# Dynamic Response of Asymmetric and Nonlinear Packaging System under Random Excitation

Authors: Song-Ping Yang; Zhi-Wei Wang
Journal: Shock and Vibration
 (2020)

---

## 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.


---

## Body

## 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 Section2. 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 Figure1.Figure 1
The proposed vibration model for transport package.In Figure1, m represents the mass of the product, δ represents the displacement of the product, andfδ and c represent the restoring force and the damping of packaging cushion materials, respectively. fδ follows relation of three types shown in Figure 2, showing the distinctions of ATNPS, AHTNPS, and ATHNPS.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.
(a)(b)(c)Hence, the differential equation of motion for transport package can be written as(1)mδ..+cδ˙+fδ=wt,where wt represents the Gaussian white noise excitation with zero mean value. Letting α=c/m，equation (1) can be written as(2)δ..+αδ˙+fδm=wtm,where wt/m=2πK and K represents the power spectral density of the excitation wt. pδ,δ˙ can be obtained through the Fokker–Planck equation [19](3)pδ,δ˙=Cexp−ασ02δ˙22+∫0δfξdξ,where σ02=πK and the normalizing constant C=∫−∞+∞∫−∞+∞pδ,δ˙dδdδ˙−1.


## 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 intensity2πK 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:(4)f1δ=2k1d1πtanπδ2d1,δ≥0,2k2d2πtanπδ2d2,δ<0,and letting k1/m=ω12 and k2/m=ω22 and substituting equation (4) into (3) result in(5)p1δ,δ˙=Cexp−ασ02δ˙22−4ω12π2d12⋅lncosπδ2d1,0≤δ<d1,Cexp−ασ02δ˙22−4ω22π2d22⋅lncosπδ2d2,−d2<δ<0.Assumingσ02=0.1,α=0.5,ω1=1,d1=2,ω2=2,andd2=4, the joint PDF of the displacement and velocity p1δ,δ˙ 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.Figure 4
Comparison of joint PDF of the displacement and velocity of ATNPO. (a) Theoretical solution. (b) Numerical solution.
(a)(b)The displacement PDF of ATNPS can also be given through the Fokker–Planck equation.(6)p1δ=C1expασ024ω12π2d12⋅lncosπδ2d1,0≤δ<d1,C1expασ024ω22π2d22⋅lncosπδ2d2,−d2<δ<0,where the normalizing constant C1=∫−∞+∞p1δdδ−1. 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.Lettingα=0.5,ω1=1,d1=2,ω2=2,andd2=4, the displacement PDFs of ATNPS can be plotted for various values of σ02 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 p1δ 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 ofp1δ from Figure 5, leading to further verify the theoretical model. It can be seen that the displacements δ corresponding to the displacement PDF p1δ of the product move higher and narrower with excitation level decreases. Meanwhile, the restoring force fδ of packaging cushion material gradually approaches the initial linear, and the response displacement is smaller so that the displacement PDF p1δ approaches Gaussian distribution with the external excitation grade σ02 decreases.Figure6 presents the displacement PDFs of ATNPS except the difference of the damping correlation coefficient α. As the cushion capacity of the system increases with the α 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 p1δ is closer to Gaussian distribution. It is interesting to find that as α tends to 0 (the system damping tends to 0), p1δ approaches a uniform distribution.Figure 6
Displacement PDFs of ATNPS for different damping coefficients.Lettingσ02=0.1,α=0.5,ω1=1,andω2=1, the displacement PDFs of ATNPS can be plotted for various values of the strains d1andd2 of the packaging cushion materials in Figure 7. Letting d1 equal to d2, with the d1andd2 increase simultaneously, the displacement PDF p1δ approaches Gaussian distributions. Otherwise, the displacement PDF p1δ presents distinct non-Gaussian character. Displacement responses are limited to less than d1andd2 because the nonlinear restoring force fδ of the system follows asymmetric tangent type with the boundary strain of d1andd2.Figure 7
Displacement PDFs of ATNPS for various values of packaging material strains.Figure8 shows the displacement PDFs of ATNPS for various ω1 and ω2 which represent the initial angular frequency of the packaging cushion material. p1δ nearly follows Gaussian distribution when ω1 = ω2 because the asymmetric tangent nonlinear packaging system begins same deformation on the both sides. We also discover that p1δ is symmetrical when ω1 exchange ω2.Figure 8
Displacement PDFs of ATNPS for various values of initial angular frequencies.Lettingn1=4αω12d12/π2σ02 and n2=4αω22d22/π2σ02, equation (6) becomes(7)p1δ=C1cosπδ2d1n1,0≤δ<d1,n1≥0,C1cosπδ2d2n2,−d2<δ<0,n2≥0.Whend1=d2=2, the amplitude δ is plotted versus displacement PDFs of ATNPS at various values of n1andn2 in Figure 9. Because n1andn2 are directly proportional to cushion performance parameters ω12, d1, ω22, and d2 of packaging materials, they are inversely proportional to the measure of the excitation level σ02. Hence, n1andn2 maybe called “inverse excitation cushion factors” (IECFs). We find that as n1andn2 tend to 0 (σ02 tends to infinity), p1δ approaches a uniform distribution. When n1andn2<9, p1δ presents non-Gaussian characters. It can be seen that as n1=n2≥9, p1δ constantly approaches Gaussian distribution; as n1andn2 tend to infinity (σ02tends to 0), p1δ can be regarded as following Gaussian distribution.Figure 9
Displacement PDFs of ATNPS for various values of IECFs.p1δ follows “piecewise cosine exponential distribution,” so the normalizing constant C1 can be obtained from(8)C1=∫−∞+∞∫−∞+∞pδ,δ˙dδdδ˙−1=∫−d20cosπδ2d2n2+∫0d1cosπδ2d1n1−1,letting y=cosπδ/2d and ∫0π/2cosyn=π/2Γn+1/2/Γn/2+1 from the table of integrals [32], so(9)C1=πd1Γn1+1/2/Γn1/2+1+d2Γn2+1/2/Γn2/2+1,n1≥0,n2≥0.The mean-square response of the massm or the variance of x is determined by σδ2=∫−∞∞δ2pδdδ.


