# The Dynamic Evaluation of Tilted Support Spring Nonlinear System with Critical Components under the Action of a Rectangular Pulse

Authors: Ningning Duan; Shuang Song; Anjun Chen
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Dimensionless nonlinear dynamical equations of a tilted support spring nonlinear system with critical components were obtained under the action of a rectangular pulse, and the numerical results of the shock response were studied using Runge-Kutta method. To evaluate the dynamic characteristics of critical components, a new concept of three-dimensional shock response spectra was proposed, where the ratio of the maximum shock response acceleration of critical components to the peak pulse acceleration, the pulse duration, and the frequency ratio were three basic parameters of three-dimensional shock response spectra. Based on the numerical results, the effects of the angle, the peak pulse acceleration, the mass ratio, the frequency ratio, and the pulse duration on the shock response spectra were discussed.


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## Body

## 1. Introduction

A tilted support spring nonlinear system has been widely applied in the vibration reduction of vehicle suspension and is an improved cushion structure, which is used to connect an inner body with an outer body by four tilted support springs. The system usually shows superior absorber effects to a linear system and is applied to protect precise instruments from damage since the 1960s. In the automotive engineering, the absorber system of an engine generally adopts the angle 70°–90° of the tilted support spring nonlinear system, which provides effective absorber protections for the engine. According to extensive researches, a product is damaged in the transport, mainly because of shock and vibration. So, to estimate dynamic characteristics of the system, the concept of a shock response spectrum is proposed and used by many specialists. Alexander [1] presented that the shock response spectrum was used to analyze the frequency response of an acceleration-time-history record to probe into the maximum dynamic response. Tuma at al. [2] proposed the calculation method of the shock response spectra, which is used to deal with employing Signal Analyzer. Xu et al. [3] analyzed the high pressure regimes of porous material in shock wave response by using a microscopic method. To evaluate the influence of the impulse and geometric parameters on shock characteristics of plates under impulse loads, the model and shock response spectra of the system were carried on by Botta and Cerri [4]. Goetz and Matous [5] presented that the model of two-layered cellular materials under impulse loads was conducted to give insights into the shock characteristics of the system and to optimize the system. A dynamic model of the corrugated paperboard cushioning packaging system was established by Wang et al. [6], and a comparison between the approximate solution of nonlinear equations and the numerical simulation results was made. Wang et al. [7] obtained a model for critical components of a nonlinear packaging system and then researched the effect of the mass ratio, the frequency ratio, and the damping ratio on three-dimensional shock spectrum. As can be seen from the above analysis, most studies mainly focus on material nonlinearity at present, but the geometric nonlinearity caused by the structure of the system has barely been reported.Shock and vibration are main factors causing damages of products in transport. It is extremely important to protect products from being destroyed by dynamic evaluation of a nonlinear system. Wang and Hu [8, 9] obtained the shock response spectra and damage boundary curve in order to investigate the shock characteristics of various nonlinear materials under the action of typical pulses. Wang and Chen [10] gained the model of suspension packaging system under rectangular pulse and further discussed the effects of the angle and the damping ratio on the shock response spectra and the acceleration response of dimensionless peak pulse.However, the shock response spectrum of the tilted support spring system with critical components has hardly been discussed until today. In this paper, the dimensionless nonlinear dynamical equations for the tilted support spring system with critical components are obtained under the action of a rectangular pulse. According to the evaluation methodology [8–11], a new concept of three-dimensional shock response spectra is gained and the effects of the peak pulse acceleration, the pulse duration, the angle, the mass ratio, and the frequency ratio on dynamic characteristics of critical components are discussed. The results provide theoretical foundations for the design of the tilted support spring nonlinear system.


