§§ WWI'llilHHHlH1WNW ' THS_ 1%M 716’“ Lust; Madison ““9 MICHIGAN STATE UNIVERSITY LIBRAfillfiml 3 1293 00569 4504 This is to certify that the thesis entitled DEVELOPING A MATHEMATICAL MODEL FOR A 45 DEGREE EDGE DROP TO PREDICT THE DYNAMIC BEHAVIOR OF A LOW DENSITY CLOSED_CELL FOAM presented by Kuang-Nan Taw has been accepted towards fulfillment of the requirements for M.S. , Packaging degree in . ;‘a‘b' / May 20, 1988 I)ate 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES ‘1 \ 5'“: ' 7:73: Ham 3 1991.. NOV $312002 RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. a)? ’/:Z§fd% ABSTRACT DEVELOPING A MATHEMATICAL MODEL FOR A 45 DEGREE EDGE DROP T0 PREDICT THE DYNAMIC BEHAVIOR OF A Low DENSITY CLOSED—CELL FOAM By Kuang-Nan Taw First, a model of a plank of foam cushion is used to develop the usual cushion curves for flat drops. The model is based on the assumption that the dynamic behavior of a low density closed-cell foam under impact is governed by the adiabatic compression of trapped air within the cells. The mathematical analysis consists of an energy balance and an application of Newton's Law. Next, the model and analysis are extended to 45° edge drops by observing the stress-strain behavior of an edge of a cushion under static compression and identifying an isolated compression region. The result is a fairly accurate model which is shown to agree with actual results to within experimental accuracy. The relationship between the thickness of the foam and a key parameter referred to as the stretch ratio of the foam is carefully examined. The limitations of the new model with respect to the assumed geometry of deformation of the foam under dynamic loading is then discussed. ACKNOWLEDGMENTS I would like to thank Dr. Gary J. Burgess for his time, friendship, understanding, encouragement, and most of all guidance throughout the study while acting as my advisor. The appreciation is beyond words and will never fade away. I would also like to thank Dr. George E. Mase for his support and tender care while serving as a member of my committee. Special thanks to Dr. Julian Lee, a member of my committee, whose thoughtfulness, advice and friendship will always be with me for the years to come. I am also grateful to Dr. Jack R. Giacin for his constant greeting and kindness which is a resource of my comfort and strength in these years. TABLE OF CONTENTS PAGE LIST OF TABLES .............................................. v LIST OF FIGURES ............................................. vi LIST OF SYMBOLS ............................................. vii CHAPTER 1 ................................................... 1 INTRODUCTION ........................................... 1 CHAPTER 2 ................................................... 4 THEORY AND ANALYSIS .................................... 4 2,1 Stress-Strain Model .............................. 4 2.2 Adiabatic Model for Flat Drops ................... 6 2.3 Adiabatic Model for the Edge Drop Situation ...... 7 CHAPTER 3 ................................................... 26 TEST PROCEDURE, MEASUREMENT ERRORS, AND RESULTS -~'--°'- 26 3.1 t Procedure co.ocu.o.Ioooo-oooo-ccaooolooooccoc 26 1 Material 0.000.0coocooo-ocoouoocoooco-ucocoo 26 2 Apparatus Icoococoo.ooooooooaoocooooco-ocoo- 26 3Test Procedure no...ooooooaoooooooooooaoonco 27 3.2 Measurement Errors 0.0.0.0....OIIOOOIOOOOIOCOOOIOO 29 3.3 Results 0'0...oocccocncoooooaococoooolcocooccoooo- 31 CHAPTER 4 aooooooooooooooooooocooooooooccoococo-ooo-0.0000000 35 DISCUSSION AND CONCLUSIONS ............................. 35 APPENDICES APPENDIX A: ADIABAIC MODEL FOR FLAT DROP -‘-'---~'00---- 40 APPENDIX B: SPECIFICATIONS ............................. 43 APPENDIX C: TABLES FOR DATA AND RESULTS -------------°-- 50 APPENDIX D: COMPUTER PROGRAM ........................... 66 REFERENCES O..00.00'....0..IOOOOIIIIOOIOCOOOOOOO0.00.00.00.00 67 TABLE LIST OF TABLES PAGE Static edge compression test results for the ....... 50 3.125" and 4.75" thick specimens. Edge contact length is 8" Data for 3.125" corner-to-corner thickness ........ 51 for Ethafoam for first drop Data for 4.75" corner-to-corner thickness -'°--°'--- 53 for Ethafoam for first drop Data for 3.125" corner-to—corner thickness --~--°-" 55 for Ethafoam; 2nd through 5th drops Data for 4.75" corner-to-corner thickness ----o'---' 58 for Ethafoam; 2nd through 5th drops Results for 3.125" corner-to-corner thickness ...... 63 for Ethafoam under edge drop Results for 4.75" corner-to-corner thickness ....... 64 for Ethafoam under edge drop LIST OF FIGURES FIGURE PAGE 1 Flat drop Onafoam plank coo-coco.ooonocntuoooooooo 18 2 Compression behavior of Dow Ethafoam 220 ........... 19 LDPE and various models [9] 3 Compressive load-deflection of 2 pcf LDPE foam [9] .. 19 4 Dynamic compression curves computed for ............ 20 2 pcf LDPE foam [10] 5 Parameters used in the flat drop analysis .......... 21 6 Comparison between the adiabatic model and .......... 22 actual results [12] 7 Parameters used in the edge drop analysis ------°--- 23 8 Deformation of a closed-cell foam under ............ 24 static edge compression 9 Force imbalance during edge drop compression ....... 25 on a corner block of foam 10 Illustration of cushion test equipment ............. 34 11 Analysis of progressive foam deformation under ----- 39 dynamic loading vi LIST OF SYMBOLS intial pressure before compression, (psi) final pressure after compression, (psi) ratio of specfic heats, (1.4 for air) initial volume before compression, (in3) final volume after compression, (in3) initial temperature, (0R) maximum temperature, (OR) maximum cushion stress, (psi) cushion static stress, (psi) maximum cushion strain, (in/in) product weight, (lbs) drop height, (in) contact area, (in2) edge contact length, (in) thickness of foam, (in) peak deceleration, (g's) acceleration due to gravity (386.4in/sec2) o radius of compression boundary before compression, (in) radius of compression boundary after compression, (in) diameter of undeformed cells, (in) length of collapsed cells, (in) stretch ratio potential energy, (lbs-ft) m dynamic deflection, (in) force, (lbs) cushion factor, (lb/in2)1/2 membrane tension, (lb/unit length) sum time, (sec) viii CHAPTER 1 INTRODUCTION Cushion design consists of two parts: the determination of the properties of cushion materials such as density, creep, stiffness, and percent recovery, and the analysis of the entire packaging system which uses the cushion based upon the laws that govern the motion of accelerated bodies in a cushioned drop [1, 2,3,4]. This thesis deals with the former and concentrates on the dynamic properties of low density closed-cell cushions used in edge drop situations. The dynamic properties of cushions have been presented in the form of "cushion curves" which relate the kind of cushion, its thickness, the weight of product on the cushion, and the contact (impact) area to the peak acceleration transmitted through the cushion. Cushion curves for most commercially available cushions based on experimental results [4,5] have been compiled. MIL-HDBK-304B and ASTM D 1596-78a [5,6] are two principal specifications outlining the experimental procedure for the construction of cushion curves. Essentially, all published cushion data obtained using a drop tester are for flat drop situations while in fact, edge or corner drops occur most often throughout the distribution (handling and storage) environment in the real world. Some researchers did offer an explanation of the effects of orientation in an edge drop on a cushion and produced some analytical results [7,8], but the mechanical behavior of the cushion itself under dynamic loading has not yet been discussed. Surprisingly, no attempt has been made to quantify edge-drop dynamics in order to develop a procedure for total cushion design of the packaging system. Because of the many types of cushions available and the numbers of parameters which determine their properties, large volumes of cushion curves are required as a data base for a thorough cushion design. It is natural at this point to attempt to reduce or even substitute the immense data base by using a mathematic model either for the cushion curves themselves or for the cushion material and then solving the dynamics problem. The material modelling approach has been the focus of recent research [9,10,11,12]. The stress-strain behavior of a cushion was first experimentally examined and an empirical model that assumes adiabatic compression of the gas in a closed-cell foam was proposed [9]. This model coupled with a simple energy balance approach can be used to predict dynamic cushion behavior and is shown to reproduce the published cushion curves fairly well [9,10]. The adiabatic assumption however has its conceptual shortcomings as a dynamic model of a closed- cell foam since heat transfer seems to be substantial and therefore cannot be ignored [12]. Nevertheless, the adiabatic model does appear to produce adequate results. Since the basic adiabatic air compression model and its refinements have previously been applied to only flat- drop conditions, the main purpose of this thesis is to extend the use of the adiabatic theory and the energy balance approach to edge drop orientations on low density closed-cell foams. Beginning with chapter 2, the basic adiabatic compression model is reviewed and the simple energy balance approach which uses this in a flat-drop situation is then introduced. Because of the uncertainty in the amount of contact area and foam volume actually involved in the dynamic compression process in an edge drop situation, it is very difficult to apply the same steps used in the flat-drop analysis to edge-drop case. By observing a simple static edge compression test on a corner block of foam, an isolated compression region can be identified. When Newton's law is applied to this region using the adiabatic model and an energy balance, an accurate empirical model for edge drop situations is produced. It is not surprising to find that contact area is no longer one of the parameters that determines shock transmission in an edge drop, but the length of the cushion in contact with the product which is important. There is a new term defined here as the "cushion factor" which is produced in the analysis and is used only in the edge-drop model. The dynamic deflection of the cushion in an edge drop situation is also discussed in the context of this empirical model. In chapter 3, the details of the experiment and the results from the drop tests are presented. Measurement errors are also identified. Finally, a discussion of the isolated compression region in relation to limiting dynamic deflections and actual cushion thicknesses is presented in order to identify the limitations of the new model and to suggest modifications for future research. CHAPTER 2 THEORY AND ANALYSIS To introduce some of the modelling features and the method of analysis that will be used in analyzing the edge drop situation, the simplest situation of a flat drop as shown in Figure 1 will be considered first. The entire flat drop impact consists of three stages: the free fall, the contact (cushion compression) phase, and the rebound phase. What we are most concerned with here is the reaction of the foam during the second stage, the contact phase, which results in the dynamic compression loading of the cushion. The analysis of this stage involves several fundamental concepts such as Newton’s law of motion, conservation of energy, and the stress-strain behavior of the cushion. The stress-strain behavior of closed-cell foams under compression is the only new information required and so will be covered first. 2.1 Stress-Strain Model The nature of foam compression under dynamic loading is very complex because of the possibility of cell rupture under even moderate loading. An energy balance between the amount of energy needed to compress the air in the foam and that needed to tear it must be considered. Since dynamic stress-strain behavior is expected to be closely related to static stress—strain behavior because of the small mass of the cushion, the best place to start is by observing the stress-strain behavior under static loading. 4 5 Throne and Progelhof [9,10] have authored a series of papers in which the stress—strain behavior of closed-cell foam under dynamic loading is examined. In their first paper [9], they carefully review and compare various stress-strain models proposed for static loading and compare them to actual results (see Figures 2 and 3). The best model proposed which accounts for the observed static compression properties is that of air trapped in closed cells which undergoes adiabatic compression according to P V — = < —°>7 (1) PO V where 7 is the ratio of specific heats of air, taken to be 1.4, PC and V are the initial pressure and cell volume, and P and V are the pressure and cell volume at any other time during compression. In their second paper [10], they apply this model to the case of an object dropped on a cushion. In a simple energy balance, the maximum potential energy available in the drop is converted into energy stored in the foam at maximum compression (see Figure 5). The energy stored in the foam per unit volume of material at maximum compression is just the area under the static stress-strain curve and the potential energy is just object weight times drop height, so this energy balance yields 6 m0(e) de = 0 At (2) where A is the flat drop contact area, t is the thickness of the foam, W is the weight of the object, H is the drop height, and 6m is the maximum cushion strain. At this point, the relationship between stress and strain is unspecified. Furthermore, since the maximum stress am corresponding to the 6 maximum strain 6m is just am - 0(em), the peak deceleration, G = a/g, may be obtained from Newton's law, a amA — W = W(—-—) (3) — 1 (4) Equations 2 and 4 allow for G to be determined directly from the static stress-strain curve of a specimen of foam material without having to model the cushion material. This has been done with actual stress-strain curves with good agreement between the peak decelerations predicted by equation (4) and actual results (Figure 4). This fact establishes that the static stress-strain curve can be reliably used in the dynamic case. I 2.2 Adiabatic Model for Flat Drops Now since the adiabatic model has been shown to accurately describe the static stress-strain curve, it may be used directly in equations (2) and (4). It is more instructive however to resolve the problem from first principles. This has been done in Appendix A. The results for the peak deceleration are G=——14.7 (1+R)3'5 (5) a where a = W/A is the static stress and a H R = 0.027 ( 1: Note that the parameters H, t and a used to determine G are exactly the parameters which are used in the traditional cushion curves. Figure 6 compares the predictions of equation (5) for the 7 adiabatic model to the actual cushion curve data for Ethafoam® 220 LDPE foam for a drop height of 24 inches. The agreement between the predictions and the actual data is quite good considering the simplicity of the model. Most refinements to the basic adiabatic model are based on arguments involving the liklihood of strain rate dependency on stress- strain properties. In an attempt to take strain rate dependency into account in the stress-strain model, Bigg [10] proposed a more accurate but more complex model based on empirical dynamic stress-strain relationships. Burgess [12] however pointed out that heat is transferred from the air to the foam cell walls during compression which places the entire adiabatic model in question. Since the net effect of heat transfer is to dissipate energy continuously over the duration of the impact, it may turn out that the observed effect of this process is equivalent to that of a mechanical strain rate dependent term. Nevertheless, in spite of the differences that exist between the different models, the basic adiabatic model still yields acceptable results. This fact and its inherent simplicity are strong arguments in favor of retaining the basic model for the edge drop situation. The main goal of this paper then is to develop the expression for the peak deceleration in an edge drop situation using this model. What complicates the analysis is the fact that the geometry of deformation of the edge of a cushion is more involved than the simple uniform compression of a plank cushion in a flat drop situation. 