Nonlinear vibration isolation via a circular ring

https://doi.org/10.1016/j.ymssp.2019.106490Get rights and content

Highlights

  • A novel nonlinear vibration isolator in the shape of a circular ring is proposed.

  • Stretching-induced tension coupled with the curvature changes produce highly nonlinear stiffness.

  • The use of stiffness nonlinearity improves vibration isolation efficiency.

  • The results of transmissibility are validated through an experiment.

Abstract

A novel nonlinear vibration isolator in the shape of a circular ring is investigated in this study. When the ring is compressed along a diametral line, it exhibits highly nonlinear geometric stiffness due to the effects of stretching-induced tension coupled with the curvature changes. The use of stiffness nonlinearity improves vibration isolation efficiency. A mechanical model of the ring under compression is derived from beam theory. Methods of direct separation of motions and harmonic balance are employed to determine the frequency response and displacement transmissibility of the device under base excitations for different geometric configurations. The analytical results are numerically validated via direct time integration of the equation of motion and via experimental data. It is shown that an increase in the pre-deformation of the ring enhances the vibration isolation performance.

Introduction

Vibration isolation is essential for protecting precision instruments from excessive vibration. Linear vibration isolators work at an excitation frequency higher than √2× peak frequency. To extend the isolation range, the isolator should have low dynamic stiffness. However, a low dynamic stiffness can cause the isolator to become unstable. Therefore, a nonlinear design of vibration isolators has recently been developed to overcome this problem; the development of this design continues to grow at a rapid pace [1], [2], [3], [41].

A promising method to improve isolation performance involves using the high-static-low-dynamic-stiffness (HSLDS) concept [4]. HSLDS vibration isolators are usually modeled using a combination of a linear positive stiffness element and a negative stiffness corrector [5]. Zhou et al. [6] designed an HSLDS vibration isolator with a cam-roller-spring mechanism. Then Zhou et al. [7] proposed a 6-DOF nonlinear vibration isolator using a quasi-zero-stiffness strut. Jing et al. [8], Bian et al. [9] and Hu et al. [10] introduced a scissor-like structure in vibration isolators for improving their performance. Lu et al. [11] designed a two-stage negative stiffness vibration isolator based on a bistable composite plate. Palomares et al. [12] proposed a negative stiffness vibration isolator using pneumatic linear actuators. Yan et al. [13] designed a geometric anti-spring isolator for half-sine pulse excitations. Buckled slender beams were also proposed as vibration isolators [14], [15]. Liu et al. [16] used pre-deformed Euler beams to induce negative stiffness at equilibrium to offset the primary positive stiffness. Huang et al. [17] molded a vibration isolation system configured by a buckled beam. Rostam et al. [18] designed a nonlinear vibration isolator that consists of a curved beam with eccentric moving loads. Ding et al. [19] studied a viscoelastic beam with a vertical elastic support boundary. The elastic support facilitated effective vibration isolation. To date, nonlinear vibration isolation has been realized using magnets, horizontal springs, post-buckled beams under uniaxial compression, curved beams with eccentric moving loads, viscoelastic beams with a vertical elastic support boundary, and bistable composite plates. Therefore, the physical realization of HSLDS is still a hot research topic. A circular ring can be used in the nonlinear design of vibration isolators, but this idea has not yet been implemented.

Buckling and post-buckling of a circular ring are classic problems that were first identified by Love [20]. He mainly focused on the problems of the equilibrium and elastic stability of a circular ring bent in- and out-of-plane. Carrier [21] was the first to study buckling of rings. Carrier’s study attracted the attention of many researchers, such as Naschie and Nashai [22], Sills and Budiansky [23], Kyriakides and Babcock [24] and Eslami [25], who made numerous attempts to mitigate the large deflections of these rings. Wu and Huang [26] investigated the elastic stability of bending and twisting curved bars near some equilibria. Tse et al. [27] studied the nonlinear spring behavior of orthotropic symmetric circular rings. Kim and Chaudhuri [28] investigated the strain–displacement relationship of the complete form for a ring in the deep post-buckling region. Wu et al. [29] determined the post-buckling response of a ring by using the harmonic balance method. Wang et al. [30] studied the load–deflection characteristics of a normal cantilever under a tip load, based on a homotopy perturbation approach. Batista [31] proposed an analytical solution to equilibria of a cantilever with loads on its tip. Kerdegarbakhsh et al. [32] explored the post-buckling characteristics of a functionally graded material ring and derived the highly nonlinear stability equations from a nonlinear equilibrium criterion. Chen and Cai [33] derived the stress–strain relationship of a tubular material from a unified ring model. Lu et al. [34] studied the equilibria and stability of a rotating ring. Lu and Metrikine [35] analyzed the stability of a rotating ring attached to a moving oscillator. Azzuni and Guzey [36] examined the sensitivity of circular rings to initial imperfections based on a perturbation technique. The above mentioned investigations focused on equilibrium, stability, and deflections of rings under static forces. The dynamic stiffness characteristic around the static equilibrium has not been addressed. However, the static deflection used to determine the static equilibrium caused by a loaded mass has significant effects on the stability of isolators. Moreover, the dynamic stiffness around the equilibrium is a key factor that must be considered in the design of nonlinear vibration isolators. Prior to the dynamic analysis of nonlinear vibration isolators, a static analysis of the HSLDS of a circular ring is necessary.