### 4.2. Response PDFs of AHTNPS

Due to some packaging cushion materials follow asymmetric hyperbolic tangent constitutive relation, the restoring forcefδ of AHTNPS can be expressed as(10)f2δ=F1tanhk1′δF1,δ≥0,F2tanhk2′δF2,δ<0,and assuming F1/m=ω12′ and F2/m=ω22′， and then substituting equation (10) into (3), pδ,δ˙ can be obtained as(11)p2δ,δ˙=C′exp−ασ02δ˙22+F1ω1′2k1′⋅lncoshk1′δF1,δ≥0,C′exp−ασ02δ˙22+F2ω2′2k2′⋅lncoshk2′δF2,δ<0.Whenσ02=0.1,α=0.5,ω1=1,ω2=2,k1′/F1=0.01,andk2′/F2=0.02, the displacement and velocity united PDF p2δ,δ˙ 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.Figure 10
Comparison of displacement and velocity united PDF of HATNPO. (a) Theoretical solution. (b) Numerical solution.
(a)(b)The displacement PDF of AHTNPS can be obtained(12)p2δ=C1′exp−ασ02F1ω1′2k1′⋅lncoshk1′δF1,δ≥0,C1′exp−ασ02F2ω2′2k2′⋅lncoshk2′δF2,δ<0,where the normalizing constant C1′=∫−∞+∞p2δdδ−1 can be calculated, as well; when α=0.5,ω1′=1,ω2′=2,k1′/F1=0.01,andk2′/F2=0.02, the displacement PDFs of AHTNPS can be plotted for various values of σ02 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 δ move lower and wider as the excitation intension increases.Figure 11
Displacement PDFs of AHTNPS for various values ofσ02. (a) Comparison of the numerical and theoretical solution. (b) The result of Gaussian fitting.Assumingσ02=0.1,ω1′=1,ω2′=2,k1′/F1=0.01,andk2′/F2=0.02, Figure 12 depicts the displacement PDFs of AHTNPS with different α. As the cushion capacity of the system decreases with decrease in α 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 p2δ presents distinct non-Gaussian characters. It is obvious that as α tends to 0 (the system damping tends to 0), p2δ approaches a uniform distribution.Figure 12
Displacement PDFs of AHTNPS for various damping coefficients.Lettingn1′=−αF1ω12′/k1′σ02 and n2′=−αF2ω22′/k2′σ02, equation (12) becomes(13)p2δ=C1′coshk1′δF1n1′,δ≥0,n1′≤0,C1′coshk2′δF2n2′,δ<0,n2′≤0.Whenk1′/F1=k2′/F2=0.01, the amplitudes are plotted versus displacement PDFs of AHTNPS for various values of n1′andn2′ in Figure 13. It is interesting to find that as IECFs n1′andn2′ tend to 0 (σ02 tends to infinity), p2δ approaches a uniform distribution. As n1′andn2′<−3, p2δ presents non-Gaussian characters. It also clearly shows that as n1′=n2′≥−3, p2δ is gradually closer to Gaussian distribution, and as n1′andn2′ tend to infinity (σ02 tends to 0), p2δ can be considered as Gaussian distribution.Figure 13
Displacement PDFs of AHTNPS for various values ofn1′andn2′.


