- Research
- Open Access
Ren-He’s method for solving dropping shock response of nonlinear packaging system
- An-Jun Chen^{1, 2}Email author
- Received: 13 August 2016
- Accepted: 19 October 2016
- Published: 28 October 2016
Abstract
In this paper, Ren-He’s method for nonlinear oscillators is adopted to give approximate solutions for the dropping shock response of cubic and cubic-quintic nonlinear equations arising from packaging system. In order to improve the accuracy of the solutions, a novel technique combining Ren-He’s method with the energy method is proposed, the maximum values of the displacement response and acceleration response of the system are obtained by the energy method, and the approximate solution is corrected. An analytical expression of the important parameters including the maximum displacement, the maximum acceleration of dropping shock response and the dropping shock duration is obtained. The illustrative examples show that the dropping shock response obtained by this method is very similar to the one by the fourth order Runge-Kutta method. The result provides a new simple and effective method for the dropping shock response of a nonlinear packaging system.
Keywords
- Ren-He’s method
- energy method
- displacement maximum value
- acceleration maximum value
- dropping shock duration
1 Introduction
It is a well known fact that the dynamic model of some engineering problems can be described by nonlinear differential equations. In the field of packaging engineering, due to the nonlinear characteristics of cushion packaging materials, the dynamics model of a packaging system usually have strong nonlinear characteristics for the theoretical analysis process of packaged product damage evaluation. Therefore, it is general difficult to obtain the theoretical analytical solution for this kind of problems. The numerical analysis method such as Runge-Kutta method is often adopted to analyze the dynamics performance and dropping shock damage evaluation of the packaging systems [1–7]. Despite the effectiveness of numerical method, the explicit expression of approximate solutions is still expected to be obtained to conveniently discuss the influence of initial conditions and parameters on the solution.
Finding approximate, and if possible in closed-form, solutions of nonlinear differential equations is the subject of many researchers. Recently, various analytical approaches for solving nonlinear differential equations have been widely applied to analyze engineering problems, such as the variational iteration method (VIM) [8–13], the energy balance method (EBM) [14–16], the homotopy perturbation method (HPM) [17, 18], He’s frequency-amplitude formulation (FAF) [16, 19–22], the Hamiltonian approach [23], and many others. Although these methods have been applied to obtain approximate solutions of nonlinear equation with large amplitude of oscillations, the solving process of these methods involve sophisticated derivations and computations, and they are difficult to implement. A simple approximate method to nonlinear oscillators is proposed by Ren and He [24] (Ren-He’s method) and the analysis process of this method is very simple, anyone with basic knowledge of advanced calculus can apply the method to finding the approximate solution of a nonlinear oscillator.
In this paper, for getting analytical results of the cubic and cubic-quintic nonlinear equation arising from packaging system, Ren-He’s method is used to solve the dropping shock dynamic equations of the system, and the approximate solutions including the displacement response, acceleration response and the dropping shock duration are obtained. In order to improve the accuracy of the solution, the correction method combining the Ren-He’s method with the energy method (EM) is developed. The research provides an effective method for solving dropping shock problems of the nonlinear packaging systems.
2 Basic idea of approximate analysis
2.1 Analysis process of Ren-He’s method
2.2 Basic idea of EM
In view of the un-damped conservative systems, when the maximum deformation of the buffer occurs, the restoring force of system will be the maximum, the velocity of system is equal to zero and the acceleration response of the system obtains the maximum value, this is the real physical process. In the following discussion, combining Ren-He’s method with EM according to the physical process, a correction method of the approximate analytic solution of Ren-He’s method is put forward for the drop impact problems in packaging engineering.
3 Approximate solution of the dropping shock response
In order to illustrate the advantages and the accuracy of Ren-He’s method, we take the cubic and cubic-quintic nonlinear equations arising from packaging system as examples. The approximate solution of the dropping shock response is obtained by using Ren-He’s method, and the correction method of approximate solutions is proposed.