Approximate analytical methods allow to quantify conveniently the performance of nonlinear vibration isolators. The harmonic balance method is often employed to predict the steady-state responses of nonlinear vibration isolation systems, such as Hu and Zheng [37], Zhou et al. [7], Lu et al. [11], Liu et al. [16] and Huang et al. [17]. Another effective approximate analytical method is the averaging method. Zhou et al. [6] applied this analytical method to study HSLDS vibration isolators with a cam-roller-spring mechanism. The peak value and starting frequency of transmissibility outperformed the linear one. Hao and Cao [38] used the same method to examine a 1-DOF stable-quasi-zero-stiffness vibration isolator with geometrical nonlinearity. In addition to the analytical works mentioned above, Blekhman [39], [40] proposed a technique for the direct separation of motions to calculate the periodic solutions of a nonlinear dynamic system. So far, this technique has not been used in vibration isolation studies. In the present work, the flexibility of this technique is explored. The resulting predictions are compared with those obtained through the harmonic balance method and are validated via numerical simulations.

In this manuscript, a circular ring is proposed to achieve HSLDS. The stiffness characteristic of the ring when bent in-plane is analyzed. In this work, focus is placed on the benefits of stiffness nonlinearity improving vibration isolation efficiency. Two approximate analytical methods are used to study the transmissibility of the diametrical vibration of the compressed ring. The results are validated through an experiment.

The manuscript is organized as follows: In Section 2, a nonlinear vibration isolator in the shape of a circular ring is modeled. In Section 3, the displacement transmissibility is analytically obtained via the direct separation of motions, the effects of geometrical parameters on the transmissibility are examined, and the approximate analytical results are validated through a numerical method. In Section 4, the analytical results are compared with the results obtained with the harmonic balance method. In Section 5, an experiment is performed to demonstrate the benefits of the ring vibration isolator, and in Section 6, concluding remarks are presented.

Section snippets

Novel nonlinear vibration isolator

Fig. 1(a) shows a nonlinear circular ring vibration isolator. The bottom of the ring is fixed to the base, and a mass is placed on top of the ring. The isolator is assumed to have linear viscous damping, which is experimentally identifiable. The ring induces geometrically nonlinear stiffness. The system is subjected to the base motion xe(t) = Xe cos(ωt), and the amplitude frequency response and displacement transmissibility are used to evaluate the vibration isolation performance. Fig. 1(b)

Method of direct separation of motions and numerical validation

Employing the method of direct separation of motions yields the following solution:xr=α(T1)+ψ(T1,T0)where the new timescales T0 and T1 are denoted as T0 = t0 and T1 = εt0. t0 = ωt, ε is a small bookkeeping parameter, α is the slow parameter, and ψ is the fast parameter.

The variables T1 and T0 are considered independent, so thatdxrdt0=ωεαT1+ωψT0+ωεψT1d2xrdt02=ω2ε2d2αdT12+ω22ψT02+2ω2ε2ψT1T0+ω2ε22ψT12

Substituting Eqs. (15), (16) and timescales T1, T0 into Eq. (13) givesmω2ε2d2αdT12+2ψ

Comparison with the harmonic balance method

The relative displacement responses were expanded into sets of Fourier series with the same orthogonal basis.xrt=a10+n=1Na1ncosnωt+a2nsinnωt

Substituting Eq. (32) into Eq. (13) yields 2N+2 nonlinear algebraic equations using the method of harmonic balance. Applying the harmonic balance method with higher harmonics yields more accurate results. In the limit of including all the harmonics, they must provide exactly the same solution with exact result. However, it is difficult to construct

Experimental validation

A schematic and photograph of the experimental test rig used to experimentally investigate the performance of a circular ring vibration isolator are shown in Fig. 13(a) and (b), respectively. The test rig contained a circular ring vibration isolator with a mass on top of the ring. The mass could be varied. Different masses m of 0.2 kg, 2.0 kg, and 5.1 kg were applied to validate the analytical transmissibility in the three stages. The isolator was attached to a shaker that created the required

Conclusions

A circular ring was proposed as a nonlinear vibration isolator. The same isolation efficiency can be achieved with much smaller static deflection and, hence, space when using a ring compared to a linear spring. Beam theory was applied to model the ring stiffness. The displacement transmissibility was used to measure the performance of the ring vibration isolator. The method of direct separation of motions was employed to analyze the frequency response and displacement transmissibility, and the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11872037 and 11572182), China Scholarship Council (No. 201706895001), the Key Research Projects of Shanghai Science and Technology Commission (No. 18010500100), and the Innovation Program of Shanghai Municipal Education Commission (No. 2017-01-07-00-09-E00019).

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