Simulation Study on the Shock Properties of the Double-Degree-of-Freedom Cushioning Packaging System

Transcription

1 Proceeding of the 7th IAPRI World Conference on Packaging Simulation Study on the Shock Propertie of the Double-Degree-of-Freedom Cuhioning Packaging Sytem Xia Zhu, Qiaoqiao Yan, Xiaoling Yao, Junbin Chen, Ping Zhao, Zhongjun Hu, Chunjiang Zhao, Congguang Lin, Chunhui Wang 2 Logitical Engineering Univerity Chongqing 40, China 2 Logitic Science Reearch Intitute of the GLD Beijing 0007, China zhuxia4@ina.com Abtract: According to the mechanic model of the double-degree-of-freedom cuhioning packaging ytem, the paper etablihe a mathematic differential equation. It et up the linear and nonlinear imulation model of hock propertie for the double-degree of freedom cuhioning packaging ytem with the principle of Matlab/Simulink. It give the model parameter value, then plot the acceleration repone curve of product and vulnerable part, and doe a comparative analyi on the acceleration repone curve of linear and nonlinear ytem. The concluion i that implifying the nonlinear ytem to a linear ytem will induce inadequate packaging. Keyword: double-degree-of-freedom; cuhioning packaging; hock propertie; imulink imulation Product uffer from different form of damage in the proce of logitic. The damage caued by impact and vibration i the mot common, and the lo i the mot eriou, therefore there i a need to tudy cuhioning packaging of product to reduce the force caued by impact and vibration. There i alo a need to combine lot of experimentation into reearching on cuhioning packaging, or it will induce inadequate or over packaging. The product will be damaged eaily and the intactne rate will decreae in the cae of inadequate packaging, while part of the wrapper will be wated and the packaging cot will increae when the product are over packaged. A experimentation on real product i uually detructive and expenive, imulation technique can be ued to ave time, labor and money.. Preent Reearch In the proce of logitic, it i hard to decribe the proce of the hock and the curve of the hock wave with mathematic formula. Therefore there i a need to tudy the hock propertie of product and the more vulnerable part, uch a acceleration, velocity and diplacement. At preent, there are many imulation tudie on the hock propertie of cuhioning packaging ytem. Yang Hong et al. [] did imulation tudy on ingledegree-of-freedom cubic nonlinear cuhioning ytem, and plotted the dropping impact curve. Li Xiaoli et al. [2] etablihed the imulation diagram of cuhioning packaging ytem at impact condition of dropping, plotted the diplacement and velocity repone curve of linear and Supported by the Academic Innovation Fund of Logitical Engineering Univerity cubic nonlinear cuhioning ytem, and then compared the two type of ytem. Liu Xia et al. [] analyzed the effect of airdrop velocity, cargo material, and cargo thickne on peak acceleration and pule duration, and then obtained the experiential formula between the correlative variable from the imulation data. The above reearche tudied ingle-degree-of- freedom ytem, but the cuhioning ytem in practice are uually double-degree-of-freedom or multi-degree-of-freedom nonlinear ytem, o if the concluion reached in thee tudie are applied to real life practice, it will induce error. Wang Chunyan et al. [4] did imulation reearch on natural frequency of the reaonable ditribution of the buffering cuhion and the vulnerable part uing Matlab with the olution method of non-decoupling, and obtained two natural frequency optional ditribution cope. But hi concluion can only be applied when the ma of the vulnerable part i much le than that of the product. Sek MA et al. [5] analyzed enhancement of cuhioning performance with paperboard crumple inert. But it failed to do adequate reearch on the nonlinear ytem. Moreover, there i lot of reearch on the double-degreeof-freedom linear cuhioning ytem. Therefore, the cuhioning packaging model baed on the preent reearch are motly ingle-degree-of-freedom model, but the cuhioning ytem in practice are uually double-degree-of-freedom or multi-degree-offreedom nonlinear ytem. Up to now, there i not any reearch on the hock proprietie of multi-degree-offreedom cuhioning ytem, o thi paper will tudy double-degree-of-freedom cuhioning ytem SciRe 8