8 2.3 Adiabatic Model for the Edge Drop Situation In a general edge drop, the center of gravity of the cushioned product will not lie directly over the edge of the cushion during impact. The potential energy converted into product kinetic energy just before impact then is not entirely converted into stored strain energy in the cushion. Some of this will go into compressing the edge and the rest will go into rotating the product. Because the worst situation as far as cushion compression is concerned is expected to occur when the product's center of gravity is directly over the edge, a 45 degree edge drop on the edge of the cushion as in Figure 7 is the only case analyzed in this research. Note that both the contact area between the edge and the ground and the actual cushion volume involved at maximum compression in Figure 7 are unknown beforehand in contrast to the flat drop situation. Before proceeding with the stress-strain model, it is therefore necessary to gain a better understanding of the geometry of deformation of the edge. Static Edge Compression Analysis By simply observing the deformation of the edge of a block of foam under an increasing static load on a compression tester, an interesting and useful result is obtained. As in Figure 8a, two points A and C at a distance Ro from edge D are first marked on the cushion and a straight line ABC is drawn before compression. When the edge is compressed until points A and C contact the platen of the compression tester, the straight line ABC is seen to bend into the quarter circular arc ABC shown in Figure 8b. This means that all of the foam within the area in 9 triangle ACD before compression gets compressed into the circular sector ABCA and line ABC in Figure 8a and forms a boundary beyond which no appreciable deformation occurs. In fact, closer observation shows that those cells just outside the boundary (arc ABC) in the unshaded area undergo very little compression. Since the arc ABC acts like a membrane composed of cell walls which separates a high pressure region (compressed sector ABCA shaded in Figure 8b) from the low pressure region outside the shaded area, nearly all of the strain energy stored in the foam occurs within this small region. The effect of the movement of cells outside the boundary ABC on the shock transmitted will be discussed in Chapter 4. Now, when the edge is compressed, the high pressure region inside sector ABCA will attempt to stretch the cells on boundary. There are two extreme cases: 1) The boundary does not elongate, which means that straight line ABC in Figure 8a bends directly into arc ABC in Figure 8b without changing length. Then the radius of the circular arc in Figure 8b satisfies n 2/2 R=J§Ro or R=-—-————R (6) N | a o 2) The cells undergo maximum stretch wherein a cell, considered to be a sphere of diameter D collapses into a thin plane as shown below. undeformed cells on straight collapsed cells on deformed line ABC (Fig. 8a) are ABC (Fig. 8b) 2 D l X 10 In the collapsed state, x is just half the circumference of the sphere, or x - «D/2. Then length of deformed arc ABC n x flat cell width length of straight line ABC n X spherical cell diameter x n =—= <7) D 2 where n is the number of cells along the boundary ABC. Now, from equation (7), the radius R in Fig. 8b must be length of deformed arc ABC (length of straight line ABC) X n/2 R = = n/2 fl/Z -= R077 (8) Combining equations (6) and (8), R0 x 2/§/« s R s ROJE'. (9) Setting R = k Ro , (10) 2f2'/1r _<. k 5 J7 (11) 0.9 s k 5 1.414 The lower limit on k is for no stretch and the upper limit is for maximum stretch. Since k is a measure of the degree of stretch of the cells along the boundary, it will be referred to as simply the "stretch ratio" and hopefully will be a specific property of the foam. Next, consider the adiabatic compression of the trapped air in region ACD at atmospheric pressure P0 in Figure 8a into the smaller region ABCA in Figure 8b where the pressure is P. Using equation (1), R 2 L w l P (———9-———)7 = P (—-— R2L — ——— RZL)7 (12) ° 4 2 where L is the edge contact length (see Figure 7). 11 Using equations (10) and (12) and solving for P gives 2 p-p[ 17 an O (« — 2)k2 This result says that the pressure P in region ABC in Figure 8b is independent of the initial volume of air compressed. At this point, there are a sufficient number of assumptions made and conclusions reached to warrant a simple static edge compression test on a cushion to check some of these before proceeding any further. The material selected was Ethaform 220®, a low density closed—cell expanded polyethylene foam made by Dow Chemical, with an edge length of L - 8" and two different thicknesses, 3.125" and 4.75", measured from the edge to the inside corner. Before compression, a series of equally spaced lines (0.2" apart) parallel to ABC in Figure 8a were drawn on both specimens. At each stage of compression where the endpoints of one of the parallel lines touched the platen, values of force F and compression x were recorded as well as the dimensions b and h of the quarter circular compression boundary ABC in Figure 8b. From these values, stress a and strain 6 were determined according to a = F/bL and e - x/t where t is the thickness. In addition, the stretch ratio k was obtained by measuring R (Figure 8b) for a given RO (Figure 8a) and using equation (10). The value of R0 is a multiple of 0.2/2 because of the spacing between the parallel lines drawn on the cushion before compression and the value of R may be determined from the values of b and h using the figure below: /’N tv2 R ‘P—H 12 R2 - (R - h)2 + (b/2)2 R - h/2 + b2/8h (14) The results of the compression test for F, x, b and h and the calculated values for a, e, and k for both specimens are shown in Table l in Appendix C. The data clearly shows that the stretch ratio does in fact appear to be a deformation property of the cushion with respect to edge compression since it is nearly constant and falls within the expected limits set in equation (11). From Table 1, the average stretch ratios are 0.91 for the 3.125" thick cushion and 0.94 for the 4.75" thick cushion. Apparently, the compression boundary stretches very little since the values for the stretch ratio are closer to the 'no stretch' limit of 0.9 than to the 'maximum stretch' limit of 1.414. The predictions of equation (13) are also verified in two ways. First, it predicts that the pressure P in the cells contacting the platen is independent of the amount of compression. Since the stress exerted by the foam on the platen is just this constant pressure less atmospheric pressure (because the load transducer was zeroed for atmospheric pressure), the stress-strain curve should be a horizontal straight line. From Table l, the stress a is evidently nearly constant with an average value of 19.4 psi for the 3.125" thick specimen and 20.5 psi for the 4.75" thick one. Second, equation (13) predicts that these constant stresses are related to the stretch ratios. Using P = a + P0 and solving for k in equation (13) gives 2 P ‘ k = J// ( ° )1/7 (15) (w-2) a + P , 13 where Po - 14.7 psi, 1 - 1.4, and a is the constant (average) measured stress value. Using a - 19.4 psi in the above for the 3.125" thick specimen gives k = 0.98 and again with a = 20.5 psi for the 4.75" thiCk specimen gives k - 0.97. Outlined above are two practical methods for determining the stretch ratio. The first involves drawing a straight line ABC on the cushion at any radius Ro as in Figure 8a and then measuring the deformed arc parameters b and h in Figure 8b. Equation (14) gives the radius R of the compression boundary and equation (10) then gives the stretch ratio k. The second method requires either a compression tester or an appropriate fixture which allows for a force F to be applied and the arc base b to be measured. Equation (15) may then be used with a = F/bL. Since the results for k from the first method (0.91 for 3.125" thickness and 0.94 for 4.75" thickness) are somewhat different from the results for the second method (0.98 and 0.97), there appear to be some minor problems either with the theory, the measurements, or both. 0f the two methods, the first is expected to be more practical and more accurate simply because of the steps involved and because the second method is based on a consequence (equation (13)) of the first. The values k - 0.91 for t = 3.125" and k = 0.94 for t = 4.75" will therefore be used in what follows. Having demonstrated that the basic ideas behind the model (compression boundary concept and adiabatic air assumption) are essentially correct for the static case, all that remains is to use the model in the dynamic case of an edge drop. To do this, the static stress-strain curve will be used and an energy balance in the drop will be performed. 