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

Qingtao Wang,Zhaojun Pang,Zhiyong Zhang,Yongheng Ying

doi:10.3390/app11167722

Qingtao Wang, Zhaojun Pang + Show 2 more

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https://doi.org/10.3390/app11167722

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Journal: Applied Sciences	Publication Date: Aug 22, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Nanjing University of Science and Technology

Abstract

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.

Highlights

The mass-spring model is one of the basic mechanical elements, which can be used to describe the interaction of deformation, energy transfer, and motion control between different bodies
The power-form nonlinear restoring force FC(x) = KC·xα is more common, where the nonlinear power-form exponent term α of the normal contact force between two elastic surfaces is in the range of 1.5 to 3.5 [3], the nonlinear force in piano hammers is 2.2 < α < 3.5 [4], and even the case of 2 < α < ∞ exists in the solitary wave propagation through particle chains [5]
For α = 1.5 and W = 0, the continuous system is power-form nonlinear without exterFnoarllαy =no1r.5maanldloWad=. 0It,sthpericmonatriynureosuosnsaynscteemexihsibpiotws hera-rfdoermninngonsplirnienagr cwhiatrhaocutetreixsttiecrsnsianlclye tnhoermpearlioloda-d1.frIetsquperinmcya-rryesrpesoonnsaenccuerveexhbiebnitdsshtaortdheenrinigghtsp(sreinegFicghuarreac2tae)r,iastnicdscsyicnlcice tfhoeldpinegriobdif-u1rfcraetqiounens cayt-rtuesrpnoinngsepcouinrvtse Abe1nadnsdtoAt4hcearnigbhrtin(sgeejuFmigpuirneg2pah),eannodmceyncalictofotlhdeisnygstbeimfu.rcFaotironαs=at0t.6urannindgWpo=in0ts, Aas11 asnhdowAn44 ciannFbirginugreju2mb,ptihneg pprhiemnaormyerneasotonathnecesyosftetmhe

Summary

Introduction

The mass-spring model is one of the basic mechanical elements, which can be used to describe the interaction of deformation, energy transfer, and motion control between different bodies. Micro, and nano scales, linear spring models with a linear relationship between force and displacement depicted by hook’s law F(x) = kx are widely observed [1]. Many interactions are not suitable for linearization. The classical three-body problem mentioned by Newton in 1687 promotes the development of nonlinear science [2]. The nonlinear load-deformation relationship is generally expressed as F(x) = an·xn + an−1·xn−1 + .

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