Theory of Machines (TOM) — GATE Formula Sheet & Theory Cards

01

Kinematics of Mechanisms

HIGH PRIORITY

🔗 Links, Pairs & Degrees of Freedom GATE FAVOURITE ▼

Mechanism = links connected by pairs. Kinematics studies motion WITHOUT considering forces. DOF (Degrees of Freedom) batata hai ki mechanism kitne independent inputs se move karta hai. GATE mein Gruebler's equation seedha aata hai.

Gruebler's EquationKutzbachDOFInversion

Gruebler's Equation (Planar)

F = 3(n−1) − 2j₁ − j₂

n = no. of links, j₁ = full joints (1 DOF), j₂ = half joints (2 DOF). Ground link bhi count hota hai.

Kutzbach Criterion

M = 3(l−1) − 2p − h

l = links, p = lower pairs (full), h = higher pairs. Same as Gruebler.

4-Bar Mechanism DOF

F = 3(4−1) − 2(4) = 1

4 links, 4 revolute pairs → F=1. 1 input needed for fully constrained motion.

Grashof's Condition

s + l ≤ p + q

s = shortest, l = longest link. If satisfied → at least one link can rotate fully (crank exists).

Lower Pair

Surface contact → 1 DOF

Revolute, Prismatic, Screw, Cylindrical, Spherical, Planar — 6 types. Area contact.

Higher Pair

Line/Point contact → 2 DOF

Cam-follower, gear teeth — higher pair. More complex motion possible.

🧠

Inversions yaad karo: 4-bar ke 4 inversions hain — crank-rocker, double-crank (drag link), double-rocker, crank-crank. Slider-crank ke 4 inversions → Pendulum pump, Oscillating cylinder, Rotary engine, etc.

💡

GATE Trick: F = 0 → structure (rigid), F = 1 → mechanism, F = 2 → bi-stable (needs 2 inputs), F < 0 → over-constrained/redundant. Sab cases GATE mein aate hain.

02

Velocity & Acceleration Analysis

HIGH PRIORITY

🏃 Instantaneous Centre & Relative Velocity HIGH MARKS ▼

Velocity analysis mein 2 main methods hain — Instantaneous Centre (IC) method aur Relative Velocity (graphical) method. GATE mein IC method zyada easy hai number-wise. Acceleration analysis mein Coriolis component extra hai sliding pairs ke liye.

No. of Instantaneous Centres

N = n(n−1) / 2

n = no. of links. For 4-bar → N = 4×3/2 = 6 ICs.

Relative Velocity (two points on same link)

V_AB = ω × r_AB

Perpendicular to link AB. ω = angular velocity of link.

Velocity using IC

V_A / V_B = I_A / I_B

Distances from instantaneous centre I₁₃ of the link. Ratio = ratio of distances.

Centripetal (Radial) Acceleration

aʳ = ω²r = V²/r

Directed towards centre of rotation. Always present during rotation.

Tangential Acceleration

aᵗ = α × r

α = angular acceleration. Perpendicular to link. Only when angular velocity is changing.

Coriolis Acceleration

aᶜ = 2ω × V_r

Only for sliding links (slider on rotating link). Direction: rotate V_r by 90° in direction of ω.

⚠️

Common Mistake: Coriolis acceleration sirf tab hoti hai jab ek link dusre pe slide karta ho AND dusra link rotate kare. Pure rotation ya pure sliding mein Coriolis = 0. GATE isko twist karke poochta hai.

🧠

Kennedy's Theorem: Any 3 ICs of 3 links must be collinear. Yeh rule use karke unknown ICs locate karo. 4-bar mein 6 ICs hain, 4 directly milte hain, 2 Kennedy's theorem se.

03

Cams & Followers

MEDIUM PRIORITY

🎯 Cam Terminology & Follower Motions CAM-FOLLOWER ▼

Cam converts rotary motion → specific follower motion. GATE mein displacement diagrams aur SHM/Uniform velocity ke formulas directly pooche jaate hain. Pressure angle critical hai for smooth operation.

SHM Follower — Displacement

x = (h/2)[1 − cos(πθ/β)]

h = total lift (stroke), β = angle of action, θ = cam angle from start.