14 The energy stored in the entire volume of foam in Figure 8b at maximum compression during the impact consists of two parts: 1) the energy stored in the compressed shaded sector ABCA, Ul - — f P dV. Since P is independent of volume, U1 - P (V From equation (13), initial — vfinal) 2 R02 L n l 2 U1-%[ ZIU————-(——-—HRM (« - 2)k 2 4 2 or, using the stretch ratio, 2 2 l r l 2 U1 - PoRo L I 2 17[-—— - <——— — ———>k 1. (16) (n - 2)k 2 4 2 2) the energy in remainder of the foam outside arc ABC, Since the region outside ABCA is essentially a much large volume of air at normal atmospheric pressure P0 = 14.7 psi, U2 - POAV where AV is just the decrease in volume of the air outside straight line ABC. But this is just the volume associated with region ABCA in Figure 8b, so that n 1 2 2 U - Po(-— - -——)k Ro L . (l7) 2 4 2 The total energy stored in the entire system then is U = U1 + U2 or 2 U - PoRo L X Q (13) 15 2 1 fl 1 2 n 1 2 where Q - [ 2 171- - (— - —)1< 1+<— - —)k . (19) (x - 2)k 2 4 2 4 2 Having obtained the expression for the energy stored in the foam in a 450 edge compression situation in terms of degree of compression R0, edge contact length L, and stretch ratio k, an energy balance for the drop may now be performed. Equating the initial potential energy WH to the stored strain energy (equation (18)) gives POROZL x Q - WH WH so that R - (20) O 0 Equation (20) predicts the degree of deformation associated with dropping a weight W from height H onto a cushion corner as in Figure 7. A better measure of the degree of deformation is the dynamic deflection dm defined to be the maximum vertical compression of the cushion, 2WHk2 dm - R 72 - k R072 - —_P_;E- . (21) o If the predicted dynamic deflection is close to the actual thickness of the cushion from outside corner to inside corner, then the arguments used in arriving at this result are likely to break down. Next .consider the force imbalance at maximum compression in Figure 9. Using Newton's law, {FT-(P—P0)Rf2‘L—W=WXC. From equations (10) and (20), WE R - k (22) P LQ O and from equation (13), 16 2 7 kL ZWH ] — P l-—-- ° w G - {P [ — 1 (23) ° (N — 2)k2 POLQ HL or G - c -—— , (24) f N where C, referred to as the 'cushion factor' in this thesis, is 2 2P ———————-§]7 — l} (n - 2)k Q c = k{[ ° (25) Note that C depends only on the stretch ratio (k) and air properties (Po and 7). The last term (-1) in equation (23) has been dropped in equation (24) because it is small compared to the term shown. Equations (24) and (25) form the cushion curves for 45 degree edge drops using the adiabatic air compression model for the stress-strain characteristics of closed-cell foams. For air, P0 = 14.7 psi and 7 = 1.4. Using these values in equations (19) and (25) gives 22.25k—1'8 — 10.15k 1b 0 = 2 (26) in 7/3.841k'2°8 - 2.1921.<’°°8 + k2 In view of the limits on k from equation (11), the limits on C are evidently —l.815 s C 5 9.382 where the lower limit on C corresponds to the upper limit on k. Since C cannot be negative as this gives physically meaningless results for G, the upper limit on k may be reduced from its maximum stretch value of 1.414 to a value consistent with physical reality. Setting C = 0 in equation (26) gives this new upper limit as k = 1.324. In other words, the geometric limits on k (0.9 s k 5 1.414) are less restrictive than 17 the dynamic limits (0.9 S k 5 1.324), which ultimately require that 0 S C 5 9.382 (27) I I 1.324 2 k z 0.9 Because of the limits on k and the corresponding limits on C, the average variation in C with respect to k is (9.382—0)/(0.9—1.324) = —22.13, which means that C is fairly sensitive to k. For example, if the expected error in the calculated value of k based on measuring the deformed arc parameters b and h in Figure 8b is 0.04- (see Table 1), then the error in C is as much as 0.04(22.13) = 0.89. Since the actual value of k is very near 0.9, the corresponding value of C is near 9.382, so that the percent error in C is approximately (0.89/9.382) x 100 z 10%, which then translates into a 10% error in G. This being the case, it may be more appropriate for design purposes to simply use the upper limit of C - 9.382 in equation (23) in order to establish an upper limit on G in the edge drop. The predictions of equation (23) will now be checked against experimental results in Chapter 3 with emphasis on the following properties: 1) G varies as JR, JL and 1/JWI 2) Actual G's are less than 9.382 JHL/W. 18 Stage 2: Contact (Compression) closed cell aw foam cushion Stage 3: Rebound b-------q Stage 1: Free Fall product weight V dm, surface Figure 1: Flat drop on a foam plank. //7//////7 19 1.0 Rusch 0 actual 0.8- (LDPE) -— theory Throne ‘ ° 6 0.6. (LDPE) 0.4. Gent/Thomas (Natual Rubber) 0.2- 0 j fl 1 I l 1 l 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 f(e) Figure 2: Compression behavior of Dow Ethaform 220 LDPE and various models. [9] 1.0 0.8- q 0'4 . actual O ./ O I V F T 1 I l 1 1 O 10 20 30 40 50 60 70 80 90 100 0.24 Stress 0, (psi) Figure 3: Compressive load-deflection of 2 pcf LDPE foam. [9] G, Average deceleration (g) 20 h: drop height t: thickness of foam 180 h/t 160- ' 24 140 120- 100- 80 60‘ 40. 20. 0 .4 .5 1.2 1:6 2fo 214 2.8 Static stress 0, (psi) Figure 4: Dynamic compression curves computed for 2 pcf LDPE foam. [10] 21 BLOCK W \\ AREA A / Pi FOAM Figure 5: Parameters used in the flat drop analysis. 22 H - 24" t = 1 inch 120 _ Adiabatic Model ,\ 100 ' ---’-_“- (0 :9 Actual Results g; 80 ‘ S -H 2 inch 1; / p 60" w H o o 8 40- .x w o 9.. 20 ‘ 0 I 1 I I I r“ 0 .5 1 0 1.5 2.0 2 5 3.0 Stress a , (psi) Figure 6: Comparison between the adiabatic model and actual results. [12] 23 W center of gravity inside to outside L ._______. corner ‘\\~l thickness 3: Figure 7: Parameters used in the edge drop analysis. 24 A \ f/f/ / D \’7 fixed platen \/RO (a) Contact fixed platen R (b) Maximum Compression Figure 8: Deformation of a closed-cell foam under static edge compression. 25 //// f/f/ 111th Figure 9: Force imbalance during edge drop compression on a corner block of foam. CHAPTER 3 TEST PROCEDURE, MEASUREMENT ERRORS, AND RESULTS ,;;l_lg§t Procedure 3.1.1 Material The cushion material used in this research was Ethafoam® 220, a low density expanded polyethylene foam with a density of 2.2 lb/cu ft made by Dow Chemical. Drop tests from heights ranging from 16" to 55" were performed on right angle corner blocks of cushion as in Figure 7 with edge contact lengths of 4", 6" and 8" and corner to corner cushion thicknesses, hereafter referred to as just thickness, of 3.125" and 4.75" corresponding to nominal plank thicknesses of 2" and 3 5". 3.1.2 Apparatus The test apparatus used was standard drop tester consisting of a free falling platen and rigid impact surface to simulate impacts onto hard surfaces. The signal from a piezoelectric accelerometer mounted on the falling platen was fed into an amplifier and then into an oscilloscope in order to obtain the complete history of the impact shock pulse for later use in analyzing peak deceleration. The arrangement is shown in Figure 10. The relevant specifications of the equipment used are briefly described below. 1) Cushion Test Machine: the drop tester used was a Lansmont Model 23 cushion tester equipped with a free falling platen, weight ballast attachments, platen brakes and a lifting mechanism. The free falling platen may be loaded with various weights, hoisted to 26 27 a designated drop height and then released. The loaded platen was also fitted with a corner block of wood around which a cushion of uniform thickness made from two flat planks was placed. The entire arrangement was then dropped onto the stationary rigid base platen of the cushion tester. 2) Accelerometer: a Kistler model 818 piezoelectric accelerometer was mounted on the free falling platen. A piezoelectric material is one which produces a charge proportional to the deceleration it experiences thereby allowing the entire time history of deceleration of the free falling platen to be recorded. The advantages of a piezoelectric accelerometer compared to other types such as a strain gage are high frequency response, large output, ruggedness and ease of operation. 3) Coupler: the role of the coupler in relaying the accelerometer output to the oscilloscope is to supply the piezoelectric accelerometer with a constant excitation current. A model 5116 AC coupler made by Kistler Instrument Corporation was used. 4) Oscilloscope: a model COS 5020-ST by Kikusui International Corp. equipped with storage capability was used to capture the shock pulse. More details of the specificatiOns for these instruments are presented in Appendix B for later use in the discussion on measurement errors. 