Vibrations · Theory of Machines · GATE ME
Marks 1
1
A linear spring-mass-dashpot system with a mass of 2 kg is set in motion with viscous damping. If the natural frequency is 15 Hz, and the amplitudes of two successive cycles measured are 7.75 mm and 7.20 mm, the coefficient of viscous damping (in N.s/m) is
GATE ME 2024
2
The figure shows a block of mass m = 20 kg attached to a pair of identical linear springs, each having a spring constant k = 1000 N/m. The block oscillates on a frictionless horizontal surface. Assuming free vibrations, the time taken by the block to complete ten oscillations is _________ seconds. (Rounded off to two decimal places)
Take π = 3.14.
GATE ME 2023
3
For a dynamical system governed by the equation,
$\ddot{x}(t)+2 ζ \omega_n \dot{x}(t)+\omega_n^2x(t)=0$
the damping ratio ζ is equal to $\frac{1}{2\pi}\log_e2$. The displacement x of this system is measured during a hammer test. A displacement peak in the positive displacement direction is measured to be 4 mm. Neglecting higher powers (> 1) of the damping ratio, the displacement at the next peak in the positive direction will be _______ mm (in integer).
GATE ME 2022 Set 2
4
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 r2 = R2/2, and the acceleration due to gravity is g. If the time period of small amplitude simple harmonic motion is given by $T = β π \sqrt{R/g} $ where π is the ratio of circumference to diameter of a circle, then β = ________ (round off to 2 decimal places).
GATE ME 2022 Set 1
5
The damping ratio for a viscously damped spring mass system, governed by the relationship $$\,m{{{d^2}x} \over {d{t^2}}} + c{{dx} \over {dt}} + kx = f\left( t \right),\,\,\,$$ is given by
GATE ME 2017 Set 1
6
A mass $$m$$ is attached to two identical springs having spring constant $$k$$ as shown in the figure. The natural frequency of this single degree of freedom system is
GATE ME 2017 Set 2
7
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 _____________
GATE ME 2016 Set 3
8
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.
GATE ME 2016 Set 1
9
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 __________
GATE ME 2015 Set 2
10
Which of the following statements are TRUE for damped vibrations?
$$P.$$ For a system having critical damping, the value of 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.
GATE ME 2015 Set 3
11
A point mass is executing simple harmonic motion with an amplitude of $$10$$ mm and frequency of $$4$$ Hz. The maximum acceleration (m/s2) of the mass is _________.
GATE ME 2014 Set 4
12
Critical damping is the
GATE ME 2014
13
In vibration isolation, which one of the following statements is NOT correct regarding Transmissibility ($$T$$)?
GATE ME 2014
14
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
GATE ME 2014 Set 3
15
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
GATE ME 2013
16
The natural frequency of a spring-mass system on earth is $${\omega _n}.$$ The natural frequency of this system on the moon (gmoon = gearth/6) is
GATE ME 2010
17
The rotor shaft of a large electric motor supported between short bearings at both the ends shows a 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
GATE ME 2009
18
For an under damped harmonic oscillator, resonance
GATE ME 2007
19
The differential equation governing the vibrating system is
GATE ME 2006
20
There are four samples P, Q, R and S, with natural frequencies 64, 96, 128 and 256 Hz, respectively. There are mounted on test setups for conducting vibration experiments. If a loud pure not of frequency 144Hz is produced by some instrument, which of the samples will show the most perceptible induced vibration?
GATE ME 2005
21
A vibrating machine is isolated from the floor using springs. If the ratio of excitation frequency of vibration of machine to the natural frequency of the isolation system is equal to 0.5, the transmissibility ratio of isolation is
GATE ME 2004
22
In the figure shown, the spring deflects by $$\delta $$ to position A ( the equilibrium position) when a mass $$m$$ is kept on it. During free vibration, the mass is at position $$B$$ at some instant. The change in potential energy of the spring mass system from position $$A$$ to position $$B$$ is
GATE ME 2001
Marks 2
1
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 $x_A = a \sin \omega t$. 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).
GATE ME 2024
2
A spring mass damper system (mass m, stiffness k, and damping coefficient c) excited by a force F(t) = B sin ωt, where B, ω and t are the amplitude, frequency and time, respectively, is shown in the figure. Four different responses of the system (marked as (i) to (iv)) are shown just to the right of the system figure. In the figures of the responses, A is the amplitude of response shown in red color and the dashed lines indicate its envelope. The responses represent only the qualitative trend and those are not drawn to any specific scale.
Four different parameter and forcing conditions are mentioned below.
(P) c > 0 and $ω=\sqrt{k/m}$
(Q) c < 0 and ω ≠ 0
(R) c = 0 and $\omega=\sqrt{k/m}$
(S) c = 0 and $\omega \cong\sqrt{k/m}$
Which one of the following options gives correct match (indicated by arrow →) of the parameter and forcing conditions to the responses?
