Free & Forced Vibration Notes | EduRev

Theory of Machines (TOM)

Mechanical Engineering : Free & Forced Vibration Notes | EduRev

The document Free & Forced Vibration Notes | EduRev is a part of the Mechanical Engineering Course Theory of Machines (TOM).
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Mechanical Vibrations

The to and for motion of a body about its mean position is termed as mechanical vibrations or harmonic vibrations or oscillations.
[Mean Position = Equilibrium Position = Zero Position] All are same
Free & Forced Vibration Notes | EduRevType of oscillating waves

Any vibrating system is a combination of:

  • E storing device (i.e. mass = m)
  • E storing device (having stiffness = s)
  • Kinetic friction
  • Unbalanced forces
  1. Natural Vibrations
    The vibrations in which there is no friction at all (static friction may have, but no kinetic friction) as well as there is no unbalanced force after the initial release of the system are called as “Natural Vibrations”
    Free & Forced Vibration Notes | EduRevSpring-mass systemUsing D-Alembert's Principle:
    ma + sx = 0
    Free & Forced Vibration Notes | EduRevFree & Forced Vibration Notes | EduRev
  2. Energy Method to Calculate Natural Frequency
    This method is used only for natural vibrations and is used specially for Rolling problems.
    In Natural Vibrations,
    As, kinetic friction = 0
    Total energy = constant
    Free & Forced Vibration Notes | EduRev
  3. Torsional Vibrations
    The vibrations of a system about its own center of mass is termed as torsional vibrations.
    In case of torsional vibrations, At fixed Print, as. Hence, Vibration’s amplitude will also zero. This point is called ‘Node’.
    Free & Forced Vibration Notes | EduRevshaft-rotor systemFree & Forced Vibration Notes | EduRev
    (kT = Torsional stiffness of shaft)
    Note: If shaft mass MOI is also considered (= Is)
    Free & Forced Vibration Notes | EduRev
    (i) Two-Rotor System
    Free & Forced Vibration Notes | EduRev2-rotor systemAt Node Point:
    ωn1 = ωn2 (i.e. net vibrations of Node Point = 0)
    Free & Forced Vibration Notes | EduRev
    for this same shaft, G1J1 = G2 J2      [∵ G1 = G2, J1 = J2]
    ⇒ I11 = I22    (2)
    Also, ℓ1 + ℓ2 = ℓ   (3)
  4. Rayleigh’s Method to Calculate Natural Frequency (Method of Static Deflection of Mass)
    Free & Forced Vibration Notes | EduRevBasic spring Mass System∇ = static deflection of mass ′m′
    = (mg/s)  [∵ mg = s · ∇]
    Free & Forced Vibration Notes | EduRev
  5. Longitudinal-Vibrations of Beams
    Vibrations along the length of Beam are termed as longitudinal vibrations.
    Free & Forced Vibration Notes | EduRevlongitudinal vibrations of a beamAxial or longitudinal stiffness of beam,
    s = (AE/L)
    Free & Forced Vibration Notes | EduRev
  6. Transverse –Vibrations of Beams
    Vibrations in a direction perpendicular to axis of the Beam.
    Free & Forced Vibration Notes | EduRevTransverse vibrations of a beamFree & Forced Vibration Notes | EduRev
    where, δ = static defection of Beam Under transverses loads
    Example:
    For a cantilever Beam with Point load 'w' at free end δ = WL3/ 3EL
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