### 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 as(14)f3δ=2k3d3πtanπδ2d3,δ≥0,F4tanhk4δF4,δ<0,if k3/m=ω32 and F4/m=ω42, then substituting equation (14) into (3), pδ,δ˙ can be obtained(15)p3δ,δ˙=C〞exp−ασ02δ˙22−4ω32π2d32⋅ln cosπδ2d3,0≤δ<d3,C〞exp−ασ02δ˙22+F4ω42k4⋅ln coshk4δF4,δ<0.Whenσ02=0.1,α=0.5,ω3=1,d3=2,ω4=1,andk4/F4=0.1, the displacement and velocity united PDF p3δ,δ˙ of ATHNPS is shown in Figure 14.Figure 14
Comparison of displacement and velocity united PDF of ATHNPO. (a) Theoretical solution. (b) Numerical solution.
(a)(b)The numerical model is carried out, leading to the verification of the theory. The value ofp3δ,δ˙ with different δ and δ˙ is available because p3δ,δ˙ is conducted by cos and cosh functions, and p3δ,δ˙ is small percentage located on the side of δ<0 by the δ-axis and less distributed on the side of δ˙<0 by the δ˙-axis.The displacement PDF of ATHNPS can be expressed as(16)p3δ=C1′expασ024ω32d32π2⋅ln cosπδ2d3,0≤δ<d3,C1′exp−ασ02F4ω42k4⋅ln coshk4δF4,δ<0,where the normalizing constant C1′=∫−∞+∞p3δdδ−1. As α=0.5,ω3=1,d3=2,ω4=2,andk4/F4=0.1, the displacement PDFs of ATHNPS can be plotted for various values of σ02 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 ofσ02. (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.σ02 is lower as the external excitation intensity decreases; we can see that the displacements δ corresponding to p3δ of the product moves higher and narrower with excitation level decreases. Meanwhile, the restoring force fδ of packaging cushion material gradually approaches the initial linear, then the response displacement δ is smaller, and the displacement PDF p3δ is closer to Gaussian distribution with the external excitation intensity decreases.Assumingσ02=0.05,ω3=1,d3=2,ω4=2,andk4/F4=0.1, the displacement PDFs of ATHNPS can be plotted for various values of damping coefficient α in Figure 16. Damping correlation coefficient α 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 α increases, the nonlinear factors can also be compensated in the system so that p3δ is closer to Gaussian distribution. It is interesting to find that as α tends to 0 (the system damping tends to 0), p3δ approaches a uniform distribution.Figure 16
Displacement PDFs of ATHNPS for various values of damping coefficient.Lettingn3=4αω32d32/π2σ02 and n4=−αF4ω4′2/k4′σ02, equation (16) becomes(17)p3δ=C1′cosπδ2d3n3,d3≥δ≥0,n3≥0,C1′coshk4′δF4n4,δ<0,n4≤0.Whend3=2andk4/F4=0.2, the amplitude δ is plotted versus displacement PDFs of ATHNPS for various values of n3andn4 in Figure 17. It is interesting to find that as IECFs n3andn4 tending to 0 (σ02 tends to infinity), p3δ is approaching a uniform distribution. It shows that as n3/n4 following the proportion, p3δ is approaching the Gaussian distribution, else p3δ presents non-Gaussian characters. As n1′andn2′ tend to infinity (σ02 tends to 0), p3δ can be regarded as Gaussian distribution.Figure 17
Displacement PDFs of ATHNPS for various values ofn3andn4.


## 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:(4)f1δ=2k1d1πtanπδ2d1,δ≥0,2k2d2πtanπδ2d2,δ<0,and letting k1/m=ω12 and k2/m=ω22 and substituting equation (4) into (3) result in(5)p1δ,δ˙=Cexp−ασ02δ˙22−4ω12π2d12⋅lncosπδ2d1,0≤δ<d1,Cexp−ασ02δ˙22−4ω22π2d22⋅lncosπδ2d2,−d2<δ<0.Assumingσ02=0.1,α=0.5,ω1=1,d1=2,ω2=2,andd2=4, the joint PDF of the displacement and velocity p1δ,δ˙ 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.Figure 4
Comparison of joint PDF of the displacement and velocity of ATNPO. (a) Theoretical solution. (b) Numerical solution.
(a)(b)The displacement PDF of ATNPS can also be given through the Fokker–Planck equation.(6)p1δ=C1expασ024ω12π2d12⋅lncosπδ2d1,0≤δ<d1,C1expασ024ω22π2d22⋅lncosπδ2d2,−d2<δ<0,where the normalizing constant C1=∫−∞+∞p1δdδ−1. 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.Lettingα=0.5,ω1=1,d1=2,ω2=2,andd2=4, the displacement PDFs of ATNPS can be plotted for various values of σ02 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 p1δ 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 ofp1δ from Figure 5, leading to further verify the theoretical model. It can be seen that the displacements δ corresponding to the displacement PDF p1δ of the product move higher and narrower with excitation level decreases. Meanwhile, the restoring force fδ of packaging cushion material gradually approaches the initial linear, and the response displacement is smaller so that the displacement PDF p1δ approaches Gaussian distribution with the external excitation grade σ02 decreases.Figure6 presents the displacement PDFs of ATNPS except the difference of the damping correlation coefficient α. As the cushion capacity of the system increases with the α 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 p1δ is closer to Gaussian distribution. It is interesting to find that as α tends to 0 (the system damping tends to 0), p1δ approaches a uniform distribution.Figure 6
Displacement PDFs of ATNPS for different damping coefficients.Lettingσ02=0.1,α=0.5,ω1=1,andω2=1, the displacement PDFs of ATNPS can be plotted for various values of the strains d1andd2 of the packaging cushion materials in Figure 7. Letting d1 equal to d2, with the d1andd2 increase simultaneously, the displacement PDF p1δ approaches Gaussian distributions. Otherwise, the displacement PDF p1δ presents distinct non-Gaussian character. Displacement responses are limited to less than d1andd2 because the nonlinear restoring force fδ of the system follows asymmetric tangent type with the boundary strain of d1andd2.Figure 7
Displacement PDFs of ATNPS for various values of packaging material strains.Figure8 shows the displacement PDFs of ATNPS for various ω1 and ω2 which represent the initial angular frequency of the packaging cushion material. p1δ nearly follows Gaussian distribution when ω1 = ω2 because the asymmetric tangent nonlinear packaging system begins same deformation on the both sides. We also discover that p1δ is symmetrical when ω1 exchange ω2.Figure 8
Displacement PDFs of ATNPS for various values of initial angular frequencies.Lettingn1=4αω12d12/π2σ02 and n2=4αω22d22/π2σ02, equation (6) becomes(7)p1δ=C1cosπδ2d1n1,0≤δ<d1,n1≥0,C1cosπδ2d2n2,−d2<δ<0,n2≥0.Whend1=d2=2, the amplitude δ is plotted versus displacement PDFs of ATNPS at various values of n1andn2 in Figure 9. Because n1andn2 are directly proportional to cushion performance parameters ω12, d1, ω22, and d2 of packaging materials, they are inversely proportional to the measure of the excitation level σ02. Hence, n1andn2 maybe called “inverse excitation cushion factors” (IECFs). We find that as n1andn2 tend to 0 (σ02 tends to infinity), p1δ approaches a uniform distribution. When n1andn2<9, p1δ presents non-Gaussian characters. It can be seen that as n1=n2≥9, p1δ constantly approaches Gaussian distribution; as n1andn2 tend to infinity (σ02tends to 0), p1δ can be regarded as following Gaussian distribution.Figure 9
Displacement PDFs of ATNPS for various values of IECFs.p1δ follows “piecewise cosine exponential distribution,” so the normalizing constant C1 can be obtained from(8)C1=∫−∞+∞∫−∞+∞pδ,δ˙dδdδ˙−1=∫−d20cosπδ2d2n2+∫0d1cosπδ2d1n1−1,letting y=cosπδ/2d and ∫0π/2cosyn=π/2Γn+1/2/Γn/2+1 from the table of integrals [32], so(9)C1=πd1Γn1+1/2/Γn1/2+1+d2Γn2+1/2/Γn2/2+1,n1≥0,n2≥0.The mean-square response of the massm or the variance of x is determined by σδ2=∫−∞∞δ2pδdδ.


