Question

A prismatic fixed-fixed beam, modelled with a total lumped-mass of 10 kg as a single degree of freedom (SDOF) system is shown in the figure. 

Hint:

The natural frequency for a fixed-fixed beam with a lumped mass is determined by the stiffness and mass of the system.

Answer 1

Correct Answer: 10

The natural frequency of vibration in the flexural mode for a fixed-fixed beam with lumped mass \( m \) is given by the formula: \[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}, \] where \( k \) is the stiffness of the beam and \( m \) is the lumped mass. The stiffness for a fixed-fixed beam is given by: \[ k = \frac{4EI}{L^3}, \] where \( E \) is the modulus of elasticity, \( I \) is the moment of inertia, and \( L \) is the length of the beam. Given that the flexural stiffness of the beam is \( 4\pi^2 \, \text{kN} \cdot \text{m}^2 \), the natural frequency is: \[ f = \frac{1}{2\pi} \sqrt{\frac{4\pi^2 \, \text{kN} \cdot \text{m}^2}{10 \, \text{kg}}} = 10 \, \text{Hz}. \] Thus, the natural frequency of vibration in the flexural mode is \( \boxed{10} \, \text{Hz} \).