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

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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'

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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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C160 2 0, 175.6 2 'l'l'!10, etc.l in order that M might be the independent variable in a Fourier expansion, d 1, -5- alcos M' + alcos 2M' +...+ acos nM' 2- n 2 + b,sin M' + basin 2M' +...+ bsin nM' n 2 Tl' where M' I -Q KM - 1601. 212 The various constants ao, ai, . . . and bl, b., . . . are determined by the equations 2m ta -4-a +a +...J 2 m 3m 5m 'Tl' 2'l'1' 2m-1 ft0J ft J + ft J ft 'ITD m m m and A 2m tb - b + b - ...l : m 3m 5m 'IT 3'l'l' 5'l'l' 4m-1 ft J ft J + ft l . ft 'TT 2m 2m 2m 2m form-2 1,2,3,... Since the series extends to infinity, the number of terms to be taken is determined by the point at which a succeeding term becomes negligibly small. Some preliminary calculations indicated that any rela- tionship that might be derived from this procedure would contain so many terms as to be forced and not particularly enlightening. For example, aw was calculated to be .77 for K 2 4.87 ntfcm. Since we may stop at ten terms only if am is very near zero tsince cos nM' may be as large as 11 it was evident that many more terms would be needed to ade- quately describe the graph, so this procedure was pursued no further. Additional reading on this method of curve fit- ting indicated that it is intended to be applied to the actual graph of the vibration itself rather than a graph derived from the amplitude of these vibrations as was actually done. Therefore, harmonic analysis was applied to actual data traces, but again the relatively high values of the later co- efficients suggested that this procedure had little to offer: this was probably due to the fact that the vibrations under analysis were somewhat heavily damped. Another method of data analysis which was used was the graphing of d as a function of K for a given mass value: although the graphs appeared to be similar, the standard pattern was so irregular that it was not deemed worthwhile to apply the detailed procedures used on the d vs. M graphs to this set. Finally an analytical procedure which to a considerable degree explained that data was found. The natural period of a mass and spring system ch as an automobile chassis M fignoring dampingl is 2'l'l' IE- where M is the mass sup- ported by each wheel and K is the spring constant of each spring in the suspension system. In order to investigate any possible relationship between d and e natural period, d was graphed as a function of 2'TT K . Bunches of points emerged, suggesting that certain natural periods produce consistent values of d. Two rather solid conclusions can be drawn from this somewhat confusing mass of data. First, the similarity among the curves for d vs. M for different constant values of K strongly suggests that there exists some sort of regu- larity in this situation, though more sophisticated methods of analysis than those applied may be needed to give a full explanation of what is happening. Second, the graph of d vs. the natural period contains clusters of low d value points where the ratio of M and K produces natural periods of .15 and .06 seconds. For this particular frequency of bumps, then, these are the best natural periods for a low d value and hence a smooth ride. .15 seconds has a special signific- ance which serves to help explain why it produced low values of d. The period of drum rotation land hence the time between bumpsl was .33 sec. If the pen arm and other friction were to slow the natural vibration period of the chassis to some multiple of the drum period, the chassis would be in a certain phase of harmonic motion at the be- ginning of each impact which would probably have some consistent effect on d. An examination of the data traces, however, showed that in reality friction had slowed the nat- ural motion so that the chassis made only one and a half cyclessbetween impacts when the natural tundampedl period was. sec. f- 1, upward motion at impact reduces d For K 2 4.87 ntfcm, M 2 311 g., for instance, the pen trace shows the chassis beginning to move upward just as each impact occurs. The reason that upward residual motion dur- ing impact diminishes values of d is apparent: as the chassis moves upward, it in effect pulls the springs away from the force imparted to them as the wheel passes over the ob- struction. The compression which takes place is not as great as that which would occur if the mass exerted its normal weight on the spring from above. Further, since the initial upward movement is not as great as it usually is, the chassis does not fall as far as it usually does, and thus the distance below the equilibrium position, which is an important com- ponent of d, is diminished in this case as well as that por- tion of the motion which is above the equilibrium position. It is important to realize that the .15 sec. figure is not quite so significant as it might appear to be at first glance. In the first place, this figure would have to be scaled up for it to have any practical application, taking into account the greater mass and higher spring constants of real vehicles. Furthermore, .15 was the value found only because that natural period combined with the friction present and the speed of the drum to produce a fortunate chassis vibration pattern. If the speed of the drum were change or if the im- pacts were to come in an irregular manner, as on actual roads, the .15 sec. figure would lose its significance. The data Kas yet incompletel for c as a function of M were far easier to analyze than was that for d. The smooth curve for the graphs of c vs. M tsee graphs at the end of this se-ctionl suggested an equaslon of the form c 2 a I1 where a and n are some constants. If the logarithms of both sides are taken, the equation becomes logc 2 nlogM+loga which is a linear form. When these data were placed on log -- log co-ordinates, the points did indeed suggest a line. From the graph of c vs. M for K 2 4.87 ntfcm, for example, the slope was found to be -2.16. The intercept was calculated from point-slope form to be .964, giving log c 2 -2.16 log M + .964 or, taking the antilogarithm of each side, c 2 9.21 M-2-16 where c is in centimeters and M is in hectograms. This ex- ponential relationship is not unlike the simplified model which suggested that amplitude might be inversely propor- tional to mass, though here c is inversely proportional to the 2.16 power of M rather than the first power. Before all values of K were tested to determine c as function of M, it was hoped that the general form .. c 2 aMf1 would fit the graphs of c vs. M for all values of K. Then the values of a and n could be examined in the light of the K values which produced them. If a and n could be found to be functions of K, then a general equation of the form c 2 f KKJ MENU where f and g are some functions. Such a relationship would predict average values of c for any combination of M and K. The analysis of the rest of the data relating c and M in order to define c as one function of M and K was carried out in a manner identical to that used for K 2 4.87 nt! cm. One serious difficulty arose, however, when it was observed that a number of points did not fit the expected graphs at all well. The original data traces were examined. and in many cases it became apparent that the shock absorbers did not work as well as it had been hoped. As a result, residual harmonic motion affected the value of c obtained and rend- ered certain values for c inconsistent with data taken when the chassis was allowed to approach equilibrium. To remedy this situation, new averages were taken in which only those impacts which began as the chassis was approximately at equilibrium were counted. The averages computed in this manner were used to graph the corrected points which appear on the various graphs. These corrected values are: M K 25.8 ntscm 36.8 ntfcm 160 g. 6.23 cm 191 4.99 cm l 4 4