2 Proceeding of the 7th IAPRI World Conference on Packaging Figure. The mechanic model of package ytem. 2. The Mechanic Model of Cuhioning Package Sytem The cuhioning package ytem i the abtract of diverified cuhioning package part, including container ued for outer packing, and cuhioning pad ued a the medium. The theory of cuhioning packaging doen t reearch on concrete product or package, but the mechanic model. The product in the container are called package, a hown in Fig (a). When the product in the container move downward, upward elaticity will be caued by the cuhioning pad below the product. When the product in the container move upward, downward elaticity will be caued by the cuhioning pad above the product. So the function of the cuhioning pad i imilar to that of a pring. A to thoe fixed package, vibration generated by the carrier will be tranferred to the product via the container and the cuhioning pad, o the function of the container i imilar to the bearing force. According to the above analyi, the cuhioning pad can be abtracted to the pring, and the container can be abtracted to the bearing of the product-cuhioning pad ytem, and the function of diverified reitance caued by the vibration of the product can be abtracted to the damper, o the mechanic model of cuhioning package ytem can be obtained, a hown in Fig. (b)[6]. In Fig., m denote the quality of product; k denote the elaticity contant of the cuhioning pad; C denote the reitance coefficient of product-cuhioning pad ytem, epecially the inner reitance of cuhioning pad; m denote the quality of vulnerable part; k denote the elaticity contant of vulnerable part; C denote the reitance coefficient of vulnerable part. The balance point of the package i defined a the origin; the package i defined a the y axi; the product i defined a the x axi; the vulnerable part i defined a the x axi. The movement of the container i the ame with that of the carrier when the vibration of package i analyzed. The y axi i the quantity known, and the pace point of package can be known when the coordinate of the product and vulnerable part are got, o the mechanic model of cuhioning package ytem i the double-degree-offreedom ytem.. Simulation Analyi on the Shock Propertie of Double-Degree-of-Freedom Linear Cuhioning Packaging Sytem To analyze the hock propertie, the firt tep i to etablih the mathematical model of double-degree-offreedom liner cuhioning packaging ytem. When the cuhioning packaging ytem fall down from the height h, it function i equal to the free falling movement from the ame height. At the moment of touching the ground, the ytem i on the ame velocity. And the initialization condition i: x( 0) x (0) 0, x (0) x (0) gh () 2 The mathematic equation of package at impact condition of dropping i: mx C( x y ) k( x y) C ( x x ) k ( x x ) 0 m x C ( x x ) k ( x x) 0 (2) It i obviou that there are force and counterforce between product m and vulnerable part m, o it i hard to olve the equation in the mathematical method becaue it i in coupled tate, and the equation even can t be olved. Now the equation in coupled tate i olved by twotage etimation method, which implify the doubledegree-of-freedom tranport packaging ytem to two independent ingle-degree-of-freedom ytem, and it will induce error. But it could be olved directly by Matlab/Simulink imulation, and the reult will be more accurate. The ground i conidered a a concrete body, o y 0, y 0. Then the hock propertie of doubledegree-of-freedom cuhioning packaging ytem will be tudied by Matlab/Simulink. Equation (2) i tranformed a the following formula in order to etablih the Simulink imulation model. C C x m C x x m C k k k x x x x C k k x x x The imulation model of double-degree-of-freedom cuhioning packaging ytem are etablihed with the principle of Matlab/Simulink, which could get the hock propertie, a hown in Fig. 2. In Fig. 2, intructe the central imulation module to make the imulation module to be read eaily. The model parameter value are given a the following: m 80 kg, C 65 N/m, k N/m, m 2 kg, C 20 N/m, k 000 N/m. And the height h =0.8m. The ode45 of Runge-Kutta method ha lot of advan- () SciRe

3 Proceeding of the 7th IAPRI World Conference on Packaging tage, uch a high preciion and little error, and it i ued commonly, o it i choen a the arithmetic in the imulation. The imulation time i 40, for the ytem will not change obviouly and it will tabilize after 40. Run the emulator, and then the variable tout and yout of the Matlab workpace are given the imulation reult by the output module of the imulation model during the initialized time, o the reult are aved in the Matlab workpace. The acceleration repone curve of product and vulnerable part could be got by the inner function of Matlab. And the acceleration repone curve of product and vulnerable part are plotted together in one chart to compare the hock propertie of the product and vulnerable part eaily, which can be een in Fig.. caued by the nonlinear elaticity of cuhioning material or the configuration by packing. The damp of the cuhioning material will reult in the nonlinear characteritic, but the nonlinear effect caued by damp can alway be ignored. acceleration acceleration 2 velocity Figure 2. The imulation diagram of hock propertie of linear ytem. It can be een from the imulation reult that the ytem tabilize after 0 and it i periodic obviouly. It can be een from Fig. that it take for both the acceleration of product and vulnerable part to reach the maximum, and the maximum of the vulnerable part acceleration i m/ 2, while that of the product i only m/ 2. So the product alway break away from the vulnerable part, which i coincidence with the concluion about the product fragility in literature [7]. 4. Simulation Analyi on the Shock Propertie of Double-Degree-of-Freedom Nonlinear Cuhioning Packaging Sytem. The cuhioning packaging ytem in practice are uually nonlinear ytem. The nonlinear characteritic i mainly diplacement velocity diplacement Figure. The acceleration repone curve of product and vulnerable part (linear ytem). The cuhioning material i conidered a cubic material, o the following formula can be concluded according to the relation between force and deformation. F kx rx (4) So the mathematic equation of double-degree-offreedom cubic cuhioning packaging ytem i: mx Cx m x C kx rx C x x k x x x x k x x 0 The initialization condition i: x 0 x 0 0, x0 x 0 gh 2 0 (5) (6) The above formula i tranformed a the following in order to etablih the Simulink imulation model. C C x m C x x m C k k k r x x x x x m C k k x x x The imulation model of the mathematic differential equation are etablihed with the principle of Matlab/Simulink, which could get the hock propertie, a hown in Fig. 4. The model parameter value are given a the following: m 80 kg, C 65 N /m, k N/m, r N/m, m 2 kg, C 20 N /m, (7) SciRe 20

