Question

dt? . Extra Credit-Damped Oscillation (1) mei + b de fotkx = 0 Define w=ffd Divide by mi and 7= 2 & (2) de + 2 Voit twex=o to the chosen Show that if reco, X=Xmettos (w't+) where wit/w?r? Satisfies (2) abone by direet substitutions 1 Equations must be written with an equation editor (solue with pencil and paper first),

Answer 1

Sol" :- We have equation of motion of damped es cillation system, morning to a nem kx 20 par em +w2x = 0 - lij where, w= k/m = natural frequency we have the solution of differential equation (i), as 84w, lx= xm ért as (witte) where, w'=(w2.pz 3 [w? W²7 12 of vibration Now, Differentiating in with respect to ľ two times, dx = -Xme rw'si(W4 76 ) - Yxme!t Costwitto) ++o) + cos(w't to] Agarrix 3+Xmre pt casi singu't +6]+ y esku' 76]] JA -Xmet[wcos(wt +b) - yn' sin(witter]

3d x = Yment ffrwsinfu't +6)+ r?cas(w4 +63% -- {w?cosťo't +$) -rw Sinbaren From lij, bij and live substituting unli) we have, Xm e rw* sinfu't +60)7. r?cos(w'4 +6)-w% allW4te on the trailer Irwiesin (w't +6)-27 W' lsin (wt 46) - 282656w't +) #.772 as w48] 3 [2 Xw" silwf 70+ 27%.carku'+6) -27 m m 1-2 Ywsinin't toj - w 2 colhitto) twi2 cossly tolle oö It satisfied the given solution by direct substitutions