## 2. Modeling and Equations

The tilted support spring nonlinear system with critical components is shown in Figure1, where m1 and m2 denote the mass of critical components and the main body, respectively; k1 is the linear elastic coefficient between critical components and the main body; k2 is the linear elastic coefficient of the tilted support spring system; l0 and φ0 are the length and the support angle of the four tilted support springs before they are compressed.Figure 1
The model of the tilted support spring nonlinear system with critical components.Because the model of the system is principally affected by the vertical direction, the study on other directions of the system can be omitted, just like Figure1. To facilitate the numerical analysis, the coordinate system is obtained, the static equilibrium position is treated as the original points, and the downward direction is regarded as the positive direction. Then, the vertical natural vibration dynamic equations [12] can be obtained as
(1)m1d2x1dt2-k1(x2-x1)=0m2d2x2dt2+2k2(a0x2+b0l0x22+c0l02x23)+k1(x2-x1)=0,
where a0=sin2φ0, b0=-3/2sinφ0cos⁡2φ0, and c0=1/2(1-6sin2φ0+5sin4φ0), which is related to the angle of the system, namely, geometric structure nonlinearity.To evaluate the dynamic characteristics of the system under the action of a rectangular pulse, the equations of the pulse is expressed as(2)u¨0={u¨0m0≤t≤t00t>t0,
where t0 is the pulse duration. Substitute (2) into (1); then the shock dynamic equations of the system can be derived as
(3)m1d2x1dt2-k1(x2-x1)=0m2d2x2dt2+2k2[a0(x2+u0)+b0l0(x2+u0)2+c0l02(x2+u0)3]+k1(x2-x1)=0,
where x1,2(0)=0 and dx1,2(0)/dt=0 are initial conditions of the displacement and the velocity, respectively.Lettingy1=(x2-x1)/l0, y2=(x2+u0)/l0, ω1=k1/m1, and ω2=2k2/m2 and combing with (3), the dimensionless nonlinear shock dynamic equations of the system under the action of a rectangular pulse can be shown as
(4)d2y1dτ2+(a0y2+b0y22+c0y23)+λ12λ2y1+λ12y1=0d2y2dτ2+(a0y2+b0y22+c0y23)+λ12λ2y1=βu¨0,
where T=1/ω2 is taken as the period parameter; τ=t/T is the dimensionless time parameter; τ0=t0/T is the dimensionless pulse duration parameter; λ1=ω1/ω2 is the frequency ratio of the system; λ2=m1/m2 is the mass ratio of the system; β=T2/l0 is defined as a characteristic parameter of the system; βu¨0m is the dimensionless peak pulse acceleration, respectively.According to (4), initial conditions of the dimensionless displacement and velocity can be transformed as y1,2(0)=0 and dy1,2(0)/dτ=0, respectively.The expression of the dimensionless rectangular pulse is then in (4):
(5)βu¨0={βu¨0m0≤τ≤τ00τ>τ0.