## 4.2. Response PDFs of AHTNPS

Due to some packaging cushion materials follow asymmetric hyperbolic tangent constitutive relation, the restoring forcefδ of AHTNPS can be expressed as(10)f2δ=F1tanhk1′δF1,δ≥0,F2tanhk2′δF2,δ<0,and assuming F1/m=ω12′ and F2/m=ω22′， and then substituting equation (10) into (3), pδ,δ˙ can be obtained as(11)p2δ,δ˙=C′exp−ασ02δ˙22+F1ω1′2k1′⋅lncoshk1′δF1,δ≥0,C′exp−ασ02δ˙22+F2ω2′2k2′⋅lncoshk2′δF2,δ<0.Whenσ02=0.1,α=0.5,ω1=1,ω2=2,k1′/F1=0.01,andk2′/F2=0.02, the displacement and velocity united PDF p2δ,δ˙ 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.Figure 10
Comparison of displacement and velocity united PDF of HATNPO. (a) Theoretical solution. (b) Numerical solution.
(a)(b)The displacement PDF of AHTNPS can be obtained(12)p2δ=C1′exp−ασ02F1ω1′2k1′⋅lncoshk1′δF1,δ≥0,C1′exp−ασ02F2ω2′2k2′⋅lncoshk2′δF2,δ<0,where the normalizing constant C1′=∫−∞+∞p2δdδ−1 can be calculated, as well; when α=0.5,ω1′=1,ω2′=2,k1′/F1=0.01,andk2′/F2=0.02, the displacement PDFs of AHTNPS can be plotted for various values of σ02 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 δ move lower and wider as the excitation intension increases.Figure 11
Displacement PDFs of AHTNPS for various values ofσ02. (a) Comparison of the numerical and theoretical solution. (b) The result of Gaussian fitting.Assumingσ02=0.1,ω1′=1,ω2′=2,k1′/F1=0.01,andk2′/F2=0.02, Figure 12 depicts the displacement PDFs of AHTNPS with different α. As the cushion capacity of the system decreases with decrease in α 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 p2δ presents distinct non-Gaussian characters. It is obvious that as α tends to 0 (the system damping tends to 0), p2δ approaches a uniform distribution.Figure 12
Displacement PDFs of AHTNPS for various damping coefficients.Lettingn1′=−αF1ω12′/k1′σ02 and n2′=−αF2ω22′/k2′σ02, equation (12) becomes(13)p2δ=C1′coshk1′δF1n1′,δ≥0,n1′≤0,C1′coshk2′δF2n2′,δ<0,n2′≤0.Whenk1′/F1=k2′/F2=0.01, the amplitudes are plotted versus displacement PDFs of AHTNPS for various values of n1′andn2′ in Figure 13. It is interesting to find that as IECFs n1′andn2′ tend to 0 (σ02 tends to infinity), p2δ approaches a uniform distribution. As n1′andn2′<−3, p2δ presents non-Gaussian characters. It also clearly shows that as n1′=n2′≥−3, p2δ is gradually closer to Gaussian distribution, and as n1′andn2′ tend to infinity (σ02 tends to 0), p2δ can be considered as Gaussian distribution.Figure 13
Displacement PDFs of AHTNPS for various values ofn1′andn2′.


