# GATE Questions & Answers of Vibrations Mechanical Engineering

#### Vibrations 45 Question(s)

The equation of motion for a spring-mass system excited by a harmonic force is

$M\ddot x+Kx=F\;c\mathrm{os}\left(\omega t\right)$,

where $M$ is the mass, $K$ is the spring stiffness, $F$ is the force amplitude and $\omega$ is the angular frequency of excitation. Resonance occurs when $\omega$ is equal to

A machine of mass $m=200$ kg is supported on two mounts, each of stiffness $k=10$ kN/m. The machine is subjected to an external force (in N) $F\left(t\right)=50\;\cos\;5t$. Assuming only vertical translatory motion, the magnitude of the dynamic force (in N) transmitted from each mount to the ground is ______ (correct to two decimal places). In a single degree of freedom underdamped spring-mass-damper system as shown in the figure, an additional damper is added in parallel such that the system still remains underdamped. Which one of the following statements is ALWAYS true? The damping ratio for a viscously damped spring mass system, governed by the relationship $\style{font-family:'Times New Roman'}{m\frac{\mathrm d^2x}{\mathrm dt^2}+c\frac{\mathrm dx}{\mathrm dt}+kx=F(t)}$, is given by

A thin uniform rigid bar of length L and mass M is hinged at point O, located at a distance of $\style{font-family:'Times New Roman'}{\frac L3}$ from one of its ends. The bar is further supported using springs, each of stiffiness k, located at the two ends. A partical of mass $\style{font-family:'Times New Roman'}{m=\frac M4}$ is fixed at one end of the bar, as shown in the figure. For small rotatios of the bar about O, the natural frequency of the system is A mass m is attached to two identical springs having constant k as shown in the figure. The natural frequency ω of this single degree of freedom system is 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 center 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_________

A single degree of freedom spring mass system with viscous damping has a spring constant of 10 kN/m. The system is excited by a sinusoidal force of amplitude 100 N. If the damping factor (ratio) is 0.25, the amplitude of steady state oscillation at resonance is ________mm.

The spring constant of a helical compression spring DOES NOT depend on

A solid disc with radius $a$ is connected to a spring at a point $d$ above the center of the disc. The other end of the spring is fixed to the vertical wall. The disc is free to roll without slipping on the ground. The mass of the disc is $M$ and the spring constant is $K$. The polar moment of inertia for the disc about its centre is $J =Ma^2/2$ . The natural frequency of this system in rad/s is given by

A single degree of freedom mass-spring-viscous damper system with mass $m$, spring constant $k$ and viscous damping coefficient $q$ is critically damped. The correct relation among $m,k,$ and $q$ is

The system shown in the figure consists of block A of mass 5 kg connected to a spring through a massless rope passing over pulley B of radius r and mass 20 kg. The spring constant k is 1500 N/m. If there is no slipping of the rope over the pulley, the natural frequency of the system is_____________ rad/s. 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 $\mathrm g=10\mathrm m/\mathrm s^2$. The natural frequency of this spring-mass system (in rad/s) is_____________

A single degree of freedom spring-mass system is subjected to a harmonic force of constant amplitude. For an excitation frequency of $\sqrt{\frac{3k}{m}}$ , the ratio of the amplitude of steady state response to the static deflection of the spring is __________ Considering massless rigid rod and small oscillations, the natural frequency (in rad/s) of vibration of the system shown in the figure is A mobile phone has a small motor with an eccentric mass used for vibrator mode. The location of the eccentric mass on motor with respect to center of gravity (CG) of the mobile and the rest of the dimensions of the mobile phone are shown. The mobile is kept on a flat horizontal surface. Given in addition that the eccentric mass = 2 grams , eccentricity = 2.19 mm, mass of the mobile = 90 grams , g = 9.81 m/s2 .Uniform speed of the motor in RPM for which the mobile will get just lifted off the ground at the end Q is approximately.

A precision instrument package(= 1 kg) needs to be mounted on a surface vibrating at 60 Hz. It is desired that only 5% of the base surface vibration amplitude be transmitted to the instrument .Assume that the isolation is designed with its natural frequency significantly lesser than 60 Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of the required mounting pad is ________.

In a spring-mass system the mass is m and the spring constant is k. The critical damping coefficient of the system is 0.1 kg/s. In another spring mass system, the mass is 2m and the spring constant is 8k. The critical damping coefficient (in kg/s) of this system is _______.

A single-degree-freedom spring-mass is subjected to a sinusoidal force of 10 N amplitude and frequency $\omega$ along the axis of the spring. The stiffness of the spring is 150 N/m, damping factor is 0.2 and undamped natural frequency is 10 $\omega$. At steady state, the amplitude of vibration (in m) is approximately

Which of the following statements are TRUE for damped vibrations?

