Question

The system shown in the figure below consists of a cantilever beam (with flexural rigidity \( EI \) and negligible mass), a spring (with spring constant \( K \) and negligible mass) and a block of mass \( m \). Assuming a lumped parameter model for the system, the fundamental natural frequency (\( \omega_n \)) of the system is 

 

• A. \( \sqrt{\dfrac{\dfrac{3EI}{L^3} + K}{m}} \)
• B. \( \sqrt{\dfrac{\dfrac{EI}{L^3} + K}{m}} \)
• C. \( \sqrt{\dfrac{\dfrac{3EI}{L^3} + K}{2m}} \)
• D. \( \sqrt{\dfrac{\dfrac{EI}{L^3} + K}{2m}} \)

Hint:

For a system involving a cantilever beam and a spring, the natural frequency is influenced by both the bending stiffness of the beam and the spring constant.

Answer 1

The Correct Option is A

The fundamental natural frequency \( \omega_n \) for a cantilever beam with a spring and mass is derived by considering both the flexural rigidity of the beam and the spring constant. 
The characteristic equation for the system is: \[ \omega_n = \sqrt{\dfrac{\dfrac{3EI}{L^3} + K}{m}} \] Here, \( EI \) is the flexural rigidity of the beam, \( L \) is the length of the beam, \( K \) is the spring constant, and \( m \) is the mass of the block.