MECHANICAL VIBRATIONS ME65

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MECHANICAL VIBRATIONS (ME65)

• Session 8
• BY
• Dr. P. Dinesh
• Sambhram Institute of Technology
• Bangalore

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In this session 8

• Forced Vibration
• Steady state solution for a single DOF system due
  to harmonic excitation
• Vector force polygon
• Magnification factor
• Characteristic curves

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Forced Vibration
c
k
F sin?t
m
4
F sin?t
kx
cx
m
x
mx
Forces acting on the mass
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• Graphical method of analysing forced vibration
• Equation of motion is
• mxcxkx Fsin?t
• The solution of above equation is in two parts
• xcf complimentary function
• xpi particular integral
• xcf Xe ?ntsin(?d F)

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• xpi x X sin(?t F )
• F is the phase difference between impressed F
  and the displacement x
• xpi is the solution of interest as is the
  transient one.
• x ?Xcos(?t F ) ?Xsin(?t F 90º)
• x - ?2Xsin(?t F) ?2Xsin(?t F 180º)

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• Substituting x and x in eqn. of motion
• m(?2Xsin(?t F 180º)c(?Xsin(?tF 90º) k(X
  sin(?t F )) Fsin?t
• Rearranging,
• Fsin?t- m?2Xsin(?t F 180º)- c?Xsin(?tF
  90º)- kX sin(?t F ) 0
• Where,
• Fsin?t is the impressed force

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• Displacement X lags F by F
• m?2X is the inertia force lagging F by F 180º
• c?X is the damping force lagging F by F 90º
• kX is the spring force lagging F by F

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