Dynamic Performance of Stacked Packaging Units

Transcription

1 PACKAGING TECHNOLOGY AND SCIENCE Packag. Technol. Sci. 2016; 29: Published online 11 August 2016 in Wiley Online Library (wileyonlinelibrary.com).2232 Dynamic Performance of Stacked Packaging Units By Zhi-Wei Wang 1,2,3 * and Ke Fang 1,2,3 1 Packaging Engineering Institute, Jinan University, Zhuhai, China 2 Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, Jinan University, Zhuhai, China 3 Zhuhai Key Laboratory of Product Packaging and Logistics, Jinan University, Zhuhai, China Stacked packaging units is the main form of distribution packaging of products. Its dynamic performance is not fully understood. This paper investigated the influence of the constraint, input vibration, location and contact nonlinearity on the dynamic performance of three layers stacked packaging units. The dynamic contact force between surfaces and acceleration response of products were obtained. In sine sweep vibrations, the constraint to stacked packaging units has an obvious influence on the dynamic characteristics. The acceleration response of product is associated with the vibration mode. The force amplification factor is in general between 1.5 and 2, but it can close to 3 on top layer in the case of no fixed. In random vibrations, non-gaussian data of dynamic contact force appear when Gaussian data of input vibration pass through the stacked packaging units, resulting a Weibull distribution of force level-crossing. The force level-crossing diagram becomes more abrupt with the decreasing of input vibration level, smoother from the top contact surface to middle and bottom ones, and moves right and becomes smoother with the constraint strengthen. In the case of lower input level, Gaussian distribution of force level-crossing appears. The force power spectral density (PSD) between bottom box and table is much larger than that between boxes, which is significantly influenced by the first resonance frequency. However, the acceleration PSD of product is significantly influenced by both the first and second resonance frequencies, and controlled by the vibration mode. It depends much on the value of input acceleration PSD around the resonance frequencies. Received 5 January 2016; Revised 26 April 2016; Accepted 12 July 2016 KEY WORDS: dynamic performance; stacked packaging units; random vibration; pressure-mapping system; power spectral density INTRODUCTION Stacked packaging units is the main form of distribution packaging of products. Its dynamic performance in transportation is concerned. Although the vibrational performance of single-unit packaging has been thoroughly studied, the behaviour of stacked packaging units is not fully understood. The complexity of the problem is compounded when the effects of vertical restraints and nonlinearity are taken into account. 1 To understand deeply the behaviour of stacked packaging units and provide a helpful reference to distribution packaging of products, this paper presents a study on the dynamic performance of stacked packaging units of products. The influence of every factor, such as the constraint to stacked packaging units, input vibration level, input acceleration power spectral density (PSD) shape, location in stacked packaging units is investigated. The nonlinearity of contact is discussed in detail. The outline of this paper is as follows: * Correspondence to: Zhi-Wei Wang, Packaging Engineering Institute, Jinan University, Zhuhai , China. wangzw@jnu.edu.cn Copyright 2016 John Wiley & Sons, Ltd.

