11 | Continuous Systems – Transverse
At the end of this section you should be able to:
Identify homogeneous boundary conditions
Solve PDE with homogenous BCs for natural frequencies for transverse vibrations
Transverse Beams (Beams)
Transverse Beams (Beams)
For Euler-Bernoulli beams with only axial displacement, homogeneous material, constant cross-section and free vibration, the EOM is
where
The eigenfunction will be in the general form of
where the natural frequencies can then be calculated using
Analytically Finding the Natural Frequencies of a Fixed-Free Beam in Transverse Vibration
Analytically Finding the Natural Frequencies of a Fixed-Free Beam in Transverse Vibration
Analytically Finding the Natural Frequencies of a Fixed-Pinned Beam in Transverse Vibration
Analytically Finding the Natural Frequencies of a Fixed-Pinned Beam in Transverse Vibration
Common Boundary Conditions
Common Boundary Conditions
Limitations
Limitations
Constant
Cross-section (Area and MOI)
Material (Density and Young’s Modulus)
Autonomous/Free Vibration
Homogeneous boundary conditions
Euler-Bernoulli (“slender”) beam
Linear-Elastic Deformation
Code
Code