Abstract
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 ω and clearance δ, 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 δ is small or small enough, the transition from 1–p–p to 1–(p+1)–(p+1) motion (the fundamental group of motions, p≥1) 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 ω 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.
Highlights
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
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 relationship 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 [59,60,61], 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 [32,33,34,68,69,70,71], etc
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 periodicallyforced system with a clearance represented by two symmetric rigid stops
Summary
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 vibro-impact systems display many of the complex dynamical features near critical points of high codimension bifurcations; in particular, different types of codimension two bifurcations and parameter unfolding associated with the interaction of saddle-node and grazing bifurcations [23], double flip bifurcation [36], Hopf-flip bifurcation [37], Hopf-pitchfork bifurcation and double Neimark–Sacker bifurcation [38], codimensiontwo grazing bifurcations [39,40], grazing-sliding bifurcation [41], etc., were studied by numerical simulations and qualitative analyses in recent years. 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. Based on the sampling ranges of parameters, the influence of system parameters on impact velocities, existence regions and correlative distribution of different types of periodicimpact motions of the system is emphatically analyzed in Sections 4 and 5
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