GATE ME 2022 Set 2
3
Consider a forced single degree-of-freedom system governed by $\rm \ddot x(t) + 2 ζ ω_n \dot x (t) + ω_n^2 x(t) = ω_n^2 \cos (ω t)$, where ζ and ωn are the damping ratio and undamped natural frequency of the system, respectively, while ω 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/ωn. The peak amplitude of this response occurs at a frequency ω = ωp. If ωd denotes the damped natural frequency of this system, which one of the following options is true?
GATE ME 2022 Set 1
4
A thin uniform rigid bar of length $$L$$ and mass $$M$$ is hinged at point $$O,$$ located at a distance of $${L \over 3}$$ 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 $$m = {m \over 4}$$ 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 systems is
GATE ME 2017 Set 1
5
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 _________.
GATE ME 2017 Set 2
6
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.
GATE ME 2016 Set 2
7
A single degree of freedom spring-mass system is subjected to a harmonic force of constant amplitude. For an excitation frequency of $$\sqrt {{{3k} \over m}} ,$$ the ratio of the amplitude of steady state response to the static deflection of the spring is __________
GATE ME 2016 Set 3
8
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 $$J = {{M{a^2}} \over 2}$$
The natural frequency of this system in rad/s is given by
GATE ME 2016 Set 1
9
A single-degree-freedom spring mass system 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 the undamped natural frequency is $$10$$$$\omega $$. At steady state, the amplitude of vibration (in m) is approximately
GATE ME 2015 Set 2
10
Considering massless rigid rod and small oscillations, the natural frequency (in rad/s) of vibration of the system shown in the figure is
GATE ME 2015 Set 1
11
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 = $$9.81$$ $$m/{s^2}.$$ Uniform speed of the motor in $$RPM$$ for which the mobile will get just lifted off the ground at the end $$Q$$ is approximately
GATE ME 2015 Set 1
12
A precision instrument package (m = 1kg) 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 60Hz, so that the effect of damping may be ignored. The stiffness (in N/m) of the required mounting pad is _____________.
GATE ME 2015 Set 1
13
Figure shows a single degree of freedom system. The system consists of a mass less rigid bar $$OP$$ hinged at $$O$$ and a mass $$m$$ at end $$P.$$ The natural frequency of vibration of the system is
GATE ME 2015 Set 3
14
A single degree of freedom system has a mass of 2kg, stiffness 8 N/m and viscous damping ratio 0.02. The dynamic magnification factor at an excitation frequency of 1.5 rad/s is ___________
GATE ME 2014 Set 4
15
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 $${k_1}$$ and $${k_2}$$ as shown below.
GATE ME 2014 Set 2
16
Consider a cantilever beam, having negligible mass and uniform flexural rigidity, with length $$0.01m$$. The frequency of vibration of the beam, with a $$0.5$$ kg mass attached at the free tip, is $$100Hz$$. The flexural rigidity (in $$N.{m^2}$$) of the beam is _________.
GATE ME 2014 Set 1
17
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
GATE ME 2014 Set 1
18
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 _____________
GATE ME 2014 Set 3
19
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 40mm. The damping coefficient (c, in N.s/m) of the dashpot is ______________
GATE ME 2014 Set 3
20
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
GATE ME 2013
21
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 un damped natural frequency of the system is
GATE ME 2012
22
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
GATE ME 2011
23
A mass of $$1$$ kg is attached to two identical springs each with stiffness $$k=20kN/m$$ as shown in the figure. Under frictionless condition, the natural frequency of the system in $$Hz$$ is close to
GATE ME 2011
24
A mass $$m$$ attached to a spring is subjected to a harmonic force as shown in the figure. The amplitude of the forced motion is observed to be $$50$$ mm. The value of m (in kg) is
GATE ME 2010
25
An automotive engine weighing $$240$$ kg is supported on four springs with linear characteristics. Each of the front two springs have a stiffness of $$16$$ MN/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
GATE ME 2009
26
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 ($$\xi $$) and damped natural frequency ($${f_n}$$) respectively are
GATE ME 2009
27
A uniform rigid rod of mass $$m =1kg$$ and length $$L=1$$ m is hinged at its centre & laterally supported at one end by a spring of spring constant $$k=300N/m.$$ The natural frequency $${\omega _n}$$ in rad/s is
GATE ME 2008
28
The natural frequency of the spring mass system shown in the figure is closest to
GATE ME 2008
29
The equation of motion of a harmonic oscillator is given by $${{{d^2}x} \over {d{t^2}}} + 2\xi {\omega _n}{{dx} \over {dt}} + \omega _n^2\,x = 0\,\,\,$$ and the initial conditions at $$t=0$$ are $$\,x\left( 0 \right) = X,{{dx} \over {dt}}\left( 0 \right) = 0.\,\,$$
The amplitude of $$x(t)$$ after $$n$$ complete cycles is
The amplitude of $$x(t)$$ after $$n$$ complete cycles is
GATE ME 2007
30
The natural frequency of the system shown below is
GATE ME 2007
31
A vibratory system consists of a mass $$12.5$$ kg, a spring of stiffness $$1000$$ $$N/m,$$ and a dashpot with damping coefficient of $$15$$ $$Ns/m.$$
The value of critical damping of the system is:
GATE ME 2006
32
A vibratory system consists of a mass $$12.5$$ kg, a spring of stiffness $$1000$$ $$N/m,$$ and a dashpot with damping coefficient of $$15$$ $$Ns/m.$$
The value of logarithmic decrement is
GATE ME 2006
33
A machine of $$250$$ kg mass is supported on springs of total stiffness $$100$$ kN/m. Machine has an unbalanced rotating force of $$350$$ N at speed of $$3600$$ rpm. Assuming a damping factor of $$0.15,$$ the value of transmissibility ratio is
GATE ME 2006
34
In a spring-mass system, the mass is 0.1 kg and the stiffness of the spring is 1 kN/m. By introducing a damper, the frequency of oscillation is found to be 90% of the original value. What is the damping coefficient of the damper?
