Abstract

This paper focuses on the dynamic response of three asymmetric and nonlinear packaging systems (ANPSs) of practical transport package under random excitation. Next to presenting the displacement probability density functions (PDFs) of ANPSs derived through FPK equations, the influences of the excitation grades and the characteristic parameters (such as the damping, the nonlinear stiffness, and the strain) of packaging cushion materials on the response are discussed in detail. Then, the generalized PDFs of the response peaks are defined and examined by three common distributions. The investigation shows that “inverse excitation cushion factors” (IECFs) have a significance on effecting the displacement PDFs approaching different distributions. Most PDFs of the unilateral response peaks approach Rayleigh distributions and present non-Gaussian characters. The present methods are verified through the validation numerical solutions. Furthermore, the application of the scheme for fatigue damage evaluation of the transport package is carried out.

1. Introduction

The spring-mass-damping system was introduced to describe transport package by Mindlin [1], who pioneered packaging dynamics study. The simplest package system was considered linear damping and linear stiffness; one of the biggest benefits of this was that you can use the superposition principle [2]. However, the packaging materials always follow nonlinear constitutive relations [3]. Peleg [4], Wang et al. [5], Wang and Hu [6], and Wang and Khan [7] did beneficial exploring on the packaging system with cubic nonlinear, tangent nonlinear, and tangent nonlinear stiffness under shock and vibration excitation, but the external excitation was always considered as deterministic harmonic excitation.

Since the external excitation during the transportation has always been random [8], an increasing effort was presently being expended to study the response of a spring-mass-damping packaging system under stochastic excitation [9–12]. Rountree et al. [13] studied the fragility of missile instrument under random vibration. Thakur et al. [14] established a nonlinear theoretical model for the packaging system to obtain displacement response under random vibration. Song [15] gave the definition of vibration fragility for the packaging system under random excitation and illustrated the difference with the impact fragility. Liaudet et al. [16] studied a random vibration model with fractional order stiffness and derived the probability density functions (PDFs) of displacement. Gan et al. [17] used the random average method to establish FPK equation for the packaging system with cubic nonlinear stiffness under stochastic excitation. Wang et al. [18] established a vibration model of the mass-spring-damping system by considering two stiffnesses of the system and the constraint condition under acceleration random excitation in order to evaluate fatigue failure of the packaging paperboard box.

The scholars always consider nonlinear stiffness of the packaging system following the unidirectional function relation, such as cubic unidirectional function relation. However, due to the otherness of the packaging cushioning material or the asymmetries caused by packaging process, the stiffness of the packaging system presents asymmetric nonlinear characters.

Chen and Chen [19] discussed a single-degree-of-freedom (SDOF) damped system with the piecewise linear stiffness under Gaussian white noise. Zhuang et al. [20] explored the SDOF isolation system with the antisymmetrical and nonlinear stiffness, but only the approximate solution of the displacement mean square root was given by resolving the restoring force into odd function and even function. Fu et al. [21] obtained the response of a SDOF asymmetric piecewise linear system. Urbanik [22] studied the nonlinear vibration of the corrugated carton package through the piecewise nonlinear method. Yang et al. [23] used computer simulation to analyze the performance on the piecewise linear transit packaging system under oscillatory surroundings. Fang et al. [24, 25] introduced the SDOF nonlinear packaging oscillator with piecewise cubic nonlinearity, only obtained the joint displacement and velocity probability distribution. The systems with the piecewise asymmetric tangent or hyperbolic tangent stiffness under random excitation have seldom been investigated.

However, many packaging cushion materials such as the neoprene, preloading expandable polystyrene (EPS) foam, expandable polyethylene (EPE) foam, etc., follow tangent or hyperbolic tangent types of the force-displacement relation [1, 3, 26], and the cushion materials are around the product after packaging; the different materials sometimes are adopted as upper and lower layers cushioning material, so the piecewise asymmetric and nonlinear packaging systems (ANPSs) represent more accurate models of actual transport package.

