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GATE ME Theory of Machines & Vibrations Previous Year Questions

60 solved GATE ME questions on Theory of Machines & Vibrations, drawn from 8 exam years and grouped by year. Every question shows the official answer and a step-by-step solution.

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Theory of Machines & Vibrations: trace the motion from input to output

Identify kinematic pairs, count degrees of freedom, and track velocity through linkages and gear trains before analyzing forces or vibrations. These selected foundations cover mechanisms, gear trains, and single-DOF vibrations. They are not a complete syllabus; balancing, governors, and multi-DOF systems need separate study.

Our study notes and original examples support the PYQs below; they are not official exam questions or a replacement for the current syllabus.

Before you start

  • Trigonometry, angular velocity (rad/s), and the relationship v = rω for circular motion.
  • Second-order ordinary differential equations with constant coefficients.

Concepts to revise before solving

Degrees of freedom and Gruebler's equation

For a planar mechanism, DOF = 3(n - 1) - 2j₁ - j₂, where n is the number of links (including the fixed frame), j₁ is the number of lower pairs (revolute, prismatic), and j₂ is the number of higher pairs. A four-bar linkage has n = 4, j₁ = 4, j₂ = 0, giving DOF = 3(3) - 2(4) = 1.

Check yourself: Have you counted the fixed link and identified all kinematic pairs correctly?

Velocity analysis using instant centers

The instant center of two links is the point at which they have the same velocity. For a four-bar linkage, Kennedy's theorem locates instant centers at the intersection of lines connecting known pairs. The velocity of any point on a link equals ω x (distance to the fixed pivot's instant center).

Check yourself: Are you measuring the distance from the correct instant center?

Gear trains and speed ratios

For a simple gear train, the speed ratio equals the product of driving teeth divided by driven teeth. An odd number of external meshes reverses direction. In an epicyclic (planetary) gear train, use the tabular method: fix the arm, apply relative rotations, then superpose arm rotation to match boundary conditions.

Check yourself: Have you correctly fixed the arm and applied tooth ratios in the relative-motion step?

Free vibration of a single-DOF system

A mass-spring system has natural frequency ωₙ = √(k/m) rad/s. The general undamped solution is x(t) = A sin(ωₙt + φ). With viscous damping c, the damping ratio ζ = c/(2√(km)). Underdamped (ζ < 1) response oscillates with exponentially decaying amplitude at damped frequency ωd = ωₙ√(1 - ζ²).

Check yourself: Is ζ less than 1, confirming underdamped oscillatory motion?

Forced vibration and resonance

Harmonic forcing F₀ sin(ωt) on an undamped system gives steady-state amplitude X = F₀/k / |1 - (ω/ωₙ)²|. Resonance occurs when ω = ωₙ, causing unbounded amplitude in the undamped model. With damping, peak amplitude is finite and occurs near ω = ωₙ. Transmissibility measures force transmitted to the foundation.

Check yourself: Are you using the forcing frequency ω or the natural frequency ωₙ in the amplitude formula?

Mistakes to avoid

Forgetting to count the fixed frame as a link in Gruebler's equation.
The ground/frame is link 1; total links n includes it. Subtract 1 for the fixed link inside the formula.
Using the wrong tooth count for the driving or driven gear in an epicyclic train.
In the tabular method, the gear meshing with the sun relative to the arm must use the correct tooth ratio. Label each gear and its mating pair.
Applying the undamped resonance formula to a damped system.
With damping, steady-state amplitude is F₀/k / √[(1 - r²)² + (2ζr)²], where r = ω/ωₙ. Peak does not occur exactly at r = 1 unless ζ = 0.

Original teaching example · not a PYQ

Work through the reasoning

Original example: a mass-spring system has m = 4 kg and k = 900 N/m. No damping is present. Find the natural frequency in rad/s and in Hz. If the mass is displaced 0.02 m from equilibrium and released from rest, find the vibration amplitude.

  1. Natural frequency: ωₙ = √(k/m) = √(900/4) = √225 = 15 rad/s.
  2. In Hz: fₙ = ωₙ/(2π) = 15/(2π) ≈ 2.39 Hz.
  3. Initial conditions: x(0) = 0.02 m, v(0) = 0. Solution form: x(t) = A sin(ωₙt + φ).
  4. At t = 0: x(0) = A sin φ = 0.02 and v(0) = Aωₙ cos φ = 0, so cos φ = 0, giving φ = π/2. Then A sin(π/2) = A = 0.02 m.

ωₙ = 15 rad/s (≈ 2.39 Hz). Amplitude A = 0.02 m.

Try it before reading the answer

An epicyclic gear train has a sun gear S with 20 teeth and a ring gear R with 80 teeth. The arm A carries a planet gear P. If the sun is fixed and the arm rotates at 100 rpm clockwise, find the ring gear speed.

