UIESNVR :11111 1111111111111111111111 3800 761607 This is to certify that the thesis entitled AN ALTERNATIVE METHOD FOR DETERMINING THE DAMAGE BOUNDARY CURVE OF A SHOCK-SENSITIVE PRODUCT presented by GARY ALLEN LI-ERNAN has been accepted towards fulfillment of the requirements for _HASIER___degree in W jor professor Date FEBRUARY 5, 1991 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution s «fag LIBRARY Mlchlgan State University PLACE IN RETURN BOX to remoue this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE __‘ MSU Is An Affirmative ActiorVEqual Opportunity Institution . omens-oi -._ _ AN ALTERNATIVE METHOD FOR DETERMINING THE DAMAGE BOUNDARY CURVE OF A SHOCK?SENSITIVE PRODUCT BY Gary Allen Lieberman A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE School of Packaging 1991 ABSTRACT AN ALTERNATIVE METHOD FOR DETERMINING THE DAMAGE BOUNDARY CURVE OF A SHOCK-SENSITIVE PRODUCT BY Gary Allen Lieberman Determining the fragility of a shock-sensitive product is the first step towards meeting packaging requirements. The primary goal of fragility testing is to develop the product's Damage Boundary Curve. Problems with the current methodology which uses a vertical shock machine to construct the Damage Boundary Curve are examined. An alternative method is proposed which does not require the use of a shock machine. With the aid of minimal instrumentation and a computer program, the fragility of a shock-sensitive product can be inexpensively determined from simple drop tests. Copyright by Gary Allen Lieberman 1991 Dedication This thesis is dedicated to Shelia, Jeremy, and Tim. iv ACKNOWLEDGMENTS I wish to thank Dr. Gary James Burgess, Professor - School of Packaging, who served as my major professor and was instru— mental in the development of this thesis. Without Dr. Burgess's instruction, this thesis would not have been possible. I also wish to extend my gratitude to my two other commit- tee members: Dr. Susan Elizabeth Selke, Professor - School of Packaging, and Dr. George E. Mase, Professor - Department of Metallurgy, Mechanics, and Materials Science. TABLE OF CONTENTS LIST OF TABLES ............OOOOOOOOOOOOOOOOOO...... Viii LIST OF FIGURES .................................... ix Chapter One: Introduction and Overview of Shock and Fragility ................ 1 I. Impact and Damage ............................ 2 A. The Shock Pulse ............................ 2 B. Fragility and the Damage Boundary Curve ... 4 II. The Vertical Shock Machine, ASTM D 3332 and the Conventional Method of DBC Construction ... 15 A. The Vertical Shock Machine ................ 15 B. ASTM D 3332 - Mechanical Shock Fragility of Products Using Shock Machines and the Conventional Method of BBC Construction ... 19 III. Limitations of Damage Boundary Theory and Shock Machines .................... 20 A. Damage Boundary Theory .................... 20 B. Limitations of the Shock Machine .......... 20 Chapter Two: An Alternative Method for Determining the DEC of a Shock—Sensitive Product .. 27 I. The Alternate Method .......................... 27 Chapter Three: Simulation Test Results, Sensitivity Analysis, Experimental Procedure, and Concluding Comments ................. 36 I. Simulation Test Results ...................... 36 II. Sensitivity Analysis ......................... 38 III. Experimental Procedure ....................... 43 vi TABLE OF CONTENTS CONTINUED IV. conCluding coments ......OOOOOOOOOOOOOOOOO... List Of References 00......00............OOOOOOOOOOOO Appendix Appendix Appendix Appendix Appendix Appendix Appendix A B C D vii 48 49 50 51 52 53 55 56 57 Table Table Table Table Table Table LIST OF TABLES Combinations of peak acceleration and duration (Columns 1 and 5) or peak accel- eration and velocity change (Columns 5 and 6) for half—sine shocks to the product which just damage the critical element Combinations of peak acceleration and duration (Columns 1 and 5) or peak accel- eration and velocity change (Columns 5 and 6) for square-wave shocks to the product which just damage the critical element Shock Machine Calibration Chart for 2 ms Half-Sine Programmers (bare table) Ability of the Program to Predict fce 20 Hz and Gcr = 100 6'3 Using Border- line Damage Half-Sine Shock Pairs from Appendix D Ability of the Program to Predict the Properties of fce and GC]: for Various Critical Elements Effect on fce and Gcr Determinations When V and G Values of prod prod Shock Pairs #12 and #43 from Appendix D Are Perturbed 10%. viii 9 12 37 Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES Three parameters associated with a sinusoidal shock pulse ................... Rectangular (a) and Trapezoidal (b) ShOCk pUISes ..0......00......0...0.00.00.00 Model of product protected by a cushion .... DBC for half-sine shocks to a product containing a critical element with f = 20 H2 and Gce = 100 G'S oooooooooooooo ce DBC for square-wave shocks to a product containing a critical element with f =20 Hz and Gee: 100 G's 0.0.0.0000000. C8 An example of a vertical shock machine ..... ASTM D 3332 method of fairing asquare-wave ShOCk pUlse 0.000000000000000. ASTM D 3332 method of fairing a half-sine shock pulse Haversine shock pulse ix 11 13 16 21 23 39 Chapter One: Introduction and Overview of Shock and Fragility The primary function of packaging is to preserve and protect the product as it is transported through the distri- bution environment. Specifically, a function of packaging materials is to reduce the intensity of an external shock transmitted from the environment to the product inside the package. Due to the high cost of package cushioning materi- als, it is important that the fragility of a shock-sensitive product be accurately determined so that packaging costs can be minimized. The conventional method used to determine the fragility of a shock-sensitive product relies on a vertical shock machine with a programmable impact surface. The purpose of this thesis is to eliminate the need for a shock machine in fragility testing and to promote an alternative method that is much less expensive. First, the need for fragility testing of shock-sensitive products and the role of shock machines for this purpose are examined. Next, the capabilities of a conventional shock machine are explored. Finally, an alternative method is presented that requires minimal instrumentation to determine the damage boundary curve for a shock-sensitive product. I. Impact and Damage A. The Shock Pulse The potential for damage to a product in an impact is related to certain features of the shock pulse to the product. The shape of the shock pulse in most situations is very nearly sinusoidal as shown in Figure 1. It has become customary to concentrate on the peak acceleration, the pulse duration, and the velocity change. The peak acceleration is usually expres- sed in "G's"; a reported value of 20 6'8 for example means that the peak acceleration is 20 times that of normal gravity, g = 386.4 in/sec’. A c 2 Area = Velocity g Change u o .... o o o m .x m m m |e—- Duration fl time Milliseconds Figure 1. Three parameters associated with a sinusoidal shock pulse. A less common shock pulse which can be produced by a vertical shock machine is the "trapezoidal-wave" and its relative, the "square-wave", both of which are shown in Figure 2. _1 L._/ \_ (a) (b) Figure 2. Rectangular (a) and Trapezoidal'(b) shock pulses. All of these pulses are characterized by the same three parameters; peak acceleration, pulse duration, and velocity change. The‘Uuee can be related to each other using the definition for instantaneous acceleration; acc = -QX— (1) dt where acc is the acceleration, dv is the differential change in velocity, and dt is the differential change in time. Rearranging and integrating over the duration T of the shock pulse gives T T dv = acc*dt (2) 4 The integral on the left in equation(2) is the velocity change .AW’and the integral on the right is the area under the acceleration vs time curve. Expressing this area as a shape factor multiplied by the base and height of the wave- form gives ZSV = (shape factor)*(peak acceleration)*(duration) (3) where the shape factor is 1.0 for a square-wave and Zfln’ = 0.636 for a half-sine wave. The velocity change in equation (3) is the sum of magnitudes of the impact and rebound veloc- ities. B. Fragility and the Damage Boundary Curve It is not obvious which features of the shock pulse are associated with damage. Is