SHM — Velocity

v = (πhω/2β) sin(πθ/β)

ω = angular velocity of cam. Max at θ = β/2.

SHM — Acceleration (max)

a_max = π²hω² / 2β²

At start and end of stroke. Finite — no jerk discontinuity unlike uniform velocity.

Uniform Velocity — Acceleration

a = ∞ at start/end

Infinite acceleration → shock/jerk. Not used in high-speed machines. Only theoretical.

Pressure Angle (α)

tan α = (dy/dθ) / (r_b + y)

r_b = base circle radius. Max pressure angle should be ≤ 30° for translating, ≤ 45° for oscillating followers.

Undercutting Condition

r_b ≥ R_min − r_f

R_min = min radius of curvature of pitch curve, r_f = follower radius. Undercutting avoided by increasing base circle radius.

💡

Follower Comparison: Uniform velocity → infinite acceleration (bad). SHM → finite acceleration (better, used for moderate speeds). Cycloidal → zero jerk at ends (best for high speed machines). GATE mein iska order poochha jaata hai.

04

Dynamics — Turning Moment & Inertia

HIGH PRIORITY

⚙️ Turning Moment Diagram & Equivalent Masses CORE CONCEPT ▼

Turning moment diagram dikhata hai crankshaft pe torque ka variation. Energy fluctuation determine karta hai flywheel ka size. Equivalent mass system 2-mass ya 3-mass representation hai complex links ke liye.

Dynamic Force on Piston

F_p = P·A − m_r·a_r

P = gas pressure, A = area, m_r = reciprocating mass, a_r = acceleration of piston.

Piston Acceleration (approx)

a = rω²[cosθ + (r/l)cos2θ]

r = crank radius, l = connecting rod length, n = l/r (obliquity ratio). First term = primary, second = secondary.

Primary Force

F₁ = m_r·r·ω²·cosθ

At crank frequency (1× RPM). Along cylinder axis.

Secondary Force

F₂ = m_r·r·ω²·(r/l)·cos2θ

At twice crank frequency (2× RPM). Smaller, but significant at high speeds.

Dynamically Equivalent 2-Mass System

m₁ + m₂ = m
m₁l₁ = m₂l₂
k² = l₁·l₂

k = radius of gyration of original link. 3 conditions must be satisfied.

Torque on Crankshaft

T = F_p × r·[sinθ + (r/2l)·sin2θ]

F_p = piston force. TMD is plotted for full cycle (0 to 4π for 4-stroke).

💡

Real World: Mere 8 saal ke experience mein — TMD aur flywheel calculation sabse zyada design interviews mein poochhe jaate hain. Press machines, compressors, IC engines — sab mein yeh concept use hota hai.

05

Flywheel

HIGH PRIORITY

🌀 Coefficient of Fluctuation, Energy & Rim Design DIRECT QUESTIONS ▼

Flywheel energy store karke speed fluctuation ko control karta hai. Coefficient of fluctuation of speed (Cs) design parameter hai — lower Cs = more uniform speed = heavier flywheel. GATE mein energy, Cs aur rim design directly aate hain.

Coefficient of Fluctuation of Speed (Cs)

Cs = (ω_max − ω_min) / ω_mean

Typical values: Crushing machine → 1/5, Pumps → 1/10, Lathes → 1/30, DC generators → 1/100.

Energy Fluctuation (ΔE)

ΔE = I·ω²·Cs

ΔE = max − min energy from TMD (area between mean torque line and curve). I = moment of inertia of flywheel.

Moment of Inertia (Solid Disc)

I = mR²/2 = mk²

k = R/√2 for solid disc. For rim: k ≈ R (mean radius, since rim is thin).

Rim Flywheel — Energy

ΔE = m·v²·Cs

v = mean peripheral velocity = ωR. m = mass of rim. Used when rim mass >> hub/arms mass.

Hoop Stress in Rim

σ_h = ρ·v²

ρ = density of material, v = rim velocity. Max safe rim speed for CI ≈ 30 m/s.

Coefficient of Fluctuation of Energy

Ce = ΔE / W_mean

W_mean = mean work per cycle. Different from Cs — energy ratio vs speed ratio.

