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  HW#1 SDOF Free Response Assign on 1/24/2013 recommend Due on 2/5/2013 1.  Prob. (1.33). (i.e. Derive eq. (1.31) from eq. (1.28) in Inman’s book.) 2. Prob. (1.38). (i.e. Verify eq. (1.46) in Inman’s book.) 3. Prob. (1.39). (i.e. Derive eq. (1.48) from eq. (1.47) in Inman’s book.) 4.  Plot the following solutions (or responses, signals, output) of a SDOF m/k vibration system by Matlab, with m = 1 kg, c = 0, k = 9 N/m. (Turn in both of your m files  and figures .) t  jt  j nn eaeat  x  ω ω   −+ += 21 )(   t  At  At  x nn sincos)( 21  +=   )sin()(  φ  += t  At  x n  (Where a 1 , a 2 ,  A 1 ,  A 2 ,  A, and φ   can all be calculated with proper Initial Conditions; since the I.C.'s are not given, you can just choose your numbers.) EXTRA CREDIT :  m = 1 kg, c = 0, k = 9 N/m, the mass is moved 100 mm below its equilibrium position and released with a velocity of 500 mm/s upwards at t=0. Calculate a 1  & a 2 ,  A 1 &  A 2 ,  A  & φ  , and plot these 3 responses by Matlab. Turn in both of your m files  and figures . (3 signals should look EXACTLY the same.)   5.  Solve the following standard SDOF m/k vibration system: m = 1 kg, c = 0, k = 9 N/m, the mass is moved 100 mm below its equilibrium position and released with a velocity of 500 mm/s upwards at t=0, a.   Define your coordinate system clearly. b.   Write down the E.O.M. of the system. c.   Calculate period, amplitude, and undamped natural frequencies (both  f  n   and   w n ). d.   Determine the position of the mass as a function of time, and PLOT in Matlab. (i.e. the solution, the response, the signal, or the output of the system.) e.   Calculate the earliest time t>0 when the acceleration of the mass = 0. m    k    c    GROUND    6.  Solve the following standard SDOF m/c/k vibration system: m = 1 kg, c = 2 N.s/m, k = 9 N/m, the mass is moved 100 mm below its equilibrium position and released with a velocity of 500 mm/s upwards at t=0, a.   Define your coordinate system clearly. b.   Write down the E.O.M. of the system. c.   Is this system overdamped, underdamped, or critically damped? d.   Does this system oscillate? e.   Calculate ζ , w d ,  f  d ,  T  d  if they exist. f.   Determine the position of the mass as a function of time. (Recommend plot in Matlab.) g.   Determine the position of the mass at t=2 second. 7.  If c = 12 N.s/m, a.   Write down the E.O.M. of the new system. b.   Is this system overdamped, underdamped, or critically damped? c.   Does this system oscillate? d.   Calculate ζ , w d ,  f  d ,  T  d  if they exist. e.   Determine the position of the mass as a function of time. (Recommend plot in Matlab.) 8.  To design a new damper to make this a critically damped system, a.   Calculate this critically damping c value. b.   Write down the E.O.M. of the new system. c.   Does this system oscillate? d.   Calculate ζ , w d ,  f  d ,  T  d  if they exist. e.   Determine the position of the mass as a function of time. (Recommend plot in Matlab.) If you plot Problems 5, 6, 7, and 8 responses all together in Matlab, you will see the different characters between undamped and damped systems.
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