2 492 Z.-W. WANG AND K. FANG In section 1 the background and objective are described In section 2 the literature is reviewed In section 3 the packaged product and experiment equipment are introduced In section 4 the sine sweep and random vibration experiments of three layers stacked packaging units are described, and two kinds of constraint to stacked packaging units (no fixed, fixed with two grades of pressures to the box), two kinds of input acceleration PSD shape and four levels of input vibration are used in experiments In section 5 the acceleration response of products and dynamic force between contact surfaces in sine sweep vibration experiments are obtained and discussed. The probability density function of force level-crossing, dynamic contact force PSDs and acceleration PSDs of products in random vibration experiments are obtained and discussed. In section 6 the conclusions are reached. LITERATURE REVIEW Godshall investigated the frequency response, damping and transmissibility characteristics of toploaded corrugated containers, and experimentally identified the resonance of containers at Hz with a resonant transmissibility amplification ratios as high as With the aim of understanding the dynamics of stacked packaging units, Urbanik devoted to the numerical analysis of a stack of loaded boxes by taking the stacked boxes as a multi-degree of freedom system, and plotted the transmissibility in each container over a range in frequencies. 3 The laboratory tests were conducted to characterize the response of stacked corrugated containers by using a force plate combined with a single accelerometer. 4 Rouilard et al. developed a numerical model to predict the dynamic response of stacked package systems when subjected to vertical vibrational excitation. The effects of vertical restraint tension and stiffness were taken into account. 1 Bernad et al. investigated the dynamics of different corrugated stacked packaging units by means of operational modal analysis technique and a 6 degrees of freedom multi-axis shaker table to demonstrate the necessity of multiaxis testing. 5,6 When the nonlinear effect of corrugated packaging is taken into account, the dynamics analysis of stacked packaging units is more complex. Thakur and Peng developed a non-linear model to simulate the complex loading patterns in the stack of packages, and a modified Runge Kutta method of numerical technique was used in calculation. 7 The dynamic contact force of stacked packaging units is most concerned because it can be used for the evaluation of the strength and fatigue life of boxes. Marcondes and Schueneman measured the dynamic forces within a stack of packages when subjected to vertical random vibration, and revealed that the level of dynamic force within the package stack was dependent on the frequency of vibration. 8 To understand the dynamic interaction between wood pallet and corrugated container during resonance, a computer model of a palletized bulk bin loaded with fruit was developed using finite element method. 9 Recently, Jamialahmadi et al. studied the dynamic interaction of two layers stacked corrugated boxes by using a pressure-mapping system (I-Scan system) to measure the real-time dynamic pressure data in vibration experiment. The dynamic contact forces between boxes in both vertical and horizontal directions were measured, and the position of the instantaneous centre of force was traced. The level-crossing diagrams of the contact forces shown a Rayleigh distribution for the vertical contact between boxes and a Gaussian distribution for the horizontal contact between upper boxes. 10 When the multilayer stack is concerned, the dynamic interaction between boxes is more complicated. Wang et al. investigated the dynamic contact force and its distribution of three layers stacked packaging units when subjected to sine sweep vibration and random vibration. A typical change of the amplitude and distribution of dynamic contact force was illustrated in a cycle. Around the resonance frequency, the PSD of product acceleration is much larger than that of the vibration table, and the PSD of dynamic contact force is much larger. 11

3 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 493 PACKAGED PRODUCTS AND EXPERIMENT EQUIPMENT Packaged product Three layers stacked packaging units of products was used in experiments. A concrete block with size of 300 mm 140 mm 180 mm was used to model the product. The weight of the concrete block is 17.5 kg. Cushion pads of ethylene vinyl acetate copolymer were used in the cushioning packaging of product. The double-wall corrugated box with outer dimensions of 350 mm 190 mm 230 mm was filled with the product and the cushion pads. Figure 1 gives the detail of packaged product. Vibration equipment An electrohydraulic vibrator (Model Vibration Test Machine, Lansmont, USA) with a frequency range of Hz, and a maximum load capacity 998 kg was used in vibration experiments. I-Scan system To obtain real-time pressure data in experiments, the I-Scan pressure-mapping system (Tekscan Inc., South Boston, MA, USA) was used. The pressure-sensitive films were model 3150 and model 5270 (Tekscan Inc., South Boston, MA, USA). Model 3150 provides an array of 2288 (52 44) forcesensitive cells with a saturation pressure of MPa, and model 5270 an array of 1936 (44 44) with a saturation pressure of MPa. Each cell is mm 2 (8.4 mm 8.4 mm) in size. The I-Scan system is based on a pressure-sensitive film with an upper limit for its sampling frequency 100 frames/s. Figure 1. Packaged product. Figure 2. Vibration experiment.

4 494 Z.-W. WANG AND K. FANG VIBRATION EXPERIMENTS Three-layer stacked packaging units of products were placed on the vibration table with two types of constraint, no fixed and fixed with resilience ropes, as shown in Figure 2. Two grades of pressures to the box induced by the ropes tension were carried out in experiments, respectively, 78.4 N and N. The pressure-sensitive films 1, 2 and 3 of the I-Scan system were, respectively, placed in the contact areas to record the dynamic force distributions. Three accelerometers (Ch2, 3 and 4) were, respectively, attached on three products to measure the acceleration of each product. Another accelerometer (Ch1) was attached on the vibration table. The sine sweep vibrations and random vibrations were performed. The sine sweep vibration tests were performed from frequency 3 Hz to 100 Hz with sweep rate 15 Hz/min, and the acceleration amplitude in experiments was 0.5 g. The input acceleration PSDs of modified (band limited) ASTM D4169 truck levels I, II, III 12 and customized level IV from frequency 1 Hz to 50 Hz, and Figure 3. Input acceleration PSDs. Table 1. Frequency resolutions of PSDs for modified (band limited) ASTM D4169 truck levels I, II, III and customized level IV. Frequency (Hz) D4169 Truck I Power spectral density level (G 2 /Hz) D4169 Truck II D4169 Truck III Custom IV G rms (g, m/s 2 ) Frequency (Hz) Table 2. Frequency resolutions of PSDs for band limited white noise levels I, II, III and IV. noise I Power Spectral Density Level (G 2 /Hz) noise II noise III noise IV G rms (g, m/s 2 )

