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E ‘___ 2 __¥:_:_.:a_.:_:_.‘__.h___f_:.___L_:_____‘._3233;“ IIWWWWWWWI ' ’ -:i 3 1293 01087 9025 V‘I'i‘ ." 71r-':-—"-.--s “a. l 4‘ This is to certify that the thesis entitled AN ANALYSIS OF MOTION WITH VELOCITY SQUARED DAMPING BY METHODS OF FINITE DIFFERENCES presented by THOMAS E. JONES has been accepted towards fulfillment of the requirements for _M~_§_-___deqree. mQIEILJLGR. (Applied Mechanics) MO. NW Major professor March 13, 1952 Date - -_‘<-._'—.~Y ‘. 4 1.1.565 :‘.1 _'!'__ '.II mceummumummmm. TOAVODMMMUW‘.“ ”MT DATE DUE DATE DUE DATE DUE FEB 2 [mm-1.13 =T= \ w i MSU Is An Affirmative Action/Equal Oppommlty Instltulon m AN ANALYSIS OF MOTION WITH VELOCITY SQUARED DAMPING BY METHODS OF FINITE DIFFERENCES By Thomas B. ignes, Jr 4-“) A THESIS Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science ’ in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1952 mutt: ”45 I- .l t I I. I .I\dl In! ...!:L§V. mi. . AU INN-nu. “9““ gt‘ ' J / fl :’ v 92: ‘ 4/! egg; 1’- (’13): I. '. 4 J'L) ..’ . '. If f} TABLE OF CONTENTS Statement of Problem . . Scope of the Analysis . Free A. Linear Vibrations o 0 Degree of Freedom . . . . . . . . . . Development of the Differential Equation Application of methods of Finite Differences . . . . . Supplementary Data . . . . . . . . . . . Comments and Conclusions . . . . Page WNNl-‘l-J 18 STATEMENT OF PROBLEM The problem, herein, is to analyze the free vibration of a spring supported body subjected to a damping force proportional to the square of the velocity of the body. SCOPE OF THE ANALYSIS The scope of this analysis is limited in three general respects. All considerations are restricted to free, linear vibration with one degree of freedom. 1. ‘E239 In each case, the motion is considered as being free of any external exciting force once it has been started by an initial displacement. This limi- tation thus eliminates the problems of an analysis concerning forced vibrations and resonance."K 2-W The analyses of linear vibrations and torsional vibrations would involve considerations of similar differential equations which would have similar solutions. In considering only linear vibrations there is, therefore, a certain saving in nomencla- ture with only a limited sacrifice of generality. ‘T*A condition effected when the frequency of the external exciting force equals the natural frequency of the system and resulting in very large amplitudes. 3. Degree of Freedom The number of degrees of freedom is equal to the number of coordinates required to specify completely the position of the body or system at any time.I If, therefore, to describe the motion of a vibrating body, only one independent coordinate is necessary, the body is said to have but one degree of freedom and can vibrate in only one manner. A. H. Church,'Elgmgntggzngchgnigal V br t , 1948. p. 7. p DEVELOPMENT OF THE DIFFERENTIAL EQUATION Consider now a vibrating body as shown in Figure la. A free body diagram of this body would show either the forces in Figure lb, or the forces in Figure 1c, depending on whether the displacement is positive or negative. The symbol, 6“, is the static deflection of the spring due to the weight W. New, the basic equation of motion is, ZFyamay which in this case becomes may: -ky or 2 d y (1) m ‘ Fat 4- ky- 0 2 where m is the mass weight of the body, a = .373;— 13 its acceleration, k is a spring constant (unit force per unit length), and y is the displacement of the body. Let us now introduce into the system a third force which we shall require to oppose the velocity of the body and thereby produce a damping effect on the motion. The particular damping force Fd, selected here is pro- portional to the square of the velocity of the vibrating body. Hence ’ 2 dy f Fd a + CV : c'_dt_lf when V is negative, 01‘ Fd = -CV2'when V is positive. . ‘ ' .