Show answer and reasoning

Ring gear speed = 125 rpm clockwise.

Use the standard relation: (N_R - N_A)/(N_S - N_A) = -T_S/T_R = -20/80 = -1/4. Sun is fixed: N_S = 0. Arm speed N_A = 100. So (N_R - 100)/(0 - 100) = -1/4. N_R - 100 = (-1/4)(-100) = 25. N_R = 125 rpm clockwise (same direction as the arm).

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Supplemental reading, not an official GATE reading list or an endorsement of these notes.

Apply this to the previous-year questions

Previous-year questions by year

This page shows 60 recent questions from the released archive, newest first. For older questions and complete papers, browse all GATE ME papers. Questions can carry more than one subject tag; counts are not marks weightage.

GATE ME 20265 questions

  1. Set 1 Q17A gear train with five gears is shown in the figure below. The number of teeth on each gear is N1,N2,N3,N4N_1, N_2, N_3, N_4, and N5N_5. The idler gear in this gear…MCQ · +1 marks · Easy
  2. Set 1 Q18In a single degree of freedom vibrating system with only viscous damping, the critical damping coefficient is 350 N s/m and the damping coefficient is 35 N…MCQ · +1 marks · Easy
  3. Set 1 Q20The rotor of an aeroplane engine has a mass moment of inertia 1.0 kg m21.0 \text{ kg m}^2. The engine rotates at a speed of 500 RPM500 \text{ RPM} in the clockwise…MCQ · +1 marks · Medium
  4. Set 1 Q22Worm gearsets are used for transmitting rotary motion betweenMCQ · +1 marks · Easy
  5. Set 1 Q42A 5 kg mass is suspended at the free end of an overhanging massless beam, having a pin support and a roller support, as shown in the figure below. Young’s…MCQ · +2 marks · Medium

GATE ME 20255 questions

  1. Set 1 Q29In the context of balancing of rotating and reciprocating masses, which one of the following options is true?MCQ · +1 marks · Easy
  2. Set 1 Q35A 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…NAT · +1 marks · Medium
  3. Set 1 Q42The system shown in the figure below consists of a cantilever beam (with flexural rigidity EIEI and negligible mass), a spring (with spring constant KK and…MCQ · +2 marks · Medium
  4. Set 1 Q59A pair of spur gears is required to maintain a velocity ratio of 1:2. The module of the gears is 10 mm and the addendum is 10 mm. If the operating pressure…NAT · +2 marks · Hard
  5. Set 1 Q60An offset slider-crank mechanism is shown in the figure below. The length of the stroke of the slider is ______ mm (rounded off to nearest integer). [figure]NAT · +2 marks · Medium

GATE ME 20245 questions

  1. Set 1 Q23The change in kinetic energy ΔE\Delta E of an engine is 300 J300\text{ J}, and minimum and maximum shaft speeds are ωmin=220 rad/s\omega_{min} = 220\text{ rad/s} and…MCQ · +1 marks · Medium
  2. Set 1 Q25A 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…MCQ · +1 marks · Medium
  3. Set 1 Q52The Levai type-A train illustrated in the figure has gears with module m=8m = 8 mm/tooth. Gears 2 and 3 have 19 and 24 teeth respectively. Gear 2 is fixed and…NAT · +2 marks · Medium
  4. Set 1 Q54At the instant when OP is vertical and AP is horizontal, the link OD is rotating counter clockwise at a constant rate ω=7\omega = 7 rad/s. Pin P on link OD…NAT · +2 marks · Hard
  5. Set 1 Q55A vibratory system consists of mass mm, a vertical spring of stiffness 2k2k and a horizontal spring of stiffness kk. The end A of the horizontal spring is…NAT · +2 marks · Hard

GATE ME 20233 questions

  1. Set 1 Q29Two meshing spur gears 1 and 2 with diametral pitch of 8 teeth per mm and an angular velocity ratio ∣ω2∣/∣ω1∣=1/4|\omega_2|/|\omega_1| = 1/4, have their centers 30 mm…NAT · +1 marks · Medium
  2. Set 1 Q30The figure shows a block of mass m=20 kgm = 20\text{ kg} attached to a pair of identical linear springs, each having a spring constant k=1000 N/mk = 1000\text{ N/m}. The…NAT · +1 marks · Easy
  3. Set 1 Q38The figure shows a wheel rolling without slipping on a horizontal plane with angular velocity ω1\omega_1. A rigid bar PQ is pinned to the wheel at P while the…MCQ · +2 marks · Hard