## 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 as(14)f3δ=2k3d3πtanπδ2d3,δ≥0,F4tanhk4δF4,δ<0,if k3/m=ω32 and F4/m=ω42, then substituting equation (14) into (3), pδ,δ˙ can be obtained(15)p3δ,δ˙=C〞exp−ασ02δ˙22−4ω32π2d32⋅ln cosπδ2d3,0≤δ<d3,C〞exp−ασ02δ˙22+F4ω42k4⋅ln coshk4δF4,δ<0.Whenσ02=0.1,α=0.5,ω3=1,d3=2,ω4=1,andk4/F4=0.1, the displacement and velocity united PDF p3δ,δ˙ of ATHNPS is shown in Figure 14.Figure 14
Comparison of displacement and velocity united PDF of ATHNPO. (a) Theoretical solution. (b) Numerical solution.
(a)(b)The numerical model is carried out, leading to the verification of the theory. The value ofp3δ,δ˙ with different δ and δ˙ is available because p3δ,δ˙ is conducted by cos and cosh functions, and p3δ,δ˙ is small percentage located on the side of δ<0 by the δ-axis and less distributed on the side of δ˙<0 by the δ˙-axis.The displacement PDF of ATHNPS can be expressed as(16)p3δ=C1′expασ024ω32d32π2⋅ln cosπδ2d3,0≤δ<d3,C1′exp−ασ02F4ω42k4⋅ln coshk4δF4,δ<0,where the normalizing constant C1′=∫−∞+∞p3δdδ−1. As α=0.5,ω3=1,d3=2,ω4=2,andk4/F4=0.1, the displacement PDFs of ATHNPS can be plotted for various values of σ02 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 ofσ02. (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.σ02 is lower as the external excitation intensity decreases; we can see that the displacements δ corresponding to p3δ of the product moves higher and narrower with excitation level decreases. Meanwhile, the restoring force fδ of packaging cushion material gradually approaches the initial linear, then the response displacement δ is smaller, and the displacement PDF p3δ is closer to Gaussian distribution with the external excitation intensity decreases.Assumingσ02=0.05,ω3=1,d3=2,ω4=2,andk4/F4=0.1, the displacement PDFs of ATHNPS can be plotted for various values of damping coefficient α in Figure 16. Damping correlation coefficient α 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 α increases, the nonlinear factors can also be compensated in the system so that p3δ is closer to Gaussian distribution. It is interesting to find that as α tends to 0 (the system damping tends to 0), p3δ approaches a uniform distribution.Figure 16
Displacement PDFs of ATHNPS for various values of damping coefficient.Lettingn3=4αω32d32/π2σ02 and n4=−αF4ω4′2/k4′σ02, equation (16) becomes(17)p3δ=C1′cosπδ2d3n3,d3≥δ≥0,n3≥0,C1′coshk4′δF4n4,δ<0,n4≤0.Whend3=2andk4/F4=0.2, the amplitude δ is plotted versus displacement PDFs of ATHNPS for various values of n3andn4 in Figure 17. It is interesting to find that as IECFs n3andn4 tending to 0 (σ02 tends to infinity), p3δ is approaching a uniform distribution. It shows that as n3/n4 following the proportion, p3δ is approaching the Gaussian distribution, else p3δ presents non-Gaussian characters. As n1′andn2′ tend to infinity (σ02 tends to 0), p3δ can be regarded as Gaussian distribution.Figure 17
Displacement PDFs of ATHNPS for various values ofn3andn4.


## 5. Peak PDFs of ANPSs

The times of sample function cross zero with positive slope were given by To [33]:(18)v0+=∫0+∞δ˙p0,δ˙dδ˙,and after substituting equation (18) into pδ,δ˙, 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]:(19)ppeakhdh=−1v0+dvh+dhdh,generally, v0+ only concerns positive numbers by the equation (18), considering −δ when δ<0 , and combining the equations (19) and (18), we can obtain the generalized peak PDF(20)ppeakh=−dpδ/dδδ=hpδδ=0.


