Grazing and Hopf Bifurcations of a Periodic Forced System with Soft Impacts

Based on the research of a periodic forced system with soft impacts, the piecewise properties of the softimpacts system, such as asymmetric motion and singularity, were analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and grazing bifurcation, were investigated exactly. In the large constraint stiffness case, the Hopf bifurcation exists in the periodic forced system with soft impacts. The clearances of the system are the main reasons for influencing the chaotic motion. For small clearances, the grazing bifurcations bring about asymmetric motion and singularity. The steady 1-1-1 period orbits will exist within a wideband frequency range when appropriate system parameters are chosen.


INTRODUCTION
The soft impacts induce extensive oscillation in mechanical systems.Such piecewise linear systems are capable of exhibiting classically non-linear behavior such as grazing bifucations.For example, impact dampers, shakers, etc., are based on the impact of moving bodies.With other s, mechanisms with clearances, gears, wheel-rail interaction of railway coaches, etc., impacts also occur, but they are undesirable as they bring about increased wear and noise levels.It is necessary to be able to accurately model the dynamics of mechanical systems with soft impacts and clearances, to enlarge profitable effects and minimize adverse effects.The grazing bifurcation, which is a key to determine the motions switching in discontinuous dynamical systems, has been discussed by a number of researches in the past several years.Shaw and Holmes [1] studied a singledegree-of-freedom vibro-impact system by using the traditional approaches for analyzing periodic-impact responses in the system.The results showed all types of typical nonlinear behavior: saddle node and flip bifurcations, multiple coexisting attracting solutions and chaos, etc. Nordmark [2] developed systematic methods for investigating grazing dynamics and attendant bifurcations of the piecewise linear and vibro-impact systems.A study De Souza and Wiercigroch [3] focused on the grazing transitions from no impact to impact motion and investigated parameter space region around the grazing bifurcations.The qualitative changes that can be associated with grazing and corner-collision bifurcations were observed through a combination of experiments and numerical calculations *Address correspondence to this author at the School of Mechatronic Engineering, Lanzhou Jiaotong University, Box no.406, 88 West Anning Road, Lanzhou, Gansu Province, Postcard: 730070, P.R. China; Tel: 0086-931-4938143; E-mail: zhuxf@mail.lzjtu.cnLong et al.,Luo and Gegg [4,5] carried out a comprehensive investigation on grazing motions in the dry-friction oscillator for a better understanding of the grazing mechanism of a discontinuous system.Experimental study on piecewise linear oscillator was performed by Sin and Wiercigroch [6].Luo and O'Connor [7,8] presented an idealized, piecewise linear system to model non-smooth vibration of gear transmission appearing from impacts between the gear teeth.For example, in wheel-rail impacts of railway coaches Luo et al. [9], Jeffcott rotor with bearing clearance Karpenko et al. [10], gears transmissions Alshyyab, and Kahraman [11], small vibro-impact pile driver Luo and Yao [12], etc., impacting models have been proved to be useful.A periodic forced system with soft impacts and clearances has been established.The main purpose of the present study was to analyze the piecewise properties of such system, including stability, Hopf bifurcation, grazing bifurcations period doubling bifurcation, etc.The routes from quasi-periodic impact motions, grazing motions or period-doubling cascades to chaos were observed by using the Poincaré map and numerical simulation.Finally, the influences of clearances on periodic motions and bifurcations of the periodic forced system are discussed in detail.

MECHANICAL MODEL
The mechanical model of a periodic forced system with soft impacts is shown in Fig. (1).Displacements of the masses M 1 , M 2 and M 3 are described by X 1 , X 2 and X 3 , respectively.The mass M 1 and M 2 are connected by linear spring with stiffness K 1 and linear viscous dashpot with damping constant C 1 , and M 2 and M 3 are connected by K 2 and C 2 analogously.The mass M 3 is attached to the supporting base by the linear spring with stiffness K 3 and linear viscous dashpot with damping constant C 3 .The excitations on masses are harmonic with amplitudes P 1 , P 2 and P 3 .! is the excitation frequency, and ! is the phase angle.The mass M 2 begins to hit the right (left) soft impact represented by linear spring with stiffness K 4 ( K 5 ) and linear viscous dashpot with damping constant C 4 ( C 5 ) when the displacement X 2 of mass M 2 equals the clearances B , i.e.X 2 (t) = B ( X 2 (t) = !B ).The motion processes of the system when the absolute value of displacement X 2 is less than the clearances are considered.The condition of the periodic forced system, just immediately after impact, is observed as initial condition in the subsequent process of the motion.The non-dimensional differential equations of motion are given by Eq.( 1)~(4).
where the non-dimensional quantities are given by Periodic-impact motions of the system are described by the symbol n-p-q, where n denotes the number of excitation periods and p (q) denotes the number of impacts with right (left) soft impacts, during one impact motion period, respectively.In order to establish the Poincaré map of the periodic forced system, we chose the Poincaré section: x 2 > 0 } .The disturbed map of period n single-impact motion is expressed briefly by where