Hence, this paper focuses on the dynamic response of asymmetric tangent nonlinear packaging system (ATNPS), asymmetric hyperbolic tangent nonlinear packaging system (AHNPS), and asymmetric tangent-hyperbolic tangent nonlinear packaging system (ATHNPS) under random excitation based on the FPK method which has been prospering as a method to obtain the system steady state solution [27, 28]. The study can be used as guidelines for first pass or fatigue damage evaluation of the transport package or the critical component.

The structure of this article is as follows. The transport package is modeled as ANPS in Section 2. In Section 3, the technology roadmap of numerical simulation process is illustrated. In Section 4, the response PDFs for three types of ANPSs under random excitation are derived through FPK equations, and the validation numerical models are carried out to verify the present method. Furthermore, the generalized peak PDFs are defined and examined by three common distributions. The influences of the system parameters, such as the damping correlation coefficient, the nonlinear stiffness, and the strain and the incentive intensity on the non-Gaussian characters of above responses are analyzed in detail in Section 5. Section 6 gives the application of the scheme for fatigue evaluation. Finally, the main conclusions are summarized.

2. Vibration Model for Transport Package

During the logistics process, the complete transport package is subjected to the stochastic excitation in the horizontal or vertical direction while the gravity effect is always ignoring. The packaging cushion materials follow tangent or hyperbolic tangent types of the stress-deformation relations. Hence, the modeling for transport package regarding as SDOF ANPS is shown in Figure 1.

Figure 1 

The proposed vibration model for transport package.

In Figure 1, represents the mass of the product, represents the displacement of the product, and and represent the restoring force and the damping of packaging cushion materials, respectively. follows relation of three types shown in Figure 2, showing the distinctions of ATNPS, AHTNPS, and ATHNPS.

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Figure 2 

Three types of restoring forces of packaging cushion materials. (a) Antisymmetry tangent type. (b) Antisymmetry hyperbolic tangent type. (c) Asymmetric tangent-hyperbolic tangent type.

Hence, the differential equation of motion for transport package can be written aswhere represents the Gaussian white noise excitation with zero mean value. Letting ，equation (1) can be written aswhere and represents the power spectral density of the excitation . can be obtained through the Fokker–Planck equation [19]where and the normalizing constant .

3. Numerical Verification

In order to verify the theoretical solution, the numerical simulation is conducted based on the following steps. Firstly, the Gaussian white noise excitation is conducted by digital simulation, and the excitation intensity is corresponding with the theoretical motivation. Secondly, through the four-stage Runge–Kutta method, the response of the packaging system characterized by digital simulation can also be obtained. The piecewise programs are considering in Runge–Kutta calculation steps. Thirdly, the Monte Carlo approach, which was first proposed by Metropolis et al. [29] and generalized by Hastings [30], is used to solve complex integrals [31]. The technology roadmap of numerical simulation process is shown in Figure 3, and afterwards the comparisons between the theoretical solution and numerical result can be given.

Figure 3 

Technology roadmap of numerical verification.

4. Response PDFs of ANPSs

Three packaging systems are concerned with the different piecewise forms of stiffnesses. They can be expressed as three forms of restoring forces. PDFs of the displacement and velocity for three systems are investigated.

4.1. Response PDFs of ATNPS

Some packaging cushion materials follow tangent force-displacement relation, so the nonlinear restoring force of ATNPS considering the asymmetric tangent stiffness is as follows:and letting and and substituting equation (4) into (3) result in

Assuming , the joint PDF of the displacement and velocity of ATNPS is obtained to compare with numerical result shown in Figure 4. Both have the nearly same shape and peak, leading to the verification of the theory.

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Figure 4 

Comparison of joint PDF of the displacement and velocity of ATNPO. (a) Theoretical solution. (b) Numerical solution.

The displacement PDF of ATNPS can also be given through the Fokker–Planck equation.where the normalizing constant . In order for examining the influences of the external excitation and the system parameters, such as the damping correlation coefficient, the nonlinear stiffness, and the strain on the response, the figures are plotted of different characteristic parameters.