 P. For a system having critical damping, the value of the damping ratio is unity and system does not undergo a vibratory motion. Q. Logarithmic decrement method is used to determine the amount of damping in a physical system. R. In case of damping due to dry friction between moving surfaces resisting force of constant magnitude acts opposite to the relative motion. S. For the case of viscous damping, drag force is directly proportional to the square of relative velocity.

Figure shows a single degree of freedom system. The system consists of a massless rigid bar OP hinged O and a mass m at end P. The natural frequency of vibration of the system is Critical damping is the

Consider a cantilever beam, having negligible mass and uniform flexural rigidity, with length 0.01 m. The frequency of vibration of the beam, with a 0.5 kg mass attached at the free tip, is 100 Hz. The flexural rigidity (in N.m2) of the beam is _______

A rigid uniform rod AB of length L and mass m is hinged at C such that AC = L/3, CB = 2L/3. Ends A and B are supported by springs of spring constant k. The natural frequency of the system is given by In vibration isolation, which one of the following statements is NOT correct regarding Transmissibility (T)?

What is the natural frequency of the spring mass system shown below? The contact between the block and the inclined plane is frictionless. The mass of the block is denoted by m and the spring constants are denoted by k1 and k2 as shown below. Consider a single degree-of-freedom system with viscous damping excited by a harmonic force. At resonance, the phase angle (in degree) of the displacement with respect to the exciting force is

The damping ratio of a single degree of freedom spring-mass-damper system with mass of 1 kg, stiffness 100 N/m and viscous damping coefficient of 25 N.s/m is _______

A mass-spring-dashpot system with mass m = 10 kg, spring constant k = 6250 N/m is excited by a harmonic excitation of 10 cos(25t) N. At the steady state, the vibration amplitude of the mass is 40 mm. The damping coefficient (c, in N.s/m) of the dashpot is _______ A single degree of freedom system has a mass of 2 kg, stiffness 8 N/m and viscous damping ratio 0.02. The dynamic magnification factor at an excitation frequency of 1.5 rad/s is _______

If two nodes are observed at a frequency of 1800 rpm during whirling of a simply supported long slender rotating shaft, the first critical speed of the shaft in rpm is

A single degree of freedom system having mass 1 kg and stiffness 10 kN/m initially at rest is subjected to an impulse force of magnitude 5 kN for 10-4 seconds. The amplitude in mm of the resulting free vibration is

A concentrated mass m is attached at the centre of a rod of length 2L as shown in the figure. The rod is kept in a horizontal equilibrium position by a spring of stiffness k. For very small amplitude of vibration, neglecting the weights of the rod and spring, the undamped natural frequency of the system is A mass of 1 kg is attached to two identical springs each with stiffness k = 20 kN/m as shown in the figure. Under frictionless condition, the natural frequency of the system in Hz is close to A disc of mass m is attached to a spring of stiffness k as shown in the figure. The disc rolls without slipping on a horizontal surface. The natural frequency of vibration of the system is The natural frequency of a spring-mass system on earth is ωn. The natural frequency of this system on the moon (gmoon = gearth /6) is

A mass m attached to a spring is subjected to a harmonic force as shown in figure. The amplitude of the forced motion is observed to be 50mm. the value of m (in Kg) is The rotor shaft of a large electric motor supported between short bearings at both deflection of 1.8 mm in the middle of the rotor. Assuming the rotor to be perfectly balanced and supported at knife edges at both the ends, the likely critical speed (in rpm) of the shaft is

An automotive engine weighing 240 kg is supported on four springs with linear characteristics. Each of the front two springs have a stiffness of 16MN/m while the stiffness of each rear spring is 32 MN/m. The engine speed (in rpm), at which resonance is likely to occur, is

A vehicle suspension system consists of a spring and a damper. The stiffness of the spring is 3.6 kN/m and the damping constant of the damper is 400 Ns/m. If the mass is 50 kg, then the damping factor (d) and damped natural frequency (fn), respectively, are

The natural frequency of the spring mass system shown in the figure is closest to A uniform rigid rod of mass m = 1kg and length L = 1m is hinged at its centre and laterally supported at one end by a spring of spring constant k = 300 N/m. The natural frequency ωn in rad/s is

For an underdamped harmonic oscillator, resonance The equation of motion of a harmonic oscillator is given by $\frac{{d}^{2}x}{d{t}^{2}}+2\zeta {\omega }_{n}\frac{dx}{dt}+{\omega }_{n}^{2}x=0$, and the initial conditions at t = 0 are $x\left(0\right)=X,\frac{dx}{dt}\left(0\right)=0$. The amplitude of x(t) after n complete cycle is