Abstract
The nonlinear dynamical equations of tilted support spring packaging system with critical components were obtained under the action of half-sine pulse. To evaluate the shock characteristics of the critical components, a new concept of three-dimensional shock response spectrum was proposed. The ratio of the maximum shock response acceleration of the critical components to the peak pulse acceleration, the dimensionless pulse duration, and the frequency parameter ratio of system or the angle of tilted support spring system were three basic parameters of the three-dimensional shock response spectrum. Based on the numerical results, the effects of the peak pulse acceleration, the angle of the tilted support spring, the frequency parameter ratio, and the mass ratio on the shock response spectrum were discussed. It is shown that the effects of the angle of the tilted support spring and the frequency ratio on the shock response spectrum are particularly noticeable, increasing frequency parameter ratio of the system can obviously decrease the maximum shock response acceleration of the critical components, and the peak of the shock response of the critical components can be decreased at low frequency ratio by increasing mass ratio.
1. Introduction
The main dynamic factors to cause the damage of product are shock and vibration in the transportation; it is extremely important to protect product by the study on shock characteristics and corresponding evaluations of packaging system. Newton’s damage boundary concept [1] is the foundation for the present packaging design. In this theory, the product packaging system was considered to be single degree of freedom undamped linear system. These may not be valid because of the complexity of products configuration and the diversity of cushioning material. The fatigue damage boundary concept was presented by Burgess [2] to describe the effect of multiple shocks on damage of products. The concept of displacement damage boundary was applied by Wang et al. [3] to study the deformation of cushioning materials. The dropping damage boundary concept has been proposed by Wang [4] to evaluate the dropping damage of products for nonlinear packaging system and expand to the two degrees of freedom of packaging system with critical components [5, 6]. The concept of bruising fragility and bruising boundary for fruits was established by Lu and Wang [7], which can be applied to other similar viscoelastic products. The theory of shock response spectrum [8] was introduced into packaging design from another engineering. The shock response spectrum can predict the response of packaged product with a known natural frequency, which did not focus on the input shock impulse but on the response of product. In recent years, packaging researchers have become increasingly interested in shock response spectrum for packaged product, for example, [9–12].
The absorber effect of tilted support spring system is superior to linear system composed of vertical suspension [13], mainly applied to product transportation, providing absorber protection for precision instruments with low fragility. The absorber system of engine adopts the angle 70°–90° of tilted support spring system, which provides effective absorber protections for the engine. On the basis of the nonlinear vibration equations of tilted support spring system, the nonlinear natural vibration characteristic of the system and the influence of different angle, amplitude, and so on are discussed by Wu et al. [13, 14]. In [11], the nonlinear dynamical equations of tilted support spring are obtained under the action of rectangular pulse; the ratio of the maximum shock response acceleration of system to the peak acceleration of pulse, the dimensionless pulse duration, and the angle of system are three basic parameters of the three-dimensional shock response spectra, and the effects of the peak pulse acceleration and the damping of system on the shock response were introduced. However, the shock response characteristic of the tilted support spring system with critical components has hardly been discussed until today. In this paper, the dimensionless nonlinear dynamical equations of tilted support spring system with critical components were obtained under the action of half-sine pulse. Based on the numerical results, to evaluatethe shock response characteristics of critical components, the three-dimensional shock response spectrum of the critical components is established and the effects of the peak pulse acceleration, the pulse duration, the angle, the mass ratio, and the frequency parameter ratio on dynamic characteristics of critical components are discussed. The results provide theoretical foundations for the design of the tilted support spring system.
2. Dimensionless Shock Dynamic Equations
The model of tilted support spring system with critical components is shown in Figure 1. The packaged product is supported by two springs which own the same stiffness and length , is the angle of primary support position, and denote the mass of critical components and the main body, is the coupling stiffness of the critical component, and and represent the displacement of the critical component and the main body, respectively. To facilitate the numerical analysis, the coordinate system is obtained, the static equilibrium position treated as the original points and the downward direction regarded as the positive direction. Then, the vertical natural vibration approximate dynamic equations can be expressed as where , , and , which is related to the angle of tilted support spring, namely, geometric nonlinear.