3.1.3 Test Procedure The drop test procedure was performed according to the standardized procedure outlined in MIL-HDBK-304B. Two test specimens 28 with same dimension (edge contact length and thickness) were prepared for dropping under same weight and height. A series of five drops were performed for each sample at a predetermined drop height and weight unless one of the specimens was broken in earlier drops. A one minute waiting period was allowed between drops in order to let the cushion recover. The same procedure was repeated with several different weights and drop heights. The series of weights and drop heights for the two thicknesses chosen are shown below. thickness 3.125" 4.75" weight (lbs) drop height (in) drop height (in) 17.5 16.625 13.9875 36.25 34.125 52.875 53.125 17.0 17.25 20.3875 33.625 32.75 51.625 52.0 16.625 17.0 26.7875 33.75 33.75 ------ 51.75 The experimental results for the peak decelerations produced during impact under these conditions are presented in Tables 2 through 5 in Appendix C. Tables 2 and 3 are the data for first drops only and Tables 4 and 5 list the peak decelerations for the second through the fifth drops. Only the first drop data will be compared to the predictions of the adiabatic model. There are two reasons for this. First, it is customary to do so for design purposes since if damage occurs, it is likely to do so on the first drop and second, the effect of dropping an object onto a cushion in an edge drop is to 29 progressively deteriorate the cushion by breaking cell walls. This phenomenon is not yet understood well enough to attempt to model it. 3.2 Measurement Errors The measurement errors associated with this experiment come from the cushion tester, the accelerometer, the coupler and the oscilloscope. The individual percent errors associated with each instrument from Appendix B are: Accelerometer: i2% This means that the rated output voltage of 10.09 mV/g is really 10.09 i2% mV/g. Coupler: i5% This means that the signal from the accelerometer may be modified by as much as i5% by the coupler. Oscilloscope: 13% This means that the oscilloscope displays an input voltage as a vertical height on the CRT grid with an accuracy of i3% on height. An additional source of error comes from the fact that the width of the beam on the screen when the oscilloscope is operated in the storage mode is approximately 0.1 division. Then the peak height of the shock pulse may be in error by the same amount and since the typical vertical sensitivity used was 200mV/div., the error incurred in just reading the peak height off the grid is i0.1(200) = i20 mV. Using 50 g's as one of the higher G-levels recorded which would show up on the oscilloscope as a 50(10.09) = 504.5 mV signal, the reading error becomes i20/509.5 = 0.04 mV/mV or 4%. 30 Total error; Since the pulse height, oscilloscope vertical sensitivity, and accelerometer/coupler calibration values are multiplied together to get peak deceleration, the accumulated pulse measurement error is just the sum of the individual percent errors. The total error in measuring peak deceleration level can therefore be as much as i(2+5+3+4)% = 114%. Drop Tester Indicted Drop Height: -2% The indicated drop height obtained from a scale mounted on the drop tester is not a true free fall drop height because of friction between the guide columns on the drop tester and the falling platen. The amount of friction depends on the alignment of the columns and on the amount of lubrication. By measuring the impact velocity Vi using two microswitches separated by approximately one inch which engage the falling platen just before impact, the true drop height may be calculated as h - Viz/2g. Experience has shown that the true free fall drop height is generally less than an inch smaller than the platen drop height indicated on the machine. A conservative measure of the drop height error then is —1"/50" or -2%. In other words, the true drop ht is between hi - 2% and hi where hi is the indicated drop height. Cushion Material' ? Probably the biggest source of uncontrollable error is the variation between cushion samples. Since Ethafoam 220_ consists of closed cells which range in diameter from 0.01" to 0.1"(by inspection), it is highly likely that one sample may contain predominantly larger 31 cells while another contains predominantly smaller cells. The effect of cell size is to increase G-levels for smaller cell size samples since more of the cushion material itself is compressed which in turn extracts more energy from the falling mass than would a sample with less material (large cell size). Because of the uncertainties involved in the foam manufacturing process which directly affects cell size and the complexities that arise in determining how cell size affects peak deceleration, no specific error may be associated with the material. In view of the errors associated with measuring the peak decelerations (i 14%), the uncertainty in true drop height (—2%), and the variation in cushion material, it is entirely likely that the experimental G—values reported in Tables 2—5 in Appendix C may be in error by as much as i 16%. This in turn means that as much as 16% of the variation between the actual data and the predictions of equation (23) may be the result of measurement error. One of the criteria for judging accuracy of the model will therefore be taken as agreement between the actual and predicted values to within i16%. 3.3 Results The actual data for the first drop peak deceleration in Tables 2 and 3 will be compared to the predictions of equation (24) repeated here for reference, G - c ———— (24) where C, the cushion factor, depends only on the stretch ratio k 32 through equation (26) and lies within the range 0 to 9.382. The first prediction which will be checked is the dependence of G on the parameters H, L and W. To do this, a power law equation, 0 - CHXLVWZ (28) will be fitted to the data in Tables 2 and 3 by minimizing the variance between the natural logs of the actual C values and those obtained from equation (28), N VAR - 2 [In C + xln Hi + yln L + zln W. - 1n 0.]2 - minimum (29) 1-1 1 1 i where N is the number of data, (Hi’ Li’ Wi, Gi)’ i = l to N. Minimization requires that the partial derivitives of the variance with respect to each of the unknowns, C, x, y, 2, be zero, which leads to the system of linear equations shown below in matrix form, / \ N Zln(Hi) 21n(L.) 21n(w.) \ ’ln(C) r[2111(0) \ 2 l 1 1 Z(ln(Hi)) Zln(Hi)ln(Li) 21n(Hi)1n(wi) 2 x = Zln(Hi)ln(Gi) (30) 2 Z(ln(Li))‘\\‘\~Zln(Li)1n(Wi) y Zln(Li)ln(Gi) (Symmetric) Z(ln(W.))2 z 21n(w.)1n(c.) \ 1 J - K 1 1J The solutions to this system of equations for the two samples studied are t - 3.125" : C=3.846 x-0.688 y=0.l85 z= —0.300 t - 4.75" : C-3.182 x=0.763 y=0.071 z= —0.185 The expected solution for both cases is from equation (24), x-y=0.5 and z- —0.5, with C somewhere between 0 and 9.382. The discrepancy in the powers x,y,and z is not necessarily an indication that the model is 33 incorrect because of the fact that the experimental values for G which enter the right hand side of equation (30) may each be in error by as much as i16% and the values for H on the left hand side by as much as —2%. Since it is difficult to assess the effects of independently varying each of the N G1 in equation (30) by up to il6% and the N Hi by up to —2% on the solution (C,x,y,z), no further remarks concerning the power law fit to the data will be made. Instead, the model itSelf will be compared to the data in Tables 2 and 3 using equation (24). From the previous static edge compression tests (Table l), the stretch ratios for the 3.125" and 4.75" thick cushions were determined to be 0.91 and 0.94 respectively. From equation (26) then, C = 9.2 for the 3.125" thick cushion and C = 8.64 for the 4.75" thick cushion. The program in Appendix D uses these values in equation (24) to compare the model to the actual results by determining an average percent error, E |(Gcal — Gexp)/Gexp| / N. The results are shown in Tables 6 and 7 along with the calculated values for R0 and dm. The program results for the 3.125" thick cushion show that the average percent error of 11.92% is well within the limits of experimental error (il6%) set earlier but that 32.5% of the actual C values exceeded the theoretical upper limit on calculated C values using C — 9.382 in equation (24). This latter discrepancy may however be the result of the il6% measurement error on G. For the 4.75" thick cushion, the average percent error is 21% which is still within experimental error since the total error range corresponding to il6% is 32%. The model is therefore considered to be accurate within reason based on these results. 34 H1] i“* , 1 lifting mechanism 2 weight attachments 3 free falling platen 4 wooden corner prototype 5 test specimen 7 (corner shaped foam) 6 stationary impack base I 2 I / {4? 