## 3. Dynamic Evaluation

Based on the analysis of (4), it demonstrates that the dimensionless acceleration response of a product and its critical components is relevant to the angle of the system, the dimensionless peak pulse acceleration, the dimensionless pulse duration, the mass ratio, and the frequency ratio of the system. This section will chiefly concentrate on three-dimensional shock response spectra of critical components, which can clearly reflect the dynamic characteristics of the tilted support spring nonlinear system caused by the geometry nonlinearity of the structure.To evaluate the dynamic characteristics of the system under the action of a rectangular pulse, a new concept of three-dimensional shock response spectra is proposed, where the ratio of the maximum shock response acceleration of critical components to the peak pulse accelerationγ=(x¨1)m/u¨0m is regarded as the response index of the system, namely, dynamic amplification coefficient. Integrating the dimensionless parameters of the system y1=(x2-x1)/l0 and y2=(x2+u0)/l0 with (4), the dynamic amplification coefficient can be written as
(6)γ=(d2x1/dt2)md2u0m/dt2=((d2y2/dτ2)-(d2y1/dτ2)-(β(d2u0/dτ2)))mβ(d2u0m/dτ2)=(λ12y1)mβu¨0m.The dimensionless pulse durationτ0 and the frequency ratio of the system λ1 are considered to be two basic parameters of the three-dimensional shock response spectra. Based on the numerical results, the effects of the angle, the mass ratio, the frequency ratio, the dimensionless peak pulse acceleration, and the dimensionless pulse duration on shock response characteristics of critical components are discussed.The dimensionless shock dynamic equations (4) are solved using fourth-order Runge-Kutta method. At the same time, set the dimensionless pulse duration 0≤τ0≤40 and the frequency ratio of the system 0≤λ1≤6. When we choose the mass ratio λ2=0.01 and the angle of the system φ0 = 60°, the effect of the dimensionless peak pulse acceleration on the three-dimensional shock response spectra of critical components is given in Figure 2.The effect of the peak pulse acceleration on three-dimensional shock response spectra of criminal components when (a)βu¨0m=0.01, (b) βu¨0m=0.05, (c) βu¨0m=0.1, and (d) βu¨0m=0.15. γ: the ratio of the maximum shock response acceleration of critical components to the peak pulse acceleration, λ1: the frequency ratio of the system, and τ0: the dimensionless pulse duration.
(a)
(b)
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(d)As is shown in Figure2, when it is low frequency ratio, under the action of a rectangular pulse, increasing the dimensionless peak pulse acceleration can lead to noticeable fluctuation of the maximum shock response acceleration of critical components and arising more peaks of shock response spectra.According to the previous researches, the discussions about the tilted support spring nonlinear system use the range of 60°–90°. Therefore, under the conditions of the mass ratioλ2=0.01 and the dimensionless peak pulse βu¨0m=0.15, the effect of the angle (60°, 70°, 85°, 90°) on three-dimensional shock response spectra of critical components is revealed in Figure 3.The effect of the angle of the tilted support spring system on three-dimensional shock response spectra of criminal components when (a)φ0 = 60°, (b) φ0 = 70°, (c) φ0 = 85°, and (d) φ0 = 90°. γ: the ratio of the maximum shock response acceleration of critical components to the peak pulse acceleration, λ1: the frequency ratio of the system, and τ0: the dimensionless pulse duration.
(a)
(b)
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(d)As is shown in Figure3, comparing with linear system (φ0 = 90°), decreasing the angle can remarkably aggravate the fluctuation of maximum shock response acceleration of critical components at low frequency ratio. Wu et al. [12] have researched the fact that the damping effect of the tilted support nonlinear spring is superior to the linear system, while the shock resistance function of the system decreases.Figure4 presents the effect of the mass ratio on three-dimensional shock response spectra of critical components, setting the angle of the system φ0 = 60° and the dimensionless peak pulse βu¨0m=0.05.The effect of the mass ratio on three-dimensional shock response spectra of criminal components when (a)λ2=0.01, (b) λ2=0.05, (c) λ2=0.1, and (d) λ2=0.2. γ: the ratio of the maximum shock response acceleration of critical components to the peak pulse acceleration, λ1: the frequency ratio of the system, and τ0: the dimensionless pulse duration.
(a)
(b)
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(d)According to analysis of numerical results in Figure4, it reveals that increasing the mass ratio can effectively restrain the maximum shock response acceleration of critical components at low frequency ratio.Figures2–4 show the effect of frequency ratio on three-dimensional shock response spectra. In terms of tilted support spring nonlinear system with critical components, the maximum shock response acceleration of critical components under the action of a rectangular pulse is sensitive to low frequency ratio. That is to say, the three-dimensional shock response spectra of criminal components exist in sensitive areas.


## 4. Conclusions

By analyzing the three-dimensional shock response spectra of the tilted support spring nonlinear system with critical components, Figures2-3 demonstrate that increasing the peak pulse acceleration or decreasing the angle can rapidly increase the maximum shock response acceleration of critical components, and the effect of low frequency ratio on shock response of the system is particularly sensitive. It shows in Figures 2–4 that increasing the frequency ratio of the system can obviously decrease the maximum shock response acceleration of critical components, and the peak of the shock response for critical components can be evidently decreased at the low frequency ratio by increasing the mass ratio of the system. In addition, Figures 2–4 can clearly show that, with the increase of the dimensionless pulse duration, at low frequency ratio (λ1<5) the shock response acceleration of critical components can arise multiple peaks, while at high frequency ratio the shock response acceleration of critical components tends to be stable (γ=2). Therefore, it is extremely necessary to control the above parameters adequately for decreasing the maximum shock response acceleration of critical components.On the basis of the numerical analysis, the frequency ratio of the system is an important parameter. It is particularly significant for advancing shock resistance characteristics of critical components to control the frequency ratio suitably and increase the frequency ratio of the system as possible in the permissive condition (it is suggested that frequency ratio isλ1>5).



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*Source: 404163-2014-03-25.xml*