Illinois Junior Academy of Science - Yearbook (Urbana, IL)

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PROCEDURE Six pairs of springs of varying spring constants were mounted in turn on the chassis. The lead weights were added to the chassis in pairs up to a total of fourteen weights for each spring constant, giving forty-eight combinations of weights and springs Cincluding no weights with each of the pairs of springsl that were tested. The large drum on which the rear wheels of the chassis rest was allowed to run at a constant speed of 36 scale m.p.h. and the effect of the bump mounted on the drum on the chassis was recorded by the pen arm attached to the chasis. The pen recorded a series of regular oscillations in most tests. Care was taken to be sure that the chassis assumed a regular pattern before the re- corder was turned on if at all possible and that the pen arm touched the recording drum at the same point on every trial. Trials were made with the obstacle on the drum a long bar of 56 square spruce encountered by both wheels at the same time and also with a shorter bar encountered only by the right wheel. The deflection of the chassis in each particular encounter with the obstruction was recorded by the pen arm, measured, and averaged with other values taken under the same con- ditions. Since each impact produced one immediate high peak and then a minimum height as the chassis fell back to the road and the springs were compressed, the deflection to be measured, hereafter referred to as d, was arbitrarily but meaningfully defined to be the vertical distance from 1. . H-. . H, M- wr-U-8 Q 'kb'-'5EuTZ5FETi'8'w '-' the highest point after impact to a line drawn between the two adjacent minima. CRefer to figure.J Generally, one run consisted of one rotation of the recording drum and produced between fifteen and twenty separate values of d to be averaged. In trials where many weights were added, the additional drag on the large drum occasionally caused the motor to slow. In these instances, the motor was assisted to maintain standard speed through the use of a hand crank. The disappointing results of this first series of experi- ments led to a re-examination of the degree of control which had been exercised over the variables. Two possibilities arose: the tires on the vehicle were elastic to some degree: perhaps they absorbed enough of each impact to throw off the data. The nature of the obstruction used also came under scrutiny. The possibility that a more gradual obstruc- tion for the wheels, perhaps with a semicircular rather than a square cross-section would provide better data was also considered. Accordingly, the rubber tires were replaced with metal wheels and the square obstruction with a semicircular one. Again the mass was varied while spring constant re- mained at 4.87 ntfcm, but the amplitudes which were av- eraged failed to suggest any meaningful relationship between the mass of the chassis and d for a given spring constant. Then it was observed that, due to the varying natural fre- quencies attained by the chassis as mass increased, the har- monic motion which the chassis undergoes between impacts, was placing t.he chassis in a slightly different phase of har- monic motion at each impact, depending on the mass, thus producing different amounfts of residual vibration which were added to the fresh impact. Thus, changing mass values introduced another variable, the amount of residual harmonic motion, which was not being controlled. To remedy this situation, additional friction was intro- duced into the suspension system in order to gradually de- crease the residual harmonic vibrations after the initial im- pact and bring the chassis back to equilibrium before the following impact. In this way, the state of the chassis rela- tive to the suspension, which had been varying and possibly obscuring meaningful data, could be made constant before each new impact. The method chosen to introduce this fric- tion was the use of wood strips which pressed against the axles and rubbed against the axles as the chassis vibrated. Tests with varying masses and spring constants were re- peated in the manner previously described after the chassis had been thus modified. However, measurements of d still did not show any clear pattern even after the major chassis modifications were made. It was noted, though, that the distance from the high- est point reached Cimmediately after impactl to the position of the chassis immediately before impact CAfter the instal- lation of the shock absorbers or wood strips, the chassis usually was nearly at equilibrium, that is, without any resi- dual motion, just before