ENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15


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1 NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound at the other end on a drum of varyng radus. The drum rad at the two ends are r 1 and r as shown. The end of vew of the drum s shown n the nset, whch shows the gears that rotate the drum. The radus of the drver gear A s r A and that of the drven gear B s r B. Gear B and the concal drum are attached rgdly to the same shaft. The gear rad are as shown and the axal length of the drum s L = 10R. Intally, the cable s completely unwound,.e., the cable s n contact wth the drum at radus r 1. Startng from rest, the drver gear rotates n counter clockwse drecton at a constant angular acceleraton of = 1 rad/s durng tme T = 40 s, followed by a constant deceleraton to rest. Cable dameter = d r r 1 r drum r A = R r B = 4R r =6R r 1 =4R R = 10 cm d=1 cm m
2 () erve expressons for the angular speed of gear B B as a functon of tme t for 0 t T and T t T and plot B as a functon of tme for 0 t T. Show the maxmum value of B on the plot. What s the drecton of rotaton of the drum? () erve expressons for the angle of rotaton of the drum () as a functon of tme t for 0 t T and T t T. Plot as a functon of tme t for 0 t T. What are the values of at t = T and t = T? () Note that the pont where the cable separates from the drum (let s call ths pont the cable contact pont ) advances by a dstance d along the drum surface for each complete revoluton of the drum. erve a relaton between the radus of the drum at the cable contact pont (r) as a functon of the rotaton angle. (v) erve expressons for the speed of the mass as a functon of tme t for 0 t T and T t T. (v) What s the total dstance traveled by the mass m?. The fgure shows a crankrocker mechansm. It conssts of three rgd members AB, BC and C connected by pn onts. Pns at A and are attached to the ground and reman fxed. AB s the crank and C s the rocker. As the crank AB rotates wth a constant angular velocty ω, the pont C rocks back and forth along an arc. The lnk AB rotates at a constant angular velocty of ω = 10 k rad/s. The lnk C s vertcal at the nstant shown. () Calculate the velocty vector of pont B at the nstant shown n the fgure, expressng your answer as components n the {, } coordnate system shown. () etermne the angular veloctes of members BC and C, and the velocty vector of pont C. () What are the angular acceleratons of BC and C? 0.4m C 0.1m B 0.08m A 0.06m
3 3. Trflar pendulum analyss A trflar pendulum s used to measure the mass moment of nerta of an obect. It conssts of a flat platform whch s suspended by three cables. An obect wth unknown mass moment of nerta s placed on the platform, as shown n the fgure. The devce s then set n moton by rotatng the platform about a vertcal axs through ts center, and releasng t. The pendulum then oscllates as shown n the anmaton posted on the man N40 homework page. The perod of oscllaton depends on the combned mass moment of nerta of the platform and test obect: f the moment of nerta s large, the perod s long (slow vbratons); f the moment of nerta s small, the perod s short. Consequently, the moment of nerta of the system can be determned by measurng the perod of oscllaton. The goal of ths problem s to determne the relatonshp between the moment of nerta and the perod. As n all free vbraton problems, the approach wll be to derve an equaton of moton for the system, and arrange t nto the form d x 0 n x dt Snce we are solvng a rgd body problem, ths equaton wll be derved usng Newton s law F ma COM, and the momentangular acceleraton relaton Mk IZk. Here, a COM s the acceleraton of the center of mass; Mk s the net moment about the center of mass (COM); I Z s the moment of nerta about the zaxs; k s the angular acceleraton of the platform. Note that, by symmetry, the center of mass s the center of the platform. You are already famlar wth the frst of these equatons (F = ma) for a partcle. The second equaton s ust an analog for rotatons, that relates the net moment wth the angular acceleraton. It wll be derved n the class soon (may be on Tuesday, 4/13). But, untl then, have fath n your nstructors and ust accept t. Before startng ths problem, watch the anmaton posted on the N40 homework page closely. When you are feelng sleepy, emal your Swss bank account number to Professor Guduru. Then, notce that () () The table s rotatng about ts center, wthout lateral moton If you look closely at the platform, you wll see that t moves up and down by a very small dstance. The platform s at ts lowest poston when the cables are vertcal.
4 F L k B 10 0 C B A C F R A 10 0 (a) The fgure above shows the system n ts statc equlbrum poston. The three cables are vertcal, and all have length L. The platform has radus R. Take the orgn at the center of the dsk n the statc equlbrum confguraton, and let {,,k} be a Cartesan bass as shown n the pcture. Wrte down the poston vectors r, r, rf of the three attachment ponts n terms of R and L. F L z b c k a b F c a (b) Now, suppose that the platform rotates about ts center through some angle, and also rses by a dstance z, as shown n the fgure. Wrte down the poston vectors ra, rb, rc of the three ponts where the cable s ted to the platform, n terms of R, z and. (c) Assume that the cables do not stretch. Use the results of () and () to calculate the dstance between a and, and show that z and are related by the equaton:
5 R (1 cos ) z( z L) 0 / Hence, show that f the rotaton angle s small, then z R L. (Hnt use Taylor seres). Snce z s proportonal to the square of, vertcal moton of the platform can be neglected f s small. (d) Wrte down formulas for unt vectors parallel to each of the deflected cables, n terms of L, ra, rb, r c and r, r, r F. (It s not necessary to express the results n {,,k} components). (e) raw a free body dagram showng the forces actng on the platform and test obect together. (f) Assume that the tenson has the same magntude T n each cable. Hence, use (e) and (d) and Newton s law of moton to show that (remember that the center of mass, COM, s at the center of the platform) d z r r rf ra rb rc x y ( ) ( ) m a a k T mgk dt L (g) Note that ( r r r ) / 3 s the average poston of the three ponts where the cables connect to a b c the platform. By nspecton, ths pont must be at the center of the platform. Usng a smlar approach to determne a value for ( r r r ) / 3, show that F d z m ax ay k 3 T (1 z / L) mg k dt (h) For small, we can assume z 0, d z / dt 0. Hence, fnd a formula for the cable tenson T. () Fnally, consder rotatonal moton of the system. Use the rotatonal equaton of moton to show that (agan, remember that the center of mass, COM, s at the center of the platform)
6 d ( ra zk) ( r ra ) ( rb zk) ( r rb ) ( rc zk) ( rf rc ) I k T T T dt L L L ther by usng MAPL to evaluate the cross products, (or f you are maplephobc try to fnd a clever way to evaluate the cross products by nspecton you mght lke to do ths as a challenge even f you love MAPL. Then agan, you may prefer to have your wsdom teeth pulled.), show that d 3R T I sn 0 (1) dt L () Hence, fnd a formula for the frequency of smallampltude vbraton of the system, n terms of m, g, R, L and I. (k) Wrte a MATLAB scrpt that wll solve equaton (1) (do not lnearze the equaton), wth ntal condtons d / dt 0 0 t 0. Use your code to compute the perod of vbraton for 0 / 4 and plot the results on a graph. What s the maxmum allowable ampltude of vbraton 0 f the approxmate formula derved n part () must predct the perod to wthn 5% error? How about 1%?
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