5 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 495 Figure 4. Accelerations of products in sine sweep vibration experiments. (a) No fixed. (b) resilience ropes pressure 78.4 N to box. (c) resilience ropes pressure N to box.

6 496 Z.-W. WANG AND K. FANG of band limited white noise levels I, II, III and IV from frequency 4 Hz to 50 Hz for random vibration experiments were, respectively, used in this study. They are given in Figure 3, and the corresponding frequency resolutions of PSDs are given in Tables 1 and 2. The reason for the frequency of the PSDs limited to 50 Hz is that the sampling frequency of the pressure-sensitive film is 100 frames/s, which limits the information from the higher-frequency domain. 10,11 The PSD levels of the band limited white noise are determined so that the acceleration root mean square values G rms in the band limited white noise are, respectively, equal to that in the ASTM D4169 truck levels. All the time series data recorded from the I-Scan system and accelerometers were analysed in frequency domain by using MATLAB (MathWorks, Natick, MA, USA). RESULTS AND DISCUSSION Sine sweep vibration experiment Figure 4 gives the measured accelerations of products in sine sweep vibration experiments, which clearly shows the moving of the first, second and third resonance frequencies of the system because of the constraint. The accelerations of products at the first resonance frequency are increased with the constraint strengthen. In the vicinity of the first resonance frequency, the acceleration of top product is the highest, and that of bottom product the lowest. However, in the vicinity of the second resonance frequency, the acceleration of middle product is the highest, and those of top and bottom products are almost equal; in the vicinity of the third resonance frequency, the acceleration of bottom product is the highest, and that of top product the lowest. This implies that the acceleration response of product is associated with the vibration mode of the system. In the case of no fixed, a bit of jumping of products at the first resonance frequency had been seen in experiment, which is corresponding to the chattering phenomenon in Figure 4(a). The resonance frequencies and corresponding accelerations in sine sweep vibration experiments are shown in Table 3. The total dynamic force (the measured force) between contact surfaces is the sum of the forces registered at all active sensels of the I-Scan sensor at a given time. It was scaled by the corresponding initial static pressure in which the pressure to the boxes induced by the ropes tension was included. Figure 5 gives the scaled dynamic forces on each contact surface as a function of frequency in sine sweep vibration experiments. The peak dynamic forces in Figure 5 correspond to the three resonance frequencies of the system. It is observed in Figure 5 that the constraint (no fixed or fixed with resilience ropes) has an obvious influence on the dynamic characteristics of the stacked packaging units of products. Because of the constrain, the first, second and third resonance frequencies of the system are increased, respectively, from 11, 33 and 60 Hz (no fixed) to 13, 41 and 64 Hz (fixed with resilience ropes pressure N to the box). Table 3. Resonance frequencies and corresponding accelerations in sine sweep vibration experiments. Frequency (Hz) First resonance Second resonance Third resonance Acceleration (g, m/s 2 ) Frequency (Hz) Acceleration (g, m/s 2 ) Frequency (Hz) Acceleration (g, m/s 2 ) No fixed Top Middle Bottom N N Top Middle Bottom Top Middle Bottom