impact.l exhibited a definite de- creasing trend as the mass increased. This quantity was designated as c and was measured for as many combina- tions of springs and weights as was possible. -9- Quatre-tom DEFHHTQDN or Q. . RESULTS Average values of d for the various combinations of spring constants and chassis mass supported by each spring and wheel Cone half the total chassis mass supported over the drum road J with the obstruction a long square cross- sectioned bar encountered simultaneously by both wheels were as follows: Chassis mass C8-J No. of over weights each Spring Constant, K lntlcml used wheel 3.78 4.87 6.38 25.8 36.8 110 0 160 1.72cm 2.42cm 1.91cm 1.92cm 2.92cm 5.88cm 2 191 1.35 0.90 2.92 1.78 2.60 4.85 4 221 1.46 2.42 4.03 1.62 1.57 4.85 6 251 1.37 1.26 5.92 1.79 1.89 5.62 8 280 2.66 1.12 8.28 2.35 1.37 3.64 10 311 3.28 0.85 7.85 2.49 4.28 3.93 12 342 3.34 1.23 8.22 2.44 5.19 3.73 14 372 4.44 1.65 8.85 2.52 6.03 3.54 Before the friction-inducing wood strips Cwhich function in the same capacity as shock absorbers on an actual autol, metal wheels, and semicircular obstructions were added to the apparatus, data was taken in which the square cross- section bump was encountered by only the right wheel. The pen arm, moving through an arc, made traces from which d was measured. -5----'. - 13' KH HRM PKC ul-ni Smart GDM? These measurements yielded the following average values: No. of weights Chassis mass fg.l d Ccml used over each wheel lK:36.8 ntfcml 0 160 6.10 2 191 5.93 4 221 5.63 6 251 5.93 8 280 5.85 10 311 5.81 12 342 5.58 14 372 5.69 Data were measured for only one value of K as no meaning- ful pattern emerged, suggesting that laboratory time might be better spent varying conditions in order to hit upon co- hesive data rather than exhaustively investigating an un- promising situation. Data traces were made for all springs, but an examination of the traces confirmed the suspicion that measuring these data would be a waste of time. Data taken last year under apparently identical circumstances conformed to a smooth exponential curve: however, inability to repro- duce these values strongly suggests that the validity of these previous data is at best doubtful. After the metal wheels and semicircular obsruotion were added to the apparatus, data traces were made with the standard combinations of weights and springs, with the obstruction striking either both wheels or only the right. As before, an examination of the data traces suggested no trends or regularities as mass was increased for a given spring, so time-consuming measurements were dispensed with and the shock absorbers mounted. Data taken with the shock absorbers in place exhibited a definite decreasing trend as chassis mass was increased, so careful measurements of the data traces and averages were computed. It should be noted that the quantity which showed this regularity was not d, which had been measured in previous experimentation, but

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1 g5.sr. x PNG..- u.'.5 'M - JS.. U., 1 16.501 1 5 5 o U UI eu RSSB C5106 umm H o CNRS ' Mo X The magnitude bfwffhe deflection at the rear of the chassis when the wheels actually do strike the obstruction on the ro- tating drum is recorded by means of a spring aluminum pen arm attached to the rearmost chassis member. The pen arm is reinforced with wood except at the very end to reduce tor- sion caused by the friction of the pen on the recording paper. The writing instrument is a felt marker with a very fine point. It records on paper mounted on a rotating drum placed to the side of the chassis. lnqflah uasem N--' I 1 gqaflnca ..L'!' Gam QQ MOTU'- , - Posrpuu GNU r.oNs1ilvc1 9? fl-Grefkbws DY-Oh frfc-r g To QQCALQX The recording drum consists of a hollow sheet metal cyl- inder, open the bottom, which is fastened to a rotating shaft. The shaft is supported by a bearing and is driven through an 8:1 gear reduction from a small, low speed electric motor. The chassis itself is a simple perimeter type with cross bracing. It is constructed primarily of 56 square spruce strips glued and nailed together. The two wheels riding on the drum turn on two half-axles which are independently sprung, each pivoting about a pin slightly offset from the center of the chassis. Each half-axle is sprung by a coil spring and located fore-and-aft by wood strips forming a channel which limits axle motion to the vertical. wana str-WS X SPW5' Quoy SUSPGNSIDIJ frm 10 seated AX,-Q gent WWW Sine who In later experimentation, the rubber tired wheels which were used originally were replaced with metal wheels. The half-axles used with these new wheels were thinner than the old ones so the simple channel which had previously located the axles for-and-aft was no longer effective. The suspen- sion was therefore modified slightly with the introduction of control arms to locate the axles. These control arms pivoted with the axles about a point directly in front of the axle pivot. nv Wm svsveusmvl cemtut Aan Pm I MDIHI'-ICFTIUNS 1' get mee- no-r 10 stud ' mt Qwot- U game as HW?