1.4 Using Newton s laws, show that r satisfies the differential equation 2 2

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1 EN40: Dnmics nd Vibtions Homewok 3: Solving equtions of motion fo pticles School of Engineeing Bown Univesit. The figue shows smll mss m on igid od. The sstem stts t est with 0 nd =0, nd then the od begins to otte with constnt ngul speed. The sstem ottes in the veticl plne, nd gvit should be included. Fiction m be neglected.. Wite down fomul fo the ngle s function of time.. Wite down fomul fo the cceletion vecto of the mss m in tems of nd, epessing ou nswe s dil-pol components in the bsis e, e (ou don t need to deive the esult just use the fomul)..3 Dw fee bod digm fo the mss..4 Using Newton s lws, show tht stisfies the diffeentil eqution d g sint.5 Use Mthemtic to solve the diffeentil eqution (Mthemtic will give the ect, nlticl solution).6 Show tht the diffeentil eqution in.4 cn be epessed in MATLAB fom s d v v g sint.7 Wite MATLAB scipt tht will plot s function of time t given ppopite vlues fo nd the vlues of nd v t time t=0. If ou like, ou cn downlod the.m file used in clss tht will nimte the motion of the sstem fom the HW web pge. To use the scipt, put the.m file in the sme diecto s ou homewok MATLAB, nd mke sue the file is nmed nimte_impelle.m Then, inset line in ou code tht looks like nimte_impelle(time_vls,sol_vls,omeg) fte ou ODE solve, whee time_vls, nd sol_vls e the solution found in ou ODE solve, nd omeg is the vlue of the ngul velocit of the b. Thee is no need to submit solution to this poblem.8 Use ou MATLAB scipt to plot s function of time t fo 0<t<. sec, nd initil conditions 0.m, v 0 t time t=0. T solutions with () 7.d/s nd (b) 6.9 e e j i

2 Skie in wind-tunnel Flight-pth of tpicl ski jump. In this poblem ou will implement compute-model of ski-jumpe in flight. Fo detiled bckgound to this poblem see Wolfm Mülle, Diete Pltze, Benhd Schmölze, Dnmics of humn flight on skis: Impovements in sfet nd finess in ski jumping, Jounl of Biomechnics, 9, 996, pp , ( ). Both figues shown bove e tken fom this ticle. Duing flight, thee foces ct on the thlete (the e shown on the figue): (i) An eodnmic lift foce, which cts pependicul to he pth (ii) A dg foce, which cts pllel to the pth, opposite to the diection of motion (iii) Gvit Wind tunnel epeiments show tht the lift nd dg foces depend on the ngle of ttck (the ngle between the skis nd the iflow eltive to the skie). Mulle et l show tht the mesued lift nd dg foces cn be ppoimted b FL V 4 FD V whee is the i densit, V is the skie s ispeed, is the ngle of ttck in degees (SI units must be 0 used fo ll othe quntities). Both epessions e vlid onl fo 5 dg nd lift fo smll, which is nonsense). (the fomul pedicts negtive. Let v vi vj denote the velocit vecto of the jumpe while in flight. Find epessions fo unit vectos pllel to the lift nd dg foces (which ct pllel nd pependicul to the jumpe s pth) in tems of v, v. Hint to clculte the vecto pllel to lift, ou could use coss poduct. Wite down Newton s lw of motion fo the skie, nd show tht the cn be e-nged in MATLAB fom s

