Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 379068, 5 pages http://dx.doi.org/10.1155/2013/379068

Research Article Drop Impact Analysis of Cushioning System with an Elastic Critical Component of Cantilever Beam Type

De Gao,1 Fu-de Lu,2 and Si-jia Chen1

1 Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China 2 School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

Correspondence should be addressed to Fu-de Lu; [email protected]

Received 25 March 2013; Revised 16 April 2013; Accepted 17 April 2013

Academic Editor: Jun Wang

Copyright © 2013 De Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In some electronic products and sculpture crafts, there are possibly vulnerable elements that can be idealized as cantilever beam type, failure of which will certainly lead the whole product to lose function. Based on the critical component of cantilever beam type, a nonlinear coupling dynamics model between the critical component and the item was established. The computing procedure of the model was designed using finite difference scheme. A numerical example shows that the acceleration changes notably withthe length of the critical component, and the cantilevered end of the critical component is liable to be damaged, because the dynamic stress there is the largest. In this case, the maximum acceleration just cannot serve as the damaged criterion of the packaged goods, only with the maximum stress value. In order to avoid making a mistake, it is necessary to consider the critical component as an elastic element. The maximum stress of the cantilever beam surpasses the proportional limit of elastic components or is notan effective structural strength to determine whether the product loses its functions.

1. Introduction vulnerable element of some products for which this simplified method is not suitable. Aiming at crafts, Xi and Peng [12]ana- The fragility of the products must be determined by a fragility lyzed the response of the products with a vulnerable element assessmentprocedurebeforecushioningisprovided.The that was idealized as a simple supported beam, and thought maximum acceleration versus static stress is mostly used in that the simple mass-spring system was not suited for evaluat- cushionpackaging,becausethevalueoftheaccelerationisthe ing the structural strength. The cantilever beam, simply sup- criterion to assess whether the items are destroyed [1]ornot. ported beam, and bar element are also available in the micro- Newton first introduced fragility damage boundary curves andportableelectronicproducts[13–15]. Subir [16]certifi- by linking the input environment and the characteristics cated the criterion of the maximum stress that is responsible of critical component of packaged item [2], assuming the for structural strength, and the analytical modeling demon- product as a spring-mass system. Newton’s model is only strated the application of the measured acceleration as a suitable for brittle materials such as glass and hard plastics. criterion of the dynamic strength can be misleading if applied Furthermore, inspection of many products after failure on to elements of different dimensions, weight, and materials. the shock motion revealed plastic deformation [3, 4]like Gao and Lu [17] studied the responses of the electronic prod- soft plastics, aluminum, and soft steels failed in a ductile ucts with bar type critical component, which is simplified as mode. Burgess developed a rigid-plastic model to account uniform and elastic parts. Abovementioned publications on for progressive failure and then obtained the elastic-plastic crafts and portable electronic products have shown that it is damage boundary theory. Many papers [5–11]developed the maximum stress, not the maximum acceleration, which damage boundary curves based on previous research results. is responsible for the dynamic strength of a structure. The studies cited above are all for mass-spring system; This paper presents the dynamics model of expanded some researchers have studied the impact response of elastic polyethylene foam cushioning system for products with 2 Mathematical Problems in Engineering

𝑙

𝑥 𝑥𝑑𝑥

𝑦2 𝑜 𝑥

𝑚

𝑦2(𝑥, 𝑡) 𝑦1 𝑦2 𝑓(𝑦1) (a) Coordinate system of the critical component 𝜕𝑀 𝑀 𝑀+ 𝜕𝑥 𝐻

𝑄 𝜕𝑄 𝑄+ 𝜕𝑥 Figure1:Packagingsystemwithcantileverbeamtypecritical component. 𝑑𝑥

(b) Force diagram of infinitesimal cantilever beam type vulnerable element, which is suitable section for some portable electronic products or crafts with cantilever Figure 2: Coordinate system of the critical component and force beam critical components, under drop impact loadings dur- diagram of infinitesimal section. ing transportation to introduce its application in cushioning packaging design.

