Theory of Machines — Kinematics
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Four-bar mechanism: Grashof criterion: S + L ≤ P + Q → at least one link can rotate fully. S = shortest link, L = longest link.
Instantaneous Centre (IC): Point in mechanism where velocity of two bodies is equal. Number of ICs = n(n–1)/2 for n links.
Kennedy’s Theorem: Three centrodes always have a common point. Useful for finding velocity by locating ICs.
Velocity Analysis: v = ω × r (perpendicular to link, from IC). For relative velocity: v_B = v_A + v_B/A.
Slider-crank: Offset creates secondary forces and changes in piston velocity. Use relative velocity method.
Cams: For knife-edge follower: minimum follower radius = zero. For roller follower: base circle + roller radius = prime circle.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Four-Bar Linkage Analysis
Grashof Criterion
For a four-bar mechanism with links of lengths S (shortest), P, Q, L (longest):
Grashof mechanism: S + L ≤ P + Q
- At least one link is a revolute (full rotation possible)
- If S + L = P + Q exactly → change point (full rotation possible for one link at specific positions)
Non-Grashof: S + L > P + Q
- All joints are rocker or drag-link — no link completes full revolution
Classification:
| Condition | Type | Motion |
|---|---|---|
| S + L < P + Q | Crank-rocker | Crank fully rotates, rocker oscillates |
| S + L = P + Q | Change point | Transition state |
| S + L > P + Q | Double-rocker | Both rockers oscillate |
| S+L ≤ P+Q + special cases | Drag-link | Crank drags other link fully |
⚠️ GATE trap: Students often forget that Grashof doesn’t guarantee continuous rotation — practical issues like joint size, clearances, and link interference can prevent full rotation.
Velocity in Four-Bar Mechanisms
Angular velocity relationship: ω₃ / ω₂ = (AB/BC) × sin(θ₃) / sin(θ₂) — derived from relative velocity at joint B
More practically, use instantaneous centre method:
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Identify all ICs: For a 4-bar with links 1-2-3-4 (1=ground):
- IC₁₂ at joint A (ground link)
- IC₂₃ at joint B ( coupler)
- IC₃₄ at joint C (output link)
- IC₁₃ at intersection of extensions of links 1 and 3
- IC₂₄ at intersection of extensions of links 2 and 4
- IC₁₄ at joint D (ground link)
-
Kennedy’s Theorem: The IC of any two links lies on the line joining the ICs of those links taken two at a time.
Velocity calculation using IC: v_B = ω₁₄ × r_(B,IC₁₄) where r is perpendicular distance from IC to point B.
Slider-Crank Mechanism
Geometry
Slider-crank: Link 1 (ground/crank), Link 2 (crank), Link 3 (connecting rod), Link 4 (slider).
Crank angle θ: Angle of crank from line of stroke. Offset e: Perpendicular distance between slider axis and crank center. Stroke: 2 × crank radius if e = 0.
Slider position: x = r cos θ + √(l² – (r sin θ + e)²)
Where r = crank length, l = connecting rod length.
When e ≠ 0 (offset slider-crank):
- Piston doesn’t follow simple harmonic motion
- Retardation > acceleration during expansion stroke
- Acceleration > retardation during compression stroke
- This asymmetry causes secondary unbalanced forces in engines
Velocity of Piston
v_piston = dx/dt = –rω sin θ – (r²ω sin 2θ)/(2l √(1 – (r/l)² sin² θ))
Approximation when r/l is small: v ≈ –rω sin θ (first term dominates)
Maximum piston velocity: Occurs at θ ≠ 90° for offset crank. Solve dv/dθ = 0: cos θ_max = (–1 + √(1 + 16(r/l)² cos² φ_e))/(2r/l) … complex for GATE.
For exam purposes: v_max ≈ rω × (0.75 to 0.85) occurs slightly before/after 90° depending on r/l ratio.
Acceleration of Piston
α_piston = d²x/dt² = –rω² cos θ – (r²ω² cos 2θ)/(l(1 – (r/l)² sin² θ)^(3/2))
Simplified (r/l small): α ≈ –rω² cos θ
💡 GATE Pattern: Slider-crank questions often ask about velocity at specific crank angles or about the ratio of connecting rod angular velocity to crank angular velocity. Focus on the relative velocity method.
Relative Velocity Method
Procedure
- Draw velocity polygon at the point of interest
- Identify known velocities and directions
- Apply: v_B = v_A + v_B/A (where v_B/A is velocity of B relative to A, perpendicular to link AB)
- Solve polygon graphically or analytically
Example: Four-bar slider-crank velocity at B:
- v_B = ω₂ × AB (perpendicular to AB, away from A)
- v_C = v_B + v_C/B (v_C/B perpendicular to BC)
- v_C direction is horizontal (slider guide)
Magnitude of v_B/A = ω_link × length(link)
Acceleration in Mechanisms
Coriolis acceleration appears in slider-crank and other mechanisms with sliding:
a_Coriolis = 2 × ω × v_sliding
Direction: rotate v_sliding by 90° in direction of ω.
⚠️ GATE frequently tests Coriolis acceleration — students forget the factor of 2 or rotate in the wrong direction. Remember: Coriolis always appears when you have simultaneous rotation and sliding.
Instantaneous Centre (IC)
Properties
- Velocity property: Two bodies have equal velocity at IC (relative velocity = 0)
- No velocity transmission through IC — instantaneous
- IC of rotating bodies lies on perpendicular bisector of line joining their velocity centroids
Locating ICs — Kennedy’s Theorem
Kennedy’s Theorem: The three ICs of three bodies moving in plane motion always lie on a straight line.
