N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?. Bcycle gears In a bcycle, the front gear (r 1 =10mm) s rgdly attached to the petals (r P =40mm), and the rear gear (r =45mm) s rgdly attached to the bac wheel (r W =330mm). A cyclst, startng from rest, pedals (clocwse) wth an angular acceleraton shown below. x y
.1. Plot (ether by hand or wth matlab/mathematca) the angular velocty of the front gear, 1 (t)... Plot the angular velocty of the rear gear, (t)..3. How many revolutons does the cyclst pedal?.4. What s the total dstance travelled?.5. Plot the bcycle s velocty, v(t). 3. Helmet mpact A football player s wearng an nstrumented helmet that calculates the net acceleraton and force of the head at mpact. The traner (on the sdelne) receves data wrelessly from the accelerometers mounted on the helmet. A smple Matlab routne computes the mpact force, and the traner can decde f the player should be taen out, or stay n the game. In ths problem, we wll wor out the nematcs equatons used n ths software. 3.1. In the fgure above, an accelerometer s mounted flush to the head at a poston r 1 and measures the acceleraton at that locaton as a 1. Wrte a vector expresson for a 1 assumng we now the rotatonal acceleraton and velocty and the lnear acceleraton of the center of mass (CM), a CM. 3.. However the accelerometer s a sngle-axs accelerometer and t can only measure the acceleraton n a sngle drecton, thus the orentaton of the accelerometer s mportant! et s say that the accelerometer s mounted such that t has a unt vector e 1, and t reads a magntude a 1. Wrte an expresson for the magntude, a 1, assumng we now,, and a CM. 3.3. Once we measure the poston and orentaton of the accelerometer, and t records the acceleraton of an mpact, how many unnowns (degrees of freedom) are n ths equaton? What are they? If we want to solve for a CM, how many accelerometers do we need? 3.4. In order to reduce the complexty of the equaton we derved n 3., we wll mount the accelerometers so that they record the acceleraton tangent to the helmet surface. Ths way, one of the terms n 3. becomes much smaller than the others, and we can neglect ths term. Wrte the smplfed vector equaton and explan why we can neglect ths term. 3.5. Wrtng the vectors n component form, formulate a scalar expresson for a 1.
r 1 = x + y + z a CM = a x + a y + a z = x + y + z e 1 = e x + e y + e z 3.6. You wll now wrte the code that calculates a CM gven the output of 6 accelerometers. Onlne there s a m-fle that has a lst of poston vectors, orentaton vectors, and acceleratons. Usng ths data, wrte a matlab routne that wll solve the system of 6 equatons. What s the magntude of the lnear acceleraton of the center of mass? 4. Trflar pendulum 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 unnown 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 homewor 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 ma COM, and the moment-angular acceleraton relaton M IZ. Here, a COM s the acceleraton of the center of mass; M s the net moment about the center of mass (COM); I Z s the moment of nerta about the z-axs; 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 ( = 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. But, untl then, have fath n your nstructors and ust accept t. Before startng ths problem, watch the anmaton posted on the N40 proect descrpton page closely. When you are feelng sleepy, emal your Swss ban account number to Professor ranc. Then, notce that () () The table s rotatng about ts center, wthout lateral moton If you loo 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.
B 10 0 C B A C 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. The platform has radus R. Tae the orgn at the center of the ds n the statc equlbrum confguraton, and let {,,} be a Cartesan bass as shown n the pcture. Wrte down the poston vectors r, r, r of the three attachment ponts n terms of R and. z b 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 r, r, r of the three ponts where the cable s ted to the platform, n terms of R, z and. b c a a b c (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: R (1 cos ) z( z ) 0 Hence, show that f the rotaton angle s small, then z R /. (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, r, r, r and r, r, r. (It s not necessary to express the results n {,,} components). a b c
(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 r ) ( ra rb rc ) m ax ay T mg dt (g) Note that ( ra rb rc ) / 3 s the average poston of the three ponts where the cables connect to 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 d z m ax ay 3 T(1 z / ) mg dt (h) or small, we can assume z 0, d z / dt 0. Hence, fnd a formula for the cable tenson T. () nally, 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) d ( ra z) ( r ra ) ( rb z) ( r rb ) ( rc z) ( r rc ) I T T T dt ther by usng Mathematca to evaluate the cross products, (or f you can try to fnd a clever way to evaluate the cross products by nspecton you mght le to do ths as a challenge even f you love Mathematca. Then agan, you may prefer to have your wsdom teeth pulled.), show that d 3R T I sn 0 (1) dt () Hence, fnd a formula for the frequency of small-ampltude vbraton of the system, n terms of m, g, R, and I. () Recall that you wrote a MATAB scrpt as part of HW5 to solve equaton (1), and have used your code to plot a graph of the perod of vbraton wth ntal condtons d / dt 0 0 t 0. Use your code to compute the maxmum allowable ampltude of vbraton f the approxmate formula derved n part () must predct the perod to wthn 5% error? How about 1%?