3 v v d v FDv / ( mv ) FLv / ( mv ) v FDv / ( mv ) FLv / ( mv ) g whee V v v nd F, F e computed fom the complicted fomuls given elie..3 Wite MATLAB scipt to clculte,, v, v L D s functions of time, given vlues fo the following quntities: The ngle between the skis nd the i diection Skie mss m Ai densit The gvittionl cceletion g The initil position of the skie nd the initil velocit of the skie. Note tht the ngle of ttck cn be clculted fom 80 / tn ( v / v ), whee is the ngle between the skis nd the i diection in dins. You should find tht ou cn just modif the scipt shown in clss (lso in online notes) tht clculted the motion of pojectile with i esistnce (Thee is no need to submit solution to this pt)..4 Run ou simultion with the following pmetes nd initil conditions (tken fom Mulle et l): (i) Mss m = 70 kg (ii) Initil velocit v 8.7 v. m/s nd initil position ==0. (iii) Ai densit.03 kg/m 3 (iv) Ski oienttion (ssume tht the skie does not otte duing flight this is not quite coect, s epet 0 skies otte thei skis to mimize lift) 0 Plot the tjecto ( s function of ) fo 0<t<5 sec. Once ou code is woking, dd n event function tht will stop the clcultion when the skie lnds on the slope (fo simplicit, ssume tht the hillside is just stight line with slope 3 0 ). Hint: note tht tn ( / ) 3 /80 0 when the skie lnds. Hnd in: (i) A plot of the tjecto pedicted b ou finl code (with the event function) (ii) A plot of the mgnitude of the skie s velocit s function of time (iii) The pedicted length L of the jump.5 Fo compison, compute (b hnd) the pedicted length L of the jump without i esistnce (ou need to use the tjecto equtions to do this follow the pocedue used fo the shoot the elmo demo in clss)..6 Repet (.5) fo skies with msses of m=60kg nd m=80kg. (epot the pedicted jump lengths onl, thee is no need to hnd in plots of the tjectoies o velocities)..7 OPTIONAL et cedit poblem Find the vlue of tht mimizes the jump length. L 3 0

4 3. The gol of this poblem is to test poposed guidnce sstem fo n intecepto, whose pupose is to impct n steoid. Fo simplicit, we will neglect gvit in this poblem, nd ssume tht both steoid nd intecepto move in the (,) plne. 3. When the steoid is fist detected, it hs position ( 0, 0) nd hs constnt velocit ( V, V ). Wite down fomuls fo the position 0 0 vecto of the steoid s function of time (just use the stight line motion fomuls ) j Intecepto p F Asteoid p i 3. At time t=0 the intecepto is t est nd t the oigin. The intecepto is poweed b ocket eeting constnt thust F. The intecepto will be steeed b lteing the diection of the thust. The guidnce sstem will detect the instntneous position of the steoid ( ( t), ( t )), nd djust the diection of the thust so tht it lws points towds the steoid. The gol of this poblem is to detemine whethe this pocedue will esult in n impct. Let (, ) nd ( v, v ) denote the components of the position vecto of the intecepto nd its velocit. Wite down the esultnt foce vecto cting on the intecepto, in tems of F, ( ( t), ( t)) nd (, ) (stt b witing down unit vecto pllel to the thust vecto). 3.3 Using Newton s lws, show tht the equtions of motion fo the intecepto cn be e-nged into the stndd MATLAB fom s follows v v d v F( ) / ( dm) v F( ) / ( dm) Whee d ( ) ( ) nd m is the intecepto mss. Note tht, e known functions of time (fom pt 3.) ou will hve to ente fomuls fo these functions in ou MATLAB scipt. 3.4 Wite MATLAB scipt to compute the pth of the intecepto. Add n event function to ou code tht will stop the clcultion if the steoid nd intecepto e less thn km pt (wok with units of km, kn nd s). You cn downlod mtlb.m file tht will nimte the tjecto of both the intecepto nd the steoid. To the scipt, put the.m file in the sme diecto s ou homewok MATLAB. Inset line tht looks like nimte_pojectiles(time_vls,sol_vls,[x0,y0],[v0,v0]) fte ou ODE solve, whee time_vls, nd sol_vls e the solution found in ou ODE solve, nd X0,Y0, V0,V0 e the initil position nd velocit of the steoid. Thee is no need to hnd in solution to this poblem

