Analysis of the Dynamic Stiffness, Hysteresis Resonances and Complex Responses for Nonlinear Spring Systems in Power-Form Order

Power-form nonlinear contact force models are widely adopted in relatively moving parts of macro (e.g., rolling bearings considering Hertzian contact restoring force between rolling elements and bearing raceways) or micro (e.g., the micro cantilever probe system of atomic force microscopy) scale mechanical systems, and contact resonance could cause serious problems of wear, contact fatigue, vibration, and noise, which has attracted widespread attention. In the present paper, the softening/hardening stiffness characteristics of continuous and one-sided contact power-form nonlinear spring models are addressed, respectively, by the analysis of the monotone features of resonant frequency-response skeleton lines. Herein, the period-n solution branch and its stability characteristics are obtained by the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory. Compared with previous studies, this paper will furtherly clarify the influences of externally normal load, the power form exponent term, and excitation amplitude on the softening/hardening stiffness characteristics of general power-form spring systems. In addition, for a power-form system with a one-sided contact, the phenomena of primary and super/sub-harmonic hysteretic resonances inducing period-doubling, folding bifurcation, the coexistence of multiple solutions are demonstrated. Besides, it gives the evolution mechanism of two types of intermittency chaos in a one-sided contact system. The overall results may have certain basic theoretical significance and engineering values for the control of vibration and noise in contact mechanical systems.

Dynamic Behavior Analysis of an Axially Loaded Beam Supported by a Nonlinear Spring-Mass System

Dynamic analysis of an Euler–Bernoulli beam with nonlinear supports is receiving greater research interest in recent years. Current studies usually consider the boundary and internal nonlinear supports separately, and the system rotational restraint is usually ignored. However, there is little study considering the simultaneous existence of axial load, lumped mass and internal supports for such nonlinear problem. Motivated by this limitation, the dynamic behavior of an axially loaded beam supported by a nonlinear spring-mass system is solved and investigated in this paper. Modal functions of an axially loaded Euler–Bernoulli beam with linear elastic supports are taken as trail functions in Galerkin discretization of the nonlinear governing differential equation. Stable steady-state response of such axially loaded beam supported by a nonlinear spring-mass system is solved via Galerkin truncation method, which is also validated by finite difference method. Results show that parameters of nonlinear spring-mass system and boundary condition have a significant influence on system dynamic behavior. Moreover, appropriate nonlinear parameters can switch the system behavior between the single-periodic state and quasi-periodic state effectively.

Vibration Characteristics of Rolling Mill System under Constraints of the Nonlinear Spring Force and Friction Force from Hydraulic Cylinder

Considering the two kinds of nonlinear constraints of rolling mill hydraulic cylinder, spring force and friction force, the vibration model of rolling mill system is established. The amplitude frequency response equations are obtained by using the average method. Comparing the time history curves of vertical vibration displacement of rolling mill system under the nonlinear spring force and friction force, the amplitude frequency characteristic curves are simulated. The external excitation amplitude is viewed as the bifurcation parameter, and the system bifurcation response changing with the external excitation amplitude is analyzed. The influence of the external excitation amplitude on the system stability is studied. The results indicate that the increase of the nonlinear spring force makes the rolling mill system’s unstable area to become wider, and the influence on the rolling mill system of nonlinear friction force behaves as the damping characteristics; the vibration of rolling mill system is alternating between the periodic, period-doubling, and the chaotic motion. The research results provide a theoretical support for restraining the vibration of the rolling mill system in the actual production process.