He Chengtian’s Inequalities for a Coupled Tangent Nonlinear System Arisen in Packaging System

Abstract

He Chengtian’s inequalities from ancient Chinese algorithm are applied to strong tangent nonlinear packaging system. The approximate solution is obtained and compared with the solution yielded by computer simulation, showing a great high accuracy of this method. The suggested approach provides a novel method to solve some essential problems in packaging engineering.

1. Introduction

In order to avoid some restrictions of perturbation method [1], some other methods are developed, such as the homotopy perturbation method (HPM) [2, 3], the variational iteration method (VIM) [4–6], the homotopy analysis method (HAM) [7], and He Chengtian’s inequalities which cannot be found in the literature but recently reported in [8]. The max-min approach which is developed from the idea of ancient Chinese mathematics is demonstrated to be of convenient application, less calculation, and high accuracy. Among current researches about He Chengtian’s inequalities and their applications [9–11], few involved coupled nonlinear problems. In our previous research [12], He Chengtian’s inequalities were introduced to study the nonlinear dropping shock response for coupled cubic nonlinear packaging system, showing the effectiveness of the method. In packaging system, many cushioning materials behave as the tangent nonlinear characteristics [13, 14], and the dropping shock response of tangent packaging system with critical component is also studied [15]. In this paper, He Chengtian’s inequalities are applied to the coupled nonlinear tangent packaging system with critical component, and the obtained analytical solution is compared with the solution of computer simulation. The aim of this research is to suggest a new and simple mathematical method for solving the nonlinear dropping shock equations arisen in packaging system.

2. Modelling and Equations

The governing equations of tangent nonlinear cushioning packaging system with the critical component can be expressed as [15] 𝑚₁𝑥^′′+𝑘₁𝑚𝑥−𝑦=0,₂𝑦^′′+2𝑘₂𝑑_𝑏𝜋𝜋tan2𝑑_𝑏𝑦−𝑘₁𝑥−𝑦=0, (1) where 𝑥(0)𝑦(0)𝑥=0,=0,_′(0)=𝑦2𝑔ℎ,_′(0)=2𝑔ℎ. (2)

Here the coefficients 𝑚₁ and 𝑚₂ denote the mass of the critical component and the main part of the product, respectively, while 𝑘₁ and 𝑘₂ are the coupling stiffness of the critical component and that of cushioning pad, respectively, 𝑑_𝑏 is the compression limit of the cushioning pad, and ℎ is the dropping height. Equation (1) can be equivalently written in the following forms: 𝑋̈+𝜔²₁()̈1𝑋−𝑌=0,𝑌+𝑌+3𝑌³+2𝑌15⁵+1−𝜔²₂(𝑋−𝑌)=0, (3) where 𝑥𝑋=2𝑑_𝑏,𝑦/𝜋𝑌=2𝑑_𝑏,𝑡/𝜋𝜏=𝑚₂/𝑘₂,𝜔₁=𝑘₁𝑚₂𝑘₂𝑚₁,𝜔₂=𝑚1+¹𝜔²₁𝑚₂,𝑋(0)𝑌(0)𝑋̇(0)==0,=0,𝑚₂/𝑘₂2𝑑_𝑏/𝜋𝑌̇(0)=2𝑔ℎ,𝑚₂/𝑘₂2𝑑_𝑏/𝜋2𝑔ℎ. (4)(5)(6)(7)(8)(9)(10)(11)(12)

3. Application of He Chengtian’s Inequalities

From (3), we can easily obtain 𝑌⁽⁴⁾+𝜔²₁+𝜔²₂+𝑌²+23𝑌⁴𝑌̈+𝜔²₁1𝑌+3𝑌³+2𝑌15⁵=0. (13)

Rewrite (13) in the form 𝑌⁽⁴⁾𝜔=−²¹+𝜔²₂𝑌+𝑌²+23𝑌³𝑌̈+𝜔²₁11+3𝑌²+2𝑌15⁴𝑌. (14)

According to He Chengtian’s inequalities, we choose a trialfunction in the form 𝑌=𝐴sin(Ω𝜏), (15) which meets the initial conditions as described in (10) and (12).