it peak acceleration, duration, (XV, or a combination of these? Since inertial forces are directly proportional to acceleration, it has become customary to associate the fragility of a product with the maximum accel— eration that it can withstand without breaking [Newton, 1976]. Unfortunately, this is too simplistic. The duration of the shock is equally important. Consequently, a complete asses- sment of the fragility of a product must account for the combi- nations of peak acceleration, duration, and velocity change which damage a product [Brandenburg and Lee, 1985]. The analysis begins with a model of the product as a rigid mass containing a critical element which is elastically 5 connected to the product, as indicated in Figure 3 [Newton, 1976]. The product is assumed to fail when the critical ele- ment breaks. cushion Figure 3. Model of product protected by a cushion. Four important assumptions are made about the nature of the critical element: (1) it is lightweight compared to the bulk of the product; (2) it acts as an ideal (linear) spring/ mass system; (3) it fails when its acceleration exceeds a certain amount regardless of duration; and (4) it is the most fragile component of the product. The shock that is transmitted to the critical element in an impact is not the same as the shock to the product. Depending on the duration of the shock, G = Ce Am*G prod (4) where Gce is the shock transmitted to the critical element, 6 Am is an amplification factor, and G is the shock to the prod product. Am is a function of the ratio of the natural fre- quency of the critical element fce to the natural frequency of the critical element fce to the natural frequency of the product on its cushion f . The value of A also depends prod m on the shape of the shock pulse to the product. For half—sine shock pulses, AI“ is determined by [Brandenburg and Lee, 1985]: * Am = 2‘“fee/fprodwcofifa!*fce/2 fprod) (5) 3 1 ' (fee/fprod) when f /f <1-0; ce prod -— * * A = f /f sin Tr N 2 + 1 i] (6) * m ce prod f /f when f /f ce prod>1’O where N is the integer between 1.0 and (1 + fee/fprod)/2 which maximizes Am A table of Am versus frequency ratio for half—sine shocks is shown in Appendix A. For square-wave shock pulses, Am is determined by [Brandenburg and Lee, 1985]: fce’Vfl’ * 2 fprod when fee/fprod SE1.0; A = 2*sin (7) A = 2.0 when fee/fprod )>1.0 (8) 7 A table of Am versus frequency ratio for square-wave shocks is shown in Appendix B. To illustrate the use of equation (4) in determining the complete fragility picture, consider a hypothetical product containing a critical element which has a natural frequency of 20 Hz and a fragility of 100 6'3 (see Figure 3). The pro- duct itself could be a television set and the critical element a transistor or integrated circuit mounted on a circuit board inside the TV. In the following simulation, the product will be "sub— jected" to a series of cushioned drops which produce half- sine shock pulses with different durations. The combinations of peak G and duration which cause the critical element to fail will be tabulated. For each shock, it is necessary that the natural frequency of the product on its cushion be known in order that the correct amplification factor be selected to obtain the acceleration of the critical element. As the dur- ation of each shock varies, the natural frequency of the pro- duct on its cushion does also. Since the duration of the shock is one half of the natural period of vibration [Brand- enburg and Lee, 1985], the natural frequency (cycles per sec- ond) is fprod = 1/(2*duration) (9) 8 In Table 1, the duration of the shock is arbitrarily cho- sen and the corresponding shock and velocity change to the product which just damage the critical element are calculated. Column 1 contains the shock duration chosen and Column 2 shows the natural frequency of the product on its cushion for the given duration obtained from equation(9). Column 3 lists the ratio of f /f ce prod' Since the natural frequency of the criti- cal element is known to be 20 Hz, this ratio is always equal to 20/f Column 4 contains the amplification factor for prod' this frequency ratio using either equations(5) and (6) or the table in Appendix A. Column 5 shows the required shock to the product using equation(4). Since the critical element is known to break at 100 G's (commonly depicted as Gcr = 100 6'5 where Gcr refers to the critical acceleration of the critical element), is always equal to 100/Am. Finally, the last Gprod column shows the corresponding velocity change in accordance with equation(3), Av Av shape factor x acc x duration (3) 0.636*(Column 5 * 386.4)*(Column 1/1000) Table 1. Combinations of peak acceleration and duration (Columns 1 and 5) or peak acceleration and velocity change (Columns 5 and 6) for half- sine shocks to the product which just damage the critical element. (1) (2) (3) (4) (5) (6) duration fprod fee/fprod Am Gprod [y (ms) (Hz) (in/sec) 100.0 5.00 4.00 1.268 78.9 1941.0 50.0 10.00 2.00 1.732 57.7 710.0 40.5 12.35 1.62 1.769 56.5 563.0 25.0 20.00 1.00 1.571 63.7 391.0 20.0 25.00 0.80 1.373 72.8 358.0 15.0 33.33 0.60 1.102 90.7 335.0 10.0 50.00 0.40 0.771 129.7 319.0 5.0 100.00 0.20 0.396 252.5 311.0 2.5 200.00 0.10 0.200 500.0 307.5 These results merit some discussion. It is interesting to note that despite the fact that the critical element's fragility is 100 G's, the smallest shock to the product that is needed to damage the critical element is only 56.5 G's because the largest Am possible is 1.769. Also, as foe/f prod * becomes small, the value of Am approaches 2 fee/fprod’ This result follows directly either from taking the limit of equation(S) for small frequency ratios or by inspection from the table in Appendix A. This leads to the result that G prod becomes inversely proportional to the duration; from equations (4) and (9), _ G — * g * = Gprod ' ce Gce Gce fprod Gce 1'0 Gce (10) * * * * * * * Am 2 fce 2 fee 2 fce 2 d 4 fce d fprod 10 where d represents the shock duration. This also leads to the result that the velocity change approaches a critical lower limit; from equations (3) and (10), = * * = I [fiver -2L(Gprod) (duration) 1 ‘* Gcr*386.4 (11) ’rr' 2*Tr f C8 This limiting value of velocity change is referred to as "the critical velocity change", Avcr' The significance of [chr is that if the actual velocity change in a drop is less than this value, the critical element will not break regard— less of the shock to the product. The reason for this is that the duration of the shock is so short that the critical ele- ment cannot fully respond to the shock. The results in Table 1 are shown graphically in Figure 4. vs [XV combinations of shock and velocity change to the product The locus of points on G axes representing prod prod which just damage the critical element is known as the Damage Boundary Curve (DBC). The smallest product velocity change required to damage the critical element is the critical veloc- ity change [five to damage the critical element is the critical acceleration r and the smallest shock to the product required Gcr' Note that the DEC in Figure 4 is for half—sine shocks to the product (Figure 1). 11 350 325. 11 275. 250. 225. I 1(;(3€3 “cool-batten (c a) u s: 1' f” R“... '“ (0 b & I l I I l I l T l 7 0 5 1W0 15% 2000 25W 3W0 3500 4000 4500 5000 55006000 Velocity Change (in/sec) Figure 4. 038 for half-sine shocks to a product containing a critical element with _ _ 0 fce - 20 Hz and Gcr - 100 G s. The most important conclusion to be drawn from this exam- ple is that a knowledge of the fragility SCI and natural fre- quency fee of the critical element completely determine the DEC. This was expected since it is the critical element which is assumed to break. The two most notable features of the DEC, Ava equation(11) and the table in Appendix A, Av and Gcr' for half-sine shocks to the product are from * or 61.5 Gcr/fce (11) 12 Ger/1°769 (12) Again, information about the critical element determines both these quantities. For this particular product, [chr = 307.5 in/sec and G ' =56.S G's. ce For shocks to the product which are not half-sine, the shape of the DEC is different. The procedure for tabulating the information is the same as for Table 1 except that a dif- ferent Am is used in Column 4 and a different shape factor in Column 6. For example, for square-wave shocks as in Figure 2a, the results in Columns 5 and 6 in Table 1 would change as a result of using Am from either equations(7) and (8) or from the table in Appendix B and a shape factor of 1.0 in Column 6. The results are shown in Table 2 and Figure 5. Table 2. Combinations of peak acceleration and duration (Columns 1 and 5) or peak acceleration and vel- ocity change (Columns 5 and 6) for square-wave shocks to the product which just damage the critical element. (1) (2) (3) (4) (S) (6) duration fprod fee/fprod Am sprod Av (ms) (Hz) (in/sec) 100.0 5.00 4.00 2.000 50.00 1932.00 50.0 10.00 2.00 2.000 50.00 966.00 40.5 12.35 1.62 2.000 50.00 782.46 25.0 20.00 1.00 2.000 50.00 483.00 20.0 25.00 0.80 1.902 52.58 406.31 15.0 33.33 0.60 1.619 61.77 358.00 10.0 50.00 0.40 1.176 85.03 328.57 5.0 100.00 0.20 0.618 161.80 312.75 2.5 200.00 0.10 0.313 319.62 307.50 13 350 325. 