7 accelerometer \5 8 coupler or amplifier 9 oscilloscope [6 ' *2;— W J I __ Figure 10: Illustration of cushion test equipment. CHAPTER 4 DISCUSSION AND CONCLUSIONS Although the adiabatic model applied to the dynamic compression of foam in a 45 degree edge drop impact predicts peak deceleration levels fairly well, there are still a few points which need to be discussed. In general, the thicker the foam, the softer and more flexible it tends to be and therefore the more energy it can absorb. In an impact involving identical weights and drop heights, the thicker foam will undergo the smaller cell compression. The compression boundary itself expands in the thicker foam which allows more cells to move and absorb the impact energy. It is therefore expected that the stretch ratio k will be a function of thickness with the cells along the boundary under less tension as the thickness is increased. To show this, consider the compression boundary AB in Figure 11 which separates the high pressure region (shaded) from the low pressure region by assumption. From equation (13), the pressure in the compressed area contained within the arc AB is 2 P = P [ 17 V (13) O (n — 2)k2 regardless of the size of the arc. This means that arc A'C' in Figure 11 formed under higher loading conditions contains cells under this same pressure. The region outside either arcs AC or A'C’ is still under atmospheric pressure P0' The tension along the boundary may now be estimated by treating the boundary itself as a thin membrane under 35 36 tension separating high and low pressure regions. From the statics of membranes [13], the relationship between the pressures on either side of the membrane and the membrane tension T lbs per unit edge length is T - (P - Po)R . Substituting for P from equation (13) and R from equation (22), and using equation (19) gives 27 WHP 2 — l] x k x 0 (1r - 2)"k 7 L T - (31) 2 1 x l 2 fl 1 2 [ 2 17[- - <— — —>k 1+(— — —)k (1r — 2)k 2 4 2 4 2 It can be shown that T is a decreasing function of k. This means that as k increases, T decreases. We therefore expect that since the thicker foam will have a larger compression boundary (Ro greater) than a thinner cushion under the same loading and drop conditions and since this boundary is under less tension, it will have a larger stretch ratio. Table 1 verifies this. As mentioned in the previous paragraph, a thicker foam tends to be softer. That is, a thicker foam will accomodate a larger dynamic deflection compared to a thinner one under same conditions because of the larger available compression boundary radius. Tables 6 and 7 again show that this prediction is correct, thickness length drop height weight dm 4.75" 8" 32.75" 20.3875 lbs 1.807" 3.125" 8" 33.625" 20.3875 lbs 1.456" The facts that average percent errors fell within the expected 37 range consistent with experimental errors on the measurement of G and that the stretch ratio and dynamic deflection are related to the thickness as expected, both strengthen the conclusion that the adiabatic air model used in conjuction with the compression boundary concept is correct. The compression boundary AC in Figure 11 cannot be expanded indefinitely however since it will intersect the inside corner of the foam. When line AC is pushed up by the higher pressure underneath to the point where this happens as marked by arc A"C", any further compression of the edge should produce a compression boundary which is no longer circular, but has the shape of boundary A"'C"' in Figure 11. Ultimately, if the compression is so severe that the inside corner nearly touches the ground (the combined thicknesses of the collapsed cell walls actually separates the outside and inside corners), the boundary will probably reach A""C"" under which all of the foam originally below the horizontal line XYZ in Figure 11 is compressed into the region XYZC""YA""X. The only difference in the model then becomes the assumption made with regard to deformation geometry. At this point however, the analysis becomes much more complicated since now the major part of the resistance of the foam to further compression comes not from compressing trapped air within the closed cells in the compressed region but from compressing solid PE (in the case of Ethafoam) in the form of a vertical stack of cell walls formed from the collapsed cells below the inside corner Y. Even if an adequate model for this case could be developed, the results would be of little use since by this time the edge of the product has most likely undergone crushing. 38 If we look at the flat drop cushion curves in Figure 6, there exists a minimum G level located somewhere between minimum and maximum stresses which is a result of the fact that the cushion gets stiffer with increasing compression (nonlinear hardening). Unlike this behavior in a flat drop situation, the new model indicated that there is no minimum G-level. The deceleration is directly but not linearly related to the edge contact length (L). From the data in Tables 6 and 7, this fact is borne out by the actual G-levels. It is expected that a drop angle other than 45 degree will result in a different deformation geometry which means that Figure 8 and equation (10) will no longer be applicable. In fact, the dimension Ro has no meaning anymore. A much more complicated relationship can be foreseen resulting in an irregular deformation pattern. From the results of this research, it is apparent however that the adiabatic assumption can still be used for a low density closed-cell cushion, as long as the volume change of the irregular compressed region under a particular drop angle is accounted for. 39 segment of are forming compression boundary \ 1IPO / ‘3’ \ T R T Figure 11: Analysis of progressive foam deformation under dynamic loading. APPENDICES APPENDIX A ADIABATIC MODEL FOR FLAT DROPS (from reference 12) For any polytropic process involving an ideal gas, the relationship between absolute pressure, absolute temperature,and volume are: n pV = constant (A-l) n-l TV - constant (A-2) For the limiting cases of constant pressure, constant temperature, and constant volume, n - 0, l and m respectively. For an adiabatic process, n is the ratio of specific heats which for air is n = 1.4. Consider a rectangular cushion of area A and thickness t as shown in Figure 5, Chapter 2. 0n impact, a force imbalance in the vertical direction requires that W EFT = PA — w = (—) x acc (A-3) a where 'acc' is the acceleration of the block, acc = (d2V/d12)/A. The weight term will be discarded for simplicity because it is normally small compared to the pressure term. If the cushion is modelled as a confined quantity of air, then P = Po(Vo/V)n from equation (A-l) where Po and V0 are the initial pressure and volume: P0 = 14.7 psi under standard conditions and V0 = At from Figure 5. Equation (A-3) now requires that 2 (A-4) d12 W which is a second order nonlinear differential equation in V and 7 with 40 41 the initial conditions that at r - 0, V a V0 and dV/dr = —AJ2gH. Multiplying both sides of equation (A-4) by dV and integrating gives 1 dV PogA2Von v P gA2V -——( )2 + ( ) X —————— - constant - A2gH + ——2————3— (A-S) 2 dr W n — 1 (n—l) W where the constant has been evaluated using the initial conditions. Rearranging, dV 2 Azx 2gH Vo n-l ( )= [R+1-( ) ] (A-6) dr R V (n - l) a H R = (A-7) P t o with a = W/A, the commonly used 'static loading'. The maximum acceleration occurs when dV/dr a 0 at which time the cushion reaches its minimum volume Vm' From equation (A—6), ( ° )n‘1 - 1 + R (A-8) V m and from equation (A-1) and (A-2). the corresponding maximum pressure and temperature are n Pm = P0 (1 + R) “’1 (A-9) Tm = To (1 + R) (A-lO). When equation (A-9) is used in equation (A-3), the peak deceleration defined as G = acc/g is found to be n Pm P l c = (1 + R) “’ (A-ll) 0' U which can now be used to generate the so-called cushion curves, C vs. 42 a, for varius drop heights H and cushion thicknesses t (refer to Figure 6, Chapter 2). l. Cushion Test Machine APPENDIX B SPECIFICATIONS Lansmont Model 23 Cushion Test Machine Machine type: Platen size: Platen weight: Table rebound brakes: Table lifting/positioning: Controls: Instrumentation: Ballast kits: Maximum equivalent free-fall drop height Power requirements: Pneumatic requiremnets: Free-fall 9.06" X 9.06" 12.81bs. Electrically controlled, pneumatically actuated, brake pistons integral with heavy weight platen. By electric hoist at 32 ft/min CM series 626 model A hoist 24 vdc control system. Hand-held control box (standard). Instrumentation trigger Contains the following weights: 1 @ 6.4 lbs. 1 @ 12.8 lbs. 2 @ 25.6 lbs. 2 @ 32 lbs. 60 inches (with standard base) 115 vac, 60 Hz, 8 amperes Plant air or bottled nitrogen at 80-120 psi 43 44 2. Accelerometer Kistler Piezotron Accelerometer Model: 818 Series No.: 2187 Resonant frequency .............. 34.0 kHz Transverse sensitivity .......... 4.5% Reference voltage sensitivty .... 10.09 mV/g at 100 Hz, 75°F & ilOg, with 24 DC source through 10K ohms Range ........................... 