Melnikov′s Method of Periodic Solutions with Double Impacts for a 2-DOF Vibro-impact Quasi-Hamiltonian System

ZHANG Sijin, LIU Yu, JI Dezan


Perturbation method and Poincaré mapping method were used to derive the generalized Melnikov function of the periodic solution for a two-degree-of-freedom vibro-impact system with cubic non-linearity and external excitations. By using the Melnikov′s method, the characteristics of periodic motions with double-impact of the 2-dof system were studied, and the existence condition of period-2 motions with double-impact was determined as a critical curve in the parameter domain. The results of numerical simulations show that the regions below the critical curve are the period-2 motions with double-impact, the upper regions of the critical curve are not period-2 motions with double-impact;Meanwhile,increasing the force amplitude and keeping the other parameters unchanged, the motion state of the system changes from multi-period motions with multi-impact to period-2 motions with double-impact, while increasing the system restitution coefficient and keeping the other parameters unchanged, the motion state of the system changes from period-2 motions with double-impact to multi-period motions with multi-impact.


Keywords: vibro-impact system,  generalized Melnikov′s method,  period-2 motion,  Poincaré maping,  generalized Melnikov′s function

Full Text:



TIAN R L, ZHAO Z J, YANG X B,et al. Subharmonic bifurcation for a nonsmooth oscillator [J]. International Journal of Bifurcation & Chaos, 2017, 27 (10):1750163.

SHEN J, LI Y, DU Z. Subharmonic and grazing bifurcations for a simple bilinear oscillator [J]. International Journal of Non–linear Mechanics, 2014,60(2):70—82.

ZHANG S J, DU B X, YIN S. Anti –control of double Hopf bifurcation of vibration rating griddle system [J]. Journal of Hunan University(Natural Sciences), 2O17,44(10):55—61 (In Chinese)

BU Q, COLE C, LUO S, et al. A review of dynamics modeling of friction draft gear [J]. Vehicle System Dynamics, 2014, 52 (6): 733—758.

FAN X X, BANG Q. Research on modeling and simulation of longitudinal vehicle dynamics based on non –cmooth dynamics of multibody systems [J]. Chinese Journal of Theoretical & Applied Mechanics,2O15,47(2),3O1—3O9 (In Chinese)

CAO D, CHU S, LI Z. Study on the non–cmooth mechanical models and dynamics for space deployable mechanisms [J]. Chinese Journal of Theoretical & Applied Mechanics, 2O13, 45(1):3—15.

KUKUCKA P. Melnikov method for discontinuous planar systems [J]. Nonlinear Analysis Theory Methods & Applications, 2007, 66 (12):2698—2719.

SHEN J, DU Z. Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple cones [J]. Zeitschrift Angewandte Mathematik Und Physik, 2016, 67 (3):1—17.

CHAVEZ J P, WIERCIGROCH M. Bifurcation analysis of periodic orbits of a non–smooth Jeffcott rotor model [J]. Communications in Nonlinear Science & Numerical Simulation, 2013, 18 (9):2571— 2580.

XU J, LI Q, BANG N. Existence and stability of the gracing periodic trajectory in a two –degree –of –freedom vibro –impact system [J]. Applied Mathematics & Computation, 2011, 217 (12): 5537—5546.

L –SHUDEIFAT M A, BIERSCHEM N, QUINN D D,et al. Numerical and experimental investigation of a highly effective single –sided vibro –impact non –linear energy sink for shock mitigation [J]. International Journal of Non –linear Mechanics, 2013, 52 (6):96—109.

LUO G B, ZHU X F, SHI Y Q. Dynamics of a two –degree –of freedom periodically–forced system with a rigid stop: Diversity and evolution of periodic –impact motions [J]. Journal of Sound & Vibration, 2015, 334 (334):338—362.


  • There are currently no refbacks.