🧠

Formula chain GATE mein: TMD se ΔE nikalo (area) → ΔE = I·ω²·Cs se I nikalo → I = mk² se mass nikalo. Yeh 3-step chain direct GATE questions mein aata hai.

⚠️

Confusing point: ω = 2πN/60 mein convert karna mat bhulo. GATE mein N (RPM) dete hain, formula mein ω (rad/s) chahiye. ω² = (2πN/60)² — yeh squaring bhoolna common mistake hai.

06

Governors

MEDIUM PRIORITY

🎛️ Watt, Porter, Proell & Hartnell Governors FORMULAS DIRECT ▼

Governor engine speed ko regulate karta hai by controlling fuel supply. Centrifugal governors (Watt, Porter, Proell) pe balls ka centrifugal force balance hota hai. Hartnell = spring-controlled governor. GATE mein height (h) aur speed (N) relationships direct pooche jaate hain.

Watt Governor — Height

h = g/ω² = 895/N²

h in metres, N in RPM. Simple pendulum analogy — no central load. Low speed governor only.

Porter Governor — Speed

ω² = g(m+M)/mh or N² = 895(m+M)/mh

M = central load (sleeve), m = ball mass, h = height. Porter is Watt + central load = more stable.

Proell Governor

ω² = g(m+M)·FM / (m·BM·h)

Balls on extension of lower links. FM/BM = extension ratio (>1). Higher speed for same h than Porter.

Hartnell Governor (Spring)

Fc = m·ω²·r (centrifugal)

Spring force balances centrifugal force via bell-crank lever. F_c2 − F_c1 = s·(x+y) where s = spring stiffness.

Coefficient of Insensitiveness

= (N₂−N₁)/N_mean × 100%

Lower = more sensitive governor. Friction causes insensitiveness.

Isochronous Governor

N₁ = N₂ (for all radii)

Speed constant regardless of radius position. Ideal but unstable — not used practically. Only theoretically possible.

💡

Governor vs Flywheel: Flywheel = controls speed variation in ONE cycle (same load). Governor = controls speed variation due to LOAD CHANGE between cycles. Dono ki function alag hai — GATE mein comparison question aata hai.

🧠

Speed order (same h): Watt → Porter → Proell (increasing speed). Watt mein koi central load nahi, Porter mein sleeve load hai, Proell mein extension bhi hai → sabse high speed.

07

Balancing of Rotating & Reciprocating Masses

HIGH PRIORITY

⚖️ Static, Dynamic Balancing & Multi-Cylinder Engines HIGH MARKS ▼

Unbalanced masses cause vibrations, bearing loads, and noise. GATE mein static & dynamic balancing conditions direct pooche jaate hain with graphical methods. Reciprocating mass balancing is always partial — tradeoff between primary and secondary forces.

Static Balancing Condition

Σ(m·r) = 0

Vector sum of all centrifugal forces = 0. Only for masses in same plane.

Dynamic Balancing Condition

Σ(m·r) = 0 AND Σ(m·r·l) = 0

Both force AND couple polygon must close. Dynamically balanced → statically balanced (not vice versa).

Balancing Mass (single plane)

m_b·r_b = m·r (resultant)

Balancing mass placed 180° to resultant unbalance. r_b = radius of balancing mass.

Partial Balancing of Reciprocating Masses

m_b·r_b = c·m_r·r

c = fraction balanced (typically 2/3). Full primary balance → transverse secondary unbalance. Always a tradeoff.

Unbalanced Primary Force (Residual)

F = (1−c)·m_r·r·ω²·cosθ

Along cylinder axis after partial balancing. Transverse force = c·m_r·r·ω²·sinθ (new!).

Multi-Cylinder — Direct & Reverse Cranks

Primary balance: Σm_r·r·e^(jθ) = 0

For complete primary balance, direct crank polygon must close. Secondary: use 2θ angles.

⚠️

Key Fact: Reciprocating masses ko completely balance nahi kiya ja sakta without introducing other imbalances. Partial balancing deliberately kiya jaata hai — yeh limitation hai, mistake nahi.