where 𝜌 is the density of the beam and 𝐴0 is the section area 2. Coupling Packaging Dynamic Model of the beam. Based on the moment balance of the differential element, The cantilever beam type structure mostly exists in electronic the following equation can be obtained: products, such as an electrical lead or intersect, and some crafts, such as the tail of sculpture animals, which is usually 𝜕𝑀 𝜕𝑄 (𝑀 + 𝑑𝑥) − 𝑀 − (𝑄+ 𝑑𝑥)𝑑𝑥=0. damaged functionally or physically due to the large-scale 𝜕𝑥 𝜕𝑥 (3) deformation of the critical component. Figure 1 shows a packaging system with the tangential type cushion pad, in Equation (3) can be further simplified as follows4 ( ): which 𝑚 is the mass of the main part; 𝑦1 is the displacement 𝑓(𝑦 ) 𝐻 of the packaging item and 1 is the restoring force; is 𝜕𝑀 the dropping height of the packaging system. 𝑄= . (4) 𝜕𝑥 A local coordinate system of the critical component was built, with origin o at the cantilever end, taking the axial direction of the beam as 𝑥 direction and the deflection as the Then insert (4)into(2) and get the following expression: 𝑦2,asshowninFigure 2(a). Take the differential element 𝑑𝑥 2 𝜕2𝑦 to analyze the movement of the critical component, as shown 𝜕 𝑀 2 =𝜌𝐴0 . (5) in Figure 2(b),where𝐸 and 𝐼 are elastic modulus and section 𝜕𝑥2 𝜕𝑡2 moments of inertia, respectively; 𝑄 and 𝑀 are, respectively, shear force and bending moment of the section. The relationship between the bending moment 𝑀 and the By Newton’s second law, (1)isobtainedbasedon deflection 𝑦2 is expressed as Figure 2(b) as follows: 2 𝜕 𝑦2 2 𝑀=−𝐸𝐼 . (6) 𝜕 𝑦2 𝜕𝑄 𝜕𝑥2 𝜌𝐴 𝑑𝑥 =−𝑄+(𝑄+ 𝑑𝑥) . (1) 0 𝜕𝑡2 𝜕𝑥 Combining (5)and(6), the movement equation of the critical Simplify the above as component is given as 2 𝜕 𝑦2 𝜕𝑄 4 2 𝜌𝐴 = , 𝜕 𝑦2 𝜕 𝑦2 0 2 (2) 𝐸𝐼 +𝜌𝐴 =0. (7) 𝜕𝑡 𝜕𝑥 𝜕𝑥4 0 𝜕𝑡2 Mathematical Problems in Engineering 3

The vibration equation of the main part is given by as follows:

