9 | Continuous Systems – Axial
At the end of this section you should be able to
Identify homogeneous boundary conditions
Solve PDE with homogenous BCs for strings
Solve PDE with homogenous BCs for beams in axial deformation
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