🧠

Dynamic ≠ Static: Static balance = single plane, no couple. Dynamic balance = two planes, both force and couple zero. Engine crankshafts always need dynamic balancing.

08

Free Undamped Vibrations

HIGH PRIORITY

🌊 Natural Frequency — Springs & Systems MOST IMPORTANT ▼

Free vibration = system vibrates at its own natural frequency (ωₙ) after initial disturbance, no external force. Yeh resonance avoid karne ke liye samajhna zaroori hai. GATE mein different configurations ke liye ωₙ nikalna sabse common question hai.

Equation of Motion

mẍ + kx = 0

SHM equation. ωₙ = √(k/m). Solution: x = A·cos(ωₙt) + B·sin(ωₙt).

Natural Frequency (linear)

ωₙ = √(k/m) rad/s
fₙ = (1/2π)√(k/m) Hz

k = stiffness (N/m), m = mass (kg). Also: fₙ = (1/2π)√(g/δ_st).

Static Deflection Method

fₙ = (1/2π)√(g/δ_st)

δ_st = mg/k (static deflection). Very useful — no need to find k separately if δ_st known.

Springs in Series

1/k_eq = 1/k₁ + 1/k₂

Same force, different deflections. Weaker than individual springs.

Springs in Parallel

k_eq = k₁ + k₂

Same deflection, forces add up. Stiffer than individual springs.

Simple Pendulum

ωₙ = √(g/L)
T = 2π√(L/g)

L = pendulum length. Mass doesn't matter — same result for any mass (for small angles).

Compound Pendulum

ωₙ = √(m·g·h / I_O)

h = dist from pivot to CG, I_O = MI about pivot = I_G + mh². Equivalent simple pendulum length L_eq = I_O/mh.

Torsional Vibration

ωₙ = √(q/I)

q = GJ/l (torsional stiffness), I = polar MI of disc. Analogous to linear: k→q, m→I.

🧠

Analogy yaad karo: Linear ↔ Torsional: m↔I (MI), k↔q (torsional stiffness), x↔θ (displacement), F↔T (force/torque). Ek samjh gaye toh dono clear.

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GATE Shortcut: δ_st = mg/k diya ho, seedha fₙ = (1/2π)√(g/δ_st) use karo. k alag nikalane ki zaroorat nahi — time bachata hai exam mein.

09

Damped Vibrations

HIGH PRIORITY

🎚️ Damping Ratio, Critical Damping & Logarithmic Decrement NUMERICALS DIRECT ▼

Damping system ki energy dissipate karta hai. Critical damping woh minimum damping hai jahan system oscillate nahi karta. Damping ratio (ζ) ek non-dimensional parameter hai — GATE mein sab kuch ζ ke through express hota hai.

Equation of Motion (Damped)

mẍ + cẋ + kx = 0

c = damping coefficient (N·s/m). Three cases depend on ζ.

Critical Damping Coefficient

c_c = 2√(km) = 2mωₙ

At critical damping, system returns to rest in minimum time without oscillation.

Damping Ratio (ζ)

ζ = c / c_c = c / 2mωₙ

ζ < 1: underdamped (oscillates), ζ = 1: critical, ζ > 1: overdamped (no oscillation).

Damped Natural Frequency

ω_d = ωₙ√(1 − ζ²)

Always ω_d < ωₙ. For ζ = 0 (undamped): ω_d = ωₙ. For ζ = 1: ω_d = 0.

Logarithmic Decrement (δ)

δ = ln(x₁/x₂) = 2πζ/√(1−ζ²)

Ratio of successive amplitudes. Used to experimentally find ζ from vibration data.

Damping Factor from δ

ζ = δ / √(4π² + δ²)

For small ζ (< 0.3): ζ ≈ δ/2π. This approximation often used in GATE.

Type	Condition	Response	Example
Underdamped	ζ < 1	Oscillates, amplitude decays	Car suspension
Critically Damped	ζ = 1	Returns fastest without oscillation	Door dampers
Overdamped	ζ > 1	Slow return, no oscillation	Safety buffers
Undamped	ζ = 0	Oscillates forever	Theoretical only

⚠️

Common error: ω_d aur ωₙ confuse karna. Damped frequency hamesha kam hoti hai undamped se. GATE problem mein dhyan raho — "natural frequency" poochhe toh ωₙ, "damped frequency" poochhe toh ω_d = ωₙ√(1−ζ²).

