Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsToday’s Objectives:Students will be able to:a) Find the steady state response of 2DOF forced systemVibration absorbers (9.11 in Rao)Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsExamples of adding an auxiliary mass to reduce vibrationsTorsional damper, harmonic balancer (vibration absorber)Stockbridge dampers on power lines.Tuned mass damper atop the Taipei 101Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsVibration absorbertsinFxxkkkkkxxmmω⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎦⎤⎢⎢⎣⎡−−++⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎦⎤⎢⎢⎣⎡000021222212121&&&&This has the EOM:()⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎦⎤⎢⎢⎣⎡−−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧01021221212222221FmkkkmkZXXωωωAfter assuming x1=X1sinωt and x2=X2sinωt we get (see yesterday’s notes)So() ()02202221 and FZkXFZmkXωωω=−=How do we choose k2and m2to reduce the amplitude of x1?k2k1x1x2m1m2F0sinωtRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsWorking model simulationsRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsVibration absorbers (cont.)()()()()02222221212202222221212221 FkmkmkkkXFkmkmkkmkX−−−+=−−−+−=ωωωωωSummary of equationsSo when k2/m2= ω2we find: X1= 0 and X2= - F0/k0 0.5 1 1.5 212ωωaω1ω2Operating rangeωX1Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsDesign considerations1. The values of k2and m22. The value of x2may have a maximum allowable valuek2k1x1x2m1m2F0sinωt3. You may be given information on the desired natural frequency locations • Equation 9.140 gives a general equation and Example 9.10 Eq. E.2 for a specific case.• You may also use Maple to determine the frequencies in terms of m2and k2.4. You may be given a frequency range where x1is less then some specified valueRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsNotes and problems• Depends on knowing ω exactly (only effective in a small range of working frequencies)• Single frequency device• If ω shifts it could end up exciting a system natural frequency (resonance)• Damping, which always exists to some degree, spoils the absorptionRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsWhy not just pick a really small m2and k2?• Amplitude of X2may be very large• Operating rangeAbsorber mass = 1/10 primary massAbsorber mass = 1/100 primary massRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsMaple worksheet for
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