-6. Another modification which was made in later experi- ments was the addition of shock absorbers or more cor- rectly, dampers, consisting of strips of wood which introduced additional friction into the suspension by pressing against the axle. There were located on the outside of the wheels and pressed on that part of the axles which extended beyond the wheels themselves. The dampers were located by a yoke which ran across the width of the chassis. L., I ibut! TOY elw.vnS1?4Nv.:bvL vue-J 9 lv 5 HJ: R... 1 .Sa After a small tray of weights was added to the left side of the chassis to balance the pen arm on the right, a box-like carrier was mounted on the chassis into which weights could be taped to vary the mass of the chassis. A number of nearly identical lead weights were cast and numbered for identification. A light bar was passed through the axle channels and either end was placed on the pan of a balance at equal distances from the center of the chassis. The front of the chassis was supported, and the weights added in pairs in a standard order. The mass supported by each balance for increasing numbers of weights were as follows: Weight No.'s Mass on each balance none added 160 g. 1-2 191 1-4 221 1-6 251 1-8 280 1-10 311 1-12 342 1-14 372 Fourteen weights was the maximum load used in experimen- tation because greater numbers of weights placed dangerous strain on the apparatus as the chassis crossed the obstruc- tion. Since for a coil spring AQ Q K AF where K is some constant,A1.. is the change in length of spring, and A F is the force applied to the, spring, a single measurement of the compression caused by a known force will give the spring constant. The following apparatus was used to determine the values for K for springs used in the experiment: -1 mnovnw- 'LGU :muff WU? Masons 9 Mmpu. RT gawutvauw-5 wsmuasic -T-1...-- M555 gf,,,.,u. muowh' The springs which were tested in this manner were found to have spring constants of 3.78, 4.87, 6.38, 25.8, 36.8, and 110 newtons! centimeter. Springs which were shorter than the standard length of 3.00 cm. were lengthened with wood blocks glued onto one end, while the one that was longer than 3.00 cm was cut. 3.00 cm was chosen for the standard length because it gave the least camber variation from zero, i. e., slightly positive under light load and slightly negative under heavy load. Springs were mounted by passing a loop of adhesive tape through the last coils at either end of the spring. The tape in turn was passed around the chassis member directly above the axle just inside the wheel and also around the axle to hold the spring securely in place. 1596 TAPE -stew. M00 5Tl ' 'Q

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rather c, which has been previously defined. For ease in measurement, the pen arm was allowed to make a line around the drum while the chassis was at rest at equilibrium before each data trace was made. At this time, average values for all combinations of weights and springs have not been determined. The follow- ing table lists those values which- have been determined: No. of Chassis mass Spring constant, K tntfcml weights used over each wheel tg.J 4.87 6.38 25.8 36.8 0 160 2.93cm 3.18cm 5.92cm 6.07am 2 191 2.13 2.69 5.45 3.62 4 221 1.90 1.99 4.08 2.72 6 251 1.40 1.93 4.30 2.83 8 280 1.12 2.36 ' 3.52 2.71 10 311 0.86 2.40 4.19 2.50 12 342 0.58 2.32 3.26 2.71 14 372 0.51 2.76 3.08 2.74 4' Bottoming occurs, due to the short length of this particu- lar spring, thus causing inconsistent results. ANALYSIS OF DATA A number of procedures were followed in an attempt to organize the data obtained for d before any modifications were made on the apparatus. After the values of average d for each combination of chassis mass and spring constant were computed, this quantity was graphed as a function of the mass supported by each of the rear wheels tthat is, half the mass resting on the rear wheelsl for each particular spring constant. In the case where both wheels met the square cross section bumped simultaneously, it was observed that each of these graphs had the same general form al- though their dimensions varied. Each graph fell at first, then reached a minimum and began to increase for larger mass values. After reaching a maximum, the graphs again fell, reached a minimum, and were again increasing at the maximum mass value tested. tRefer to graphs at the close of this section.J W Qencanc r-mm, .L dum One exception to this rule was the graph of d vs. mass for K:6.38 ntfcm, which failed to exhibit the 'initial de- crease. However, the graph suggests that this may be due merely to the fact that mass values low enough to produce the expected shape of the graph could not be tested. The only other exception to the general shape was K:110 nt! cm, in which the last two points fall off rather than continue the expected final increase. Perhaps the graphs for the other springs would show this characteristic if higher mass values were tried. The characteristics of these curves suggested a general equation which might fit each of the deflection-mass rela- tionships, namely A d:-M-+B+CM+DM'-I-EM3+FM' where d is the d defined previously, M is the chassis mass supported by each wheel, and A, B, C, D, E, and F are con- stants. A The M term was introduced so that lim d 200 M -+0 This was suggested by experimental results in the form of the high d values obtained from low M values for each spring. The notion that d should approach infinity as M approaches zero is also suggested by a simplified theoretical model in which the chassis is thought of simply as a mass being ac- celerated and moved by a standard force. If for the moment we ignore the fact that the upward deflection of the chassis is limited by the springs, standard Newtonian mechanics tell us that d:56at' and F a:-. M Substitution yields Ft' az- 2M Since the time over which the force is applied to the wheel by the obstruction and the magnitude of the force applied to the wheel are determined by the speed of the drum, which is held constant, Ft' k limdflim-.- I lim- 290 2M M M-yo M-yO M-yO The nature of the remaining portion of the equation, a fourth degree relationship in M, was dictated by the presence of two minima and a maximum in the general graph, since an equation with n maxima and minima is of a degree at least n+1. In order to determine the necessary constants in each particular equation defining d in terms of M for a given K, .experimental values of M and d were inserted into the equation A M-P-B-I-CM+DM'+EM'+FM'-':d for each point on each graph for a given K. M was put in terms of hectograms to simplify computations. For each value of K, then, this yielded eight equations tsince eight values of M were tried! in six unknowns, A, B, C, D, E, and F. In using the method of least squares to get the best ap- proximate solution of these simultaneous equations which will determine the curve of the given form which will pass closest to the experimental points, each of the eight equa- tions for a given K was multiplied by its A coefficient and these eight equations added to form the first normal equa- tion. The five other normal equations were formed by using the B, C, D, E, and F coefficients in a similar manner. The six normals were then solved simultaneously to find the best values for A, B, C, D, E, and F. This procedure was repeated for the graphs for each of the six K values tested. The equations derived in this manner were as follows: For K I 3.78 ntfcm, d 2 54.66 T - 3.30 - 36.68M + 16.37M' - 2.06M' + .056M' For K I 4.87 ntfcm, d I 442.91 1- -1- 750.86 - 3Ol.24M -4- .OOOM2 + 9.293 + .271M' M 93 Sgor K I 6.38 ntfcm, d I T + 132.90 - 45.osM - 1.33M' + 1.67M' + .040M' For K I 25.8 ntfcm, d I-' 14.37 T -I- .780 - 10.29M + 4.68M' - .598M' + .013M' For K I 36.8 ntfcm, d I 83.29 T 10.49 - 68.72M + 30.95M2 - 3.94M' + .l08M' 70F6or K 2 110 ntfcm, d I 4- + 3.37 - 2.2sM + .esnvr - .142M8 + .0O4M' M tln each case, d is expressed in centimeters and M in hecto- grams.l Unfortunately, the coefficients in each of these equations except that for springs of K I 110 ntfcm contained coeffi- cients so large that if accuracy to the hundredth or even tenth centimeter is desired in the value of d calculated from any of these equations, more significant digits than the maximum of three that can be obtained from experimenta- tion must be used in the calculation of the constants so that they will be accurate to the necessary four or five significant digits. As they stand, however, these equations turned out to be several centimeters off in their predictions of d for a given Mg clearly this degree of accuracy is not sufficient when the values of d being dealt with are usually from one to six centimeters. The inability of the least squares method of analysis to organize the data considered necessitated the application of another method, harmonic analysis. The range from M 1' 160 g. to M 2 372 g. was set up to correspond with 0 to 2'l'l'