Derive an expression for the natural frequency - beam mass, Science

The simply-supported uniform beam shown in the ?gure below supports a mass m at a distance aL from the left support. Assume that the beam is relatively mass-less compared to the mass m. Gravitational acceleration is "downward" as shown in the ?gure. The transverse beam de?ection at the location of the mass is D.

54_Derive an expression for the natural frequency - Beam Mass.png

Derive an expression for the natural frequency of this beam-mass system in terms of D and g (only). For what value of a is this natural frequency the smallest?

 

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