- .u L‘ In. H -. ‘ a. '"‘ p. . - c. . . . . . p . 4 ‘7 .. .oo'b Ox“ in“? * L__J F/g. /0 Body and Lip/7779' ””079 A’Cé/ T 6”) F/g. /b Force: on body when disp/acemenf /3 pas/fires . Mix-y) F/g. /c Forces 0/7 body Wee/7 d/sp/oceMe/yr A: fi?gO’7‘/,k€ It then follows that the equation of motion for a free, linear vibration.with damping of this sort is 2 + 2 d y _* _ dy ‘2’ ”fir-1” “(‘37) If, from.this differential equation, we can.now derive an expression that enables us to represent, graphically, the displacement y as some function of time t, a considerable knowledge of the nature of this type motion may be gained by varying the damping coefficient C. APPLICATION OF METHODS OF FINITE DIFFERENCES We proceed now to establish a method of plotting curves showing displacement y as an ordinate against time t, as an abscissa by a method of finite differences. From Equation 2, 2 - a 2 ‘5 2 ( ) "31%— :Un -_£L ) 3 y m at where f: : Ct) 2 and wn, as known from previous studies of m n vibrations‘, is the natural frequency of the system mg . sec Now, lim Ay dy 4t—->--o a ‘3'?" At and lim Azy day AM 0 = -"'"z" A—Tt dt Hence, if we considered, not infinitely small increments of t, but rather, increments of finite magnitude we can replace the differential equation (3) by its equivalent difference equation 2 'l' 2 (4) A .. .. 00 - Aty .. nzy g (:3) , an approximation which, obviously, becomes more exact as At—Ho * See any text on W Mechanicgl Vibggtiggs. 7 2 The notations A z and Ar will be Atz At defined in this wise: In the diagram, Figure 2, y u f (t). y i r-f A7" —.--«— A/fi‘W-M—fi- 59.2 The slope of the curve from ym to ym + 1 is A 45* __m_____ =- W (first difference At ' A of ym) and the slope from ym + 1 to ym + 2 is Aym-rl- . 4 -y* Al: Al; It then follows, that 2 A (arm )= Arm sat Afit zit? ( m + 2 - m )‘(l - y ) +1 n+1 m 2 At2 AYE _Ym+2'23'm+1+ym At2 zit2 (second difference of ym ) . Therefore, at any point ym, Equation (4) becomes 2 ' . - 2 + .. (5) yin-r2 yme1:f£13~wn2ym:fi. ymva-l ym‘ Atz At: / Some care must now be exercised in choosing a suitable increment of time 43t: it must be small enough to assure reasonable accuracy of the curves. As previously stated, can is the natural frequency ESL). sec Clearly then, the total time elapsed per cycle for an undamped motion is 2"$T . rn'W ‘We now divide this cycle into, say,forty increments. Then : 40 At a 7; ( ) _%é£:_ and At = 40 (L’n Rewriting equation (5) _ 2 2 y + ym r 2 Zym * 1 + y a ( ) CL) m (Fm * 1 ym) m. n ‘E .. 1' 2 wilt-gm w) one x a. c 0.. ‘ .go i ' _ . . , . .u. u. .. o- 4 no v v . o o - u. o e . , g or - a ." , u a . a 0 ' c - .‘i -. 7‘ .. a no. i .— - c ~ 3 — ‘ a ‘ - § . - n - a . v . . . u . a > a n ' . . . n ‘ Q H I . " . . ' O O , . n'si. - r ’3 I. ‘04 . . . n .- the 0 v e ‘u . . 