Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsToday’s Objectives:Students will be able to:a) Find the homogeneous or steady state solution using analytical modal analysis with modal dampingAnalytical Modal Analysis with Modal Damping0 1 2 3 4 5 6 7 810-310-210-1100101Frequency (Hz)Magnitude of FRF.Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsSystems with damping[]{}[]{}[]{}{}FxKxCxM=++&&&{}[]{}qxΦ=Our equation of motion is:If we apply the coordinate transformation (using mass normalized modes for the undamped system)Substituting in and multiplying by [Φ]Twe get[][ ][]{}[][][]{}[][][ ]{}[]{}4342143421&43421&&43421⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡Φ=ΦΦ+ΦΦ+ΦΦQFqdiagKq???CqIMTiTTT2ωRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsTwo special cases:1. Proportional dampingSo the decoupled equations areProportional damping[][][]KMCβα+=[][][]ΦΦ CT[][][][][][]()[][][ ][] [][][][]()[]2iTTTTdiagIKMKMCωβαβαβα+=ΦΦ+ΦΦ=Φ+Φ=ΦΦIn general the termwill not be diagonal!The equations don’t decouple!()iiiiiiQqqq =+++22ωβωα&&&soRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsModal dampingiiiiiiiQqqq =++22ωωζ&&&()2222212 where21iiiiiiiiiiirrtantsin/KqP−=−⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−=ζφφωωωζωωω()side handright on the depends =+=−PiiiiHididitiqtsinBtcosAeqωωωζ2. Modal damping. Let’s not worry about α and β and just assume we can assign damping to each mode. This gives decoupled equations:We know how to solve this!If Qi(t) = Kisinωt we would getApply initial condition to ()()()tqtqtqPHiii+=Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsObservations[][][]()[]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡==ΦΦnniiTdiagCωζωζωζωζ2000002000222211LOMML0 1 2 3 4 5 6 7 810-310-210-1100101Frequency (Hz)Magnitude of FRF.1. The damping we identified in lab is the modal damping2. Modal damping is often an input in finite element programs3. To find a damping matrix (for example if you want one for Simulink) that gives you particular values of modal damping you can
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