GATE ME 202210 questions

  1. Set 1 Q17The figure shows a schematic of a simple Watt governor mechanism with the spindle O1O2O_1O_2 rotating at an angular velocity ω\omega about a vertical axis. The…MCQ · +1 marks · Medium
  2. Set 1 Q32A 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…NAT · +1 marks · Hard
  3. Set 1 Q38A planar four-bar linkage mechanism with 3 revolute kinematic pairs and 1 prismatic kinematic pair is shown in the figure, where AB⊥CEAB \perp CE and…MCQ · +2 marks · Hard
  4. Set 1 Q39Consider a forced single degree-of-freedom system governed by x¨(t)+2ζωnx˙(t)+ωn2x(t)=ωn2cos⁡(ωt)\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2 x(t) = \omega_n^2 \cos(\omega t), where…MCQ · +2 marks · Medium
  5. Set 1 Q53A schematic of an epicyclic gear train is shown in the figure. The sun (gear 1) and planet (gear 2) are external, and the ring gear (gear 3) is internal. Gear…NAT · +2 marks · Medium
  6. Set 2 Q17A massive uniform rigid circular disc is mounted on a frictionless bearing at the end E of a massive uniform rigid shaft AE which is suspended horizontally in…MCQ · +1 marks · Hard
  7. Set 2 Q31For a dynamical system governed by the equation, x¨(t)+2ζωnx˙(t)+ωn2x(t)=0,\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t) + \omega_n^2x(t) = 0, the damping ratio ζ\zeta is equal to…NAT · +1 marks · Medium
  8. Set 2 Q39A rigid body in the XX-YY plane consists of two point masses (11 kg each) attached to the ends of two massless rods, each of 11 cm length, as shown in the…MCQ · +2 marks · Medium
  9. Set 2 Q40A spring mass damper system (mass mm, stiffness kk, and damping coefficient cc) excited by a force F(t)=Bsin⁡ωtF(t) = B \sin \omega t, where BB, ω\omega and tt are…MCQ · +2 marks · Hard
  10. Set 2 Q52In the configuration of the planar four-bar mechanism at a certain instant as shown in the figure, the angular velocity of the 2 cm long link is ω2=5\omega_2 = 5…NAT · +2 marks · Medium

GATE ME 202111 questions

  1. Set 1 Q26Consider a reciprocating engine with crank radius RR and connecting rod of length LL. The secondary unbalance force for this case is equivalent to primary…MCQ · +1 marks · Easy
  2. Set 1 Q34The figure shows an arrangement of a heavy propeller shaft in a ship. The combined polar mass moment of inertia of the propeller and the shaft is 100100…NAT · +1 marks · Medium
  3. Set 1 Q35Consider a single degree of freedom system comprising a mass MM, supported on a spring and a dashpot as shown in the figure. [figure] If the amplitude of the…NAT · +1 marks · Medium
  4. Set 1 Q42A tappet valve mechanism in an IC engine comprises a rocker arm ABC that is hinged at B as shown in the figure. The rocker is assumed rigid and it oscillates…MCQ · +2 marks · Hard
  5. Set 1 Q62The Whitworth quick return mechanism is shown in the figure with link lengths as follows: OP=300 mmOP = 300\text{ mm}, OA=150 mmOA = 150\text{ mm}, AR=160 mmAR = 160\text{ mm},…NAT · +2 marks · Medium
  6. Set 2 Q25Consider the mechanism shown in the figure. There is rolling contact without slip between the disc and ground. [figure] Select the correct statement about…MCQ · +1 marks · Medium
  7. Set 2 Q26The controlling force curves P, Q and R for a spring controlled governor are shown in the figure, where r1r_1 and r2r_2 are any two radii of rotation. [figure]…MCQ · +1 marks · Medium
  8. Set 2 Q42A power transmission mechanism consists of a belt drive and a gear train as shown in the figure. [figure] Diameters of pulleys of belt drive and number of…MCQ · +2 marks · Medium
  9. Set 2 Q43A machine of mass 100 kg is subjected to an external harmonic force with a frequency of 40 rad/s. The designer decides to mount the machine on an isolator to…MCQ · +2 marks · Hard
  10. Set 2 Q44Consider the system shown in the figure. A rope goes over a pulley. A mass, mm, is hanging from the rope. A spring of stiffness, kk, is attached at one end…MCQ · +2 marks · Medium
  11. Set 2 Q64The torque provided by an engine is given by T(θ)=12000+2500sin⁡(2θ) N.mT(\theta) = 12000 + 2500 \sin(2\theta)\text{ N.m}, where θ\theta is the angle turned by the crank from inner…NAT · +2 marks · Medium