10

Forced Vibrations & Resonance

HIGH PRIORITY

📡 Magnification Factor, Resonance & Vibration Isolation NUMERICALS COMMON ▼

Forced vibration = external periodic force F₀sin(ωt) pe system ka response. Resonance = ω = ωₙ → amplitude → ∞ (undamped). Magnification factor (MF) batata hai static deflection se kitna zyada amplitude hai. Vibration isolation transmissibility se measure hoti hai.

Magnification Factor (MF / D)

MF = 1 / √[(1−r²)² + (2ζr)²]

r = ω/ωₙ (frequency ratio). At resonance (r=1, ζ=0): MF → ∞.

Frequency Ratio at Peak MF

r_peak = √(1 − 2ζ²)

MF_max = 1 / [2ζ√(1−ζ²)]. For ζ < 1/√2 ≈ 0.707 only.

Phase Angle

φ = tan⁻¹[2ζr / (1−r²)]

At r < 1: φ < 90°. At r = 1: φ = 90° (always, for any ζ). At r > 1: φ > 90°.

Transmissibility (TR)

TR = √[(1+(2ζr)²) / ((1−r²)²+(2ζr)²)]

Force transmitted to foundation / applied force. TR < 1 = isolation. Good isolation: r > √2, low ζ.

Vibration Isolation Condition

r > √2 i.e. ω > √2·ωₙ

TR < 1 only when r > √2. Machine speed must be much higher than natural frequency of isolator.

Rotating Unbalance Amplitude

MX/me = r² / √[(1−r²)²+(2ζr)²]

m = total mass, e = eccentricity, M = unbalance mass. Different from direct force excitation.

💡

Resonance se bachna: Machine speed (ω) ko ωₙ ke paas kabhi mat rakhna. Rule of thumb: ω < 0.5ωₙ (below) ya ω > 1.5ωₙ (above) — dono safe zones hain. ωₙ ke +/- 20% range avoid karo.

🧠

Phase shift yaad karo: r = 1 pe phase = 90° → yeh resonance ki defining characteristic hai. GATE mein "at resonance phase angle kya hai?" — answer hamesha 90° regardless of damping.

11

Gear Trains

MEDIUM PRIORITY

⚙️ Simple, Compound & Epicyclic Gear Trains TABULAR METHOD KEY ▼

Gear trains speed aur torque transmit karte hain. Simple aur compound trains seedhe hain. Epicyclic (planetary) gear train mein ek ya zyada gears ka axis move karta hai — yahi confusing hota hai. GATE mein tabular method use karo — guaranteed correct answer.

Simple Gear Train — Velocity Ratio

VR = N_driven / N_driver = T_driver / T_driven

T = no. of teeth. Speed inversely proportional to teeth count. Idler gears don't change VR — only direction.

Compound Gear Train

VR = (T_A × T_C) / (T_B × T_D)

Product of driving teeth / product of driven teeth. Used for large speed reductions in compact space.

Epicyclic — Train Value (x)

x = (ω_last − ω_arm) / (ω_first − ω_arm)

x = (−1)ⁿ × (product of driver teeth / product of driven teeth). n = no. of external meshes.

Tabular Method — Step 1

Arm fixed, first gear +1 rev → find all

Row 1: arm=0, gear A=+1, find B,C... from gear ratios. Multiply whole row by 'x' as needed.

Tabular Method — Step 2

Add 'y' to entire row (arm free)

Row 3 = Row 2 + y (for all). Now apply given conditions: e.g. arm speed = y, input speed = x+y. Solve for x and y.

Gear Efficiency

η = T_out·ω_out / (T_in·ω_in)

For GATE: ideal assumption η = 100% unless stated. Worm gears can have low η (20–50%).

💡

Tabular Method is FOOLPROOF: Epicyclic ke sabhi GATE questions tabular method se solve honge — koi shortcut formula nahi chahiye. Ek baar practice karo, har problem 2 min mein. Conditions: arm fixed ya free, input gear given, output gear find karo.

