Question

2. (30 Points) For the figure shown below, find the equations of motion by Lagrange's equations. Assume that all variables are measured from static equilibrium. (20 Points) Determine the condition under which the steady-state displacement of the mass m will be zero. Assume the Disc of mass M is described by the coordinate and has mass moment of inertia J. Disc, mass M Rolls without slipping Pulley Cord Fisin (0) Figure 2: Schematic for Problem 2

Please use Lagrange's Equation and solve both parts.

Answer 1

polley Fosinut I ault) Equation of motion maus for Kinetic energy, KE = 1 mx² 2 RO KE: MR² 0² 2 Potential PE : Energy 2 1 KRO)² + 1 K2 (RO)² 1 K, R²0² K₂ R²0² PE : + 1 2 2 By lag range's Equation,

(3) + d. (KE) - alte doj DE : 0 dej doj tp Сое d a lk.E) = MR It zoj 2 KE) :0 tae dPF) Kirio + K₂ R²0 (3कर өe = 0 . MR² ö tk, R²O + R₂ R² e + (ki+K2) 8 + (kitkai 0 I () Equation q motion for mass 2 K.E : z? WT 2 NDAY P.E: I kne? 2 Dissipation Energy (DE) il cre² 2 F = fosinat

s, abunebar har Equation Р (7+) e + (2.0) + (7.-1) e dt dej dCD.E.F. dxj dxj daj = mx d a(kE) dt dij ə KE) =D daj (7.d) e ka dxj (3.0) e ca dij . mä + ka + cà fo sin ut ; ms + C + Kx = fo sin wt o + C + Kalo to sinut m m m 2 c=2&wn K won - m 2 있 + 2 Ee wn x + una 2 imuis of

+wn X(S): 665) W S را دی Applying caplace Trams form, Considering s’ X55) + 28 Wns x(s) twn *(57 - xloro 2002:0 Fors) w s²tw² Xis) [ 3²+ 2 Econstn" on?] focs) w Iw? X(S) FIS) 2 2 s²+w² s+2&ons & wn 2 2 stw 5 +26wnst & 'wn twn & wn X(5) F(S) s²te2 (stwnst sawn)² + (wn 1-4)