300. 275. 250. 1 1 1 ad Vcr am Hi we 1 1 100. 75. 50. 25. ‘T 11 JfSIZEB 00 l I’ l l ‘T l i l 1 1 0 5 1000 1500 2000 2500 3000 3500 4000 4500 5000 55006000 Velocity Change (in/sec) acceleration (Ga: ) Figure 5. DEC for square- -wave shocks to a product containing a critical element with f = 20 Hz and so = 100 G's. ce Again, note that a knowledge of the properties of the critical element (Gcr and fce) completely determines the shape of the DEC for a specified shape of the shock pulse to the pro- duct. For a square-wave shock, the shape of the DEC is much simpler, a rectangle with a rounded corner. The critical vel- ocity change and critical acceleration are obtained as follows: for extremely short duration shocks, fprod is large and f is small. From equation(7), Am approaches a value of Am = (363 Inflfce/ fprod) so that Gprod approaches a value of Gprod = 14 (Gcr*f d)/(’D’*fce). The velocity change then approaches a pro limiting value of Ave. which is exactly the same as for half—sine shocks (equation * 61.5 Gen/foe (13) (11)). Since the largest Am is 2.00, the critical accelera- tion for a square-wave shock to the product is Gce" = Ger/2 (14) The DBC for square-wave shocks in Figure 5 is by far the more commonly used description of fragility since square-wave shocks to the product are considered to be the most severe [Newton, 1976] even though they rarely occur in practice. The problem with the approach used to generate either DBC is that it requires a knowledge of the properties of the cri- tical element. If the product were a light bulb and critical element the filament inside, it would be difficult to obtain information about Gcr and fce' This is usually true of all products which have the model in Figure 3. Information per- taining to the critical element is nonexistent. What is needed therefore is a procedure which eliminates the need to know anything about the critical element and provides for the construction of the DBC directly. For square-wave shocks, this is all made possible by the fact that the shape of the DBC is basically rectangular. The entire curve is therefore determined by only two numbers, [Eve r and Ger/2' These two 15 values may be obtained by subjecting the product to drop tests on a shock machine which can produce both long and short dura- tion square-wave shocks as outlined below. In fact, for short duration shocks, it is not necessary that the shape be a square— wave since the critical velocity change is the same for both square and half-sine shocks (see equations(11) and (13)). The procedure for performing this type of fragility test is summed up in a standard method which will be covered next. II. The Vertical Shock Machine, ASTM D 3332 and the Conventional Method of DBC Construction A. The Vertical Shock Machine The conventional method, ASTM D 3332: Mechanical-Shock Fragility of Products Using Shock Machines, that is used to determine a product's DBC utilizes a vertical shock machine, an example of which is illustrated in Figure 6. The purpose of the shock machine is to control the nature and intensity of the shock by adjusting the hardness of the impact surface (which controls the shock duration) and by varying the table drop height (which controls the velocity change). Dual hydrau- lic hoists lift the table up the guide columns. When the table is lifted to the desired drop-height, drOp controls release pneumatic brakes that hold the table and it falls (under the force of gravity minus friction effects due to the guide posts) onto the programmers. After each drop, the pneumatic brakes stops the table after it rebounds off the programmer surface to prevent multiple impacts. Can-1e Cohan. .7- 1 '1 Hannah: 1 " 1 link" Model 846.361 Shock Telling symm wam 05:06.1. ‘3?.‘-. ' Figure 6. An example of a vertical shock machine. When a half-sine shock pulse is desired, the table is programmed to fall onto 'half—sine programmers' that are made of a special epoxy resin. As the table drop height increases, both the acceleration and velocity change increase while the duration of the shock pulse is kept very nearly constant at about 0.002 seconds. When a square—wave shock pulse is desired, the table is programmed to fall onto the 'gas programmers'. As indicated in Figure 6, the table has two piston—type plungers attached 17 to its undercarriage. Nitrogen gas is pumped into two cylin- ders located directly below these plungers. As the gas pres- sure in the cylinders is increased, the duration of the shock decreases and the acceleration increases. The velocity change is held constant by maintaining the same drop height. The plungers are coated with an elastic material which act as a spring/mass system. The shock pulse rise and fall is nearly linear which makes the shock trapezoidal as in Figure 2b. The plateau region for this type of shock pulse is intended to be flat because the plunger travel is small which changes the volume of the gas in the cylinders very little. Since the gas volume is nearly constant, so is the gas pressure which leads to nearly constant deceleration. An oscilloscope may be used to record the shock pulse delivered to the product using an accelerometer attached to the shock machine table. A waveform analyzer may be used to determine the peak acceleration, duration, and velocity change. A hard copy of the signal can be obtained by simply tracing the signal with tracing paper, photographing the signal, or by means of a digital plotter. Not shown in Figure 6 are electronic filters whose func- tion is to condition the resulting complex signal so that the shock wave can be convieniently displayed. The shock pulse that results whenever the table drops onto the programmers is complex because of "ringing effects" superimposed onto the shock pulse by the table, hydraulic lift cables, and guide 18 column vibrations. The filtered signal is a very smooth curve. Finally, the shock machine is attached to a seismic base whose purpose is to absorb the shock and isolate it from the shock machine area. Shock machines may be supplied with calibration tables from the manufacturer. A calibration table would be required if a shock machine were purchased without an oscilloscope or waveform analyzer. An example of a calibration table is shown in Table 3. Table 3. Shock Machine Calibration Chart for 2 ms Half-Sine Programmers (bare table) Drop Height [XV G's (inches) (in/sec) 2 55 160 5 91 305 10 128 455 14 153 560 17 169 620 The information presented in Table 3 is only a portion of the entire calibration chart. A calibration chart also exists for the gas programmer as well. The shock machine is used to determine the DBC of a product in conjunction with ASTM D 3332. The application of this standard in determining a product's DBC is the subject of the next section. 19 B. ASTM D 3332 - Mechanical Shock Fragility of Products Using Shock Machines and the Conventional Method of DBC Construction A widely used standard which delineates the procedure for fragility testing using a shock machine is ASTM D 3332. A summary of the procedure follows. The test begins with determining the critical velocity change needed to damage the product. Since [XVCr is associ— ated with short duration shocks, the product is mounted onto the shock machine table and the table is programmed to fall onto the plastic programmers. Beginning with a low drop (typically 2"), the table is dropped and the product is inspected for damage. If there is none, the height of each subsequent drop is increased until the product is damaged. The velocity change which just causes damage to the product is defined to be the critical velocity change for the product. In the second part of the test, a new product is mounted onto the table of the shock machine and the table is raised to a height that will produce a velocity change which will exceed the critical velocity change by at least 57%. The gas pressure is gradually increased for subsequent drops until the product is damaged. When the product becomes damaged, the acceleration that just causes damage is defined to be the critical acceleration of the product, and is loosely referred to as its fragility. 