4 Proceeding of the 7th IAPRI World Conference on Packaging k 000 N/m. And the height h 0.8 m. Though the mathematic equation of double-degreeof-freedom cubic cuhioning packaging ytem in the proce of dropping ha an additional module rx than that of linear ytem, the imulation model of cubic ytem i much more complex than that of linear ytem. The acceleration repone curve of product and vulnerable part are plotted together in Fig. 5 to compare their hock propertie eaily. -K -K K Figure 5. The acceleration repone curve of product and vulnerable part nonlinear ytem). diplacement velocity acceleration cube of product -K 2 diplacement velocity acceleration -K - Figure 4. The imulation diagram of hock propertie of nonlinear ytem. It can be een from the imulation reult that the ytem get tabilized a the time goe on. The concluion could be made that the maximum of the vulnerable part acceleration repone i bigger than that of the product, and it coincide with the linear ytem and the fact, o it i neceary to protect the vulnerable part with cuhioning material. It can be een from Fig. 5 that it take for the acceleration of product to reach the maximum, while it take for the acceleration of vulnerable part to reach the maximum. That i to ay, the vulnerable part reach the acceleration maximum later than product in the nonlinear ytem for the function of the nonlinear chrematitic, and it i the difference between the nonlinear ytem and the linear ytem. The acceleration repone curve of linear ytem and nonlinear ytem with the ame parameter are compared in Fig. 6 and 7, and the imulation time i 0. Two concluion can be made from the repone curve. The firt concluion i the cycle of the nonlinear ytem i horter than that of the linear ytem; the econd concluion i that the maximum of the nonlinear Figure 6. The acceleration repone curve of product between linear and nonlinear ytem. Figure7. The acceleration repone curve of vulnerable part between linear and nonlinear ytem SciRe

5 Proceeding of the 7th IAPRI World Conference on Packaging ytem acceleration repone i bigger than that of linear ytem. When the nonlinear ytem i implified a a linear ytem, the acceleration repone will decreae. If the reult i ued for deigning cuhioning packaging ytem, it will caue big mitake, and it could lead to inadequate packaging, and then the intactne rate of product will decreae eriouly. 5. Concluion Thi paper doe a imulink imulation reearch on the hock propertie of double-degree-of-freedom linear and nonlinear tranport packaging ytem, and the concluion can be made that the nonlinear ytem could not be implified to linear ytem in the proce of deigning cuhioning packaging ytem, or it will caue inadequate packaging. The hock propertie of the multidegree-of-freedom cuhioning packaging ytem could alo be olved with the ame method of double-degreeof-freedom cuhioning packaging ytem. Though the model i more complex, the baic theorie are the ame. Reference [] Yang Hong, Zheng Quancheng, and Guo Hongju, Simulation tudy of hock propertie of nonlinear chhioning packaging ytem baed on Matlab/Simulink, Packaging Engineering, vol. 29, pp. 6-8, March [2] Li Xiaoli, Simulation tudy on hock property of package cuhioning ytem with non-linearity, Xi an univerity of cience and technology,pp.29-40,2004 [] Liu Xia, Qi Huan, Cuhion performance analyi of the impact repone of airdropping cargo. Packaging Engineering, vol.27, pp October [4] Wang Chunyan, He Banggui, Simulation reearch on the optimal ditribution of natural frequencie of cuhion packaging ytem baed on Matlab Packaging Engineering, vol.29, pp , May [5] Sek MA, Rouillard V, Tarah H, et al. Enhancement of cuhioning performance with paperboard crumple inert. Packaging Technology and Science. 2005, 8(5): [6] Su yuan, Tang boen. Fundamental and application of cuhioning packaging. Beijing, pp.7-9, [7] Yan Qiaoqiao, Zhu Xia, Zhou Xiaomin, Dicuion on dynamic reliability of tranportation packaging Packaging Engineering, vol. 0, pp. 50-5, Dec SciRe 22