500 g Amplitude Linearity (vibration) Amplitude Sensitivity Deviation ig mV/g % 1 10.09 0.0 2 10.09 0.0 5 10.09 0.0 10 10.09 REF. 20 10.09 0.0 50 10.09 0.0 100 10.08 -0.1 200 10.07 -0.2 3. Coupler lnput Characteristics: 45 [Constant Feed Current mA 2 - 13 Full Scale Signal Vpp 20 Transfer Characteristics: Gain 1 Frequency Response 35% Hz 0.5 to 250 kHz (with l meg load 8 10 Vpp) Input/Output Coupling input is AC coupled to output buffer amplifier Output Characteristics: Full Scale Signal Voltage Vpp 20 Current mA flO Impedance 0 20 DC output offset V 0 NOise Vrms 18011 Power Source ' AC line llO/ZZOV, 50/60 Hz 3 VA 46 Physical Size LxHxW in 5-9X2-9X3.2 Weight g 585 Connectors Input BNC (neg) Output BNC (neg) Power 3-wire w/ gnd Front Panel Meter yes circuit integrity check yes battery check - DC output offset check - Environmental Operating temperature °C 0 to 50 47 4. Oscillosc0pe CRT Circuit Item Specification = =W Type 6—inch rectangular, direct-viewing bistable storage tube Phosphor P31 Effective screen 8 X 10 DIV size Acceleration Approx. 3.15 kV voltage Writing speed 40 usec/DIV (25 DIV/msec) During ENHANCE mOde operation, can be increased to up to 20 usec/DIV (50 DIV/msec) or higher Read time Approx. 60 minutes Erase time Approx. 0.5 sec Storage section NOR-bi: Normal OSCillOSCOpe . Operation Operation mode STORAGE: Storage oscillOSCOpe Operation ' ENHANCE: Operation with enhanced writing speed. Adjusta- ble with ENHANCE LEVEL control. ERASE: To erase the written wave— form. During the SINGLE SWEEP mode operation, the RESET Operation is done after the ERASE Operation and the oscillosc0pe is reset to the READY state. HOLD: The waveform on the screen is instantaneously held and stored. The store state is released if changed to the NORM mode. 48 Item AUTO ERASE @ Specification J= ”Write + hold + erase” actions are done for each sweep cycle. The hold time is adjustable with the ERASE INTERVAL control for l - 30 seconds. Vertical axes Item I Specification Sensitivity NORM: 5 mV — 5 V/DIV x5 MAC: 1 mV - l V/DIV Sensitivity NORM: t3% or better accuracy XS MAC: :5% or better Vernier vertical To l/2.5 or less of panel- sensitivity indicated value Frequency NORM: DC - 20 MHz, within -3 dB bandW1d‘h x5 MAG: DC - 15 MHz, within -3 dB AC coupling: Low limit frequency 10 Hz Rise time NORM: Approx. 17.5 nsec 5 MAC: Approx. 23 nsec Input impedance -1 MO :22, 25 pF :2 pF Square wave characteristics Overshoot: Not greater than 5% Other distortions: Not greater than 3% (At 10 mV/DIV range) DC balance shift 10.5 DIV 12.0 DIV NORM: x5 MAG: 49 Item Specification Display modes CH1: CH1 single channel CH2: CH2 single channel DUAL: CHOP: 0.5 sec - l msec/DIV ALT: 0.5 msec - 0.2 psec/ DIV ADD: CH1 + CH2 ChOpping Approx. 250 kHz repetition frequency Input coupling AC/CND/DC Maximum allowable input voltage 400 V (DC + AC peak) Common mode rejection ratio 50:1 or better.at 50 kHz, sinusoidal wave Isolation between channels 1000:1 or better at 50 kHz 30:1 or better at 40 MHz CH1 signal output Approx. 100 mV/DIV when Open; approx. 50 mV/DIV when 50-ohm termination CH2 INV BAL Balanced point variation, at the center of screen: 1 DIV or less APPENDIX C TABLES FOR DATA AND RESULTS Table 1: Static edge compression test results for the 3.125" and 4.75" thick specimens. Edge contact length is 8". t = 3.125" Force Compression Arc Base Arc Height Stress Strain Stretch Ratio F(lb) x (inches) b(inches) h (inches) a(psi) e(in/in) k (in/in) 50 0.2 0.3 0.05 20.8 0.064 0.88 110 0.5 0.7 0.14 19.6 0.160 0.90 170 0.8 1.1 0.22 19.3 0.256 0.94 230 1.1 1.5 0.33 19.2 0.352 0.90 280 1.35 1.9 0.41 18.4 0.432 0.92 350 1.45 2.25 0.47 19.4 0.464 0.93 410 1.5 2.7 0.60 19.0 0.480 0.92 t = 4.75" Force Compression Arc Base Arc Height Stress Strain Stretch Ratio F(lb) x (inches) b(inches) h (inches) 0(psi) e(in/in) k (in/in) 60 0.35 0.38 0.08 19.7 0.74 0.94 110 0.70 0.65 0.11 21.2 0.147 0.95 160 1.1 1.0 0.18 20.0 0.232 0.92 250 1.6 1.6 0.35 20.8 0.337 0.96 285 1.85 1.9 0.40 21.0 0.390 0.94 340 2.2 2.2 0.44 20.2 0.463 0.94 50 Table 2: Data for 3.125" Ethafoam for first drop Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample NH NH reed r014 H‘ h‘ hard 14 H l—" NH H Drop Height (inch) 17.5 17.5 17.5 36.25 36.25 36.25 52.87 52.87 51.625 33.625 33.625 33.625 17.0 17.0 17.0 51 Weight (lbs) 13. 13. 13. 13. 13 13. 13. 13. 20. 20. 20. 20. 20. 20. 20. 9875 9875 9875 9875 .9875 9875 9875 9875 3875 3875 3875 3875 3875 3875 3875 Edge Contact Length (in) 8.0 corner-tO-corner thickness for 22. 19. 22. 18. 21. 20. 33. 35. 31. 30. 31. 31. 43. 47. 43. 44. 44. 48. 28. 31. 30. 28. 33. 32. 19. 20. l6. 18. 17. 17. 00 OO Table 2: Sample Sample' Sample Sample Sample Sample Sample Sample Sample Sample NH NH NH (continued) 16.625 16.625 16.625 33.75 33.75 52 26.7875 26.7875 26.7875 26.7875 26.7875 8. 6. 0 0 4.0 8 6. .0 0 16. 17. 15. 16. l6. 14. 33. 33. 38. 32. 53 Table 3: Data for 4.75" corner-tO-corner thickness for Ethaform for first drop Drop Height Weight Edge Contact G (inch) (lbs) Length (in) Sample 1 16.25 13.9875 8.0 22.0 Sample 2 19.0 Sample 1 16.25 13.9875 6.0 21.0 Sample 2 18.0 Sample 1 16.25 13.9875 4.0 15.0 Sample 2 15.0 Sample 1 34.125 13.9875 8.0 28.0 Sample 2 26.0 Sample 1 34.125 13.9875 6.0 27.0 Sample 2 25.0 Sample 1 34.125 13.9875 4.0 21.5 Sample 2 22.0 Sample 1 53.125 13.9875 8.0 33.0 Sample 2 43.0 Sample 1 53.125 13.9875 6.0 36.0 Sample 2 39.0 Sample 1 53.125 13.9875 4.0 33.0 Sample 2 31.0 Sample 1 52.0 20.3875 8.0 32.0 Sample 2 33.0 Sample 1 52.0 20.3875 6.0 32.0 Sample 1 52.0 20.3875 4.0 32.0 Sample 2 30.0 Sample 1 32.75 20.3875 8.0 24.0 Sample 2 22.0 Sample 1 32.75 20.3875 6.0 22.0 Sample 2 22.0 Sample 1 32.75 20.3875 4.0 20.0 Sample 2 20.0 Sample 1 17.25 20.3875 8.0 17.0 Sample 2 16.0 Table 3: Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample H BDPJ naha h‘ NH n>rJ 17. 17. (continued) 25 25 17.0 17.0 17.0 33. 33. 33. 51. 51. 51. 75 75 75 75 75 75 20. 26. 26. 26. 26. 26. 26. 26. 26. 26. 54 .3875 3875 7875 7875 7875 7875 7875 7875 7875 7875 7875 17. 16. 14. 12. 14. 16. 14. 13. 13. 11. 22. 21. 22. 20. 27. 22. 36. 31. 31. 34. 37. 39. 55 Table 4: Data for 3.125" corner-tO-corner thickness for Ethaform; 2nd through 5th drops. Drop Height Weight Edge Contact G (inch) (lbs) Length (in) Sample 1 17.5 13.9875 8.0 25. 26. 27. 27. Sample 2 17.5 13.9875 8.0 22. 23. 24. 25. 00000000 Sample 1 17.5 13.9875 6.0 24. 25. 26. 26. Sample 2 17.5 13.9875 6.0 20. 21. 21. 21. OOOOOOOO Sample 1 17.5 13.9875 4.0 22. 23. 25. 25. Sample 2 17.5 13.9875 4.0 29. 24. 26. 27. 00000000 Sample 1 36.25 13.9875 8.0 37. 40. 44. 44. Sample 2 36.25 13.9875 8.0 42. 46. 47. 47. 00000000 Sample 1 36.25 13 9875 6.0 37. 43. 44. 47. Sample 2 36.25 13 9875 6.0 37. 1 40. 42. 44. 00000000 Table 4: Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Samele Sample Sample Sample Sample 1 2 36. 36. 52. 52. 52. 52. 51. 51. 33. 33. 33. 33. 33. .625 33 17. (continued) 25 25 875 875 875 875 625 625 625 625 625 625 625 17.0 56 13.9875 13.9875 13.9875 13.9875 13.9875 13.9875 20.3785 20.378 20.3785 20.3785 20.3785 20.3785 20.3785 20.3785 20.3785 20.3785 39. 47. 40. 43. 46. 47. 55. 60. 61. 57. 62. 64. 47. 57. 62. 66. 67. 000000 000000 00000 52.0 70. O 63.0 36. 51. 40. 44. 45. 47. 000000 39.0 36. 35. 54. 48. OO 00 58.0 22. 24. 24. 24. 22. 22. 23. 24. 00000000 5.7 Table 4: (continued) Sample 1 17.0 20.3785 6.0 20. 22. 22. 23. Sample 2 17.0 20.3785 6.0 18. 21. 23. 22. 00000000 Sample 1 17.0 20.3785 4.0 21. 22. 23. 22. Sample 2 17.0 20.3735 4.0 21. 24. 25. 25. 00000000 Sample 1 16.625 26.7875 8.0 23. 24. 24. 25. Sample 2 16.625 26.7875 8.0 22. 23. 25. 25. OOOOOOOO Sample 1 16.625 26.7875 6.0 19. 21. 22. 22. Sample 2 16.625 26.7875 6.0 19. 22. 22. 22. 00000000 Sample 1 16.625 26.7875 4.0 22. 25. Sample 2 16.625 26.7875 4.0 19.0 00 Sample 1 33.75 26.7875 8.0 47. 48. Sample 2 33.75 26.7875 8.0 43. 48. 50. 51. Sample 33.75 26.7875 36. Sample 2 33.75 26.7875 6.0 36. 44. l'-‘ 0‘ 0 000000000 58 Table 5: Data for 4.75" corner-tO-corner thickness for Ethafoam; 2nd through 5th drops. Drop Height Weight Edge Contact (inch) (lbs) Length (in) Sample 16.625 13.9875 8.0 Sample 16.625 13.9875 8.0 Sample 16.625 13.9875 6.0 Sample 16.625 13.9875 6.0 Sample 16.625 13.9875 4.0 Sample 16.625 13.9875 4.0 Sample 34.125 13.9875 8.0 Sample 34.125 13.9875 8.0 Sample 34.125 13.9875 6.0 Sample 34.125 13.9875 6.0 23. 21. 22. 24. 22. 23. 24. 23. 23. 22. 21. 22. 20. 22. 22. 22. 17. 18. 18. l7. 19. 19. 19. 20. 31. 32. 35. 39. 32. 30. 29. 31. 31. 36. 34. 37. 29. 30. 30. 32. OOOOOOOU‘ OOOOOOOO OOOOOU‘IOO OOOOOOOO OOOOOOOU‘I Table 5: Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample Sample (continued) 34.125 34.125 53.125 53.125 53.125 53.125 53.125 53.125 52.0 52.0 52.0 52.0 52.0 13. 13. 13. l3. 13. 13. 13. 13. 20. 20. 20. 20. 20. 59 9875 9875 9875 9875 9875 9875 9875 9875 3875 3875 3875 3875 3875 28. 29. 31. 29. 26. 27. 28. 28. 37. 40. 41. 41. 44. 43. 43. 45. 43. 44. 48. 46. 40. 44. 46. 46. 40. 43. 45. 44. 40. 43. 44. 43. 40. 44. 47. 39. 43. 48. 49. 42. 44. 48. 49. 00000000 00000000 OOU’IOLfiOOO 00000000 O 0000 0000000 60 Table 5: (continued) Sample 1 32.75 20.3875 8.0 28. 30. 32. 32. Sample 2 32.75 20.3875 8.0 26. 28. 29. 28. 00000000 32.75 20.3875 26.0 32.75 20.3875 6.0 30. 