20 As indicated in Figure 5, the critical velocity change and critical acceleration completely determine the DBC since the entire DBC can be effectively described by the intersec- tion of two lines, the locations of which are determined by the critical velocity change and critical acceleration. In the next section, limitations of damage boundary theory and shock machines will be examined. III. Limitations of Damage Boundary Theory and Shock Machines A. Damage Boundary Theory The damage boundary curve has been constructed under the assumption that a product contains a fragile critical ele- ment(s) that behaves as an ideal spring/mass system(s). However, many types of products do not have critical elements. An example is an apple. Furthermore, even if a product has a critical element, it may not behave as an ideal linear spring/ mass system. Therefore, the DBC for the product may not have the shape shown in Figure 5. As a consequence, cushioning which is selected based on product fragility shown in the curve may lead to unexplained product failure. This may be attrib- uted to the inability of the model to account for fatigue failure as a result of repeated shocks. Even if the product can be adequately modeled as a rigid mass containing an ideal critical element, there are still practical problems associated with using the shock machine to deduce the DBC. 21 B. Limitations of the Shock Machine When a shock machine is used to produce a square-wave, it is assumed that the gas programmer generates the shock pulse shown in Figure 2a. Upon closer examination, however, the shock wave that is produced by the gas programmer is clearly not square in shape but appears as shown in Figure 7. The complex shape complicates the velocity change and peak acceleration determinations. 1 90% .. \y E 32’ \ I-Q—ATP—E" .11P L‘_ Rise Time(TRI-—>" 1......wa Time (To); f * Fail TimeITF) -d-—Pulse Duration (TP) Figure 7. ASTM D 3332 method of fairing a square-wave shock pulse. If a waveform analyzer is not used during the test, then determining the peak acceleration and velocity change may be difficult due to the presence of ringing effects discussed earlier. According to ASTM D 3332, either signal conditioning (electronic filtration) or "fairing the pulse" can be used to smooth out the shock pulse so that the values of velocity 22 change and peak acceleration can be conveniently determined. ASTM's method for fairing the pulse however is very confusing. It begins by stating that two horizontal lines be drawn through the shock pulse at levels that purportedly represent 90% and 10% of the maximum faired acceleration. But how can a tech- nician get 90% and 10% of something that they do not know yet (the maximum faired acceleration is what they are looking for!)? Nonwithstanding, ASTM D 3332 goes on to state that two perpendicular lines are then drawn from the intersection of these two lines and the shock pulse curve at the bottom of the shock pulse. As indicated by Figure 7, these two sets of ver- tical lines define the dwell time T and Pulse Duration T . D P TP and TD are used to construct an average pulse duration vari- able that is used to determine the velocity change of the shock pulse according to: - 'k * [Xv - 386.4 AM [(TR/Z) + (TE/2)] (15) where [XV has already been defined, 3’ ll maximum-faired acceleration, M TR = rise time in seconds, and TF = fall time in seconds The half-sine shock pulse produced by the plastic program- mers is faired in a similar fashion, as indicated in Figure 8. 23 ran 1 1 r$ Paired Acceleration 5 g 3 t; .3 S VELOCITY 5 ‘2 § CHANGE d E g (SHADED 8 g s PORTION) 1! a 8 2 < \ 10% ’ | Pulse Duration l I 11' "TP 7 L (1'9) 7| - p Figure 8. ASTM D 3332 method of fairing a half-sine shock pulse. The velocity change is determined by Av It has been assumed that the high frequency components * 0.636*AM TP (16) of a complex wave can be ignored, thus justifying fairing or filtration. This assumption has not been rigorously investi— gated. It may turn out that high frequency components of a complex wave could have had a detrimental effect on critical elements that have high natural frequencies. These effects would of course be covered up after filtration. Without ex- tensive laboratory investigation, such filtration appears to be unjustified. The effect of fairing and filtration has also been masked in the shock machine calibration charts. Another problem asso- ciated with the use of these calibration charts is that the values listed are valid only for a bare table. As weight is 24 added to the table, the actual G's that are experienced by both the product and table change, so that a correction factor is needed. The significance of this is that if the correction factor is not applied to the shock machine's calibration chart, then it would appear that the product has a different fragil- ity rating than it really has. As a result, a thinner cushion would be selected and the product would be under-protected, possibly leading to failure. Another important consideration in fragility testing is the mounting location of the product on the shock table. ASTM D 3332 stresses that the choice of mounting points strongly influences the dynamic response of the product and consequently may affect the fragility rating of a product. This reduces the reliability of the test results. In view of all of these considerations, possibly erro— neous fragility determinations appear to be an inevitable consequence of using the shock machine. Because of these problems, the fragility of a product may be over-estimated, underestimated, or result in an ambiguous value, all of which results in increased cost to the manufacturer. In addition to all of the problems associated with shock machines, these instruments remain very expensive investments. One manufacturer [Church, 1990] indicated that a small shock machine (65 cm x 81 cm table surface area) equipped with 25 minimal instrumentation would cost in the range of $55,000 to $65,000. As the table size increases and as the instrumenta- tion becomes more SOphisticated (signal conditioners, ampli- fiers, waveform analyzers, etc.), the total cost of a shock machine system can easily be many tens of thousands of dollars more o These machines also require frequent maintenance that can become expensive. Plastic programmers need to be replaced periodically, as do compression pads, cable wires, and so on. The gas programmers require compressed gas, as do the pneuma~ tic brakes, so gas cylinders need to be refilled. Finally, the machine itself needs to be lubricated and adjusted peri- odically, requiring the services of a skilled technician. Since the base of a shock machine is attached to a large seismic mass that is anchored to the floor wherever it is located, shock machines tend to be not very portable. Expensive, dedicated lab space must be set aside for the pur- pose of maintaining a shock machine, and so is lost for any other purpose. Expensive as this may be, greater costs may be incurred due to the inappropriate use of protective pack- aging materials, the faulty design of which is a direct conse- quence of erroneous fragility determinations. Yet, for all of the problems that are associated with shock machines, shock machines tend to be very popular 26 instruments. This is a good example of how technical people falsely place their faith in expensive instrumentation with- out really knowing what it is that the instrument is measuring. Given all of the problems of a shock machine, it would appear that the acquisition of such a machine for the purpose of DBC determination is a questionable investment. For this reason, an alternative method of determining a product's DBC is presented. 