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
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
Show Hint
- \( \sqrt{\dfrac{\dfrac{3EI}{L^3} + K}{m}} \)
- \( \sqrt{\dfrac{\dfrac{EI}{L^3} + K}{m}} \)
- \( \sqrt{\dfrac{\dfrac{3EI}{L^3} + K}{2m}} \)
- \( \sqrt{\dfrac{\dfrac{EI}{L^3} + K}{2m}} \)
The Correct Option is A
Solution and Explanation
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.
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