0‘ ' .eq ‘ ‘ ..o .. ., r. as . w“ .a . «c -o a -. (6) + 2 2 ym * 2 ‘ - [i + '§%I) .] ym : ?Xm + 1 -‘%’(im + 1 - xgl- For the sale of generality it is advantageous to write Equation 6 in nondimensional form. If yo is the initial amplitude, then (7) y y + 2 _,m,:2, = -l.0246 m +2ym*1-K(ym:1-Zn) yo Yo Yo Yo 70 C yo where K : , a nondimensional parameter. m. N : ow qu-g f(ym*1s ym) and the system may be set in motion by one of three starting conditions: 1. 'With an initial velocity, but no initial displacement. yo = o and y1 - some finite quantity. 2. ‘With an initial displacement, but no initial velocity, let yd : yl . 3. An initial displacement and an initial velocity. Since K is a function of ya, the second starting condition is selected. The displacements yo and yl are obviously not the same. From the definition of first differences it can be seen that the difference of ym and ym is a measure of velocity in + l the interval between the points m and m + 1. ‘we know, however, that a condition of zero velocity exists only at the points of L... . ext-4".“ in. m. up -~ 0 I o O. .4 10 maximum or minimum displacements. It could not exist for any finite length of time. win this regard, by choosing 7h : 4093tlwe effect less error than had we chosen, for example, '75 n 20 (4 t). The graphs of the curves, when K a 0.1, l, 2, 5, 10, and 20 are shown in Figures 3-8. Calculations of the displacements proceeds step by step from the displacement at tn to the displacement at tn + l‘ 11 ++++ oamm Illlllllll'el'lI'elLI'll-'Islelolell-lllllsTlllull-II". mmmomm. «memooo.- Hmmao.+ meomm. mmmm..£ moose. ammmm.u wmmm. mommm. mmoooo.n mmoo.+ mmmm.u meow. . whom. . mmmm.u doom. mmmm. meoooooo.u meooo.+ mmmm. now. + momm.u Home. mmoom. 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A h . . a. ’1 /4 b QR N N Y Q 5C 113 NW E K v.» ,i 0.». \\Q¥J [JT‘l QV quyx w /7 % 5Q x43 \q QNnxf 7 L_4F. i /& 3%? 18 SUPPIMEENTARY DATA From the curves shown in Figures 3-8, the die-away curves may be approximated by sketching a curve through the calculated points of maximum displacement and the reflection, about the t axis, of the calculated minimum displacement. These curves are shown in Figure 9. For two of the curves where (dd (damped frequency) and (On are still within reasonable accord (K a l, K s 2) a com- parison can be made between viscous damping (-cv) and velocity squared damping. As a basis for comparison a choice is made of $ = c (a nodimensional damping coefficient for 2 mm 11 viscous damping), such that the envelope or die-away curve for viscous damping, y— . e';%t, shows a decrement equal y o to that of the envelope of the curve where K g l and K a 2 When (A’nt = 2 II. . For K = 1, y g .323 at nt 3 2 11. For an equival- yo ent effect with viscous damping then .323 g é—S'(2II) and S = .13 0 Similarly, for K .-.- 2, y a .19 at Wnt .- 2II. yo Hence, § 3 .2648. A comparison of these curves can be made from Figure 10. It is apparent from the curves shown in Eigures 3-8 a o c . n d . . . a .-. . .- - . . I I 0 U I H .a M ‘ ' '5 . . . U‘ n . . . . . . _ . I . . . . . x . v I fl. . ‘ ~. . . O a v! ’ ‘ . n. . o a. on. no . I a. . o. . \ o ‘ . . . C. . on . I . 6 , . /.9 \c3 .%\1\ WV .13 NW NW km. 34‘ NW 9» «on. ‘b . . Nfll.’ I: :1. / . lull! Ill-Iii. /... . // QM mm 21 that the greatest decrement in amplitude occurs in the first _ cycle of motion. In light of this fact, a graph, Figure 11, has been prepared showing the percent reduction in amplitude for the first cycle as the ordinate and the damping factor K as the abscissa. It is also apparent that, with increasing K, the time required for the first quarter cycle tI,(positive displace- ment and negative velocity) increases. Figure 12 shows the manner of this increase with respect to K. Figures l3, l4, and 15 are similar but show increase in time elapse for the first half cycle a 2, first three quarter cycle a , and 3 first full cycle q4, respectively, as an ordinate. .. .. . I .c.... 0.. . . . . < . .. . 4 .9. . . .o. . . . . . . «.1..- ‘w‘. .._ .... .. L . . . . . .. i... . . . . ... . .o- ._ . . . . r > rlv r! . .. . 4 v-.. r. ... ..v. .... ... _ . .1... .. .«1. .. -2... n. _ _ .. .771“ .a. *1. .a ..v\.. . *1 *.....Oev Q..<.w... . . I'd... . . > 1 4 .12”. .— . .. u. ..v«)¢ no. _ _ . a . . . . .J.>.~ . +. :9 ...o. . .... . . . _ .1 u .o. .c. . .c... .1 .I .. C . . _ F . . T .... r.» e . . Ayn! . _ w . . . . ... .. ... .... . .. a _ . .. 1 I. .T L-.. .. . . . T .. . . .. T... n . P I» I!!!" 4. $:.... .. Tl-.. . . H. H . u... Ta a .. +. . ..c. n > . . . .. .. .. . .. ”J... u u: . a .f. ... ... . . . 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I... .. . ..|.. . . .v. < . . .. . . . .vu. . . . .. . .. . . . . . ....|4 .cn...o..o..... - e.. . . o .A ‘ . . , . . . . . . . . pl . {#:1141. IL I? . . . n n . [11 It}- I 14 J - . _ a .2 r . . . ...A. . . . . . t - . . . o . . . . . . § . . . . . . . . . I . . . . .. .9..v ._... . .. .w..- .. .. a . . l . . . . l . .. .31.. . _ .. . .. . . . . . . . . . . . . . . v. .... . . . . . . .4--- a ..w... .o .11.... r v .. u m n . . - 9|. . .. .. . u .n .. n . .. . . n . _ a _ a . .. . . . . . ~ . . . . . . ._ . . p. _ ... . v . n . .. . . . . T... . i.. . .:. 4.. . . . . . . . . o F . . . _ . . . . . . . . .. . A . o . .. . . o I . v > it nl . . 0 ll i‘i‘lll‘il VIIII..-|-l|.‘ . Iifilii. 0:. l . . lulu. ...| . Illlu .. _ l 4 VI . p I?" I‘Ivrtll. lllll‘l'l’n' Itjtl .n. {E}! I r... K . b . 4 ) s - G. ”H. .1 F. L Q L. I is ,d- {in 1' 0 you anp a‘y 23 ON. N\ am? «9&0»ka .ohbun\§\ 1.3m A o 8k? .m;\ 9\ 26‘ Q 5Q 0N .xo\ 0\ . H O\ \ \- WV \\.-: . ...o-..t . N . .\ ._ . . \ - i :, ., 25' ma .mxk .fi\ 0\ .h.\ chm. 26 %\ 50. .b;\ Q\ 5: .48 bx 27 COMMENTS AND CONCLUSIONS Consideration of the data presented permits the follow- ing comments and conclusions: 1. The nondimensional damping coefficient K is a function of the mass m, of the vibrating body and the initial displacement yo. In this regard, the rate of decay of motion is affected by m, and yo, and is not affected by the spring constant k. 2. The damped frequency wd, varies. It also ap- proaches Cd; as t increases. 3. Apparently, the damping factor producing the greatest rate of decay (in the first cycle at least) is a factor which still permits periodic motion. (With viscous damping the factor producing the greatest rate of decay and larger factors prescribe aperiodic motion). 4. In the range of damping factors examined, the motion does not tend toward a change to motion of an aperiodic nature. 5. The motion is not sinusoidal in nature: in a single quarter cycle the acceleration can change from positive to negative or vice versa. 28 BIBLIOGRAPHY Church, Austin H., 1e t r M cha c l W. New York: Pitman Pub isher, 94 . Hartog J. P. Den, W Vibrations. New York and Lon on: McGraw o mpanv, nc., 1947. Karman, Theodor von and Biot, Maurice A.,r%g§hmm1 Mgthggg 1.3 W. New York and ndon: McGraw Hill Book Company, Inc., 1940. Timeshenko, Stephen. W 1:1 We in W. New York: D. Van Nostrand Inc., 1928. ””Ijifljjjlll”Fifi“ 79025