27 Chapter Two: An Alternative Method for Determining the DBC for a Shock—Sensitive Product I. The Alternate Method If one is convinced that their shock-sensitive product can accurately be modeled as in Figure 3, an alternative method to ASTM D 3332 can be used to determine the DBC of the product without requiring the use of a vertical shock machine. As such, this method will be much less expensive because min- imal instrumentation is needed for this method. Since the shock machine will not be used, fragility tests must be conducted by dropping the product onto various surfaces and inspecting for damage. For almost all impact surfaces, the shock produced will be half-sine in shape. This is parti- cularly true for drops onto cushions. The DBC constructed from the results of these drop tests will therefore have the shape shown in Figure 4. Suppose now that a new product with an accelerometer attached to it is dropped onto a very hard surface such as concrete and that this shock just damages a critical element inside the product. From the half-sine shock pulse recorded by the accelerometer, the peak acceleration G and velocity change AV may be determined. This gives us one point ([AV,G) on the DBC in Figure 4. Next, suppose that another new specimen of the same product is dropped onto a softer surface like a cushion from a much higher height and that this shock also just damages the product. From the accel- erometer attached to the product, a second point ([CXV,G) on 28 the DBC in Figure 4 is obtained. At this point then, we have only two points in space (on G vs [XV axes) which we know lie on a curve which as the shape shown in Figure 4. Before proceeding any further, consider the related re- sults obtained by using the shock machine. The hard surface is replaced by the plastic programmers and the softer surface is replaced by the gas programmer. The results of the drop test also give the same information with the important excep- tion that the two ( [AV,G) points produced when the two new products are damaged are points on the vertical and horizontal lines of the rectangular shaped DBC in Figure 5. The signif- icance of this fact is that at this point, the entire DBC is determined since only two points are required to construct the two lines (one vertical and one horizontal) which form the rectangular DBC. The velocity change from the drop onto the plastic programmers is the critical velocity change and the peak acceleration from the drop onto the gas programmer is the critical acceleration. The critical element properties fce and Get then follow directly from equations(13) and (14). When we now reconsider the results of the drop tests in which we did not use the shock machine, how do we pass a curve with the shape shown in Figure 4 through only two points in space? Since we have no way of knowing whether these points lie on a peak or a trough of the wavy portion of the curve, or even inbetween, it appears that there could be an infinite 29 number of ways to pass such a curve through two points. This bothersome situation is a direct consequence of the fact that damage was produced with a half-sine shock, hence the shape of the DBC shown in Figure 4. Note that this problem is nonexistent when the shock machine is used since damage is produced with square-wave shocks which leads to the shape of the DBC shown in Figure 5 (no peaks, no .troughs, just two straight lines). As it turns out, the situation is not as hopeless as it seems. Although it may appear that an infinite number of curves with the shape shown in Figure 4 could be drawn through two points, this cannot be the case due to the simple fact that only two parameters, fce and Gcr' determine the shape of the entire DBC. When viewed this way, any point on the curve must contain information about these two parameters. Therefore, if we have two points on the curve, then we must have two separate pieces of information about fce and Gcr’ In mathematical parlance, we have two equations in two unknowns. In theory, these two equations may be solve simul- taneously for fce and Gcr' Once these parameters are known, the remainder of the half-sine DBC may be constructed as can the entire square-wave DBC! The first step in the analysis is to use equation(4): = * Gce Am Gprod (4) 30 Since the amplification factor for half-sine shocks (see table in Appendix A or equations(S) and (6)) is a function of fre- quency ratio only, this can be written more simply as (17) Am = funct(fce/f prod) Solving equation(9) for the duration gives: duration = 1/(2*f ) (18) prod Inserting this into equation(3) gives _ * ° * [xvprod - 0.636 peak acceleration/(2 fp (19) rod) Expressing the peak acceleration in equation(19) in in/seca and solv1ng for fprod gives *386.4) d (20) 2* AVprod 0.636(Gpro fprod Sustituting this into equation(17) and simplifying yields * Am = funct fce [XV 123*G P prod (21) rod where fce is in Hz,[AV is in in/sec, and G is in G's. prod prod The argument of the function (the quantity inside parentheses) in equation(21) is just the frequency ratio used to find the amplification factor. As a quick check on the algebraic manip- ulations up to this point, the values fCe = 20 and any choice of values for [AV and G from the same row in Table 1 prod prod may be used in the argument of the function in equation(21). 31 The result should be the frequency ratio in the same row in Table 1. For example, from the third row of Table 1, Avprod = 563 and Gprod = 56.5. The argument in equation(21) then is (20*563)/(123*56.5) = 1.62 which is seen to be the frequency ratio (Column 3) in the same row. The result in equation(21) is by no means restricted to the particular critical element parameters used to generate Table 1. The final step in the analysis is to substitute equation (21) into equation(4) (it should be noted that when the cri- tical element breaks, Gce = Gcr): 'r f*AV I = = * ce rod Gce Gcr Gprod funct P (22) 123*G prod Equation(22) is the key to constructing the DBC for half- Sine and square—wave shocks using only two ([varod'Gprod’ points. This equation provides the shock Gce to a critical element with natural frequency fce when the product receives a half-sine shock with peak acceleration G change [XV prod and velocity prod' This is a Powerful tool for two reasons: a) If information about the critical element is available (fce and Gcr are known), then equa- tion(22) may be viewed as a relation between Gprod and [xyprod' Combinations of G and prod [XV which satisfy this relation form the prod border of the half-sine damage boundary region. 32 Hence, equation(22) defines the DBC once fce and Gcr are known. b) If information about the critical element is unavailable (typically the case), then fce and Gcr may be deduced by collecting experi- mental data on two different half-sine shocks which are known to just damage the product: we will call these (AV ) and prod1'Gprod1 (zxvprodZ'Gprod2)' Specifically, fce and Gcr are the solution to the two simultaneous equations below: f *Av G = G *funct Ge Prod‘ (23) ..cr prod1 123*G prod1 ~ f *Av _l = * Gcr Gprodz funct ce prodZ (24) 123*Gprodz The solution to‘equations(23) and (24) for fce and Gcr is difficult since the function referred to here is the complex amplification factor relationship from equations(S) and (6). An iterative solution is required: Step 1. Guess a value for fee“ Start with 1 Hz. Step 2. Using the known [AV '5 and G I prod prod S for the two shocks which just damage the product, determine the agruments of the function in equations(23) and (24). 