motions, at which different saddle-node bifurcation boundaries of $1-(p+1)-(p+1)$ motion, real-grazing and bare-grazing boundaries of $1-p$-$p$ motion alternately and mutually intersect. Consequently, two types of transition regions (hysteresis and tongue-shaped regions) are created, by which neighboring fundamental impact motions are separated. The hysteresis regions are limited at the upper bound by the saddle-node bifurcation boundaries $SN_{1-(p+1)-(p+1)}^{1-p-p}$ of $1-(p+1)-(p+1)$ motion and at the lower bound by the real-grazing boundaries $G_{1-p-p}$ of $1-p$-$p$ motion. The tongue-shaped regions are limited at the upper bound by the bare-grazing boundary $G_{1-p-p}^b$ of $1-p$-$p$ motion and at the lower bound by the saddle-node bifurcation boundary $SN_{1-(p+1)-(p+1)}^{1-(p+1)-p}$ of $1-(p+1)-(p+1)$ motion. A series of

zones of fundamental $1-(p+1)-p$, subharmonic $3-(3p+1)-(3p+1)$ and $2-(2p+1)-2p$ impact orbits ($p \ge 0$) dominate in the tongue-shaped regions between adjacent regions of $1-p$-$p$ and $1-(p+1)-(p+1)$ motions.

As for large clearance $\delta$, the system exhibits mainly $1-0-0$ or symmetric $1-1-1$ motion in most regions of the $(\omega, \delta)$-parameter plane except small or narrow tongue-shaped zones nearby the critical line $L_0$: $R_{1-0-0} \cap R_{1-1-1}$, in which there exist narrower zones of fundamental $1-1-0$ (or $1-0-1$), subharmonic $3-1-1$ and $2-1-0$ (or $2-0-1$) motions.

The system exhibits more complex dynamical characteristics as $\mu_m$ becomes large enough. Consequently, more types of periodic-impact motions appear and the existence region of complete chattering-impact motions is extended to larger clearance interval. It is possible for more complex sequences of subharmonic impact motions to occur in large $\mu_m$ case, such as $2-1-1, 3-1-1, 3-2-2$ and $5-3-3$, etc. As for small $\mu_k$, the existence regions of $1-p$-$p$ motions ($p \ge 2$) are extended to larger clearance interval and the existence regions of symmetric $1-1-1$ motion are divided into two parts. Many complex sequences of subharmonic impact motions are found to exist in small $\mu_k$ case. The increase of stiffness parameter $\mu_k$ brings significant expansion in the existence region of symmetric $1-1-1$ motion and obvious shrinkage in the existence regions of $1-p$-$p$ ($p \ge 2$) and complete chattering-impact motions. As the damping distribution $\mu_c$ is not large enough, its variation hardly affects the distribution and existence regions of $1-p$-$p$ and complete chattering-impact motions. As for large $\mu_c$, the exciting frequency range of symmetric $1-1-1$ motion, associated with small clearance interval, becomes correspondingly large. The serious deformation of regions of $1-p$-$p$ motions generally occurs in enough small $\zeta$ case. As a result, the existence regions of these motions mutually interweave to cause block segmentation which makes the regions of fundamental $1-p$-$p$ motions ($p \ge 2$) present irregular distribution characteristics. Small $\zeta$ badly affects dynamical performance of the system, slightly large $\zeta$ leads to more energy consumption of the system to reduce its use efficiency. Based on comprehensive consideration of large existence region and low impact velocity of $1-1-1$ motion, the damping ratio range $\zeta \in [0.1, 0.2]$ is feasible and effective to the operation reliability of the system. Variation of the exciting force $f_{20}$ has an effect on the impact velocities and existence regions of $1-p$-$p$ motions and complete chattering-impact ones, but it hardly changes general forms of a series of periodic-impact motions. No anomaly type of periodic-impact motions occurs even for the extreme case of $f_{20}=0$ or 1. As the restitution coefficient $R$ of impact is small enough, the existence region of complete chattering-impact motions is very large so that the regions of $1-1-1$ and $1-p$-$p$ ($p \ge 2$) motions are badly extruded and substantially reduced. The restitution coefficient of impact brings also great influences on the number of impacts and the sticking time of chattering motion in a motion period. As for large $R$, the impact sequence of chattering motion becomes more obvious in the motion period and the sticking time is correspondingly shorter due to slow attenuation in impacts. Conversely, as $R$ is relatively small, the impact strength is markedly attenuated so that the impact sequence of chattering motion decays quickly in the motion period and the sticking time becomes correspondingly longer.

## Acknowledgments

The authors gratefully acknowledge the support by National Natural Science Foundation of China (11172119, 11362008), Innovative Research Group Foundation of Gansu Province of China (1308RJIA006) and Research Fund for the Doctoral Program of Higher Education of China (20126204110001).
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