Vibration ME 6442
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This 0 page Class Notes was uploaded by Chloe Reilly on Monday November 2, 2015. The Class Notes belongs to ME 6442 at Georgia Institute of Technology - Main Campus taught by Aldo Ferri in Fall. Since its upload, it has received 10 views. For similar materials see /class/234240/me-6442-georgia-institute-of-technology-main-campus in Mechanical Engineering at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
ME6442 Dr Ferri Example Problem 27 In this example a block of mass ml is dropped onto a springmassdamper platform having mass m2 stiffness k and Viscous damping constant C The platform is originally in static equilibrium with the weight of m2 alone Mass ml strikes platform m2 with an initial speed of V1 and does not rebound The displacement xt is referenced to the position of the unstretched spring positive down ml 4 kg m2 08 kg k 600 Nm X C 20 N sm g 9807 msAZ zeta C2sqrtkmlm2 damping ratio 0 damped natural frequency rads n sqrtkmlm2 natural frequency rads wd wntsqrtl zetaA2 This yields zeta 01863 wn 111803 and wd 109845 Now establish initial velocityjust before v1 and just after V2 impact Vl 5 ms hO VlA22g height from which ml was dropped V2 Vltmlmlm2 This yields h0 12746 and V2 41667 ms The initial conditions for the motion a er impact are x0 m2tgk initial disp m positive down V0 V2 positive down This yields x0 00131 and V0 41667 Finally the response can be calculated as follows x0 gwnA2 C V0 zetatwntclwd t 00000l2 x gwnA2exp zetawntclcoswdtt02sinwdt xdotl zetawnexp zetawntcltcoswdtt02sinwdt xdotl exp zetawnt wdtclsinwdtt02wdtcoswdt g xtwnAZ 2zetawnxdot The constants are determined to be c1 00654 and c2 03669 note typo in text The acceleration response is shown in Figure 1 for initial speeds of V1 5 ms and V1 2 ms Also shown is the acceleration of gravity g 9807 msz Liftoff occurs whenever xddot gt g It can be shown numerically that this occurs when V1 is greater than 25 ms approximately The displacements and velocity of the combined m1m2 system are shown in Figures 2 and 3 Note that for V1 5 the response beyond t 02769 seconds is not correct unless the block adheres to the platform a er impact Also note that the initial displacement is the same for both initial speeds Since x0 is constant but V0 differs by a factor of 25 the responses do not dilTer by a simple factor of 25 acceleration msz t 02769 sec 0 02 04 06 08 1 12 14 time sec Figure 1 Acceleration Response V15ms displacement m time sec Figure 2 Displacement Response oo Example 44 Dr Ferri o0 o0 EOM o0 o0 rhoALMprimexddot EALKprimex0 o0 o0 o0 Define lambda omegaA2rhoLA2E o0 o0 The theory says that as N gt infinity the scaled natural freqs should approach multiples of pi o0 o0 N50 d main2ones1N dsub ones1N 1L Kdiagd main diagd sub1 diagd sub 1 K1137 KNN3 7 7 Kprime NK MeyeN MprimeMN philameigKprimeMprime wnsqrtdiaglam N10 gives Wn39 31287 61803 90798 117557 141421 161803 178201 190211 197538 200000 oo wnlsortsortwn Re arrange the columns of phi consistent with the sorting used for w o0 o0 phi philsort muimat phi39Mprimephi 6 for N 10 off diag terms of muimat are order 35e 016 mu diagmu7mat PHlzerosNN 6 Normalize modes for j1N PHIiljphi1jSqrtmuj end for N 10 off diag terms of PHI39MPHI are order 35e 016 Note The normalized modes could also be normalized as PHlii PHIisqrtrhoAL figurel plotlNPHIl39o 391NPHI239A 39 xlabel39DOF 39 ylabel39Modes l and 2W legend39Mode 13939Mode 23939Location3939SouthEast39 if N gtlO figure2 plotlNPHI939o 391NPHI1039A W axis0 N 2 2 xlabel39DOF 39 ylabel39Modes 9 and 10W legend39Mode 93939Mode lO3939Location3939SouthEastW figure3 plotlNwn39o 391NlNpi39r 39 xlabel39Mode 39 ylabel39Nat Freq scaled39 legend39Lumped Mass Approximation3939Exact3939Location3939NorthWest39 err lOOabswn39 lNpi1Npi figure4 plotlNerr39rA W xlabel39Mode 39 ylabel39Percent Error in Nat Freq39 grid Modes I and 2 Modes 9 and IO DOF