### 5.1. Peak PDFs of ATNPS

Equation (6) gives(21)p10,δ˙=C12πσ0exp−δ˙22σ02,and substituting equation (21) into (18), then(22)v0+=∫0+∞δ˙p0,δ˙dδ˙=C12πσ0∫0+∞δ˙exp−δ˙22σ02dδ˙,and letting z˙1=δ˙2/2σ02, after integral transform，(23)v0+=∫0+∞δ˙p0,δ˙dδ˙=−C1σ022π∫0+∞z˙1exp−z˙1dz˙1.Since the integral ∫0+∞z˙1exp−z˙1dz˙1 is equal to unity, v0+ is given as(24)v0+=−σ02d1Γn1+1/2/Γn1/2+1+d2Γn2+1/2/Γn2/2+12π.Equation (7) yields(25)dp1δdδ=−C1n1π2d1cosπδ2d1n1−1sinπδ2d1,0≤δ<d1,−C1n2π2d2cosπδ2d2n2−1sinπδ2d2,−d2<δ<0,then(26)dp1δdδδ=h=−C1n1π2d1cosπh2d1n1−1sinπh2d1,0≤h<d1,−C1n2π2d2cosπh2d2n2−1sinπh2d2,−d2<h<0,and from equation (7),(27)p1δδ=0=C1=πΓn1+1/2/Γn1/2+1+Γn2+1/2/Γn2/2+1,n1≥0,n2≥0,and substituting equations (26) and (27) into (20),(28)p1peakh=n1π2d1tanπh2d1cosπh2d1n1,0≤h<d1,n2π2d2tanπh2d2cosπh2d2n2,−d2<h<0.Lettingd1=d2=π/2, the generalized peak PDFs of ATNPS can be displayed for various values of n1 and n2 in Figure 18. As the IECFs n1 and n2 decrease, the displacement peaks corresponding to the generalized p1peakh of the product move lower and wider. Meanwhile, the response displacement peaks of the product present a wider range.Figure 18
Generalized peak PDFs of ATNPS for various values ofn1 and n2.Then, we use Rayleigh, Weibull, and Gaussian distributions to examine the generalized peak PDFs of ATNPS forh≥0, and the result is also depicted in Figure 18. At this case，p1peakh is approaching Rayleigh and Weibull distribution as n1andn2 increase. What’s more, p1peakh is closer to Rayleigh distribution than Weibull distribution. Figure 18 reveals that as n1andn2 change, p1peakh presents non-Gaussian characters.Generalized peak PDFs of ATNPS can be plotted for differentd1andd2 in Figure 19 when n1=5 and n2=5. As d1andd2 are lower, we can see that the displacement peaks corresponding to the generalized p1peakh of the product move higher and narrower. Meanwhile, the response displacement peaks of the product present a narrower range.Figure 19
Generalized peak PDFs of ATNPS for differentd1andd2.Rayleigh, Weibull, and Gaussian distributions are introduced to examine the generalized peak PDFs of ATNPS forh≥0 for different d1andd2. It is obvious that p1peakh is approaching Weibull and Rayleigh distribution from Figure 19 as d1andd2 increase. In addition, p1peakh is closer to Rayleigh distribution than Weibull distribution. On the contrary, p1peakh is only approaching Rayleigh distribution as d1andd2 decrease. Figure 19 shows that whatever d1andd2 change, p1peakh presents non-Gaussian characters.


### 5.2. Peak PDFs of AHTNPS

Equation (12) yields(29)p20,δ˙=C1′2πσ0expF12ω1′2k1′exp−δ˙22σ02,and substituting equation (29) into (18), letting z˙2=δ˙2/2σ02, after integral transform, then(30)v0+=∫0+∞δ˙p0,δ˙dδ˙=2C1′π∫0+∞z˙2exp−z˙2dz˙2.Since the integral∫0+∞z˙2exp−z˙2dz˙2 is equal to unity, v0+ is given as v0+=2πC1′/π.From equation (13), dp2δ/dδδ=h can be obtained and p2δδ=0=C1′. Substituting above equations into equation (20) result in(31)p2peakh=−k1′n1′F1tanhk1′hF1coshk1′hF1n1′,h≥0,n1′≤0,−k2′n2′F2tanhk2′hF2coshk2′hF2n2′,h<0,n2′≤0.Lettingk1′/F1=k2′/F2=1, the generalized peak PDFs of AHTNPS can be displayed except the difference in n1′ and n2′ in Figure 20. As IECFs n1′ and n2′ decrease, it can be seen that the displacement peaks corresponding to the generalized p2peakh of the product move lower and wider. Meanwhile, the response displacement peaks of the product present a wider range.Figure 20
Generalized peak PDFs of AHTNPS except the difference inn1′ and n2′.Whenh≥0, the amplitude h is plotted versus the generalized peak PDFs of HATNPO for various n1′ and n2′ in Figure 20. It is shown that p2peakh is approaching Rayleigh and Weibull distribution in Figure 20 as n1′ and n2′ increase. p2peakh is closer to Rayleigh distribution than Weibull distribution. p2peakh approaches Weibull distribution as n1′ and n2′ decrease. Figure 20 indicates that as n1′ and n2′ change, p2peakh presents distinct non-Gaussian characters.


### 5.3. Peak PDFs of ATHNPS

Equation (16) gives(32)p30,δ˙=C1′′2πσ0exp−δ˙22σ02,substituting equation (32) into (18), after integral transform, we can also obtain v0+=−C1′σ02/2π.Based on the equations (17) and (20), we can obtain(33)p3peakh=n3π2d3tanπh2d3cosπh2d3n3,0≤h<d3,n3≥0,−k4n4F4tanhk4hF4coshk4hF4n4,h<0,n4≤0.Lettingd3=π/2, k4/F4=1, the generalized peak PDFs of ATHNPS can be displayed for various values of n3 and n4 in Figure 21. With the IECF n3 increases and n4 decreases, we can see that the displacement peak h corresponding to the generalized p3peakh of the product moves lower and wider. Meanwhile, the response displacement peak h of the product presents a wider range.Figure 21
Generalized peak PDFs of ATHNPS for various values ofn3 and n4.Whenh≥0, the amplitude h is plotted versus generalized peak PDFs of ATHNPS for various values of n3 and n4 also in Figure 21. It is shown that p3peakh is approaching Rayleigh as n3 increases and n4 decreases. Figure 21 indicates that as n3 and n4 change, p3peakh presents non-Gaussian characters.