GRAZING BIFURCATIONS AND PERIODIC MOTIONS
The existence and stability of n-p-q motions have been analyzed explicitly.Bifurcations at the points of change in stability are considered, thus giving some information about the existence of different kinds of bifurcations named periodic-doubling bifurcation, Hopf bifurcation, grazing bifurcation, etc.   4c, d) showed the 3-1-3 motion at ! = 6.5.

HOPF BIFURCATION AND QUASI-PERIODIC ATTRACTOR
The system parameters ( 2 It is shown that the system exhibited stable 1-1-1 motion with !"(2.8351, 2.9).As !passed through !c =2.8351 decreasingly, the 1-1-1 motion changed its stability, and Hopf bifurcation associated with the motion occurred so that the quasi-periodic impact motions could take place .As expected, the 1-1-1 motion is represented by one fixed point in projected portrait of Poincare´ map, appeared at ! =

THE INFLUENCE OF CLEARANCES ON PERIODIC MOTIONS AND BIFURCATIONS
The influence of clearances between the system parts on dynamics is presented in the study .Taking system parameters (1):   0.2, µ c2 = 2.0, µ c3 = 2.0, µ c4 = 2.0, µ c5 = 1.0, µ m2 = 1.0, µ m2 = 1.0, f 10 = 0.0, and f 20 = 1.0, as the criterion parameters, to analyze the influence of clearances between the system parts on periodic motions and bifurcations.Fig. (6) shows the bifurcation diagrams for the impact velocity !x 2 p of the periodic forced system under the conditions of different clearance parameters.Only changed parameter is given in Fig. (6a-d), and all the other parameters, not listed in the figure's description, were the same as the criterion parameters.
The effects of changes in clearances b were analyzed by changing their value.Large or small clearances slightly influenced the velocity amplitude of the periodic motion, which can be seen in Fig. (6a-d

CONCLUSION
In this paper, the nonlinear characteristics of the periodic forced system with soft impacts were analyzed with special attention on stability of periodic motion, grazing bifurcation, period-doubling bifurcation, Hopf bifurcation and chaotic motion, etc.The piecewise properties and routes to chaos have been shown as follows. (1) The 1-1-1 motion, in most cases, underwent grazing bifurcation or Hopf bifurcation to chaos with a change in the system parameters.
(2) Grazing bifurcation led to singularity and asymmetric periodic motion, such as 4-3-4 motion, 6-3-4 motion 3-1-3 motion,. (3) The clearances of the system are the main reasons for influencing frequency range of periodic motions and chaos. (4) The steady 1-1-1 period orbits existed within a wideband frequency range and the value of velocity achieved the desired results when appropriate system parameters were chosen.
).The minimum value of clearances, i.e. b =0.0, led to small change in the topological structure of the bifurcations, as shown in Fig. (6a).As the value of b increased, the frequency range of 1-1-1 motion became narrow and period-doubling bifurcations sequence was observed, such as 1-1-1 motion and 2-2-2 motion for large excitation frequency as shown in Fig. (6b).Large value of b resulted in the narrow frequency ranges of 1-1-1 motion, moving toward small ! .The singularities and asymmetric periodic motions such as 2-1-2, 3-2-3, and 4-3-4 motions are indicated in Fig. (6c).A slightly larger value of clearances was considered as shown in Fig. (6d), and the dynamic properties of the periodic forced system were simple.As shown in Fig. (6c, d), for large clearances, the mass M 2 did not contact the soft impacts at high excitation frequency.