󵄨 𝜕3𝑦 󵄨 𝑦 (𝑡+Δ𝑡) −𝑦 (𝑡−Δ𝑡) 𝑚𝑦̈ +𝑓(𝑦, 𝑦̇ )+𝐸𝐼 2 󵄨 =0, 𝑦̇ (𝑡) = 1 1 ; 1 1 1 3 󵄨 (8) 1 𝜕𝑥 󵄨 2Δ𝑡 𝑥=0 (12a) 𝑦 (𝑡+Δ𝑡) +𝑦 (𝑡−Δ𝑡) −2𝑦 (𝑡) 𝑦̈ (𝑡) = 1 1 1 ; 1 Δ𝑡2 where 𝑓(𝑦1, 𝑦1̇ ) is the constitution relation of expanded polyethylene between the outer package and the product. 𝜕𝑦 (𝑥,) 𝑡 𝑦 (𝑥, 𝑡 )+ Δ𝑡 −𝑦 (𝑥, 𝑡 )− Δ𝑡 2 = 2 2 ; Most of the cushioning materials for electronic products is 𝜕𝑡 2Δ𝑡 EPE, and its relation [18]canbefittedbythetangential 𝜕2𝑦 (𝑥,) 𝑡 𝑦 (𝑥, 𝑡 )+ Δ𝑡 +𝑦 (𝑥, 𝑡 )− Δ𝑡 −2𝑦 (𝑥,) 𝑡 function: 2 = 2 2 2 ; 𝜕𝑡2 Δ𝑡2 (12b) 𝑦1̇ 𝑦1 𝑓(𝑦)=𝐴[𝑎1 +𝑎2 tan (𝑎3 )] , (9) ℎ ℎ 𝜕2𝑦 𝑦 (𝑥+Δ𝑥,𝑡) +𝑦 (𝑥−Δ𝑥,𝑡) −2𝑦 (𝑥,) 𝑡 2 = 2 2 2 ; 𝜕𝑥2 Δ𝑥2 where 𝐴 and ℎ are, respectively, area and thickness of the 3 𝜕 𝑦2 cushion pad; 𝑎1, 𝑎2 and 𝑎3 are the parameters determined by =(𝑦(𝑥+2Δ𝑥,𝑡) +2𝑦 (𝑥−Δ𝑥,𝑡) 𝜕𝑥3 2 2 the experimental data. With initial conditions and boundary conditions being 3 −2𝑦2 (𝑥+Δ𝑥,𝑡) −𝑦2 (𝑥−2Δ𝑥,𝑡))/2Δ𝑥 ; considered, the coupling dynamic equation between the critical component and the main party is given as follows: 𝜕4𝑦 2 =(6𝑦(𝑥,) 𝑡 −4[𝑦(𝑥+Δ𝑥,𝑡) +𝑦 (𝑥−Δ𝑥,𝑡)] 𝜕𝑥4 2 2 2 󵄨 3 󵄨 4 𝑦1̇ 𝑦1 𝜕 𝑦2 󵄨 +[𝑦 (𝑥+2Δ𝑥,𝑡) +𝑦 (𝑥−2Δ𝑥,𝑡)]) /Δ𝑥 . 𝑚𝑦̈ +𝐴[𝑎 +𝑎 (𝑎 )] + 𝐸𝐼 󵄨 =0, 2 2 1 1 ℎ 2 tan 3 ℎ 𝜕𝑥3 󵄨 󵄨𝑥=0 (12c) (10) 𝜕4𝑦 𝜕2𝑦 2 2 So the movements 𝑦1 and 𝑦2(𝑥, 𝑡) can be obtained by 𝐸𝐼 +𝜌𝐴0 =0. 𝜕𝑥4 𝜕𝑡2 combing (10)and(11), shown as 𝑧=𝑦 (𝑥,) 𝑡 −𝑦 (𝑡) . The initial conditions and boundary conditions are 2 1 (13) The maximum stress is determined as 𝑦 0 =𝑦 𝑥, 0 =0; 1 ( ) 2 ( ) 𝑀𝑟 󵄨 𝜎= , (14) 𝜕𝑦 (𝑥,) 𝑡 󵄨 𝐼 𝑦̇ (0) = 2 󵄨 = √2𝑔𝐻; 1 𝜕𝑡 󵄨 󵄨𝑡=0 where 𝑟 istheradiusofthesectionofthecriticalcompo- 󵄨 𝜕𝑦 (𝑥,) 𝑡 󵄨 (11) nent; 𝑀 is the bending moment and calculated by 𝑀= 𝑦 (0, 𝑡) =𝑦; 2 󵄨 =0; 2 2 2 1 󵄨 −𝐸𝐼(𝜕 𝑦2/𝜕𝑥 ) 𝜕𝑥 󵄨𝑥=0 . 󵄨 𝜕2𝑦 (𝑥,) 𝑡 𝜕3𝑦 (𝑥,) 𝑡 󵄨 2 = 2 󵄨 =0. 3.1. Numerical Example. In the present study, an example was 2 3 󵄨 𝜕𝑥 𝜕𝑥 󵄨𝑥=𝑙 cited to illustrate the use of the dynamic model of the cushion packaging system with critical component of cantilever beam type.Thereisanelectronicdevicewiththemass𝑚=3kg, The analytic solutions of (10)arenotavailableduetothe andtheparametersofthecriticalcomponentareasfollows: 3 nonlinear behavior of cushion material used in this paper; density 𝜌 = 0.5 g/cm ,elasticmodulus𝐸 = 100 MPa, therefore, only numerical solutions were obtained by different proportional limit 𝜎𝑝 = 3.3 MPa, length 𝑙 = 0.03 m, and schemes. 2 cross-sectional area 𝐴0 =5mm . The parameters of the It is assumed that the mass of the critical component cushion material of expanded polyethylene between the outer ismuchsmallerthantheitemsinportableproducts,so packaging box and the product are 𝑎1 = 280 Pa⋅s, 𝑎2 = the coupling between critical component and item can be 0.0894 𝑎 = 1.91 𝐴 = 0.01 2 neglected. MPa, 3 ,area m ,andthickness ℎ = 0.035 m. The packaging system drops from the height 𝐻 = 0.2 m. It should be checked whether the cushion size 3. Algorithm Design of the Model satisfies the demand of engineering. Substituting (12a), (12b), and (12c)into(10)and(11), A different method was used to solve the coupling model the deformation, acceleration, and stress distribution are shown in (10). The different schemes are introduced obtained, as shown in Figures 3–5. 4 Mathematical Problems in Engineering

60 0.02 50 0.015 40 30

(m) 0.01 20 2 𝑦 10 0.005 0 0.03 0 0.025 0.03 0.02 0.02 𝑥 0.015 0.02 0.025 (m) 0.01 0.01 0.02 0.005 𝑥 0.015 0 0 𝑡 (s) (m) 0.01 0.01 0.005 (s) 0 0 𝑡 (a) The absolute acceleration of critical component