## 5.1. Peak PDFs of ATNPS

Equation (6) gives(21)p10,δ˙=C12πσ0exp−δ˙22σ02,and substituting equation (21) into (18), then(22)v0+=∫0+∞δ˙p0,δ˙dδ˙=C12πσ0∫0+∞δ˙exp−δ˙22σ02dδ˙,and letting z˙1=δ˙2/2σ02, after integral transform，(23)v0+=∫0+∞δ˙p0,δ˙dδ˙=−C1σ022π∫0+∞z˙1exp−z˙1dz˙1.Since the integral ∫0+∞z˙1exp−z˙1dz˙1 is equal to unity, v0+ is given as(24)v0+=−σ02d1Γn1+1/2/Γn1/2+1+d2Γn2+1/2/Γn2/2+12π.Equation (7) yields(25)dp1δdδ=−C1n1π2d1cosπδ2d1n1−1sinπδ2d1,0≤δ<d1,−C1n2π2d2cosπδ2d2n2−1sinπδ2d2,−d2<δ<0,then(26)dp1δdδδ=h=−C1n1π2d1cosπh2d1n1−1sinπh2d1,0≤h<d1,−C1n2π2d2cosπh2d2n2−1sinπh2d2,−d2<h<0,and from equation (7),(27)p1δδ=0=C1=πΓn1+1/2/Γn1/2+1+Γn2+1/2/Γn2/2+1,n1≥0,n2≥0,and substituting equations (26) and (27) into (20),(28)p1peakh=n1π2d1tanπh2d1cosπh2d1n1,0≤h<d1,n2π2d2tanπh2d2cosπh2d2n2,−d2<h<0.Lettingd1=d2=π/2, the generalized peak PDFs of ATNPS can be displayed for various values of n1 and n2 in Figure 18. As the IECFs n1 and n2 decrease, the displacement peaks corresponding to the generalized p1peakh of the product move lower and wider. Meanwhile, the response displacement peaks of the product present a wider range.Figure 18
Generalized peak PDFs of ATNPS for various values ofn1 and n2.Then, we use Rayleigh, Weibull, and Gaussian distributions to examine the generalized peak PDFs of ATNPS forh≥0, and the result is also depicted in Figure 18. At this case，p1peakh is approaching Rayleigh and Weibull distribution as n1andn2 increase. What’s more, p1peakh is closer to Rayleigh distribution than Weibull distribution. Figure 18 reveals that as n1andn2 change, p1peakh presents non-Gaussian characters.Generalized peak PDFs of ATNPS can be plotted for differentd1andd2 in Figure 19 when n1=5 and n2=5. As d1andd2 are lower, we can see that the displacement peaks corresponding to the generalized p1peakh of the product move higher and narrower. Meanwhile, the response displacement peaks of the product present a narrower range.Figure 19
Generalized peak PDFs of ATNPS for differentd1andd2.Rayleigh, Weibull, and Gaussian distributions are introduced to examine the generalized peak PDFs of ATNPS forh≥0 for different d1andd2. It is obvious that p1peakh is approaching Weibull and Rayleigh distribution from Figure 19 as d1andd2 increase. In addition, p1peakh is closer to Rayleigh distribution than Weibull distribution. On the contrary, p1peakh is only approaching Rayleigh distribution as d1andd2 decrease. Figure 19 shows that whatever d1andd2 change, p1peakh presents non-Gaussian characters.


## 5.2. Peak PDFs of AHTNPS

Equation (12) yields(29)p20,δ˙=C1′2πσ0expF12ω1′2k1′exp−δ˙22σ02,and substituting equation (29) into (18), letting z˙2=δ˙2/2σ02, after integral transform, then(30)v0+=∫0+∞δ˙p0,δ˙dδ˙=2C1′π∫0+∞z˙2exp−z˙2dz˙2.Since the integral∫0+∞z˙2exp−z˙2dz˙2 is equal to unity, v0+ is given as v0+=2πC1′/π.From equation (13), dp2δ/dδδ=h can be obtained and p2δδ=0=C1′. Substituting above equations into equation (20) result in(31)p2peakh=−k1′n1′F1tanhk1′hF1coshk1′hF1n1′,h≥0,n1′≤0,−k2′n2′F2tanhk2′hF2coshk2′hF2n2′,h<0,n2′≤0.Lettingk1′/F1=k2′/F2=1, the generalized peak PDFs of AHTNPS can be displayed except the difference in n1′ and n2′ in Figure 20. As IECFs n1′ and n2′ decrease, it can be seen that the displacement peaks corresponding to the generalized p2peakh of the product move lower and wider. Meanwhile, the response displacement peaks of the product present a wider range.Figure 20
Generalized peak PDFs of AHTNPS except the difference inn1′ and n2′.Whenh≥0, the amplitude h is plotted versus the generalized peak PDFs of HATNPO for various n1′ and n2′ in Figure 20. It is shown that p2peakh is approaching Rayleigh and Weibull distribution in Figure 20 as n1′ and n2′ increase. p2peakh is closer to Rayleigh distribution than Weibull distribution. p2peakh approaches Weibull distribution as n1′ and n2′ decrease. Figure 20 indicates that as n1′ and n2′ change, p2peakh presents distinct non-Gaussian characters.