7 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 497 Figure 5. Dynamic contact forces in sine sweep vibration experiments. (a) No fixed (scaled by static force). (b) resilience ropes pressure 78.4 N to box (scaled by static force). (c) resilience ropes pressure N to box (scaled by static force). The amplification factor of dynamic force is defined here as the ratio of the maximum dynamic force and the initial static force. In the case of fixed with resilience ropes, the amplification factor is between 1.5 and 2. However, in the case of no fixed, the amplification factor on top layer is close to 3. The total dynamic force appears non-symmetrically, which results from the nonsymmetrical contact characteristics between surfaces. This will be discussed in detail in the next section. Figure 5 also indicates that the variation range of dynamic force on top layer at a given frequency is maximum, which implies that the vibration of top box is the most violent. The dynamic force decreases significantly to a constant for higher frequencies. It should be noted that the results of the dynamic force are only valid up to 50 Hz because of limitation of the sampling frequency of the pressure-sensitive film. Random vibration experiment Level crossing of dynamic force. The number of times a force level crossed is important in a random vibration of stacked packaging units, which can be used to evaluate the fatigue life of corrugated box. The number of times a force level crossed was calculated. Figure 6 shows the probability density function of force level-crossing between surfaces in the random vibration experiments under the input of modified (band limited 1 to 50 Hz) ASTM D4169 truck levels I, II, III and customized level IV, which will be used to investigate the influence of every factor, such as the constraint to stacked packaging units, input vibration level, location in stacked packaging units and nonlinearity of contact on the number of times a force level crossed. Appendix A gives the probability density function of force levelcrossing between surfaces in the random vibration experiments under the input of band limited (4 to 50 Hz) white noise levels I, II, III and IV. In the literature reported by Jamialahmadi 10 and Wang, 11 the probability density function of the force level-crossing resembled the Rayleigh distribution. However, in this study, the distributions are more different. Because of the characteristics of skewed distribution, four types of probability density functions, Gaussian distribution, logarithmic normal distribution, Rayleigh distribution and Weibull distribution were used to fit the experiment data. The maximum likelihood estimation was used to estimate the distribution parameters, and their optimal solutions were obtained. According to

8 498 Z.-W. WANG AND K. FANG the results of fitness, Gaussian distribution, Rayleigh distribution and Weibull distribution were chosen. Their probability density functions are, respectively, as follows: " # For Gaussian distribution : y ¼ fðxjμ; σþ ¼ p 1 σ ffiffiffiffiffi exp ðx μþ2 2π 2σ 2 For Rayleigh distribution : y ¼ fðxjkþ ¼ 8 < x x2 k 2exp 2k : 2 0; x 0 ; x > 0 Figure 6. Force level-crossing diagrams in random vibration experiments with modified (band limited) ASTM D4169 truck levels I, II, III and customized level IV. (a) No fixed. (b) resilience ropes pressure 78.4 N to box. (c) resilience ropes pressure N to box.

9 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 499 Figure 6. (Continued) For Weibull distribution : 8 < y ¼ fðxja; bþ ¼ ba b x b 1 exp x a : 0; x 0 b ; x > 0 In Weibull distribution, a is the scale parameter, and b is shape parameter. When b = 2, Weibull distribution becomes Rayleigh distribution. From the fitting between probability density functions and data as shown in Figure 6 and Appendix A, it can be seen that the probability density function of force level-crossing is more close to Weibull distribution than Gaussian distribution and Rayleigh distribution. It is true for every constraint, no fixed or fixed with resilience ropes, for every vibration levels I, II, III and IV, and for different input acceleration PSD shapes, both band limited ASTM D4169 truck and band limited white noise PSDs. In the case of lower input level IV of acceleration PSDs, both Gaussian distribution and Weibull distribution are more close to the experiment results of force level-crossing diagrams, and Rayleigh distribution does not give the satisfied fitting. It is noted that in some cases Rayleigh distribution may also give the accepted fitting as reported 10,11 when the shape parameter b of Weibull distribution is close to 2. In these figures, G, R and W represent, respectively, Gaussian distribution, Rayleigh distribution and Weibull distribution. To a linear system, the response is of Gaussian distribution and the probability density function of force level-crossing is also of Gaussian distribution when the excitation is of Gaussian distribution. 13 However, the probability density function of force level-crossing here is of Weibull distribution. This is induced by the nonlinearity of three layers stacked packaging units of products. Non- Gaussian data of response are produced when Gaussian data pass through this nonlinear system. Figure 7 gives the typical time series of the dynamic contact force in three layers stacked packaging units of products, which shows obviously the characteristics of non-gaussian data. In Figure 7 the non-symmetrical response of dynamic contact force results from the non-symmetrical contact characteristics between surfaces. On the contact surface, there are push force, and no pull force. One can see the time history of jump of boxes in Figure 7. In the case of lower input level IV of acceleration PSDs, there is no jump of boxes, the response of dynamic contact force is symmetrical and the system can be taken as a linear one. So, Gaussian distribution of dynamic contact force happens.