33 Step 3. Using these arguments as frequency ratios, determine the value of the function (the amplification factor Am) from either equations(S) and (6) or the table in Appendix A. Step 4. Again, using the known G '3, find prod T the predicted value for Gcr from equations(23) and (24). The two values for Gcr from equations(23) and (24) will in all probability be different (the value for Gcr is a fixed number), the guess on fce in Step 1 must have been wrong. Step 5. Choose another fce' Try 2 Hz. Repeat steps 2 through 4 until your guess on fce produces the same predicted value for GC from equations(23) and (24). e When the value of fce is chosen so that the predicted Gcr from eqn(23)" matches that from equation(24), the itera— tive solution is complete. The values of fce and Gcr so obtained are the sought after critical element parameters. At this point, the remainder of the half-sine DBC may be con- structed either as in Table 2 or as defined by equations(13) and (14). Since the iterative procedure outlined above is tedious and is unlikely to converge to the solution by simple 34 trial and error guessing on fce' a program in BASIC has been designed which follows these steps and systematically increases the guess on fce by small amounts. The Program in Appendix C begins the search for fce with an assigned value of 1.00 Hz and continues its search up to 300 Hz in increments of 0.01 Hz (Line 130). This program first requires the operator to input the values of the peak acceleration and product velocity change for both points refer- red to in equations(23) and (24) (Lines 20, 40, 80, and 100). The corresponding durations are computed (Lines 50 and 110) so that f for both points can be determined (Lines 60 and prod 120). For each guess on fce (Line 130), Am is determined for the first fee/f ratio (Lines 140 through 160) using a sub- prod routine which utilizes equations(5) and (6) (Lines 290 through 390). The procedure is repeated for the second ratio (Lines 170 through 190). At this point, the computer makes a deter- mination of whether or not the choice of fce is indeed the true value of the critical element's natural frequency by exam- - * ining the absolute value of the difference between Am1 Gprod1 * . . and Am2 Gprod2 where Am1 refers to the amplification factor associated with the foe/f ratio of the first data point prod1 and Am2 refers to the amplification factor associated with the fce/f ratio of the second data point. Since both numbers prod2 represent predictions for Gcrr the difference should be zero. Because a zero difference can never be obtained in practice, 35 a value less than or equal to 0.1 was arbitrarily chosen to end the iteration. Once fce has been determined by the condition in Line 200, then Gce is determined in Line 230 by averaging the values of *G and A G Once f and G are known (Lines ce cr * m2 250 and 270), then the procedure used to complete Table 1 will Am1 prod1 prod2' produce the rest of the DBC when half-sine shocks to the pro- duct are used in fragility testing. The procedure used to complete Table 2 will produce the square-wave DBC. In the next chapter, simulation test results, sensitivity analysis, the experimental procedure, and concluding comments will be presented. 36 Chapter Three: Simulation Test Results, Sensitivity Analysis, experimental Procedure, and Concluding Comments I. Simulation Test Results The ability of the Program to uniquely determine the natural frequency and fragility of the critical element was tested with data from an expanded version of Table 1 which is shown in Appendix D. Since the half-sine DBC data in Appendix D was derived from a prior knowledge of fce and Gcr' it would appear that deriving fce and Gcr back again from this DBC data is nothing more than a mathematical version of the game "hide and seek". But this is not the case. The purpose of the alternate method presented here is to be able to construct a ABC from limited experimental test results: specifically, from information (Gprod and £§vprod’ contained in the half-sine shock pulses produced by accelerometers attached to two new products in borderline damage situations. The data in Appendix D should be regarded as experimentally obtained borderline damage results for 50 specimens of the same product in drops onto harder and harder surfaces (as you read down the table). Since the alternateimfihod calls for the destruction of only two new products through impacts (drops) onto two different surfaces, agy two of the 50 shocks in Appendix D may be used as fragility test results. Ten different researchers could select twenty different surfaces and get twenty different shocks which just damage the product. In theory, any two of these should determine fce and Gcr' This claim will be put 37 to the test. Ten pairs of shocks were taken from Appendix to simulate the results of ten different fragility tests. The results of the Program in Appendix C are shown in Table 4. Table 4. Ability of the Program to Predict fce = 20 Hz and Gcr = 100 G's Using Borderline Damage Half-Sine Shock Pairs from Appendix D. Shock Pairs Predicted fce Predicted Gcr 12 & 43 19.96 99.97 22 & 48 19.97 99.99 5 & 38 19.96 100.01 8 & 46 19.97 99.98 31 & 50 19.97 99.98 10 & 41 19.96 99.98 19 & 45 19.91 99.72 25 & 35 19.95 100.05 1 & 50 19.96 99.94 24 & 25 19.31 102.19 As demonstrated in Table 4, the Program in Appendix C accurately determines the critical element's natural frequency and fragility when any two points ([XV 1,G ) and prod prod1 '[SvprodZ'G ) on the product's DBC are inserted in the prod2 program. It is not necessary that the two shocks correspond to very long and very short duration shocks, as indicated by shock pairs 24 8 25. Shock pairs 1 & 50 test the capability of the program for very long and very short durations, respec- tively. The Program was checked for other critical elements to 38 ensure that it worked for broad ranges of natural frequencies and critical accelerations. The results are presented in Table 5. Table 5. Ability of the Program to Predict the Properties of fce and Gcr for Various Critical Elements. Actual f Actual G Predicted f Predicted G (Hz) ce (G's) cr (Hz) ce (G's) cr 2.00 30.00 1.99 30.02 6.00 40.00 5.98 39.998 20.00 10.00 19.63 9.908 100.00 32.00 99.65 31.999 200.00 10.00 197.74 9.99 250.00 250.00 249.69 249.999 Again, the Program accurately predicted the critical element's properties of fce and Gcr over a wide range of values. II. Sensitivity Analysis Apparently, any two shocks which just damage the critical element of the product are sufficient to accurately determine the critical element parameters. But what happens if there or both? These errors is an error in either [XV or G prod prod may come from three sources: 1) Instrument Error ASTM D 3332 requires a maximum error of "5.0%" on all instrumentation (it is not clear whether this is tgtal instru- mentation error or error associated with a particular instru- ment that is used in this standard). It is known that a typ- ical accelerometer has an associated error of 12.0%. 39 The coupler (a device that amplifies the accelerometer signal) has an error of 15.0% and an oscilloscope has an associated error of 13.0%. This means that G rod may be in error by as much as t(2.0 + 5.0 + 3.0)% = 110.0%. ii) Shock Pulse Shape If damage is produced by drops onto cushions, the shape of the shock pulse may not be a perfect half-sine. In fact, in most situations, it is closer to the "haversine" pulse [Newton, 1976] shown in Figure 9. m U C o H 4.) 3 peak .9. ace 0 o o . l 4 m l<——— duration >1 time (ms) Figure 9. Haversine shock pulse. The main feature of the haversine pulse is the gradual build up and decline of the acceleration at the beginning and end of the shock pulse (the points marked "*"). The shape factor for the haversine is 0.5 [Newton, 1976] so that the velocity change (equation(3)) is [AV = 0.5*peak acceleration*duration (25) in comparison to the half-sine pulse in Figure 1 where the 40 shape factor is 0.636. The significance of this is that the entire analysis up to this point has assumed that the shape of the shock pulse was half-sine, hence the use of the half- sine shock amplification factors in equations(S) and (6). It is not known what influence such a pulse may have on the program results for fce and Gcr' The effect may be estimated, however, using a simpler argument: using the half-sine pulse . as the basis for discussion, the velocity change for a haver- sine is 100(0.636 - 0.5)/0.636 = 21.4% smaller than that for a half-sine pulse with the same duration and peak acceleration. The error in assuming that the pulse is half—sine if and when it is in fact haversine may be as much as 1/2 * 21.4 = 10.7%. An additional source of error associated with shock pulses of any shape is the determination of the duration itself. It is usually very difficult to tell where a shock pulse begins and ends. This problem affects the ASTM procedure which uses the shock machine as well. iii) Incorrect Model of Product If the product cannot be modelled as shown in Figure 3, then the product does not have a DBC. It will be possible in general to produce damage to two new products as outlined here and then plot two points on G vs [AV axes, but the remainder of the fragility picture cannot be inferred from this informa- tion. The purpose of this study was not to address the ade- quacy of the model but to provide an inexpensive alternative to ASTM D 3332 which like ASTM D 3332 assumes from the start that the product can be modelled as in Figure 3. 