(a) The absolute displacement of the critical component

40 ×10−3 6 30 5 20 4 3 10 (m) 𝑧 2 0 1 0.03 0.025 0 0.02 0.03 𝑥 0.015 (m) 0.01 0.01 0.005 (s) 0.02 0 0 𝑡 𝑥 0.025 (m) 0.02 (b) The relative acceleration of critical component 0.01 0.015 0.01 0.005 Figure 4: The absolute acceleration and relative acceleration of 0 0 𝑡 (s) critical component. (b) The relative displacement of the critical component

Figure 3: Displacements of the critical component. The changes of the stress of cantilever beam type with time and length are shown in Figure 5. The stress at the end of the cantilever beam (2.71 MPa) is less than its proportional limit (3.3 MPa); thus, no plastic deformation occurred. Use of The absolute displacement 𝑦2 and relative displacement 𝑧 are shown in Figures 3(a) and 3(b),respectively.Itwas the maximum stress as a strength criterion for the improved shownthattheabsoluteandrelativedisplacementsofthe dynamic strength of a vulnerable element can result in free end are maximum under the conditions that the other practical design decisions. end of the cantilever beam is fixed on the product. Because elastic cantilever beam is vibrated under the impact loading 4. Conclusions condition, the displacement of cantilever beam relative to product item is obvious as shown in Figure 3(b). The packaging system with a critical component of cantilever The absolute acceleration and relative acceleration are beam type was studied by considering the nonlinear behavior shown in Figure 4. It can be seen that the absolute accelera- of cushion material. Different schemes were used to obtain tion changed significantly with the length, and its maximum the numerical solution of the nonlinear packaging system value is, respectively, 20.6 g and 63.5 g when the time 𝑡= with cantilever beam critical component. 0.0065, 𝑥=0,and𝑡 = 0.0071, 𝑥 = 0.03 m, as shown in As the critical component is of cantilever beam type, the Figure 4(a). Tovisualize the changes with the length of critical displacement and acceleration of the free end are the largest; component, the relative acceleration of products and critical from the end to the free end of the cantilever, the relative component is shown in Figure 4(b). It was demonstrated that displacement and acceleration changed gradually from zero thechangelawforaccelerationresponseissimilartothatof to maximum. The beam stress reaches maximum at the end displacement response shown in Figure 3. of the cantilever. It is concluded that the acceleration value is Mathematical Problems in Engineering 5

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(MPa) 1.5 [12] D.C.XiandN.L.Peng,“Shockproofofthepackagingforelastic 𝜎 parts with simple beam,” Packaging Engineering,vol.10,no.1,pp. 1 42–48, 1989. 0.5 [13] S. H. Park and W. J. Kim, “Structural design of a Cathode-Ray Tube (CRT) to improve its mechanical shockproof character,” 0 0.03 JournalofMechanicalScienceandTechnology,vol.20,no.9,pp. 0.02 0.02 1361–1370, 2006. 0.015 𝑥(𝑚) 0.01 0.01 [14] J. W. Yi and G. J. Park, “Development of a design system for 0 0 0.005 𝑡(𝑠) EPS cushioning package of a monitor using axiomatic design,” Advances in Engineering Software,vol.36,no.4,pp.273–284, Figure 5: The relationships between the stress, time, and length. 2005. [15] W. J. Kim, D. H. Kum, and S. H. Park, “Effective design of cushioning package to improve shockproof characteristics not the criterion for judging whether the item is damaged or of large-sized home appliances,” Mechanics Based Design of not and this only depends on whether the stress exceeds the Structures and Machines,vol.37,no.1,pp.1–14,2009. proportional limit of the material. [16] E. Subir, “Is the maximum acceleration an adequate criterion of the dynamic strength of a structural element in an electronic product?” IEEE Transactions on Components Packaging and Acknowledgments Manufacturing Technology A,vol.20,no.4,pp.513–517,1997. [17] D. Gao and F. D. Lu, “Drop impact analysis of packaging system This work was supported by the Twelfth National Five- with bar type elastic critical components,” Journal of Vibration Year Science and Technology Projects (nos. 2011BAD24B01, and Shock,vol.31,no.15,pp.47–49,2012. 2012BAD32B02). [18] D. Gao and F. D. Lu, “Optimization design of MDOF package cushioning system made of polyethylene,” Journal of Vibration References and Shock,vol.31,no.3,pp.69–72,2012.

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