A flexible rotor-shaft system comprises of a $$10kg$$ rotor disc placed in the middle of a mass-less shaft of diameter $$30$$ $$mm$$ and length $$500mm$$ between bearings (shaft is being taken mass-less as the equivalent mass of the shaft is included in the rotor mass) mounted at the ends. The bearings are assumed to simulate simply supported boundary conditions. The shaft is made of steel for which the value of $$E$$ is $$\,2.1\, \times \,{10^{11}}\,$$. Pa. What is the critical speed of rotation of the shaft?

The assembly shown in the figure is composed of two massless rods of length $$l$$ with two particles, each of mass $$m$$. The natural frequency of this assembly for small oscillations is

As shown in Figure, a mass of 100 kg is held between two springs. The natural frequency of vibration of the system, in cycles/s, is

Consider the system of two wagons shown in Figure. The natural frequencies of this system are