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

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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

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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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221 5.02 3.59 251 3.99 280 3.45 These corrected points form one possible curve for K I 36.8 ntfcm, and make clearer the curve for K I 25.8 ntlcm. The equations obtained defining c for varying K values may then be summarized: 1 For K I 4.87 ntfcm, c 'J 9.21 M-2-16 For K 2 6.38 ntfcm, c 2 5.32 M-1-11 For K I 25.8 ntfcm, c I 8.97 M--82 For 'K I 36.8 ntfcm, c I 19.83 M-2-52 Iuncorrectedl c 2 9.78 M-1-01 C corrected J Unfortunately, these four equations fail to suggest some function relating a and n in the Ngeneral equation c 2 a 11 to K. Additional trials for other values of K and the inclu- sion of more individual measurements in average values of c used to determine the c vs. M equations may improve this data to make it posible to determine f and g in the ideal general equation c 2 ffKl M2410 d fcentimetersl M fhectogramsl d vs. M, Rubber Tires, Double Square Cross Section Bump, No Shock Absorbers Um Q0 wg- mn Qu. FD CD g 5.6 56 E. 33 aff 0. 3 'O 'P ' 25 31- 94 U5 5,5 U1 EW SJ el.. Ss ag HH Oll EQ Q-il Um G 5 UQ E- gn-I Hg wg Og 5 mm 260 S00 8' '85 cg. 5 5 mx lm B0 U1 PP' V' Um Q- m 7' 'X n- fDn-4Z'U- X n- On U10 Oomll Una: gg gs ganmmguz 'RUE' C ::' gg??3f1gbo og QI mf 295'-gif 5- 2 , 33 333133370 -I+? +71 LU o 3' 0 m WZ I lim? sl'55'f'i3551lD 53' I 11 :I U55 as ggxoagf-Qiggx Q,-5 g+xo+xo s A ' xr1 or+ VG. Q' ,5fg,.9,-,Q Q aim gf' gaaawam 525555 Q ,gnu mn uuuu Masai? Q Sfggw- Q 'Eg gm-sv - 'Q..m Hn. In 0--Woo m Q: . oo -1 gunna ard na 55000-J In mo 000021100 io 'zo HUD: U2 55:15 'US' 'UE' Q. 'I' Q QQ 2 QQQQ aff 9.-+ g 553 sf gegg 3 3 g a S 5 '1 -'I 36.8 ntfcm nt! cm Analysis of Error All values analyzed thus far have been averages of many measurements. The accuracy of this method might easily be questioned. Through the use of standard deviation and Stu- dent's t on typical points, an accurate estimate of the validity of the averages analyzed may be made. Standard deviation for M 2 221 g, K I 3.78 ntfcm with two obstructions using rubber tires and the square cross- section bump fwithout shock absorbersl, a value which exhibited comparatively good consistency among the indi- vidual measurements composing the average, has been com- . Sd' s I - n-1 .2884 17 2 .13 This indicates that a zone .13 on either side of the average value for this point, 1.46 cm, will include 68.2716 of the indi- vidual readings likely to occur. For one of the most incon- sistent values such as M 2 251 g., K I 110 ntfcm with two obstructions, rubber tires, squarepcross-sectioned bumps, and without shock absorbers puted to be 50.13 S.. 15 : 1.83 This is nearly a third of the average value, 5.62 cm. 1 How close, then, can we be sure that the averages given are to the true values which would be obtained by averag- ing an infinite number of infinite number of individual meas- urements, if this were possible? The equation S l A21- gives a range, A , within which a probability can be as- signed of having the true value. A is the span on either side of the average, s, is the standard deviation for a given point, n is the number of values averaged, and t is Student's t. If a A for which a 904, probability exists that the ran e extending from the mean plus A to the mean minus lg contains the true value is desired, and eighteen values are known, as for M -'I 221, K I 3.78 ntlcm for which s was Ezalculated earlier, t is found from tables to be 1.746. There- ore, - A 2 1.746 i 'JE' 2 .054 cm For the less consistent d of M I 251 g., K I 110 ntfcm, 1.83 A :- 1.753 i Al15 2 .83 cm A for M : 221 g., K 2 3.78 ntlcm is 3.7'Z. of the average: A for M 2 251 g., K I 110 ntfcm is 14.841 of the average. Since the consistency of most sets of d values lies between these two extreme cases, it would be safe to generalize regarding the accuracy of the various values for d that there is a 90'-Z1 chance for most values of d and c being within roughly 993 of what would be the average of an infinite number of individual measurements. A 9'Z: variation in certain d and c values could well account for a number of irregularities in various graphs. Of course, the possibility must not be ignored that such a variation pro- duced by new data and introduced into present averages could well yield new discrepancies. Sullllnary of project lwestinghousel My project consisted of finding equations defining the amplitude of the vibrations of a model automobile chassis as a function of the chassis mass and the spring constant in the suspension. Low Voltage DC to High Voltage DC Power Supply PAUL FROM In order to obtain 60 volts DC from a 1.2 volt battery, DC must be changed to AC, then transformed to a higher voltage and finally changed back to DC using a full wave rectifier. To change DC to AC I used a two transistor power oscillator. The design proved to be unstable, hard to start and developed high transient voltages because of such a low operating voltage. After much experimentation, the perform- ance was greatly improved, but there is still much to be desired. The following is the modified circuit performance compared to the original and desired performance. Original Modified Desired Maximum output .03 W 2W 30 W Starting only when starts at starts at Capabilities unloaded any load any load