3.1 Cubic nonlinear system
Comparison of the important parameters by RHM with R-K method for the cubic nonlinear packaging system
\(\boldsymbol{x_{m}~(\mathbf{cm})}\) | \(\boldsymbol{\ddot{x}_{m}~(\mathbf{g})}\) | \(\boldsymbol{\tau~(\mathbf{s})}\) | |
---|---|---|---|
R-K | 3.4 | 49.74 | 0.02854 |
RHM | 3.51 | 45.35 | 0.02984 |
Relative error (%) | 3.2 | 8.8 | 4.6 |
Comparison of the important parameters by CRHM with R-K method for the cubic nonlinear packaging system
\(\boldsymbol{x_{m}~(\mathbf{cm})}\) | \(\boldsymbol{\ddot{x}_{m}~(\mathbf{g})}\) | \(\boldsymbol{\tau~(\mathbf{s})}\) | |
---|---|---|---|
R-K | 3.40 | 49.74 | 0.02854 |
CRHM | 3.40 | 49.79 | 0.02811 |
Relative error (%) | 0.0 | 0.1 | 1.5 |
3.2 Cubic-quintic nonlinear system
Comparison of the important parameters by RHM with R-K method for the cubic-quintic nonlinear packaging system
\(\boldsymbol{x_{m}}~(\mathbf{cm})\) | \(\boldsymbol{\ddot{x}_{m}~(\mathbf{g})}\) | \(\boldsymbol{\tau~(\mathbf{s})}\) | |
---|---|---|---|
R-K | 3.52 | 50.35 | 0.02920 |
RHM | 3.65 | 45.13 | 0.03074 |
Relative error (%) | 3.7 | 10.4 | 5.2 |
Comparison of the important parameters by the CRHM with the R-K method for the cubic-quintic nonlinear packaging system
\(\boldsymbol{x_{m}~(\mathbf{cm})}\) | \(\boldsymbol{\ddot{x}_{m}~(\mathbf{g})}\) | \(\boldsymbol{\tau~(\mathbf{s})}\) | |
---|---|---|---|
R-K | 3.52 | 50.35 | 0.02920 |
CRHM | 3.52 | 50.32 | 0.02869 |
Relative error (%) | 0.00 | 0.06 | 1.7 |
4 Conclusions
The evaluation of the dropping impact dynamics, the maximum displacement, the maximum acceleration of the system response, and the dropping shock duration is in focus as regards the important parameters. Ren-He’s method has been successfully used for the cubic and cubic-quintic nonlinear equations arising from a packaging system; the analytical solutions of the system are obtained, the analytic expression of the above three important parameters is obtained at the same time. In order to further improve the accuracy of the solution, a new method combining the RHM with the energy method, which is defined as a correction method of the RHM (denoted CRHM), is proposed and successfully used to study the cubic and the cubic-quintic nonlinear packaging system. The numerical examples indicate that the lowest order analytical solution obtained by the CRHM has higher accuracy, which outperforms a similar analytical method. The CRHM is obviously advantageous in that it is simple and it can avoid solving the complicated nonlinear algebraic equation. The results show that the CRHM is effective and easy for solving the dropping shock problem of a nonlinear packaging system.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Wang, ZW: On evaluation of product dropping damage. Packag. Technol. Sci. 15(3), 115-120 (2002) View ArticleGoogle Scholar
- Wang, ZW: Dropping damage boundary curves for cubic and tangent package cushioning system. Packag. Technol. Sci. 15(5), 263-266 (2002) View ArticleGoogle Scholar
- Wang, ZW, Jiang, JH: Evaluation of product dropping damage based on key component. Packag. Technol. Sci. 23(4), 227-238 (2010) View ArticleGoogle Scholar
- Wang, J, Wang, ZW, Lu, LX, Zhu, Y, Wang, YG: Three dimensional shock spectrum of critical component for nonlinear packaging system. Shock Vib. 18(3), 437-445 (2011) View ArticleGoogle Scholar
- Wang, J, Duan, F, Jiang, J: Dropping damage evaluation for a hyperbolic tangent cushioning system with a critical component. J. Vib. Control 18(10), 1417-1421 (2012) MathSciNetView ArticleGoogle Scholar
- Chen, AJ: The shock characteristics of tilted support spring packaging system with critical components. Shock Vib. 2014, Article ID 496035 (2014) Google Scholar
- Duan, NN, Song, S, Chen, AJ: The dynamic evaluation of tilted support spring nonlinear system with critical components under the action of a rectangular pulse. Math. Probl. Eng. 2014, Article ID 404163 (2014) MathSciNetGoogle Scholar