Letting , the displacement PDFs of ATNPS can be plotted for various values of in Figure 5. That means the displacement response of the system only concerns with the external excitation grade. After comparing with the numerical solutions shown on the left, we use Gaussian distribution to examine shown on the right of Figure 5.

Figure 5 

Displacement PDFs of ATNPS for various values of excitation grades. (a) Comparison of the numerical and theoretical solution. (b) The result of Gaussian fitting.

We can see a good agreement is achieved between theoretical and numerical solution of from Figure 5, leading to further verify the theoretical model. It can be seen that the displacements corresponding to the displacement PDF of the product move higher and narrower with excitation level decreases. Meanwhile, the restoring force of packaging cushion material gradually approaches the initial linear, and the response displacement is smaller so that the displacement PDF approaches Gaussian distribution with the external excitation grade decreases.

Figure 6 presents the displacement PDFs of ATNPS except the difference of the damping correlation coefficient . As the cushion capacity of the system increases with the increases, then the displacement responses of the product are limited to a smaller range, and the nonlinear factors can be compensated in the system so that the displacement PDF is closer to Gaussian distribution. It is interesting to find that as tends to 0 (the system damping tends to 0), approaches a uniform distribution.

Figure 6 

Displacement PDFs of ATNPS for different damping coefficients.

Letting , the displacement PDFs of ATNPS can be plotted for various values of the strains of the packaging cushion materials in Figure 7. Letting equal to , with the increase simultaneously, the displacement PDF approaches Gaussian distributions. Otherwise, the displacement PDF presents distinct non-Gaussian character. Displacement responses are limited to less than because the nonlinear restoring force of the system follows asymmetric tangent type with the boundary strain of .

Figure 7 

Displacement PDFs of ATNPS for various values of packaging material strains.

Figure 8 shows the displacement PDFs of ATNPS for various and which represent the initial angular frequency of the packaging cushion material. nearly follows Gaussian distribution when  =  because the asymmetric tangent nonlinear packaging system begins same deformation on the both sides. We also discover that is symmetrical when exchange .

Figure 8 

Displacement PDFs of ATNPS for various values of initial angular frequencies.

Letting and , equation (6) becomes

When , the amplitude is plotted versus displacement PDFs of ATNPS at various values of in Figure 9. Because are directly proportional to cushion performance parameters , , , and of packaging materials, they are inversely proportional to the measure of the excitation level . Hence, maybe called “inverse excitation cushion factors” (IECFs). We find that as tend to 0 ( tends to infinity), approaches a uniform distribution. When , presents non-Gaussian characters. It can be seen that as , constantly approaches Gaussian distribution; as tend to infinity (tends to 0), can be regarded as following Gaussian distribution.

Figure 9 

Displacement PDFs of ATNPS for various values of IECFs.

follows “piecewise cosine exponential distribution,” so the normalizing constant can be obtained fromletting and from the table of integrals [32], so

The mean-square response of the mass or the variance of is determined by .

4.2. Response PDFs of AHTNPS

Due to some packaging cushion materials follow asymmetric hyperbolic tangent constitutive relation, the restoring force of AHTNPS can be expressed asand assuming and ， and then substituting equation (10) into (3), can be obtained as

When , the displacement and velocity united PDF of AHTNPS is shown in Figure 10. The numerical model is also carried out and compared with the theoretical solution, leading to the verification of the theory.

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Figure 10 

Comparison of displacement and velocity united PDF of HATNPO. (a) Theoretical solution. (b) Numerical solution.

The displacement PDF of AHTNPS can be obtainedwhere the normalizing constant can be calculated, as well; when , the displacement PDFs of AHTNPS can be plotted for various values of representing the excitation grade in Figure 11. In comparison with the numerical solutions of the displacement PDFs of AHTNPS, leading to further verify the theoretical model, the displacements move lower and wider as the excitation intension increases.

Figure 11 

Displacement PDFs of AHTNPS for various values of . (a) Comparison of the numerical and theoretical solution. (b) The result of Gaussian fitting.