10 500 Z.-W. WANG AND K. FANG Figure 7. Time series of dynamic contact force in random vibration experiment with modified (band limited) ASTM D4169 truck level I. (a) No fixed. (b) resilience ropes pressure 78.4 N to box. In all cases, the probability density function of force level-crossing becomes more abrupt with the decreasing of vibration from level I to II, III and IV, meaning a more narrow force levels. It becomes smoother from the top contact surface to middle and bottom contact surfaces, meaning a broader force levels. The force level-crossing diagram moves right and becomes smoother in Table 4. Weibull distribution parameter of force level-crossing in experiments with modified (band limited) ASTM D4169 truck levels I, II, III and customized level IV. D4169 Truck I D4169 Truck II D4169 Truck III Custom IV a b a b a b a b No fixed Top Middle Bottom N N Top Middle Bottom Top Middle Bottom

11 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 501 Figure 8. PSDs of dynamic contact forces with modified (band limited) ASTM D4169 truck levels I, II, III and customized level IV. (a) Force PSDs no fixed. (b) Force PSDs fixed with resilience ropes pressure 78.4 N to box. (c) Force PSDs fixed with resilience ropes pressure N to box.

12 502 Z.-W. WANG AND K. FANG general with the constraint strengthen, meaning a larger and broader force levels. Table 4 and Appendix B give the scale parameter a and shape parameter b of Weibull distribution of force levelcrossing. The scale parameter a controls mainly the scale of peak in Weibull distribution, and the shape parameter b controls mainly the shape of Weibull distribution. Power spectral density of dynamic force. Figure 8 gives the PSDs for the dynamic contact forces measured by the I-Scan system during the random vibration experiments with modified (band limited 1 to 50 Hz) ASTM D4169 truck levels I, II, III and customized level IV, obtained by Welch method (depicted in MATLAB from MathWorks) using a fast Fourier transform block size of 1024 samples. Appendix C gives the PSDs for the dynamic contact forces during the random vibration experiments with band limited (4 to 50 Hz) white noise levels I, II, III and IV. From the force PSDs, it is shown that the force PSDs between the bottom box and vibration table are much larger than that between the boxes. It becomes larger with the constraint strengthen in general. Figure 8 and Appendix C indicate that the force PSDs are significantly influenced by the first resonance frequency of the system, and somewhat influenced by the second resonance frequency. Around the first resonance frequency, the dynamic contact force PSDs are much larger. The force PSDs decline with the decreasing of vibration from level I to II, III and IV. We note that the first resonance frequency of the system moves right a little when input vibration changing from level I to II, III and IV. It may be induced by the non-linearity of packaging material and structure. Table 5 and Appendix D give the force root mean square values N rms in experiments, respectively, with the band limited ASTM D4169 truck levels and band limited white noise levels. Power spectral density of acceleration of product. Figure 9 shows the acceleration PSDs of products measured by the accelerometers in the random vibration experiments with modified (band limited 1 to 50 Hz) ASTM D4169 truck levels I, II, III and customized level IV. Appendix E shows the acceleration PSDs with band limited (4 to 50 Hz) white noise levels I, II, III and IV. It is shown that the acceleration PSD of top product is larger than that of the middle and bottom products around the first resonance frequency. However, the acceleration PSD of middle product is the highest around the second resonance frequency. It is in agreement with the results of acceleration response in sine sweep experiments, and verifies that the acceleration PSD of product is controlled by the vibration mode of the system. Figure 9 and Appendix E also indicate that the acceleration PSDs are significantly influenced by both the first and second resonance frequencies of the system. Around the resonance frequencies, the acceleration PSDs of products are much larger than that of the vibration table (excitation signal). The acceleration PSDs decline with the decreasing of vibration from level I to II, III and IV. It should be noted when comparing Figure 9 with Appendix E that there is a great difference in the response acceleration PSDs around the second resonance frequency, which is induced by the great difference of two kinds of input acceleration PSDs around Table 5. Force root mean square values N rms in experiments with modified (band limited) ASTM D4169 truck levels. N rms (N) D4169 Truck I Force root mean square values N rms D4169 Truck II D4169 Truck III Custom IV No fixed Top Middle Bottom N N Top Middle Bottom Top Middle Bottom

13 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 503 Figure 9. Acceleration PSDs of products with modified (band limited) ASTM D4169 truck levels I, II, III and customized level IV. (a) Acceleration PSDs no fixed. (b) Acceleration PSDs fixed with resilience ropes pressure 78.4 N to box. (c) Acceleration PSDs fixed with resilience ropes pressure N to box. the second resonance frequency. This indicates that the response acceleration PSDs around the resonance frequencies rely largely on the input acceleration PSDs around the resonance frequencies. Table 6 and Appendix F give the acceleration root mean square values G rms in experiments, respectively, with the modified (band limited) ASTM D4169 truck levels and band limited white noise levels.