🔧 Gear Terminology & Tooth Profile CONCEPTUAL ▼

Involute tooth profile standard hai — constant pressure angle, easy to manufacture, some centre distance error tolerable. Cycloidal is older, theoretically perfect but not practical. GATE mein involute ke advantages aur terminology direct pooche jaate hain.

Module (m)

m = d / T = p / π

d = pitch circle dia, T = teeth, p = circular pitch. Standard: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10.

Minimum Teeth to Avoid Undercutting

T_min = 2 / sin²φ

φ = pressure angle. For φ=20°: T_min = 17. For φ=14.5°: T_min = 32. Higher pressure angle → fewer teeth possible.

Centre Distance

C = m(T₁ + T₂) / 2

For external gears. Internal gears: C = m(T₂ − T₁)/2.

Contact Ratio

CR = Arc of contact / Circular pitch

CR > 1 always (otherwise no continuous meshing). CR = 1.4–1.8 typical. Higher CR = smoother running.

ℹ️

Involute advantages (GATE list): (1) Pressure angle constant throughout engagement, (2) Conjugate action maintained even with slight centre distance error, (3) Easy to manufacture (rack cutter), (4) Interchangeable gears possible for same module.

⚡

GATE Quick Reference Sheet

TOM — ONE PAGE REVISION

// Print karo · exam se 1 din pehle dekho · sab formulas ek jagah

Gruebler's DOF

F = 3(n−1) − 2j₁ − j₂

No. of ICs

N = n(n−1)/2

Grashof Condition

s + l ≤ p + q

Coriolis Acceleration

aᶜ = 2ω·Vᵣ

Radial Acceleration

aʳ = ω²r = V²/r

Piston Acceleration

a = rω²[cosθ + (r/l)cos2θ]

Flywheel Energy

ΔE = I·ω²·Cs

Coeff. Fluctuation Speed

Cs = (ωmax−ωmin)/ωmean

Hoop Stress (Rim)

σ = ρv²

Watt Governor Height

h = g/ω² = 895/N²

Porter Governor

ω² = g(m+M)/mh

Static Balancing

Σ(mr) = 0

Dynamic Balancing

Σ(mr)=0 AND Σ(mrl)=0

Natural Frequency

ωₙ = √(k/m)

Static Deflection Method

fₙ = (1/2π)√(g/δst)

Springs in Series

1/keq = 1/k₁ + 1/k₂

Springs in Parallel

keq = k₁ + k₂

Critical Damping

cc = 2√(km) = 2mωₙ

Damping Ratio

ζ = c / cc

Damped Frequency

ωd = ωₙ√(1−ζ²)

Log Decrement

δ = 2πζ/√(1−ζ²)

Magnification Factor

MF = 1/√[(1−r²)²+(2ζr)²]

Transmissibility

TR = √[(1+(2ζr)²)/((1−r²)²+(2ζr)²)]

Isolation Condition

r > √2 (ω > √2·ωₙ)

Gear Module

m = d/T = p/π

Min Teeth (Undercutting)

T_min = 2/sin²φ

Cam SHM Displacement

x = (h/2)[1−cos(πθ/β)]

Torsional ωₙ

ωₙ = √(q/I), q = GJ/l

IMPORTANT VALUES — YAAD KARO

// GATE mein directly nahi dete — apne se yaad hone chahiye

Parameter	Value / Rule	Used In
Min teeth (φ=20°) — Involute	17	Gear design, undercutting
Min teeth (φ=14.5°) — Involute	32	Older standard
Standard pressure angles	14.5°, 20°, 25°	Gear profiles
Max pressure angle — translating follower	30°	Cam design
Cs — Crushing/hammering machines	1/5 to 1/10	Flywheel design
Cs — DC generators, AC generators	1/100 to 1/200	Flywheel design
Phase angle at resonance (r=1)	90° (always)	Forced vibration
Isolation: TR < 1 when	r > √2 ≈ 1.414	Vibration isolation
MF peak occurs at (underdamped)	r = √(1 − 2ζ²)	Resonance analysis