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When a system is disturbed from its equilibrium position and allowed to vibrate without any continuous external force, the motion is called free vibration. Depending on the direction of particle motion relative to the member’s axis, free vibration is classified as longitudinal or transverse.
Free Longitudinal Vibration
In longitudinal vibration, particles of the body move parallel to the axis of the member.
Examples: vibration of rods, bars, columns, and springs along their length.
Model: Mass–Spring System (Axial)
Consider a mass (m) attached to a spring of stiffness (k). If displaced and released, it oscillates along the spring axis.
Equation of motion:
Characteristics
Motion is along the length
Restoring force due to axial stiffness
Common in rods, springs, and tie members
Free Transverse Vibration
In transverse vibration, particles move perpendicular to the axis of the member.
Examples: vibration of beams, strings, shafts, and cantilevers.
Model: Beam with Mass
The restoring force arises from bending stiffness rather than axial stiffness.
For a simply supported beam:
Where:
(E) = Young’s modulus
(I) = Moment of inertia
(\rho) = Density
(A) = Area
(L) = Length
(C) = Constant depending on boundary condition
Characteristics
Motion is perpendicular to the axis
Restoring force due to bending resistance
Seen in beams, bridges, machine frames
Key Differences
Aspect | Longitudinal Vibration | Transverse Vibration |
|---|---|---|
Direction of motion | Parallel to axis | Perpendicular to axis |
Restoring force | Axial stiffness | Bending stiffness |
Typical members | Rods, springs | Beams, shafts |
Governing property | (E, \rho) | (E, I, \rho, A) |
Deformation type | Extension & compression | Bending |
Engineering Importance
Understanding these vibrations helps in:
Avoiding resonance in rods and beams
Designing machine members for dynamic stability
Predicting natural frequencies of structural elements
Improving durability and safety of components
2 Hrs 
