Vibrations Mechanical previous year questions with answer

Ques 1 GATE 2025

A block of mass 1 kg connected to a spring of stiffness 10 N m^-1 is operating in a viscous medium such that the damping ratio (or damping factor) is equal to the ratio of the damped frequency to the natural frequency. The magnitude of the damping ratio for this system is ______ (rounded off to 2 decimal places).

Ques 2 GATE 2025

The system shown in the figure below consists of a cantilever beam (with flexural rigidity EI and negligible mass), a spring (with spring constant K and negligible mass) and a block of mass m. Assuming a lumped parameter model for the system, the fundamental natural frequency (ω_n) of the system is

Ques 3 GATE 2022 SET-1

A rigid uniform annular disc is pivoted on a knife edge A in a uniform gravitational field as shown, such that it can execute small amplitude simple harmonic motion in the plane of the figure without slip at the pivot point. The inner radius 𝑟 and outer radius 𝑅 are such that 𝑟 =𝑅²/2, and the acceleration due to gravity is 𝑔. If the time period of small amplitude simple harmonic motion is given by 𝑇=𝛽𝜋√(𝑅/𝑔), where 𝜋 is the ratio of circumference to diameter of a circle, then 𝛽= ________ (round off to 2 decimal places).

Ques 4 GATE 2022 SET-1

Consider a forced single degree-of-freedom system governed by

mẍ(t) + 2ζωẋ(t) + ω²x(t) = ω² cos(ωt),

where ζ and ω are the damping ratio and undamped natural frequency of the system, respectively, while ω_f is the forcing frequency. The amplitude of the forced steady state response of this system is given by [ (1 - r²)² + (2ζr)² ]^-1/2, where r = ω_f/ω. The peak amplitude of this response occurs at a frequency ω_f = ω√(1 - 2ζ²). If ω_d denotes the damped natural frequency of this system, which one of the following options is true?

Ques 5 Gate 2020 Set-2

A hollow spherical ball of radius 20 cm floats in still water, with half of its volume submerged. Taking the density of water as 1000 kg/m³, and the acceleration due to gravity as 10 m/s², the natural frequency of small oscillations of the ball, normal to the water surface is _________ radians/s (roundoff to 2 decimal places).

Ques 6 GATE 2017 SET-2

A mass m is attached to two identical springs having spring constant k as shown in the figure. The natural frequency ω of this single degree of freedom system is

Ques 7 GATE 2017 SET-2

The radius of gyration of a compound pendulum about the point of suspension is 100 mm. The distance between the point of suspension and the centre of mass is 250 mm. Considering the acceleration due to gravity as 9.81 m/s², the natural frequency (in radian/s) of the compound pendulum is ________.

Ques 8 GATE 2017 SET-1

The damping ratio for a viscously damped spring mass system, governed by the relationship m(d²x/dt²) + c(dx/dt) + kx = F(t), is given by

Ques 9 GATE 2017 SET-1

A thin uniform rigid bar of length L and mass M is hinged at point O, located at a distance of L/3 from one of its ends. The bar is further supported using springs, each of stiffness k, located at the two ends. A particle of mass m = M/4 is fixed at one end of the bar, as shown in the figure. For small rotations of the bar about O, the natural frequency of the system is

Ques 10 GATE 2016 SET-3

The static deflection of a spring under gravity, when a mass of 1 kg is suspended from it, is 1 mm. Assume the acceleration due to gravity g = 10 m/s². The natural frequency of this spring-mass system (in rad/s) is ________.