Introduction to Continuous Systems

Vibrating Strings

A string with only axial force, homogeneous material, constant area and free vibration has the EOM of

where

The solution to the PDE is, in general form

The eigenfunction is in the general form of

and the remaining constants are found using initial conditions

Plucked String Example

Longitudinal Vibrations (Columns)

For a column experiencing on axial deformation with constant area, homogeneous material and free vibration, the EOM is

where

Uniform Bar with Initial Displacement Example

Common Boundary Conditions

Limitations

• Vibrating String

  • Constant-

    • Tension

    • Density

  • Autonomous/Free Vibration

  • Homogeneous boundary conditions

• Longitudinal Vibrations

  • Linear elastic deformation

  • Constant-

    • Cross-section

    • Density

    • Modulus of elasticity

  • Autonomous/Free Vibration

  • Homogeneous boundary conditions

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