By simple analysis, from (14)-(15), we know that Ω⁴=𝜔²₁+𝜔²₂Ω²−𝜔²₁+𝐴²Ω²sin²2Ω𝜏+3𝐴⁴Ω²sin⁴−1Ω𝜏3𝐴²𝜔²₁sin²2Ω𝜏+𝐴15⁴𝜔²₁sin⁴Ω𝜏. (16)

The maximal and minimal value of sin²Ω𝜏 are, respectively, 1 and 0. So, we can immediately obtain 𝑓_min=𝜔²₁+𝜔²₂+𝐴²+23𝐴⁴Ω²−2𝐴⁴+5𝐴²+15𝜔15²¹<Ω⁴<𝜔²₁+𝜔²₂Ω²−𝜔²₁=𝑓_max. (17)

According to He Chengtian’s interpolation [8, 12], we obtain Ω⁴=𝑚𝑓_min+𝑛𝑓_max=𝜔𝑚+𝑛²¹+𝜔²₂Ω²−𝜔²₁+𝑘𝑀, (18) where 𝑚 and 𝑛 are weighting factors, 𝑘=𝑚/(𝑚+𝑛), and 𝑀=(𝐴²+(2/3)𝐴⁴)Ω²−((2𝐴⁴+5𝐴²)/15)𝜔²₁.

Then, the approximate solution of (13) can be written as 𝑌=𝐴sin𝜔²₁+𝜔²₂Ω²−𝜔²₁+𝑘𝑀^1/4𝜏. (19)

To determine the value of 𝑘, substituting (19) into (13) results in the following residual [8]: 𝑅(𝜏,𝑘)=Ω²−13𝜔²₁𝑌³+23Ω²−2𝜔15²¹𝑌⁵−𝑘𝑀𝑌. (20)

And by setting ₀^𝑇/4𝑅(𝜏,𝑘)sinΩ𝜏𝑑𝜏=0, (21) where 𝑇=2𝜋/Ω, we obtain the 𝑘 value as 𝑘=6𝐴²Ω²−(1/3)𝜔²₁+5𝐴⁴(2/3)Ω²−(2/15)𝜔²₁.8𝑀 (22)

Substituting (22) into (18) yields 8Ω⁴=8𝜔²₁+𝜔²₂Ω²−8𝜔²₁+6𝐴²Ω²−13𝜔²₁+5𝐴⁴23Ω²−2𝜔15²¹. (23)

From (23), we can easily obtain the frequency value Ω. Table 1 gives the values of Ω with different 𝜔₁ and 𝜔₂, and Figure 1 shows that the approximate solution, (19), agrees well with the exact solution for various different values of 𝜔₁ and 𝜔₂, where the initial velocity is assumed as 𝑌̇(0)=𝐴Ω=1.

Table 1 

Values of Ω from (23) with different values of 𝜔₁ and 𝜔₂.

(a) 𝜔1=3, 𝜔2=1

(b) 𝜔1=4, 𝜔2=3

(c) 𝜔1=5, 𝜔2=2

(d) 𝜔1=5, 𝜔2=3

Figure 1 

Comparison of the approximate solution with the exact solution (asterisk: the approximate solution; continuous line: the exact solution).

4. Conclusion

He Chengtian’s inequalities are for the first time applied to study the nonlinear response of coupled tangent packaging system. The results show that this method can be easily used in engineering application with high accuracy without cumbersome calculation.

Acknowledgments

The authors would like to appreciate the financial support of National Natural Science Foundation of China (Grant no. 51205167), Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20120093120014), the Open Fund of Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, Jinan University, and the Fundamental Research Funds for the Central Universities (Grant no. JUSRP51302A).