41 Based on reasons i) and ii) above then, the error on both Gprod and [AV high as 110%. To see how these errors will influence the fce prodfor each of the two shock pulses could be as and Gc determinations, two arbitrarily chosen shock pairs r (12 and 43 from Appendix D) were inserted into the Program and G after their [Xyp values were "perturbed" by 110%. rod prod For example, in the first entry of Table 5, shock #12 may be perturbed by increaSing both the [xvprod and Gprod values by Vprod and Gprod for shock #43 by 10%. In Table 5, this is indicated as: 12(+10%,+10%) 10% and decreasing the values of and 43(-10%,-10%). It should be stressed that a 10% error of [xvprod and Gprod values is not a fault of this alternative method, but is a consequence of limitations associated with existing tech- nology (referring to the accuracy of accelerometers, couplers, and oscilloscopes). Since shock machines also use this equip- ment, the effects of instrument error on fce and Gcr determin- ations applies equally well to shock machines. 42 Table 6. Effect on fee and Gcr Determinations When Avprod and G Values of Shock Pairs #12 and #43 from prod Appendix D Are Perturbed 10%. Set Pertubation Predicted fce Predicted Ger 1 12(+10%,+10%) 24.20 Hz 104.12 G's 43(-10%,-10%) 2 12(-10%,—10%) 14.46 83.44 43(+10%,+10%) 3 12(+10%,-10%) 16.33 89.98 43(+10%,-10%) 4 12(-10%,+10%) 24.20 109.97 43(-10%,+10%) 5 12(-10%,-10%) 19.96 89.97 43(-10%,-10%) 6 12(+10%,+10%) 19.97 109.98 43(+10%,+10%) 7 12(+10%,-10%) 18.58 87.24 43(-10%,+10%) 8 12(-10%,+10%) 21.51 110.17 43(+10%,-10%) 9 12(-10%,-10%) 15.78 87.50 43(+10%,-10%) 10 12(+10%,-10%) 19.02 86.51 43(-10%,-10%) 11 12(-10%,-10%) 19.32 90.30 43(-10%,+10%) 12 12(-10%,+10%) 25.92 109.14 43(-10%,-10%) 13 12(+10%,+10%) 21.12 108.84 43(+10%,-10%) 14 12(+10%,-10%) 15.81 90.31 43(+10%,+10%) 15 12(+10%,+10%) 23.25 105.74 43(-10%,+10%) 16 12(-10%,+10%) 14.46 83.44 43(+10%,+10%) 43 Apparently, the effect of perturbing each of the G and AV values used by the Program oarpounds the errors on fee by as much as 27.7% (sets 2 and 16) and Gcr by as much as 16.6% (also sets 2 and 16). III. Experimental Procedure The main difference between the alternative method pre- sented here and the conventional method that uses the shock machine is the technique that is used to generate the shock pulses which just damage the product. In the conventional method, the two shock pulses that just damage the product are of two different forms: a half-sine shock pulse (produced by the plastic programmer of a shock machine) and a "square-wave" shock pulse (produced by the square-wave programmer). The alternative method uses two half-sine shock pulses to dam- age the product. For the sake of convienience, it is sug- gested that one of the shock pulses be of short duration and the other shock pulse be of long duration. The short duration shock is accomplished by simply dropping the product onto a very hard surface, such as a concrete floor. The long dura- tion shock is accomplished by mounting the product on a soft cushion(s) and dropping it onto an immovable surface (again, a concrete floor, for example). The essential components of this alternative method are described in two ASTM standards: ASTM D 3332 (Method B); and ASTM D 3331 (Assessment of Mechanical-Shock Fragility Using 44 Package Cushioning Materials), now defunct. ASTM D 3331 was very similar to the procedure outlined here. According to Committee D of ASTM, ASTM D 3331 was likely dropped because there remained an unanswered question as to where the test results were located on the DBC [Church, 1990], and that the utilization of this standard for the purpose of determining the DBC was "... the poor man's method." [Church, 1990]. As demonstrated, determining exactly where the critical accel- eration point is located on the DBC is not such a difficult task. The first drop intended to produce damage to the product should be done by dropping the product onto any hard surface (concrete or steel) at successively higher drop heights until damage occurs. Since the surface is hard, the duration of the shock will be very short and the shock pulse will be reason- ably half—sine in shape. The resulting shock pulse is equiv- alent to the shock pulse produced by the plastic programmers on the shock machine. The velocity change and peak accelera— tion from the accelerometer are used as the first point used in equation(23). The next step is to drop a new product onto a surface which will create a relatively long duration shock. The [AV and peak acceleration corresponding to damage in this part of the test are used as the 2nd point in equation(24). The tech- nique suggested for this test is to mount the product on an 45 assembly of cushions and then drop this product/cushion system onto a hard surface (again, using the concrete floor). In this scheme, all of the cushions are of the same thickness and made from the same material. The cushion assembly needs to be thick enough so that the product is capable of surviving at least one drop without incurring damage, and each individual cushion needs to be thin enough such that the product will be damaged if the product is mounted on just one of these cushions and dropped from the designated drop height. As such, individ— ual cushions are removed from the assembly with each subsequent drop until damage occurs. Since the area under the shock pulse is equal to the velocity change, and given that the velocity change remains relatively constant (not a strict requirement, but the same drop height would very likely be maintained during this phase of the experiment), then the duration of the shock pulse would decrease and the acceleration would increase, cul- minating in the failure of the critical element. Of course, there are other methods that can be used to gradually decrease the pulse duration and increase the accel- eration level. Instead of decreasing the thickness of a cushion assembly, drops could be conducted on cushions whose material properties became increasingly stiffer. In this scheme, the cushions are manufactured from different polymer systems. Another system that could be employed for this phase of 46 the test would use steel springs, instead of the polymer cush- ions. A platform constructed of a very stiff material (steel plate, plywood, or other such construction) could be construc- ted in such a fashion that the platform would be capable of accomodating a number of springs. The initial test would begin with the platform mounted on just a few of these springs, per— haps three or four. After the first drop (and after each sub- sequent drop), another spring is inserted into the platform until the platform becomes stiff enough to cause an accelera- tion level that damages the product. The necessary require- ment here is that the platform be large enough to accomodate the necessary number of springs that are needed to stiffen the the platform adequately. Alternatively, the springs can be changed with springs whose spring constants are gradually stif- fer, until such time that the platform once again becomes stiff enough to cause a shock that damages the product. In the final analysis, it makes no difference how these borderline damage shock pulses are generated as long as they are half—sine in shape. As a result of these two drop tests, two shock pairs on I . the product 8 DBC are identified, 'vaprod1'Gprod1) and (Av are inserted into the Program in Appendix C, fce and Gcr are prod2' GprodZ" After the wflums of these two shock pairs determined. This is the critical information that is needed to determine the remainder of the product's DBC (which can be done by applying equation(22) in conjunction with the procedure 47 used to construct Table 1 and 2). A slight complication may arise during the first drop test. Since the product is manually dropped in the alternate method, nearly perfect flat drops will not be possible as with the shock machine. A non-flat drop means that the peak accel- eration obtained from the accelerometer will depend on where the accelerometer is attached to the product. This situation is most pronounced when the impact surface is hard. There are two remedies for this. First, a drop-tester could be deployed during this test. Since drop-testers are capable of producing near-perfect flat drops, this problem is eliminated. However, acquisition, stor— age, and maintenance costs can be considerable, and the whole point to this thesis is to simplify and reduce the costs associated with fragility testing. The second solution to this problem is to use cushions during this test. In a non-flat drop onto a soft surface such as a cushion, the surface will deform and contact the entire base of the product. This will usually happen without affect- ing the motion of the product too much. At this point, the impact is not very much different from a flat drop. Since the Program has been shown to perform very well given any two half- sine shocks, the drop procedure may be modified to replace the drop onto a different cushion or steel spring system. 