$5\mathrm{.}3$ A concrete foundation $($unit weight $=23\mathrm{.}5k\frac{N}{m^3})$ supporting a machine is $3\mathrm{.}5m2\mathrm{.}5m$ in plan and is subjected to a sinusoidal vibrating foree $($vertical$)$ having an amplitude of $10kN($not frequency dependent$).$ The operating frequency is $2000cpm.$ The weight of the machine and foundation is $400kN.$ The soil properties are unit weight $=18k\frac{N}{m^3},$ shear modulus $=38,000$kPa, and Poisson' ratio $=0\mathrm{.}25.$ Determine

a$.$ the resonant frequency of the foundation, and

b$.$ the amplitude of vertical vibration at operating frequency.