14 504 Z.-W. WANG AND K. FANG Table 6. Acceleration root mean square values G rms in experiments with modified (band limited) ASTM D4169 truck levels. G rms (g, m/s 2 ) Acceleration root mean square values G rms D4169 Truck I D4169 Truck II D4169 Truck III Custom IV No fixed Top Middle Bottom Controlled N N Top Middle Bottom Controlled Top Middle Bottom Controlled CONCLUSIONS This paper investigated the influence of the constraint to stacked packaging units, input vibration level, input acceleration PSD shape, location in stacked packaging units and nonlinearity of contact between surfaces on the dynamic performance of stacked packaging units. The dynamic contact force between surfaces and acceleration response of products of three layers stacked packaging units were obtained in sine sweep vibration and random vibration experiments. The pressure-mapping system was used to measure the dynamic contact forces. Following conclusions can be reached. 1. In sine sweep vibration experiments, the constraint (no fixed or fixed with resilience ropes) has an obvious influence on the dynamic characteristics of the stacked packaging units of products. With the constraint strengthened, the first, second and third resonance frequencies of the system are increased, respectively, from 11, 33 and 60 Hz (no fixed) to 13, 41 and 64 Hz (fixed with resilience ropes pressure N to the box). In the vicinity of the first, second and resonance frequencies, the highest response acceleration of product is, respectively, on the top product, middle product and bottom product, which indicates the response acceleration of product is associated with the vibration mode of the system. The total dynamic force between boxes appears nonsymmetrically, which results from the non-symmetrical contact characteristics between surfaces. In the case of fixed with resilience ropes, the amplification factor of dynamic force is between 1.5 and 2. However, in the case of no fixed, the amplification factor on top layer can close to The probability density function of force level-crossing between contact surfaces in random vibration is all of Weibull distribution for different constraints, input vibration levels and acceleration PSD shapes. Only in the case of lower input vibration level, it becomes Gaussian distribution. This is induced by the non-symmetrical contact characteristics between contact surfaces in stacked packaging units. Non-Gaussian data of dynamic contact force are produced when Gaussian data of input vibration pass through this nonlinear system. In the case of lower input vibration level, no jump of boxes happens. The response of dynamic contact force is symmetrical, and the system can be taken as a linear one. 3. The probability density function of force level-crossing becomes more abrupt with the decreasing of vibration level, becomes smoother from the top contact surface to middle and bottom contact surfaces, and moves right and becomes smoother with the constraint strengthen. The peak and shape of probability density function of force level-crossing are controlled by the scale parameter a and shape parameter b. 4. The force PSDs are significantly influenced by the first resonance frequency of the system, and become larger with the constraint strengthen and increasing of vibration level. The force PSD between the bottom box and vibration table is much larger than that between the boxes. The first