5 3.5 Run ou code (fo time intevl of 50 sec) with the following pmetes: X 0 km, Y 00 km, V km / s, V km / s m 50 kg, F 50 kn, d d 0 0 t t 0 X 0 km, Y 00 km, V 8 km / s, V 8 km / s m 50 kg, F 50 kn, d d 0 0 t t 0 Plot the tjecto of both the intecepto nd steoid fo ech cse. 3.6 [OPTIONAL et cedit poblem] The guidnce sstem poposed hee is clel unstisfcto. We cn impove it b mking the diection of the foce depend on the eltive velocit of the two pticles, in s well s on thei eltive position. As fist ttempt, we could t diecting the thust long unit vecto n c( v v) / c( v v) whee c is constnt tht cn be djusted to give the best pefomnce. Modif ou code nd test this ide. Use the sme pmetes s in poblem 4.5, nd t c=0.4. You could t othe vlues of c s well if ou e cuious. Hnd in plots of the tjectoies of the steoid nd intecepto fo ech cse. 4. A clss demonsttion showed tht n inveted pendulum cn be stbilized b shking its pivot veticll t coect fequenc (fo compute model see HW3, 0). The gol of this poblem is to model stbiliztion using feedbck contol mechnism. An ctuto is ttched to the pivot of the pendulum. The length of the ctuto (t) vies with time. j Actuto L m i 4. Wite down the position vecto of the mss m, in tems of nd othe elevnt vibles. Hence, clculte n epession fo the cceletion of the mss, in tems of time deivtives of, nd othe elevnt vibles. 4. Dw fee bod digm showing the foces cting on the mss m (note tht the pendulum shft is two-foce membe) 4.3 Wite down Newton s lw of motion of the mss, nd hence show tht the ngle stisfies the eqution (Hint eliminte the unknown foce in the shft of the pendulum) d g cos d sin 0 L L 4.4 One ppoch to stbilizing the pendulum is to dd senso to the sstem tht will mesue the ngle, nd then use compute-contol sstem to etend o contct the ctuto so s to t to educe. As pticull simple scheme, we could t mking popotionl to - fo emple b setting K, whee K is positive constnt. (The negtive sign hee is counte-intuitive it mens tht if

6 the pendulum swings to the ight, the ctuto must displce the pivot to the left. The vibtions section of EN40 will give some moe mthemticl insight into wh this is necess) Show tht, with this scheme, the govening equtions fo nd cn be epessed in the fom d ( g / L)sin / ( ( K / L)cos ) 4.5 Wite MATLAB scipt tht will integte these equtions of motion. If ou would like to see n nimtion of the motion of pendulum, ou cn downlod MATLAB scipt clled nimte_pendulum.m. Stoe this in the sme diecto s ou MATLAB homewok file (mke sue ou nme the file nimte_pendulum.m), nd dd line nimte_pendulum(times,sols,l,k); fte the cll to the ODE solve in ou homewok. Hee times,sols e the time vlues nd solution vible vlues computed b the ODE solve, L is the pendulum length, nd K is the gin in the feedbck contol. The nimtion looks most elistic if the solution is computed t equll spced time intevls. THERE IS NO NEED TO SUBMIT A SOLUTION TO THIS PROBLEM. 4.6 Use ou MATLAB code to plot gph of s function of time, with time intevl of 5 sec, nd pendulum length of m. T solutions with K=.8 nd K=0.5 fo the following initil conditions d 0., 0 d 0., 0. d 0.9, 0 (this cse will blow up fo K=.8 so don t t tht one!) Note tht the pendulum is stbilized in inveted position fo K>L but if the pendulum is slightl displced it will oscillte foeve. In pctice bit of i esistnce o fiction would dmp out the oscilltions, but it is bette to design the contol sstem to void this oscillto behvio. Note lso tht the pendulum is not self-ighting in fct is stble configution. 4.7 [OPTIONAL et cedit poblem] Bette behvio is chieved if the contolle is designed so tht its velocit is contolled, the thn its length. Specificll, we could t d d K ( 0 ). Hee, K,,, 0 e constnts. Show tht with this choice, the equtions of motion fo nd cn be nged into the fom d v K ( 0) g sin / ( L K cos ) ( v)cos / ( L K cos ) v 4.8 [OPTIONAL- et cedit poblem] Modif ou MATLAB code to use the contol scheme descibed in 3.6, nd use it to plot gph of s function of time, fo time intevl 0<t<5 sec, nd pendulum length of m. T solutions with 0.8, d / 0, t time t=0, with the following pmete vlues K.5,., 0. K.5, 5., 0.

7 K.5,.,. K.5, 5., 0. Designing optiml contol sstems is concen in wide nge of engineeing pplictions. Cuise contol nd utopilots e two obvious emples, but contol sstems e lso needed in chemicl plnts, nucle ectos, obotics, nd so on. Sophisticted mthemticl methods hve been developed to help design contol sstems ou would not nomll use til-nd-eo ppoch. But it is often helpful to check the pefomnce of design b integting the equtions of motion, s done hee.

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