In the figue, two cables of stiffness connect a wheel of mass M c to gound. The wheel with adius, has mass moment of inetia is I The pendulum, attached to the wheel cente, has mass M and mass moment of inetia, I, with as the distance fom the pin connection to the pendulum. The two cables of stiffness ae initially stetched with foce F asy. At time t=0 s, the pendulum is displaced angle and eleased, motion of the whole system follows. Assume thee is no fiction between the wheels and the hoizontal suface. a) Define DFs, establish kinematic constaints and select independent coodinates fo the motion of the wheel and pendulum. b) daw FDs and use Newton s aws (Foces & Moments) to deive the system EMs of the wheel and the pendulum fo t>0 s. R b) Wite system enegies and ug agangian deive the system EMs. g M M, I, gound Rolls w/o slipping o Pendulum : M, I, SP 011 Vibations Qualifying Exam 1
DIARAM of foces at STATI EQUIIRIUM PSITIN : = = 0 M, I, M, I, (W+W): weights Fasy= Fasy MStetched cables on assembly N: Nomal foce DEFINITINS: Foces: Fasyassembly foce fo spings (extension o stetched) Paametes: M, M: masses pendulum & wheel : stiffness coefficients I = pendulum mass moment of inetia I = wheel mass moment of inetia c = adius of wheel = distance - oodinates (Vaiables): 5 DF : tanslation of wheel : otation of wheel :: tanslation of pendulum cg (0a) : otation of pendulum, (Absolute fames of efeence, with oigin at ) inematic constaints: (0b)
DIARAM of foces when moving gound F F N (0) x M W M, I, y W FR Sping foces: M, I, F F R asy F F asy (1) DEFINITINS: Foces: FF : foces fom spings x, y: connection pin foces W=Mg : weight, N: nomal foce F : contact foce Paametes: M, M: masses pendulum & wheel : stiffness coefficients I = pendulum mass moment of inetia I = wheel mass moment of inetia c = adius of wheel, = distance - oodinates (Vaiables): 5 DF : tanslation of wheel : otation of wheel :: tanslation of pendulum cg : otation of pendulum Speeds and acceleations of ba M () (3) 5 DF 3 Eqns. f constaint = independent DFS P1-Deive EM SP09 3
Wheel DIARAM of foces when system moves gound F M F F F M 0 W N I Pendulum F x N (0) R x y F W y W M x M W M, M I, FR M, I, y Sping foces: F F R asy F F asy M (1) I x y (4a) (4b) (4c) (5a) (5b) (5c) DEFINITINS: Foces: FF : foces fom spings x, y: connection pin foces W=Mg : weight, N: nomal foce F : contact foce Paametes: M, M: masses pendulum & wheel : stiffness coefficients I = pendulum mass moment of inetia I = wheel mass moment of inetia c = adius of wheel, = distance - oodinates (Vaiables): 5 DF : tanslation of wheel : otation of wheel :: tanslation of pendulum cg : otation of pendulum Select and as independent vaiables 4
DIARAM of foces when system moves F W M, M I, FR Deive Equations of Motion: gound F Wheel (4a) x N (0) M F F F R x y W Sping foces: F F M, I, R asy F F asy (1) I Fasy Fasy M I M Note how assembly (static) foces cancel out. F F R asy F F asy Equations of impotance ae (4a) and (5c): tanslation of wheel and otation of pendulum. In Eqn (4a): substitute sping foces (), contact foce fom (4c), I F and and pin foce x fom eqn (5a) M x I M M M 0 (6) This is the fist EM descibing tanslation of wheel, olling w/o slipping, and connected to a swinging pendulum 5
DIARAM of foces when system moves gound F F x N (0) W y W M, M I, FR Sping foces: F F M, I, R asy F F asy (1) Deive Equations of Motion: In eqn (5c) fo otation of pendulum: substitute pin eaction foces x, y fom eqns (5a,b) x M M M W M W y (5a,b) pendulum (5c) I x y M M W I M M W M M I M M W 0 This is the nd EM descibing otation of pendulum, connected to a wheel, olling w/o slipping ancel equal tems (7) (3) 6
ENERIES fo system components inetic enegy = T = T wheel tanslation +T wheel otation +T pend tanslation +T pend otation T M I M I 1 1 1 1 I 1 1 1 M M I I 1 1 1 T M M M I M (7) inematic constaints Potential enegy =Vstain enegy in cables + V gavitational fom pendulum V W 1 (8) 1 1 Sping on ight sping on left is static deflection (stetching fom assembly) = Fasy/ Viscous dissipated powe = v 0 Extenal wok Q=0 (10) (9) No extenal foces applied. ontact foce (olling w/o slipping) does not pefom wok M, I, 7
Deive Equations of Motion: STEP : agange s equations d dt T T V 1 v Q q k q q qk k k 0 0 k k1,... n (11) STEP 3: Deivatives of potential enegies & kinetic enegies 1 1 1 V ( 1) ; V W V W I T M M M I M T I M M M d dt 1 1 1 ; T I M M M T M I M ; d T dt I M M T 0; T M 8 (1) (13)
Deive Equations of Motion: Equation fo tanslation of wheel Fom: d dt V W 1 1 1 I T M M M I M 1 1 1 T T V 0 0 Equation fo otation of pendulum d T T V 0 0 dt I M M M W 0 I M M W 0 I M M M 0 (15)=(7) Equations of motion (14) and (15) = equations (6) and (7) (14)=(6) M, I, 9