List of top Structures Questions on Free and forced vibrations of undamped and damped SDOF systems asked in GATE AE

A system comprising a bar, spring and mass is shown in the figure below.
The bar, having negligible mass, is made of a material having Young's modulus \(E = 200\) GPa, cross-sectional area \(A = 100\) mm\(^2\), and length \(L = 100\) mm. The spring stiffness \(k = 200\) kN/mm and the mass \(M = 100\) kg. The natural frequency of free vibration of the system is _______ rad/s (rounded off to the nearest integer).
The response \(x(t)\) of a freely vibrating single degree of freedom underdamped system is given below. In the equation, \(A\) and \(\phi\) are constants. The damping ratio of the system is _______ (rounded off to 3 decimal places). \[ x(t) = A e^{-5t}\sin(10t+\phi) \]

A rigid massless rod pinned at one end has a mass $m$ attached to its other end. The rod is supported by a linear spring of stiffness $k$ as shown in the figure. The natural frequency of this system is: 

A spring-mass-damper system (m = 10 kg, k = 17400 N/m) has natural frequency 13.2 rad/s. Find the damping coefficient $c$ for critical damping (round off to nearest integer).