48 IV. Concluding Comments Nearly all of the problems associated with shock machines are thus completely avoided or at least significantly reduced by this alternative method. Most importantly, this method is cost effective. Most of the costs associated with shock machines (purchasing, maintenance, and space requirement costs) are either avoided or significantly reduced. The only equip- ment and materials needed are an oscilloscope (or waveform naalyzer), an accelerometer, cables, couplers, and a cushion- ing system (polymer cushions or steel spring system, for example). As a result, the fragility testing system discussed here is portable, whereas a shock machine is not. The fragility testing method discussed here should be capable of determining the fragility of a shock-sensitive product (which meets the classical model requirements) less expensively than when a conventional shock machine system is employed in DBC determinations. Once the fragility of a product has been accurately determined, then protective packaging requirements can be more scientifically determined which leads to decreased packaging costs. 1) 2) 3) 4) 5) 6) 7) 49 List of References Annual Book of ASTM Standards. Volume 15.09. American Society for Testing and Materials. Philadelphia, PA. 1985. Brandenburg, R.K. and Lee, J.J.L. Fundamentals of Packaging Dynamics. MTS Systems Corporation. Minneapolis. 1985. Burgess, G.J. "Product Fragility and Damage Boundary Theory". Packaging Technology and Science. Volume 1, 1988. Wiley. London. Church, E. Telephone conservation April 23, 1990. Lansmount Corporation. Monterey, CA. Higdon, A., Ohlsen, E., Stiles, W., Weese, J., and Riley, W., Mechanics of Materials. 1976. Wiley. New York. Mindlin, R.D. "Dynamics of Package Cushioning". Bell System Journal. Volume 24, October 1945. Newton, R.E. "Fragility Assessment, Theory, and Test Procedure". MTS Systems Corporation. Report 160.06. Minneapolis. 1976. Frequency Ratio 50 Appendix A Amplification Factor Shock Amplification Factors for a Half-Sine Pulse fce/fprod Am .01 .020 .02 .040 .04 .080 .10 .200 .20 .396 .30 .588 .40 .771 .50 .943 .60 1.102 .70 1.246 .80 1.373 .90 1.482 1.00 1.571 1.10 1.640 1.20 1.690 1.30 1.726 1.40 1.750 1.60 1.768 1.80 1.759 2.00 1.732 2.20 1.694 2.40 1.649 2.60 1.600 2.80 1.550 3.00 1.500 4.00 1.268 5.00 1.083 6.00 1.170 7.00 1.167 8.00 1.126 9.00 1.070 10.00 1.100 11.00 1.100 m 1.000 51 Appendix B Shock Amplification Factors for a Square-Wave Pulse Frequency Ratio Amplification Factor fce/fprod Am .02 .063 .04 .126 .10 .313 .14 .436 .20 .618 .26 .794 .30 .908 .36 1.071 .40 1.176 .46 1.323 .50 1.414 .54 1.500 .56 1.540 .58 1.580 .60 1.618 .62 1.654 .64 1.689 .66 1.721 .68 1.753 .70 1.782 .80 1.902 .84 1.937 .88 1.964 .90 1.975 .92 1.984 .94 1.991 .96 1.996 .98 1.999 1.00 2.000 :>1.00 2.000 52 Appendix C BASIC Program which determines fce and Gce from from G and .AN’ data. prod prod 10 PRINT "INPUT GP1"; 20 INPUT GP1 30 PRINT "INPUT VEL1"; 40 INPUT VEL1 50 DUR1 = VEL1/(GP1*.636*386.4) 60 FP1 = 1/(2*DUR1) 70 PRINT "INPUT VELZ"; 80 INPUT GP2 90 PRINT "INPUT VEL2"' 100 INPUT VEL2 110 DUR2 = VEL2/(GP2*.636*386.4) 120 FPZ = 1/(2*DUR2) 130 FOR FCE = 1 To 200 STEP .01 140 RATIO = FCE/FPl 150 GOSUB 290 160 AMP 1 = AMP 170 RATIO = FCE/FPZ 180 GOSUB 290 190 AMP2 = AMP 200 IF ABS(AMP*GP1-AMP2*GP2) .1 THEN 230 210 NEXT 220 STOP 230 GCE = (AMP*GP1—AMP2*GP2)/2 240 PRINT "THE NATURAL FREQUENCY OF THE CRITICAL ELEMENT IS"; 250 PRINT FCE 260 PRINT "THE FRAGILITY OF THE CRITICAL ELEMENT IS"; 270 PRINT GCE 280 END 29o REM: AMPLIFICATION FACTOR CALCULATIONS 300 IF ABS(RATIO - 1) .001 THEN RATIO = .999 310 IF RATIO 1, THEN 340 320 AMP = 2*RATIO*COS(3.14159*RATIO/2)/(1-RATIO*RATIO) 330 GOTO 390 340 AMP = 0 350 FOR N = 1 To (1 + RATIO)/2 360 Q = RATIO*SIN(2*3.14159*N/(RATIO+1))/(RATIO-1) 370 IF Q AMP, THEN AMP = Q 380 NEXT N 390 RETURN 53 Appendix D Fifty shocks which just damage a product whose critical element has a natural frequency of 20 Hz and a fragility of 100 G's. Shock Number Velocity Change G's (in/sec) 1 5591.67 90.925 2 5449.874 91.346 3 5322.816 92.012 4 5211.748 92.989 5 5023.965 92.578 6 4788.569 91.189 7 4564.891 89.198 8 4352.884 88.751 9 4152.544 87.708 10 3963.915 86.805 11 3787.11 86.063 12 3622.322 85.504 13 3469.862 85.160 14 3330.189 85.067 15 3203.97 85.277 16 3092.161 85.853 17 2996.132 86.884 18 2917.863 88.489 19 2860.261 90.840 20 2724.094 90.735 21 2501.22 87.512 22 2292.283 84.386 23 2096.812 81.362 24 1914.337 78.446 25 1744.385 75.645 54 Appendix D continued. Shock Number Velocity Change G's (in/sec) 26 1586.493 72.967 27 1440.193 70.421 28 1305.026 68.018 29 1180.533 65.772 30 1066.263 63.699 31 961.773 61.819 32 866.632 60.159 33 780.418 58.751 34 702.730 57.638 35 633.191 56.879 36 571.458 56.551 37 517.236 56.767 38 470.296 57.684 39 439.510 59.540 40 397.888 62.700 41 372.540 67.728 42 353.300 75.228 43 338.802 86.081 44 328.034 101.829 45 320.228 125.265 46 314.780 161.864 47 311.193 223.635 48 309.043 341.929 49 307.500 625.935 50 307.500 1768.097 55 Appendix E Program 2. This program requires the operator input the cri- tical acceleration and natural frequency of a product's criti- cal element. The program then provides 50 coordinates of a damage boundary curve (for either a half-sine or square-wave shock pulse system) which fully describes the product's damage boundary curve. 10 20 30 35 40 50 60 70 80 90 100 110 120 130 140 150 153 160 170 175 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 DIM VELCHNG(100),GPRODHS(100),GPRODSQ(100) INPUT F1 INPUT GCE REM: GCE IS THE CRITICAL ACCELERATION To CRITICAL ELEMENT FOR K=1 TO 50 TAU2 = 5/F1*((51-K)/50)**1.5 F2 = 1/(2*TAU2) RATIO = F1/F2 GOSUB 240 GPRODHS(K) = GCE/AMP VELCHNG(K) = 245.989*GPRODHS(K)*TAU2 Q = (386.4*GCE)/(2*F1*VELCHNG(K)) FOR I = 0.01 To 20 STEP 0.01 A = 2*SIN(1.57*I) IF I>1 THEN A = 2 IF ABS(A/I-Q)>0.01 THEN 160 AMP = A NEXT GPRODSQ(K) = GCE/AMP PRINT VELCHNG(K),GPRODHS(K),GPRODSQ(K) NEXT LPRINT "VELCHNG", "GPRODHS", "GPRODSQ" FOR K = 1 TO 50 LPRINT VELCHNG(K),GPRODHS(K),GPRODSQ(K) NEXT END REM: AMPLIFICATION FACTOR CALCULATIONS IF ABS(RATIO - 1) < 0.001 THEN RATIO = .999 IF RATIO > 1, THEN 290 AMP = 2*RATIO*COS(3.14159*RATIO/2)/(1-RATIO*RATIO) GOTO 340 AMP = 0 FOR N = 1 TO (1 + RATIO)/2 o = RATIO*SIN(2*3.14159*N/(RATIO + 1))/(RATIO - 1) IF Q > AMP, THEN AMP = Q NEXT N RETURN (column b) 56 Appendix F 350 0 n1 _. ...Av 3m. cr ni“ AVpeak2 15““- uo ' [throughl '1 225. 1) mi AVpeak1 '5 mi ‘1 mi E ; Kr-—— AV30w uni 21%10 ! £_ ¢mEEEEE .9-0 ‘99 E 75‘ “##3'4-T “0'6" V? g s. W' .691 G G 32116 [1 pz 2 0 ‘Llow 1 . J ,‘l d) 500 1000 15100 2000 2500 3000 3500 4000 4500 5000 55006000 Velocity Change (in/sec) ._£°_W_<‘=1_ AV Av (”S 5" Av o o’ o column b peak2 troughl peak1 low low p1 t1 Vcr 16.976 10.8475 9.3168 1.8614 Avlow 9.12 5.8275 5.0052 Avpeak1 1 822 1.1643 Avtroughl 1 565 Gp1 0.6225 Gt1 0.6648 1.0678 sz 0.608 0.9769 0.9148 The DBC parameter ratios listed above are constant for each shock- sensitive product, regardless of what the particular values of of the critical element's natural frequency or fragility rating happen to be. ”critical/Glow 110- 185. 1894 57 Appendix G Veritical/Glou Ratio us Natural Frequency of CP 7i o-q 60. 401 35. 30. 25. 28. 15. 10 5. .. _*.-w ' ‘ ~ .-‘_- _’ -... ..- ~ur--—“ ""13 e——__._ “E4 11111111)111811111211311111511.1170119202121. Natural Frequency of critical elenent (Hz) The relationship between Aver/Glow and the natural frequency of the product's critical element is depicted here. Once two of the three parameters (AWEr, G , or natural frequency) are low known, then the remainder of the curve can be deduced using the parameter ratios as shown in Appendix F. "lllllllfilfl‘lllill?