List of top Solid Mechanics Questions on Free vibration of undamped single degree of freedom systems asked in GATE XE

A massless cantilever beam, with a tip mass \(m = 10 \, kg\), is modeled as an equivalent spring-mass system as shown in the figure. The beam is of length \(L = 1 \, m\), with a circular cross-section of diameter \(d = 20 \, mm\). The Young’s modulus of the beam material is \(E = 200 \, GPa\).

The natural frequency of the spring-mass system is .......... Hz (rounded off to two decimal places).

The natural frequency of a spring–mass system is 10 rad/s. Which of the following statements is/are CORRECT?

Consider a spring–mass system with mass \(m\) and spring stiffness \(k\) as shown in the illustration. At time \(t=0\), the mass is displaced by \(P\) units and the velocity of the mass is zero. The displacement of the mass, \(x(t)\), is measured from the equilibrium position.

Which one of the following functions represents \(x(t)\)?

A composite rod made of steel and copper is fixed immovably at its ends. Each portion has length $1 \, m$. Cross-sectional areas are the same. $E_{steel} = 200 \, GPa$, $E_{copper} = 100 \, GPa$. $\alpha_{steel} = 12 \times 10^{-6}/^\circ C$, $\alpha_{copper} = 18 \times 10^{-6}/^\circ C$. The temperature is increased by $100^\circ C$. Find axial stress (in MPa) in the steel rod.

A spring–mass system is shown in the figure below. Take the acceleration due to gravity as $g = 9.81 \, m/s^2$. The static deflection due to weight and the time period of oscillations, respectively, are:

A plane truss is simply supported at $P$ and $R$ as shown. A downward force $F$ is applied at hinge $Q$. The axial force developed in member $PS$ is: