Mechanical Vibrations

Indian Institute of Technology Guwahati

Course contents:

  • Introduction
    Overview of the course, practical applications and research trends. Harmonic and periodic motions, vibration terminology
  • Single-DOF Free Vibrations
    Vibration model, Equation of motion-Natural Frequency. Energy method, Rayleigh method. Principle of virtual work, Damping models. Viscously damped free vibration. Special cases: oscillatory, non-oscillatory and critically damped motions. Logarithmic decrement, Experimental determination of damping coefficient. Forced harmonic vibration, Magnification factor. Rotor unbalance, Transmissibility. Vibration Isolation. Equivalent viscous damping, Sharpness of resonance.
  • Two-DOF Free Vibrations
    Generalized and Principal coordinates, derivation of equations of motion. Lagrange’s equation. Coordinate coupling. Forced Harmonic vibration.
  • Vibration Absorber
    Tuned absorber, determination of mass ratio. Tuned and damped absorber, unturned viscous damper.
  • Multi-DOF
    Derivation of equations of motion, influence coefficient method. Properties of vibrating systems: flexibility and stiffness matrices, reciprocity theorem. Modal analysis: undamped. Modal analysis: damped.
  • Calculation of natural frequencies
    Rayleigh method. Stodala method. Matrix iteration method. Holzer method and Dunkerley’s method.
  • Torsional vibration
    Simple systems with one or two rotor masses. Multi-DOF systems-transfer matrix method. Geared system. Branched system.
  • Continuous systems: closed form solutions
    Vibration of strings. Longitudinal and torsional vibration of rods. Transverse vibration of beams: equations of motion and boundary conditions. Transverse vibration of beams: natural frequencies and mode shapes.
  • Continuous systems: Approximate solutions
    Rayleigh’s energy method. Rayleigh-Ritz method. Assumed modes and Galerkin’s method.
  • Finite element analysis
    Finite element formulation for beams: Galerkin’s method. Beams elemental mass and stiffness matrices, Elemental force vector,
    Global finite element assembly and imposition of boundary conditions and solution procedure. Finite element formulation for rods.
  • Signature analysis and preventive maintenance
    Vibration testing equipments: signal generation, measuring and conditioning instruments. Vibration testing equipments: signal analysis instruments. Vibration signatures and standards. Field balancing of rotors.
Course Lectures