To evaluate the dynamic characteristics of the system under the action of a half-sine pulse, the equation of the pulse is expressed as where and denote the peak pulse and the pulse duration, respectively. The shock dynamic equations of system are formulated as with the initial conditions
To simplify these equations, we introduce the new dimensionless variables as follows: where and are the dimensionless displacements, respectively, and are circular frequency parameters of the system, is taken as the period parameter of the system, is the dimensionless pulse time, is the dimensionless pulse duration, is frequency parameter ratio of system, and is mass ratio of system. is defined as the system parameter, and is dimensionless peak pulse acceleration.
The dimensionless shock dynamic equations can be written in the following form: where Equation (7) is the dimensionless equations of half-sine pulse. The dimensionless form of the initial conditions can be expressed as
3. Dimensionless Acceleration Response of Critical Components
According to (6), combined with dimensionless parameters and , the dimensionless acceleration of critical components can be expressed as The dimensionless shock dynamic equations (6) are solved using the fourth-order Runge-Kutta method. When dimensionless peak pulse acceleration , frequency parameter ratio of system , mass ratio of system , and dimensionless pulse duration are defined, respectively, the effects of angle on dimensionless response acceleration-time history of critical components are given in Figure 2. As shown in Figure 2, with the decrease of angle of system (), the peak acceleration of critical components decreases and the period extends. Compared with linear system (), the shock absorber effect of critical components is perfect under the action of half-sine pulse. When , , , and , with dimensionless peak pulse acceleration (, , increasing, dimensionless response acceleration peak of critical components increases remarkably, which is shown in Figure 3. As is revealed in Figure 4, when , , , and , with increase of dimensionless pulse duration , dimensionless response acceleration peak of critical components increases correspondingly and the period extends obviously.
4. Shock Response Spectrum of Critical Components
To evaluate shock characteristics of critical components under the action of half-sine pulse, the ratio of the maximum shock response acceleration of the critical components to the peak pulse acceleration and the dimensionless pulse duration are two basic parameters of the two-dimensional shock response spectrum of critical components, where is regarded as the response index of the system, namely, dynamic amplification coefficient. Integrating the dimensionless parameters of the systems and with (6), the dynamic amplification coefficient can be written as
The dimensionless shock dynamic equations (6) are solved using the fourth-order Runge-Kutta method. At the same time, set the dimensionless pulse duration . When we choose , , and , the effect of frequency parameter ratio on the two-dimensional shock response spectrum of critical components is shown in Figure 5. When , , and , the effect of the angle on the two-dimensional shock response spectrum of critical components is demonstrated in Figure 6. According to the analysis of Figures 5 and 6, the effect of frequency parameter ratio on the response of critical components under the action of half-sine pulse is sensitive, especially at low frequency parameter ratio.
To reflect comprehensively the effect of frequency parameter ratio and the angle on shock response of critical components, it is extremely significant to propose the concept of three-dimensional shock response spectrum; the ratio of the maximum shock response acceleration of the critical components to the peak pulse acceleration , the dimensionless pulse duration , and the frequency parameter ratio of system or the dimensionless pulse duration and the angle of system are three basic parameters of the three-dimensional shock response spectrum, respectively. Based on the numerical results, the effects of the frequency parameter ratio, the angle, the dimensionless pulse duration, the mass ratio, and the dimensionless peak pulse acceleration on shock characteristics of critical components are discussed.
The angle of tilted support spring system is about 70°–90° in the engineering application, such as automobile engine [13]. So, set the angle of system , the dimensionless pulse duration and the frequency parameter ratio of system ; the dimensionless peak pulse acceleration is 0.05, 0.1, and 0.2, respectively.
