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.
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?