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A linearly elastic beam of length 2l with flexural rigidity EI has negligible mass. A massless spring with a spring constant k and a rigid block of mass m are attached to the beam as shown in the figure. The natural frequency of this system is

1. √((kl³ + 6EI)/(ml³))
2. √((kl³ + 48EI)/(ml³))
3. √((6EIk)/(k l³ + 6EI)m)
4. √((48EIk)/(k l³ + 48EI)m)

Correct answer: √((48EIk)/(k l³ + 48EI)m)

Solution

The correct option reflects the relationship between the stiffness of the spring and the beam's flexural rigidity, incorporating the effects of both components on the system's natural frequency. The factor of 48 arises from the specific boundary conditions and mode shapes associated with the beam's deflection under the applied load, leading to a more accurate representation of the dynamic behavior.

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