The dimensionless pulse duration and the frequency parameter ratio of system are considered to be two basic parameters of the three-dimensional shock response spectrum. Based on the numerical results, the effect of the angle on three-dimensional shock response spectrum of critical components is shown in Figure 7 on the condition that mass ratio and dimensionless peak pulse acceleration . To further study the effect of angle on shock characteristics of critical components, the angle of tilted support spring system and dimensionless pulse duration are considered to be two basic parameters of the three-dimensional shock response spectrum, when mass ratio and frequency parameter ratio , , and , respectively. The effect of the peak pulse acceleration and frequency parameter ratio on three-dimensional shock response spectrum of critical components is revealed in Figure 8. On the condition that dimensionless peak pulse acceleration and frequency ratio , the effect of mass ratio on three-dimensional shock response spectrum of critical components is demonstrated in Figure 9.
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5. Discussion
5.1. The Influence Factors Analysis of the Acceleration Response for Critical Components
The angle of the tilted support spring, the dimensionless peak pulse acceleration, and the pulse duration are the key factors affecting the acceleration response of critical components.
By the analysis of Figure 2, with the decrease of angle, the maximum shock response acceleration of the critical components decreases and the corresponding period extends. Compared to linear system, tilted support system has perfect absorber effects for critical components. From Figures 3 and 4, when dimensionless peak pulse acceleration increases or dimensionless pulse duration increases, the maximum shock response acceleration of the critical components increases and the corresponding period extends.
5.2. The Influence Factors Analysis of Shock Characteristics for Critical Components
5.2.1. The Effects of Frequency Parameter Ratio
From Figures 5 and 7, under the condition of a given mass ratio and peak pulse acceleration, at low frequency parameter ratio ( in Figure 7), decreasing angle of the tilted support springs can lead to noticeable fluctuation of the maximum shock response acceleration of critical components and increase of the peak of shock response spectrum. Acceleration response of critical components is extremely sensitive to frequency parameter ratio, and the sensitive area increases by decreasing angle of the tilted support springs.
5.2.2. The Effects of Angle of Tilted Support Spring System
According to numerical analysis of Figures 6 and 8, under the condition of a given mass ratio of system, compared with linear system (the angle of tilted support spring system ), decreasing the angle of system of tilted support spring can influence the peak shock response acceleration of critical components at the same frequency parameter ratio, and it is remarkable at the sensitive area of frequency (such as and ). It is explained that, due to geometric nonlinear system, reducing the angle can decrease the vertical stiffness of system; in addition, the increasing energy of critical components is obtained from the system at low frequency ratio, which terrifically affects maximum shock response acceleration of critical components. By these conclusions, it is obvious that, in the process of designing shock absorber with tilted support spring, it should be emphasized not only to absorber effect but also to the change of shock characteristics caused by the angle of tilted support spring system.
5.2.3. The Effects of Mass Ratio
From Figure 9, we can see that, under the action of low frequency ratio, increasing mass ratio can effectively decrease maximum shock response acceleration of the critical components.
5.2.4. The Effects of Peak Pulse Acceleration
The peak of shock response acceleration of the critical components increases with dimensionless peak pulse acceleration increasing, and it is more obvious at the sensitive area of the frequency and decreasing angle of tilted support spring system.
5.2.5. The Effects of Pulse Duration
We note that if pulse duration is large, then is close to unity. On the other hand, if pulse duration is small, is different from unity.
6. Conclusions
In this paper, the research indicated that the shock response of critical component depends much on the defined frequency parameter ratio and the angle of tilted support spring system. In the tilted support spring system design procedure, frequency parameter ratio of system and the angle of tilted support spring system are important parameters. Based on the analysis, it is particularly significant that to increase frequency parameter ratio of system as possible as in the permissive condition (it is suggested that frequency ratio is ). In addition, it is necessary to increase angle as possible as on the condition of perfect vibration absorber (it is suitable to control ). By the analysis of dimensionless peak pulse acceleration , it is evident that the peak pulse acceleration is affected by surroundings; on the basis of the above equation, it is necessary to reduce the influence on dimensionless peak pulse acceleration by increasing the stiffness coefficient () of the tilted support springs or length of springs as much as possible. But this method makes increase and frequency ratio of system decrease. Therefore, it is indispensable to consider the choice of relevant parameters in the process of designing tilted support spring system so that we can obtain perfect shock resistance characteristics of tilted support spring.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.