Assuming , Figure 12 depicts the displacement PDFs of AHTNPS with different . As the cushion capacity of the system decreases with decrease in the displacement response varies over a larger range, and the system damping is considered as linear, but the nonlinear stiffness factors were expended in the system, and presents distinct non-Gaussian characters. It is obvious that as tends to 0 (the system damping tends to 0), approaches a uniform distribution.

Figure 12 

Displacement PDFs of AHTNPS for various damping coefficients.

Letting and , equation (12) becomes

When , the amplitudes are plotted versus displacement PDFs of AHTNPS for various values of in Figure 13. It is interesting to find that as IECFs tend to 0 ( tends to infinity), approaches a uniform distribution. As , presents non-Gaussian characters. It also clearly shows that as , is gradually closer to Gaussian distribution, and as tend to infinity ( tends to 0), can be considered as Gaussian distribution.

Figure 13 

Displacement PDFs of AHTNPS for various values of

4.3. Response PDFs of ATHNPS

Using the tangent and hyperbolic tangent type of packaging cushion materials together, the restoring force of ATHNPS can be expressed asif and , then substituting equation (14) into (3), can be obtained

When , the displacement and velocity united PDF of ATHNPS is shown in Figure 14.

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Figure 14 

Comparison of displacement and velocity united PDF of ATHNPO. (a) Theoretical solution. (b) Numerical solution.

The numerical model is carried out, leading to the verification of the theory. The value of with different and is available because is conducted by and functions, and is small percentage located on the side of by the -axis and less distributed on the side of by the -axis.

The displacement PDF of ATHNPS can be expressed aswhere the normalizing constant . As , the displacement PDFs of ATHNPS can be plotted for various values of in Figure 15. It shows the regularity between the displacement response of ATHNPS and the different incentive intensity.

Figure 15 

Displacement PDFs of ATHNPS for various values of . (a) Comparison of the numerical and theoretical solution. (b) The result of Gaussian fitting.

A good agreement is achieved between theoretical and numerical solutions of displacement PDFs of ATHNPS, leading to further verify the theoretical model. is lower as the external excitation intensity decreases; we can see that the displacements corresponding to of the product moves higher and narrower with excitation level decreases. Meanwhile, the restoring force of packaging cushion material gradually approaches the initial linear, then the response displacement is smaller, and the displacement PDF is closer to Gaussian distribution with the external excitation intensity decreases.

Assuming , the displacement PDFs of ATHNPS can be plotted for various values of damping coefficient in Figure 16. Damping correlation coefficient is higher as the cushion capacity of the system increases, and the displacement response is limited to a smaller range. The system damping is considered as linear; with the increases, the nonlinear factors can also be compensated in the system so that is closer to Gaussian distribution. It is interesting to find that as tends to 0 (the system damping tends to 0), approaches a uniform distribution.

Figure 16 

Displacement PDFs of ATHNPS for various values of damping coefficient.

Letting and , equation (16) becomes

When , the amplitude is plotted versus displacement PDFs of ATHNPS for various values of in Figure 17. It is interesting to find that as IECFs tending to 0 ( tends to infinity), is approaching a uniform distribution. It shows that as following the proportion, is approaching the Gaussian distribution, else presents non-Gaussian characters. As tend to infinity ( tends to 0), can be regarded as Gaussian distribution.

Figure 17 

Displacement PDFs of ATHNPS for various values of .

5. Peak PDFs of ANPSs

The times of sample function cross zero with positive slope were given by To [33]:and after substituting equation (18) into , zero crossing with positive slope of three packaging system can be obtained.

The displacement peak PDF would like to be known to determine the probability of fatigue failure; it will estimate the relative frequency of occurrence for any peak amplitude. A method to obtain the distribution was suggested by Crandall [34]:generally, only concerns positive numbers by the equation (18), considering when , and combining the equations (19) and (18), we can obtain the generalized peak PDF

5.1. Peak PDFs of ATNPS

Equation (6) givesand substituting equation (21) into (18), thenand letting after integral transform，Since the integral is equal to unity, is given asEquation (7) yieldsthenand from equation (7),and substituting equations (26) and (27) into (20),