## 5.3. Peak PDFs of ATHNPS

Equation (16) gives(32)p30,δ˙=C1′′2πσ0exp−δ˙22σ02,substituting equation (32) into (18), after integral transform, we can also obtain v0+=−C1′σ02/2π.Based on the equations (17) and (20), we can obtain(33)p3peakh=n3π2d3tanπh2d3cosπh2d3n3,0≤h<d3,n3≥0,−k4n4F4tanhk4hF4coshk4hF4n4,h<0,n4≤0.Lettingd3=π/2, k4/F4=1, the generalized peak PDFs of ATHNPS can be displayed for various values of n3 and n4 in Figure 21. With the IECF n3 increases and n4 decreases, we can see that the displacement peak h corresponding to the generalized p3peakh of the product moves lower and wider. Meanwhile, the response displacement peak h of the product presents a wider range.Figure 21
Generalized peak PDFs of ATHNPS for various values ofn3 and n4.Whenh≥0, the amplitude h is plotted versus generalized peak PDFs of ATHNPS for various values of n3 and n4 also in Figure 21. It is shown that p3peakh is approaching Rayleigh as n3 increases and n4 decreases. Figure 21 indicates that as n3 and n4 change, p3peakh presents non-Gaussian characters.


## 6. Application for Fatigue Evaluation

Some rocket component packaged by the neoprene and some foam alloy composite materials of the aircraft are subjected to the random loading during the air transport so as to cause the fatigue failure.Take ATNPO as example shown in Figure22, assuming that fatigue damage accumulates linearly (Miner’s rule) [35], the average damage ED after N1 cycles was given by [36]:(34)ED=N1μ∫0∞sβpsds,where μ is a constant of any given material, β is the slope of the s−N curve by the logarithmic coordinate, and ps represents the PDF of peak stress amplitude, which can be obtained through the generalized PDF of the response peak distribution ppeakh.(a)
The system under study can be regarded as an approximate motion description at the center of a simply supported beam in Figure22(a), so the stress-displacement relationship can be assumed linear s1=μ1h. Combining equations (31) and (34) results in(35)ED=N1μnπ2μ1d∫0μ1ds1βtanπs12μ1dcosπs12μ1dnds1,forβ≈10andn=9, solving the integration, equation (35) results in(36)EDn=9=0.153×10−3μμ1d9N1,and if the ultimate strength of a material is 50 000 psi,μ1 is 25 000 psi/in, and d=1. Equation (36) becomes(37)EDn=9=0.153×10−3×250005000009N1=2.988×10−7N1.WhenED=1, the fatigue failure occurs, so the average cycle number to failure would be Na=1/2.988×10−7=3.346×106 cps.(b)
Considering the stress at the center of a pinned-pinned beam shown in Figure22(b), the stress-displacement nonlinear relationship was given [37] s2=μ2h+μ3′/2l√3h2=μ2h+μ3h2 (where l represents the length of the beam), and then h=−μ2+μ22+4μ3s21/2/2μ3 result in(38)END=N1μnπ2d∫0ds2βtanπh2dcosπh2dndh.Similarly,END and the average cycle number to failure Na can also be evaluated at some determine parameters.Figure 22
Displacement nonlinearity due to tangent type materials.
(a)(b)


## 7. Conclusions

The displacement PDFs of three types of ANPSs have been derived through FPK equations; the generalized peak PDFs are defined and examined by three common distributions. Meanwhile, a good agreement is achieved between theoretical solutions and numerical results, leading to verify the theories. The conclusions lie in the following aspects.(a)
The responses for three types of ANPSs derived through FPK equations present different forms, but they have the generalities. The displacement PDFs for three ANPSs are closer to Gaussian distribution with the external excitation grade decreases.(b)
With the damping correlation coefficientα increases, the nonlinear factors can be compensated in the system, and the displacement PDFs for three types of ANPSs are closer to Gaussian distribution. It is interesting to find that as α tends to 0 (the system damping tends to 0), the displacement PDF approaches the uniform distribution. Displacement responses of asymmetric tangent type are limited to the boundary strain of d1andd2. IECFs have a significance on effecting the displacement PDFs approaching a uniform distribution, non-Gaussian distribution, or Gaussian distribution.(c)
Most PDFs of the unilateral response peaks approach Rayleigh distributions, and present non-Gaussian characters. The application of the scheme for fatigue evaluation is carried out, so the analysis can be used as guidelines for fatigue damage evaluation of the transport package or the critical component.



---

*Source: 8840147-2020-11-06.xml*