15 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 505 resonance frequency of the system moves right a little with the decreasing of vibration level, which may be induced by the nonlinearity of packaging system. 5. The acceleration PSDs are significantly influenced by both the first and second resonance frequencies of the system, and controlled by the vibration mode of the system. A great difference exists in the response acceleration PSDs, resulting from the great difference of two kinds of input acceleration PSDs around the second resonance frequency. This implies that the response acceleration PSD in stacked packaging units of products depends much on the value of input acceleration PSD around the resonance frequencies. Non-Gaussian data of dynamic contact force appear when Gaussian data of input vibration pass through the stacked packaging units of products, resulting a Weibull distribution of force level-crossing between contact surfaces. Further investigations both in theory and experiment will be worth doing. Because of the limitation of the sampling frequency of pressure-sensitive film, the input excitations of random vibration are limited up to 50 Hz in this research. A more widely response investigation of stacked packaging units of products on input excitations with wide frequency range and different PSD shapes is needed to carry out in the coming days. ACKNOWLEDGEMENTS The authors acknowledge the financial support of this research by the Project supported by National Natural Science Foundation of China, and by the Zhuhai Key Disciplines Enhancement Scheme. The authors also acknowledge the reviewers helpful comments, which largely improve the quality of the manuscript. REFERENCES 1. Rouillard V, Sek MA, Crawford S. The dynamic behaviour of stacked shipping units during transport. Part 1: Model validation. Packaging Technology and Science 2004; 17(5): Godshall WD. Frequency response, damping and transmissibility characteristics of top-loaded corrugated containers. United States Department of Agriculture, Forest Service, Forest Products Laboratory, Research Paper FPL 160, Urbanik TJ. Transportation vibration effects on unitized corrugated containers. United States Department of Agriculture, Forest Service, Forest Products Laboratory, Research Paper FPL 322, Urbanik TJ. Force plate for corrugated container vibration tests. Journal of Testing and Evaluation 1990; 18(5): Bernad C, Laspalas A, González D, Liarte E, Jimenez MA. Dynamic study of stacked packaging units by operational modal analysis. Packaging Technology and Science 2010; 23(3): Bernad C, Laspalas A, González D, Núñez JL, Buil F. Transport vibration laboratory simulation: on the necessity of multiaxis testing. Packaging Technology and Science 2011; 24(1): Thakur KP, Pang D. Simulating complex loading patterns in the stack of packages. Proceedings of the 10th IAPRI World Conference on Packaging, Melbourne, Australia, [accessed 01 January 2016]. 8. Marcondes JA, Schueneman H. Measurement and analysis of dynamic forces within a stack of packages. Proceedings of the 20th IAPRI Symposium on Packaging, San Jose, United states, [accessed 01 January 2016]. 9. Weigel TG. Modeling the dynamic interactions between wood pallets and corrugated containers during resonance. PhD Thesis, Virginia Polytechnic Institute and State University, Blacksburg, United states, Jamialahmadi A, Trost T, Östlund S. A proposed tool to determine dynamic load distribution between corrugated boxes. Packaging Technology and Science 2011; 24(6): Wang Z-W, Fang K, Wang L-J, Lin S-W. Dynamic load distribution of stacked packaging unit subjected to vertical vibration by using a pressure-mapping system. Proceedings of the 27th IAPRI Symposium on Packaging, Valencia, Spain, [accessed 01 January 2016]. 12. ASTM D , Standard practice for performance testing of shipping containers and systems [accessed 01 January 2016]. 13. Bendat JS, Piersol AG. Random Data Analysis and Measurement Procedures, 4th edn. Wiley: New Jersey, 2010.

16 506 Z.-W. WANG AND K. FANG APPENDIX A FORCE LEVEL-CROSSING DIAGRAMS IN RANDOM VIBRATION EXPERIMENTS WITH BAND LIMITED WHITE NOISE LEVELS I, II, III AND IV

17 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 507 APPENDIX B WEIBULL DISTRIBUTION PARAMETER OF FORCE LEVEL-CROSSING IN EXPERIMENTS WITH BAND LIMITED WHITE NOISE I, II, III AND IV noise I noise II noise III noise IV a b a b a b a b No fixed Top Middle Bottom N N Top Middle Bottom Top Middle Bottom

18 508 Z.-W. WANG AND K. FANG APPENDIX C PSDS OF DYNAMIC CONTACT FORCES WITH BAND LIMITED WHITE NOISE LEVELS I, II, III AND IV

19 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 509 APPENDIX D FORCE ROOT MEAN SQUARE VALUES N RMS IN EXPERIMENTS WITH BAND LIMITED WHITE NOISE LEVELS N rms (N) noise I Force root mean square values N rms noise II noise III noise IV No fixed Top Middle Bottom N N Top Middle Bottom Top Middle Bottom

20 510 Z.-W. WANG AND K. FANG APPENDIX E ACCELERATION PSDS OF PRODUCTS WITH BAND LIMITED WHITE NOISE LEVELS I, II, III AND IV

21 DYNAMIC PERFORMANCE OF STACKED PACKAGING UNITS 511 APPENDIX F ACCELERATION ROOT MEAN SQUARE VALUES G RMS IN EXPERIMENTS WITH BAND LIMITED WHITE NOISE LEVELS G rms (g, m/s 2 ) Acceleration root mean square values G rms noise I noise II noise III noise IV No fixed Top Middle Bottom Controlled N N Top Middle Bottom Controlled Top Middle Bottom Controlled