Kinematics and Dynamics is a core Part B2 (Mechanical) subject in the GATE Robotics and Automation (RA) 2027 paper, covering how machines move and the forces behind that motion. The high-frequency question areas are gear trains, free and damped vibrations, gyroscopic effects and velocity analysis of mechanisms — each governed by a compact formula. This guide reviews them and works through five GATE-style problems in full.
TABLE OF CONTENTS
Mechanisms & velocity analysis
A mechanism transmits and transforms motion through links and joints. The velocity of a point on a rotating link is v = ωr, directed perpendicular to the link. Angular velocity relates to rotational speed by ω = 2πN/60 (rad/s from rpm). Velocity and acceleration diagrams (or the instantaneous-centre method) resolve the motion of any point in a linkage.
Gears & gear trains
For a meshing gear pair the speed ratio is inversely proportional to the tooth numbers: N1/N2 = T2/T1. A simple train passes the ratio through idlers unchanged in magnitude; a compound train multiplies the ratios of its stages. The train value is the product of driven-to-driver tooth ratios.
Free & damped vibrations
A single-degree-of-freedom spring–mass system vibrates at its natural frequency ωn = √(k/m). Adding damping lowers this to the damped natural frequency ωd = ωn√(1−ζ²), where ζ is the damping ratio. Resonance — large amplitude when the forcing frequency approaches ωn — is the phenomenon most machine design tries to avoid.
Figure 1. Single-DOF spring–mass system oscillating at its natural frequency ωn = √(k/m).
Gyroscopic couple
A spinning rotor that is made to precess resists with a gyroscopic couple C = Iωωp, where I is the moment of inertia, ω the spin velocity and ωp the precession velocity. This effect steers ships and aircraft and stabilises two-wheelers, and its reaction couple is a common exam calculation.
Balancing
Rotating and reciprocating masses create shaking forces that must be balanced to avoid vibration and bearing wear. Static balance requires the net centrifugal force to vanish (∑mr = 0); dynamic balance additionally requires the net couple to vanish (∑mrl = 0). Questions ask for the magnitude and angular position of a balancing mass.
Worked examples (GATE-style)
Example 1 — simple gear pair
A 20-tooth pinion running at 1500 rpm drives a 60-tooth gear. Find the gear speed and the speed ratio.
Solution. N2 = N1 × T1/T2 = 1500 × 20/60 = 500 rpm.
Speed ratio = 60/20 = 3:1 reduction.
Example 2 — natural frequency
A mass of 10 kg is suspended on a spring of stiffness 1000 N/m. Find the natural frequency in rad/s and in Hz.
Solution. ωn = √(k/m) = √(1000/10) = √100 = 10 rad/s.
fn = ωn/2π = 10/6.283 = 1.59 Hz.
Example 3 — gyroscopic couple
A rotor of moment of inertia 2 kg·m² spins at 100 rad/s and precesses at 0.5 rad/s. Find the gyroscopic couple.
Solution. C = Iωωp = 2 × 100 × 0.5 = 100 N·m.
Example 4 — velocity of a crank point
A crank of radius 0.1 m rotates at 300 rpm. Find the linear velocity of the crank pin.
Solution. ω = 2πN/60 = 2π(300)/60 = 31.42 rad/s.
v = ωr = 31.42 × 0.1 = 3.14 m/s.
Example 5 — damped natural frequency
A vibrating system has ωn = 10 rad/s and a damping ratio ζ = 0.1. Find the damped natural frequency.
Solution. ωd = ωn√(1−ζ²) = 10√(1−0.01) = 10√0.99.
ωd = 9.95 rad/s — only slightly below ωn for light damping.
Key formulas
FORMULA SHEET
Velocity: v = ωr, ω = 2πN/60
Gear pair: N1/N2 = T2/T1
Natural freq: ωn = √(k/m), fn = ωn/2π
Damped: ωd = ωn√(1−ζ²)
Gyroscopic couple: C = Iωωp
Balancing: ∑mr = 0, ∑mrl = 0
Common mistakes to avoid
- Forgetting the rpm-to-rad/s conversion: ω = 2πN/60, not simply N.
- Inverting the gear ratio: speed is inversely proportional to teeth, so the larger gear turns slower.
- Confusing ωn and ωd — damping always lowers the frequency, by the factor √(1−ζ²).
- Dropping a term in the gyroscopic couple — it is the product of all three quantities I, ω and ωp.
- Treating static balance as sufficient — dynamic balance also needs the couple ∑mrl to vanish.
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Get in-depth lectures on mechanisms, gears, vibrations, gyroscopes and balancing, with PYQs and full-length tests, taught by Piyush Wairale (IIT Madras).
Frequently asked questions
What is the natural frequency of a system?
The natural frequency is the rate at which a system oscillates when disturbed and left free, ωn = √(k/m) for a spring–mass system. If a forcing input matches it, resonance occurs and amplitudes grow dangerously large.
How does damping change the natural frequency?
Damping lowers the oscillation frequency to ωd = ωn√(1−ζ²). For light damping (ζ < 0.1) the change is tiny, but as ζ approaches 1 the system stops oscillating altogether (critical damping).
Why do rotating machines need balancing?
Unbalanced rotating masses generate centrifugal forces that shake the machine and wear its bearings. Static balancing cancels the net force, and dynamic balancing additionally cancels the net couple, giving smooth, vibration-free running.
This solved-problems guide is part of the complete GATE RA 2027 Syllabus overview. For the full topic breakdown, see the Kinematics and Dynamics syllabus guide.
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