Question

1. Suppose equation of motion for one dimensional oscillator is given by:

   ?̈ + ??̇ + 9? = 0

   1. For α values of 3, 6, and 9 indicate what kind of oscillatory system it would be.

   2. Find expression for x(t) for each value with the initial conditions, x0 = 0 and vo = 5 m/s.

      Use proper ansatz to start from scratch (Check whether these initial conditions might

      be non-sense. Choose convenient initial conditions whenever necessary).

   3. Solve the equation with α = 6 and with a drive force of f(t) = 2 cos(3t)

Answer 1

d 2x The tada JE +92 = 0 2C 10) = 0 wcom - x'ro) = 5 mis Taking a=3 022 + 3d2c +9X1V=O dt2 dt char-eqa m²+3m+g (m+ 2)² + 27 m = - 3/2 + √2/4 4 Cit) - élet [ cicos 9++ (25100524+] • Syslem is oscillatory decaying (dlamped using 100=0 oscillatory) >[a=0 xt(0) 5 2 at 2 y C2 = 5 q=5 / 4 -3/2€ Hence alt=e? 557 / Sin 131 tt similarly for a=6 di tutti d²ac f 6 +9= at2 chanego m?+6m+9=0 (+372 70 2) m= -3 i m=-3:

Xlt = (6+czt) et 10) = 0 using Scio) - Tazo system is Non- oscillatory. xilo) = 5 * C2+ -3C = 5 J C2 = 5 x 9 = 5+ E3F For od=9 dan dt +9 dhe at t be char.ege m²+gmtg=0 (m+ = -9 2 2 545 * 45 4 =o me me-1.145 m= -7.854 system is over damped for a=g. 1.145 Xl6 = ce -7.8546 & ce using 810=0 (+(2-o o -1.1459-7-85462=5 @ 2110) = 5 solung o &o = 0.7453 C2 = -0.7453 -7.854t +-Q.7453e Hence OCIH = 0,7453 ē" 1457

=(atrat) et For x=6 f=26053 XIH = act Xp xp = 2 COS3 2 Cost (M43)2 missmislu ap - m (2 Cos31 1 m²-9 6 m2 Scp 1m²-g 2 Cost 6m -bsinat 6 (-9) $ Sin(36) - Sinat JCp Hence clH = Ef (2+)6347 S70 (37) using 1 X(0)=0 La G=0 scrol=5 = (2-36 +3=5 (2-36 + 1/3 = 5 7 2= 14 Hence -35 ot alt=1 tt + 1 - Sinat

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