List of top Applied Mechanics and Design Questions on Free and forced vibration of single degree of freedom systems asked in GATE ME

A vibratory system consists of mass m, a vertical spring of stiffness 2k and a horizontal spring of stiffness k. The end A of the horizontal spring is given a horizontal motion x_A = αsinwt. The other end of the spring is connected to an inextensible rope that passes over two massless pulleys as shown. Assume m = 10 kg, k = 1.5 kN/m, and neglect friction. The magnitude of critical driving frequency for which the oscillations of mass m tend to become excessively large is ____rad/s (answer in integer).

A spring–mass–damper system (mass $m$, stiffness $k$, damping coefficient $c$) excited by a force $F(t)=B\sin \omega t$ is shown in the figure. Four different responses of the system (marked as (i)–(iv)) are shown. In the figures, $A$ is the amplitude of the oscillatory response, and the dashed lines show its envelope. The responses represent only qualitative trends. Four different forcing and parameter conditions are also given:

\[ \text{(P)}\ c>0,\quad \omega=\sqrt{\frac{k}{m}}, \qquad \text{(Q)}\ c<0\ \text{and}\ \omega\ne 0, \] \[ \text{(R)}\ c=0,\quad \omega=\sqrt{\frac{k}{m}}, \qquad \text{(S)}\ c=0,\quad \omega \approx \sqrt{\frac{k}{m}}. \] Which option correctly matches each condition to its response (i)–(iv)?