GATE ME 202011 questions

  1. Set 1 Q16A single-degree-of-freedom oscillator is subjected to harmonic excitation F(t)=F0cos⁡(ωt)F(t) = F_0 \cos(\omega t) as shown in the figure. [figure] The non-zero value of…MCQ · +1 marks · Medium
  2. Set 1 Q18A four bar mechanism is shown below. [figure] For the mechanism to be a crank-rocker mechanism, the length of the link PQPQ can beMCQ · +1 marks · Medium
  3. Set 1 Q30A flywheel is attached to an engine to keep its rotational speed between 100 rad/s100\text{ rad/s} and 110 rad/s110\text{ rad/s}. If the energy fluctuation in the flywheel…NAT · +1 marks · Medium
  4. Set 1 Q31A balanced rigid disc mounted on a rigid rotor has four identical point masses, each of 10 grams10\text{ grams}, attached to four points on the 100 mm100\text{ mm}…NAT · +1 marks · Medium
  5. Set 1 Q41A rigid mass-less rod of length LL is connected to a disc (pulley) of mass mm and radius r=L/4r = L/4 through a friction-less revolute joint. The other end of…MCQ · +2 marks · Hard
  6. Set 1 Q48A cam with a translating flat-face follower is desired to have the follower motion y(θ)=4[2πθ−θ2],0≤θ≤2π.y(\theta) = 4 [ 2\pi\theta - \theta^2 ], \quad 0 \leq \theta \leq 2\pi.…NAT · +2 marks · Hard
  7. Set 2 Q17The equation of motion of a spring-mass-damper system is given by d2xdt2+3dxdt+9x=10sin⁡(5t)\frac{d^2x}{dt^2} + 3 \frac{dx}{dt} + 9x = 10 \sin(5t) The damping factor for the system…MCQ · +1 marks · Easy
  8. Set 2 Q18The number of qualitatively distinct kinematic inversions possible for a Grashof chain with four revolute pairs isMCQ · +1 marks · Medium
  9. Set 2 Q39The sun (S) and the planet (P) of an epicyclic gear train shown in the figure have identical number of teeth. If the sun (S) and the outer ring (R) gears are…MCQ · +2 marks · Medium
  10. Set 2 Q47A hollow spherical ball of radius 20 cm20\text{ cm} floats in still water, with half of its volume submerged. Taking the density of water as 1000 kg/m31000\text{ kg/m}^3,…NAT · +2 marks · Hard
  11. Set 2 Q49The turning moment diagram of a flywheel fitted to a fictitious engine is shown in the figure. [figure] The mean turning moment is 2000 Nm2000\text{ Nm}. The…NAT · +2 marks · Medium

GATE ME 201910 questions

  1. Set 1 Q15A flat-faced follower is driven using a circular eccentric cam rotating at a constant angular velocity ω\omega. At time t=0t = 0, the vertical position of the…MCQ · +1 marks · Medium
  2. Set 1 Q16The natural frequencies corresponding to the spring-mass systems I and II are ωI\omega_I and ωII\omega_{II}, respectively. The ratio…MCQ · +1 marks · Easy
  3. Set 1 Q40In a four bar planar mechanism shown in the figure, AB=5 cmAB = 5\text{ cm}, AD=4 cmAD = 4\text{ cm} and DC=2 cmDC = 2\text{ cm}. In the configuration shown, both ABAB and…MCQ · +2 marks · Medium
  4. Set 1 Q41The rotor of a turbojet engine of an aircraft has a mass 180 kg and polar moment of inertia 10 kg⋅m210 \text{ kg}\cdot\text{m}^2 about the rotor axis. The rotor…MCQ · +2 marks · Medium
  5. Set 1 Q53A uniform thin disk of mass 1 kg1\text{ kg} and radius 0.1 m0.1\text{ m} is kept on a surface as shown in the figure. A spring of stiffness k1=400 N/mk_1 = 400\text{ N/m}…NAT · +2 marks · Hard
  6. Set 2 Q17A spur gear has pitch circle diameter DD and number of teeth TT. The circular pitch of the gear isMCQ · +1 marks · Easy
  7. Set 2 Q42A slender uniform rigid bar of mass mm is hinged at O and supported by two springs, with stiffnesses 3k3k and kk, and a damper with damping coefficient cc,…MCQ · +2 marks · Hard
  8. Set 2 Q46The crank of a slider-crank mechanism rotates counter-clockwise (CCW) with a constant angular velocity ω\omega, as shown. Assume the length of the crank to be…MCQ · +2 marks · Hard
  9. Set 2 Q48Two masses AA and BB having mass mam_a and mbm_b, respectively, lying in the plane of the figure shown, are rigidly attached to a shaft which revolves about…NAT · +2 marks · Medium
  10. Set 2 Q49A four bar mechanism is shown in the figure. The link numbers are mentioned near the links. Input link 2 is rotating anticlockwise with a constant angular…NAT · +2 marks · Hard

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