- He, JH, Wu, XH: Variational iteration method: new development and applications. Comput. Math. Appl. 54, 881-894 (2007) MathSciNetView ArticleMATHGoogle Scholar
- He, JH: Variational iteration method - some recent results and new interpretations. J. Comput. Appl. Math. 207(1), 3-17 (2007) MathSciNetView ArticleMATHGoogle Scholar
- He, JH: Variational iteration method - a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 34(4), 699-708 (1999) View ArticleMATHGoogle Scholar
- Song, S, Duan, NN, Chen, AJ: Application of variational iteration method for dropping damage evaluation of the suspension spring packaging system. Abstr. Appl. Anal. 2014, Article ID 385404 (2014) MathSciNetGoogle Scholar
- Chen, AJ: Application of variational iteration method with energy method for nonlinear equation arisen from packaging system. Math. Probl. Eng. 2013, Article ID 326239 (2013) MathSciNetMATHGoogle Scholar
- Ganji, DD, Tari, H, Babazadeh, H: The application of He’s variational iteration method to nonlinear equations arising in heat transfer. Phys. Lett. A 363(3), 213-217 (2007) View ArticleMATHGoogle Scholar
- Mehdipour, I, Ganji, DD, Mozaffari, M: Application of the energy balance method to nonlinear vibrating equations. Curr. Appl. Phys. 10, 104-112 (2010) View ArticleGoogle Scholar
- Zhang, HL: Periodic solutions for some strongly nonlinear oscillations by He’s energy balance method. Comput. Math. Appl. 58, 2480-2485 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Younesian, D, Askari, H, Saadatnia, Z, Kalamiyazdi, M: Frequency analysis of strongly nonlinear generalized Duffing oscillators using He’s frequency-amplitude formulation and He’s energy balance method. Comput. Math. Appl. 59(9), 3222-3228 (2010) MathSciNetView ArticleMATHGoogle Scholar
- He, JH: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. Math. Comput. 151(1), 287-292 (2004) MathSciNetMATHGoogle Scholar
- Rafei, M, Ganji, DD, Daniali, H: Solution of the epidemic model by homotopy perturbation method. Appl. Math. Comput. 187(2), 1056-1062 (2007) MathSciNetMATHGoogle Scholar
- He, JH: An improved amplitude-frequency formulation for nonlinear oscillators. Int. J. Nonlinear Sci. Numer. Simul. 9(2), 211-212 (2008) Google Scholar
- He, JH: Comment on He’s frequency formulation for nonlinear oscillators. Eur. J. Phys. 29(4), L1-L4 (2008) View ArticleGoogle Scholar
- Fan, J: He’s frequency-amplitude formulation for the Duffing harmonic oscillator. Comput. Math. Appl. 58, 2473-2476 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Khan, Y, Kalamiyazdi, M, Askari, H, Saadatnia, Z: Dynamic analysis of generalized conservative nonlinear oscillators via frequency amplitude formulation. Arab. J. Sci. Eng. 38(1), 175-179 (2013) MathSciNetView ArticleGoogle Scholar
- Yildirim, A, Saadatnia, Z, Askari, H, Khan, Y, Kalamiyazdi, M: Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach. Appl. Math. Lett. 24(12), 2042-2051 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Ren, ZF, He, JH: A simple approach to nonlinear oscillators. Phys. Lett. A 373, 3749-3752 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Wang, J: He’s max-min approach for coupled cubic nonlinear equations arising in packaging system. Math. Probl. Eng. 2013, Article ID 382509 (2013) Google Scholar
- Jiang, JH, Wang, ZW: Dropping damage boundary curves for cubic and hyperbolic tangent package cushioning systems based on key component. Packag. Technol. Sci. 25, 397-411 (2012) View ArticleGoogle Scholar
- Wang, J, Wang, ZW, Duan, F, Lu, LX, Gao, D, Chen, AJ: Dropping shock response of corrugated paperboard cushioning packaging system. J. Vib. Control 19(3), 336-340 (2012) View ArticleGoogle Scholar
- Hong, X, Wang, J, Lu, LX: Application of homotopy perturbation method with an auxiliary term for nonlinear dropping equations arisen in polymer packaging system. Math. Probl. Eng. 2013, Article ID 982859 (2013) MathSciNetMATHGoogle Scholar