Application: When you can’t directly find an IC geometrically:
- Find two known ICs
- Draw line joining them
- Find intersection with line of another IC pair
IC locations for common mechanisms:
| Mechanism | IC locations |
|---|---|
| Four-bar | Ground-2 (link 1-2 joint), Ground-3 (1-3 joint), Ground-4 (1-4 joint), 2-3, 2-4, 3-4 |
| Slider-crank | Ground-2 (crank pin), Ground-3 (at infinity perpendicular to guides), 2-3, 2-slider, 3-slider |
| Cam-follower | Base circle center (rotating), IC at point of pure rolling if no slip |
Cams and Followers
Cam Types
Radial (disc) cam: Follower moves radially from cam center. Most common. Translating cam: Follower translates parallel to cam axis. Cylindrical cam: Groove on cylinder surface — complex 3D motion.
Follower Types
** Knife-edge follower:** Sharp edge, high stress, rarely used industrially Roller follower: Lower friction, preferred for moderate speeds Flat-faced follower: No rotating joint, can be on either side of cam
Motion Programs
| Motion | Displacement | Velocity | Acceleration |
|---|---|---|---|
| Simple Harmonic Motion (SHM) | s = (r/2)(1 – cos θ) | v = (rω/2) sin θ | a = (rω²/2) cos θ |
| Uniform Velocity | s = kθ (linear with θ) | v = constant | a = 0 (jerk = infinite) — impractical |
| Cycloidal | s = r(θ/β – sin(2πθ/β)/2π) | v = rω(1 – cos(2πθ/β))/β | a = rω² sin(2πθ/β)/β |
SHM is GATE’s most commonly tested motion program — memorize these three equations.
Pressure Angle
Pressure angle (φ): Angle between normal to cam surface and direction of follower motion.
- High pressure angle (> 30°): Excessive side thrust on follower, potential binding
- Minimum pressure angle at base circle (lowest curvature)
- Maximum pressure angle at peak of rise curve
Prime circle: Circle drawn from cam center to the point of the cam that contacts the follower when the follower is at its minimum radius position.
For knife-edge follower: Prime circle = base circle + zero (theoretically) For roller follower: Prime circle = base circle + roller radius
⚠️ GATE trap: Confusing base circle with prime circle. Base circle is the cam’s actual boundary. Prime circle is the follower-side envelope. For a roller follower, prime circle radius = base circle + roller radius.
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Complex Number Methods for Mechanism Analysis
Position analysis using complex numbers (loop closure equation):
Four-bar: AC = AB + BC → R₁e^(iθ₁) + R₂e^(iθ₂) = R₃e^(iθ₃) + R₄e^(iθ₄)
Separating real and imaginary parts gives two scalar equations. Solve for unknown angles.
Slider-crank loop equation: r e^(iθ) + l e^(iφ) = x + i y (where x is piston position)
Taking magnitude: (r cos θ + l cos φ)² + (r sin θ + l sin φ)² = x²
Coriolis Acceleration — Deep Dive
Coriolis acceleration a_C = 2ω × v_rel
When does it appear?
- Any mechanism where a link is rotating while another link slides along/through it
- Slider-crank: connecting rod rotates while slider translates
- Four-bar with sliding contact: some drag-link mechanisms
Direction determination:
- Visualize the sliding velocity v_rel
- Imagine rotating the link at ω
- Coriolis acceleration points in direction of that rotation
Magnitude: |a_C| = 2ω v_rel
GATE application: Find Coriolis acceleration at the connecting rod of a slider-crank at θ=60°, r=50mm, l=200mm, ω=100 rad/s.
Solution: v_piston = rω sin θ (approx). a_C = 2 × ω_link × v_piston. Where ω_link = ω × (AB/BC) × sin(θ₂)/sin(θ₃) … solve polygon first.
Klein’s Construction for Slider-Crank
A graphical method to find velocity and acceleration simultaneously:
- Draw crank AB at angle θ
- Construct perpendicular at B (velocity direction of B)
- Project coupler BC position
- Use geometry to find acceleration components
Klein’s acceleration triangle: a_C = a_B + a_C/B (coriolis component + tangential component + normal component)
Normal component: a_C/B_normal = ω₃² × BC Tangential component: a_C/B_tangential = α₃ × BC Coriolis: a_C/B_Coriolis = 2ω₂ × v_C/B (perpendicular to BC)
Cam Design — Sizing
Base circle diameter: Determined by mounting requirements and shaft diameter.
Minimum prime circle radius (for knife-edge follower): r_p_min = e / sin(φ_max) where e = offset, φ_max = maximum allowable pressure angle (typically 30°)
Minimum prime circle radius (for flat-faced follower): r_p_min = √((e + V_max/ω)² + (F_max/(2K)²)) … where K is follower face width factor
For roller follower: r_p_min = r_base + r_roller + … adjust for curvature
Example Problem
GATE 2020: In a four-bar mechanism, AB = 30 mm, BC = 50 mm, CD = 55 mm, DA = 40 mm. Identify if it’s Grashof and classify.
Solution:
Step 1: Identify S (shortest) = 30 mm (AB), L (longest) = 55 mm (CD) Step 2: P = 40 mm, Q = 50 mm Step 3: Check S + L = 30 + 55 = 85 Step 4: Check P + Q = 40 + 50 = 90 Step 5: S + L < P + Q → Grashof mechanism ✓
Step 6: Classify: Since S + L < P + Q and S is adjacent to L (AB adjacent to CD which is longest), this is Crank-rocker (link AB = crank, can rotate fully).
If shortest link is ground link → crank-rocker. If shortest link is coupler → drag-link. If shortest link is opposite ground → double-rocker.
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