GATE ME 2005
35
A mass M, of 20 kg is attached to the free end of a steel cantilever beam of length 1000 mm having a cross-section of 25 $$\, \times \,$$ 25 mm. Assume the mass of the cantilever to be negligible and Esteel = 200 Gpa. If the lateral vibration of this system is critically damped using a viscous damper, the damping constant of the damper is
GATE ME 2004
36
A uniform stiff rod of length $$300$$ $$mm$$ and having a weight of $$300$$ $$N$$ is pivoted at one end and connected to a spring at the other end. For keeping the rod vertical in a stable position the minimum value of spring constant $$K$$ needed is
GATE ME 2004
37
A uniform rigid slender bar of mass $$10$$ $$kg.$$ is hinged at the left end is suspended with the help of spring and damper arrangement as shown in the figure where $$K = 2kN/m,$$ $$C =500Ns/m$$ and the stiffness of the torsional spring $${K_\theta }$$ is 1 kN/m/rad. Ignore the hinge dimensions.
The damping coefficient in the vibration equation is given by
GATE ME 2003
38
A uniform rigid slender bar of mass $$10$$ $$kg.$$ is hinged at the left end is suspended with the help of spring and damper arrangement as shown in the figure where $$K = 2kN/m,$$ $$C =500Ns/m$$ and the stiffness of the torsional spring $${K_\theta }$$ is 1 kN/m/rad. Ignore the hinge dimensions.
The un-damped natural frequency of oscillations of the bar about the hinge point is
GATE ME 2003
39
A flexible rotor-shaft system comprises of a $$10kg$$ rotor disc placed in the middle of a mass-less shaft of diameter $$30$$ $$mm$$ and length $$500mm$$ between bearings (shaft is being taken mass-less as the equivalent mass of the shaft is included in the rotor mass) mounted at the ends. The bearings are assumed to simulate simply supported boundary conditions. The shaft is made of steel for which the value of $$E$$ is $$\,2.1\, \times \,{10^{11}}\,$$. Pa. What is the critical speed of rotation of the shaft?
GATE ME 2003
40
The assembly shown in the figure is composed of two massless rods of length $$l$$ with two particles, each of mass $$m$$. The natural frequency of this assembly for small oscillations is
GATE ME 2001
41
As shown in Figure, a mass of 100 kg is held between two springs. The natural frequency of vibration of the system, in cycles/s, is
GATE ME 2000
42
Consider the system of two wagons shown in Figure. The natural frequencies of this system are
GATE ME 1999
43
A mass of $$1$$ kg is suspended by means of $$3$$ springs as shown in Figure. The spring constants $$\,{K_1},\,{K_2}\,\,$$ and $$\,{K_3},$$ are respectively $$1$$ $$kN/m,$$ $$3$$ $$kN/m$$ and $$2$$ $$kN/m.$$ The natural frequency of the system is approximately )
GATE ME 1996
Marks 5
1
A spring mass $$-$$ dashpot system is shown in the figure. The spring stiffness is $$K,$$ mas is $$m$$, and the viscous damping coefficient is $$c$$. The system is subjected to a force $$\,{F_0}\cos \omega t\,\,$$ as shown. Write the equations of motion which are needed to determine $$x.$$ (No need to determined $$x.$$)
GATE ME 2001
2
The suspension system of a two-wheeler can be equated to a single spring-mass system with a viscous damper connected in series. Sketch the free body diagram and give the equations of motion. For a mass $$m = 50$$ $$kg$$ and a spring with a stiffness of $$35$$ $$kN/m,$$ determine what should be the damping coefficient (damping constant) for critical damping. What can be the damping force for a plunger velocity of $$0.05$$ $$m/s$$?
GATE ME 1997
3
A cylinder of mass 1 kg and radius 1 m is connected by two identical springs at a height of $$0.5m$$ above the center as shown in the Fig. The cylinder rolls without slipping. If the spring constant is $$30$$ $$kN/m$$ for each spring, find the natural frequency of the system for small oscillations.
GATE ME 1996