29.0 ON 0 Sample Sample NH 0 Sample 1 32.75 20.3875 4.0 26. 26. 29. 29. Sample 2 32.75 20.3875 4.0 22. 27. 27. 29. OOOOOOOO Sample 1 17.25 20.3875 8.0 18. 18. 18. 18. Sample 2 17.25 20.3875 8.0 1 19. 20. 20. 20. 00000000 Sample 1 17.25 20.3875 6.0 18. 18. 18. 18. Sample 2 17.25 20 3875 6.0 l6. l7. l7. 18. 00000000 Sample 1 17.25 20.3875 4.0 16. l6. 17. 18. Sample 2 17.25 20.3875 4.0 15. l6. l7. 16. 00000000 61 Table 5: (continued) Sample 1 17.0 26.7875 8.0 16. l7. 17. 17. Sample 2 17.0 26.7875 8.0 18. 18. 20. 19. 00000000 Sample 1 17.0 26.7875 6.0 15. 16. 16. 17. Sample 2 17.0 26.7875 6.0 15. 16. 16. 16. UIOOOOOOO Sample 1 17.0 26.7875 4.0 16. l6. l7. 17. Sample 2 17.0 26.7875 4.0 14. l6. l6. l7. OOOOOOOO Sample 1 33.75 26.7875 8.0 24. 25. 28. Sample 2 33.75 26.7875 8.0 28. 0000 Sample 1 33.75 26.7875 6.0 29. 30. 30. 30. Sample 2 33.75 26.7875 6.0 0. 00000 Sample 1 33.75 26.7875 4.0 0. Sample 2 33.75 26.7875 4.0 31. 32. 34. 34. 00000 51.75 26.7875 8.0 0. 51.75 26.7875 8.0 . 42. 46. Sample Sample NDPJ c>c>c> Table 5: Sample 1 Sample 2 Sample 1 Sample N (continued) 51.75 51.75 51.75 51.75 62 26.7875 26.7875 26.7875 26.7875 6. 6. 91> OO 0 41. 46. 48. 51. 39. 44. 46. 48. 00000000 56.0 58. O Table 6: Results for 3.125" corner-to-corner thickness for C= 9.2 HEIGHT 17.500 17.500 17.500 17.500 17.500 17.500 36.250 36.250 36.250 36.250 36.250 36.250 52.075 52.075 52.075 52.075 51.625 51.625 33.625 33.625 33.625 33.625 33.625 33.625 17.000 17.000 17.000 17.000 17.000 17.000 16.625 16.625 16.625 16.625 16.625 16.625 33.750 33.750 33.750 33.750 633 Ethafoam under edge drOp k: HEIGHI 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 13.9075 20.3075 20.3075 20.3075 20.3075' 20.3075 20.3075 20.3075 20.3875 20.3075 20.3075 20.3075 20.3075 20.3075 20.3075 26.7075 26.7075 26.7075 26.7075 26.7075 26.7075 26.7075 26.7075 26.7075 26.7075 .91 L 0 0 6 6 4 4 0 0 6 6 4 4 0 0 6 6 0 0 0 0 6 6 4 4 0 0 6 6 4 4 0 ‘0 '6 6 4 4 0 ‘0 16 6 B-CAL 29.1 29.1 25.2 25.2 20.6 20.6 41.9 41.9 36.3 36.3 29.6 29.6 50.6 50.6 43.0 43.0 41.4 41.4 33.4 33.4 20.9 20.9 23.6 23.6 23.0 23.0 20.6 20.6 16.0 16.0 20.5 20.5 17.0 17.0 14.5 14.5 29.2 29.2 25.3 25.3 G-EXP 2°01FF 00 22.0 19.0 22.0 10.0 21.0 20.0 33.0 35.0 31.0 30.0 31.0 31.0 43.0 47.0 43.0 44.0 44.0 40.0 20.0 31.0 30.0 20.0 33.0 32.0 19.0 20.0 16.0 10.0 17.0 17.0 17.0 16.0 15.0 16.0 16.0 14.0 33.0 33.0 30.0 32.0 32.30 53.19 14.57 40.04 2.00 2.91 26.94 19.69 17.03 20.93 4.45 4.45 17.66 7.64 1.09 0.42 5.09 13.73 19.35 7.00 3.53 3.36 20.39 26.16 25.06 10.01 20.61 14.32 1.16 1.16 20.59 20.12 10.36 10.96 9.40 3.54 11.49 11.49 33.43 20.95 0.870 0.070 1.004 e— O $§§§§§§ §§§§§E§§§§ “pnpfl—“-~“~’—flh 0 AVERAGE 1 Diff BEIIEEI 0-cal 0 0'01? 3 11.92041 0= 13 1 of 63x. G-exp exceeding 33:. S-tal = 32.5 00 1.352 1.352 1.561 1.561 1.912 1.912 1.945 1.945 2.246 2.246 2.751 2.751 2.350 2.350 2.713 2.713 2.003 2.003 2.262 2.262 2.612 2.612 3.199 3.199 1.600 1.600 1.057 1.057 2.275 2.275 1.023 1.023 2.105 2.105 2.570 2.570 2.590 2.590 3.000 3.000 61: Table 7: Results for 4.75" corner-to-corner thickness for Ethafoam under edge drOp 0: 0.640001 1' .94 [10117 0010117 1. 0-0111. 0-011’ I-IIFF 00 110 16.625 13.9075 0 26.6 22.0 21.10 1.067 1.605 16.625 13.9075 0 26.6 19.0 40.22 1.067 1.605 16.625 13.9075 6 23.1 21.0 9.07 1.232 1.053 16.625 13.9075 6 23.1 10.0 20.10 1.232 1.053 16.625 13.9075 4 10.0 15.0 25.59 1.509 2.270 -16.625 13.9075 4 10.0 15.0 25.59 1.509 2.270 34.125 13.9075 0 30.2 20.0 36.32 1.520 2.299 34.125 13.9075 0 30.2 26.0 46.01 1.520 2.299 34.125 13.9075 6 33.1 27.0 22.43 1.765 2.655 34.125 13.9075 6 33.1 25.0 32.23 1.765 2.655 34.125 13.9075 4 27.0 21.5 25.54 2.161 3.252 34.125 13.9075 4 27.0 22.0 22.60 2.161 3.252 53.125 13.9075 0 47.6 43.0 10.76 1.907 2.069 53.125 13.9075 0 47.6 43.0 10.76 1.907 2.069 53.125 13.9075 6 41.2 36.0 14.57 2.202 3.313 53.125 13.9075 6 41.2 39.0 5.76 2.202 3.313 53.125 13.9075 4 33.7 33.0 2.05 2.697 4.057 53.125 13.9075 4 33.7 31.0 0.63 2.697 4.057 52.000 20.3075 0 39.0 32.0 21.96 2.270 3.427 52.000 20.3075 0 39.0 33.0 10.27 2.270 3.427 52.000 20. 3075 6 33.0 32.0 5.62 2.630 3.957 52.000 20.3075 4 27.6 32.0 13.76 3.221 4.046 52.000 20.3075 4 27.6 30.0 0.01 3.221 4.046 32.750 20.3075 0 31.0 24.0 29.05 1.007 2.719 32.750 20.3075 0 31.0 22.0 40.79 1.007 2.719 32.750 20.3075 6 26.0 22.0 21.92 2.007 3.140 32.750 20.3075 6 26.0 22.0 21.92 2.007 3.140 32.750 20.3075 4 21.9 20.0 9.51 2.556 3.046 32.750 20.3075 4 21.9 20.0 9.51 2.556 3.046 17.250 20:3075 0 22.5 17.0 32.23 1.312 1.974 17.250 20.3075 0 22.5 16.0 40.49 . 1.312 1.974 17.250 20.3075 6 19.5 17.0 14.51 1.515 2.279 17.250 20.3075 6 19.5 16.0 21.67 1.515 2.279 17.250 20.3075 4 15.9 14.0 13.53 1.055 2.791 17.250 20.3075' 4 15.9 12.0 32.46 1.055 2.791 Table 7: 17.000 17.000 17.000 17.000 17.000 17.000 33.750 33.750 33.750 33.750 33.750 33.750 51.750 51.750 51.750 51.750 51.750 51.750 (continued) 26.7075 0 ‘19.5 26.7075 0‘ 19.5 26.7075 6 16.9 26.7075' 6 16.9 26.7075 4 13.0 26.7075 4 13.0 26.7075 0 27.4 26.7075 0 27.4 26.7075 6 23.0 26.7075 6 23.0 26.7075 4 19.4 26.7075 4 19.4 26.7075 0 34.0 26.7075 0 34.0 26. 7075 6 29.4 26.7075 6 29.4 26.7075 4 24.0 26.7075 4 24.0 14.0 16.0 14.0 13.0 13.0 11.0 22.0 21.0 22.0 20.0 27.0 22.0 36.0 31.0 31.0 34.0 37.0 39.0 39.06 21.67 20.43 29.69 5.09 25.14 24.60 30.62 7.90 10.70 20.16 11.04 5.65 9.57 5.11 13.40 35.09 30.42 (55 1.493 1.493 1.724 1.724 2.111 2.111 2.103 2.103 2.429 2.429 2.974 2.974 2.604 2.604 3.007 3.007 3.603 3.603 ANERAGE 1 Diff IEIIEEI 0-041 6 G-EXP = 21.04022 u- 7 l of sex. G-exp exceeding 03:. S-tal = 13.20755 2.246 2.246 2.593 2.593 3.176 3.176 3.164 3.164 3.654 3.654 4.475 4.475 3.910 3.910 4.524 4.524 5.541 5.541 APPENDIX D COMPUTER PROGRAM IO DIN H1110),N(IIO),LCIIO),G(IIO),BI(IIO),PtIIO) 20 FOR I=I TO N 30 READ H(I),N(I),L(I),6(I) 40 PRINT H11),H(I),L(I),B(I) 50 NEXT 60 PRINT 'T- 7' 70 INPUT T 80 PRINT "Ta”;T 90 PRINT "c=?“ 100 INPUT C 110 PRINT “k8?” 120 INPUT K 130 PRINT ”Ca “30; 140 PRINT ' k- '3K:PRINT 150 Q=(I.752§K“-2.8)§(.5—.285*K“2)+.285*K‘2 160 PRINT "HEIGHT “g'NEIBHT ';“L ”;“B-CAL '; 'B-EXP ”g’ZSDIFF ";"Dfl “g“RO‘ 170 980 180 FOR I - 1 TO N- 190 61(1) = C*SQR(H(I)*L(I)IN(I)) 200 P(I)=100*((GI(I)-§(I))/B(I)) 210 P(I)=ABS(P(I)) 220 S = S + P(I) 230 RO=.261*SQR(H(I)*N¢I)/L(I)IQ) 240 DflakfiRO/SQRIZ) 250 PRINT USING "00.0.0 ';H(I); 260 PRINT USING “00.0080 ';H(I); 270 PRINT USING '0 '3L4I); 280 PRINT USING '00.! “381(1); 290 PRINT USING '0... “36(1); 300 PRINT USING “00.00 ';P(I); 310 PRINT USING “0.000 ”yDH;R0 320 NEXT 330 ACCU a SIN 340 PRINT 350 PRINT “AVERAGE Z DIFF BETWEEN B-cal & B-cxp -“;ACCU 360 "-0 370 FOR I3 1 TO N 380 D=SQR(H(I)*L(I)IN(I)) 390 IF 6(1) < (9.38290) THEN BOTD 410 400 H = n+1 410 NEXT 420 PRINT 430 PRINT ”fl- '3" 440 A-H/53e100 450 PRINT 460 PRINT '1 of max. B-cxp lac-eding sax. G-cal - ”39 66 LIST OF REFERENCES 10. ll. 12. 13. . Military Standardization Handbook, MIL-HDBK-304B, "Packaging Cushion REFERENCES . R.D. Mindlin, Dynamics of Packaging Cushioning, Bell Telephone System Technical Journal Vol. 24, pg 353-461, July-Oct., 1945. Bell Telephone System Technical Publications, Monograph B-l369. . G.S. Mustin, Theory and Practice of Cushion Design, The Shock and Vibration Information Center, United States Department of Defense, 1968. F.A. Paine, Fundamentals of Packaging, Chapter 3: Fundamentals of Cushioning, Blackie & Son Litimed, 1968. C.M. Harris and C.E. Crede, Shock and Vibration Handbook Vol. 3, "Packaging Design", McGraw-Hill Book Company, Inc. Design", United States Department of Defense, 1978. . ASTM D 1596-78a, “Test Method for Shock Absorbing Characteristics of Package Cushioning Materials", American Society for Testing and Materials, 1978. . L.W. Gammell and J.L. Gretz, "Effect of Drop Test Orientation on Impact Accelerations", Physical Test Lab., Textfoam Div., B.F. Goodrich Sponge Products Div., Shelton, Conn. 1955. D.M. Chase and I. Vigness, "Dynamics of Package Cushioning Involving Combined Rotations and Translations", Naval Research Lab., Rept. 4719, March 1956. . J.I. Throne and R.C. Progelhof, "Closed-Cell Foam Behavior Under Dynamic Loading - I. Stress—Strain Behavior of Low Density Foams. Journal of Cellular Plastics, Nov./Dec. 1984. J.I. Throne and R.C. Progelhof, "Closed-Cell Foam Behavior Under Dynamic Loading - II. Loading Dynamics of Low Density Foams. Journal of Cellular Plastics, Jan./Feb. 1985. R.J. Tait and B.J. Haddow, "Finite Compression of a Foam Rubber Support System Subjected to Impact Loading", ZAMM Z. Math. Mech. 67(1987) 3, 202-204. G.J. Burgess, "Some Thermodynamic Observations on The Mechanical Properties of Cushions", Journal of Cellular Plastics, 1987. J.H. Faupel, Engineering Design, pg 225, John Wiley and Sons, New York, 1967. 67 111111111111111111111111“111111