Vibrations

Vibrations

1. Terminology

   • Damped system
     • Energy is gradually lost (Due to damping forces)
   • Undamped system
     • Damping forces are not present
   • Viscous damping
     • Damping force is proportional to velocity
   • Dry friction (Coulomb) damping
     • Damping where energy is lost via sliding friction
   • Free vibration
     • No external forces
   • Forced vibration
     • External force f(t) applied to the system
   • Number of degrees of freedom
     • The minimum number of independent coordinates required to determine completely the position of all parts of a system at any instant of time.
       • More accurate results obtained by increasing number of degrees of freedom.
     • Continuous/disributed system
       • A continuous or distributed systems has infinite number of degrees of freedom.
     • Discrete/lumped parameter systems
       • Finite number of degrees of freedom are termed discrete or lumped parameter systems.

2. Free undamped SDOF systems

   • From equlibrium of forces
     • Equation of motion of free vibration undamped SDOF:
   • Solution of the equation of motion of a undamped SDOF in free vibration
     • Using trigonometric identities, the solution can be expressed as:
   • Velocity is maximum at displacement equal to zero.
   • Displacement is maximum when velocity is equal to zero.
   • If ẋ0 = 0, then the solution for x(t) is:
   • Angular motion
     • The equation of the angular motion:
     • The circular natural frequency of the torsional systems is:

3. Free vibration of damped SDOF systems

   • Viscous damping
     • Damping force proportional to velocity.
   • Equation of motion:
   • General solution of the equation of motion:
   • Critical damping
     • The value of the damping constant c for which the radical in the equation above is zero.
   • Damping ratio
   • circular frequency of damped vibration

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