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Theory of Machines (Kinematics & Dynamics)

From transcript: 20ME310 Kinematics of Machines (D) · 20ME420 Dynamics of Machines (D)

Cheat sheet

Printable study sheet: formulas → definitions → topic notes → traps.

Formulas

Kutzbach (planar): DoF = 3(N−1) − 2J_p − J_h (N links, Jp binary joints, Jh higher)

Grashof four-bar: s+l ≤ p+q → continuous crank possible (s shortest, l longest)

Velocity ratio gears: ω2/ω1 = −T1/T2 (external); same sign internal

Train value = product of driver teeth / driven (with signs for directions)

Cam: lift vs θ; SHM, uniform accel, cycloidal — cycloidal lowest jerk peak classically

Balancing rotating: Σ m r = 0 (static); dynamic needs couples in planes

Reciprocating primary unbalanced ~ m ω² r cosθ ; secondary ~ m ω² r cos2θ / n

Gyroscopic: τ = I ω ω_p

Governor: sensitiveness = (N1−N2)/N ; effort, power, controlling force curves

Flywheel: ΔE = I ω_mean · Δω ; Cs = (ωmax−ωmin)/ωmean

Gear contact ratio > 1.4 typical spur for smooth transfer

Epicyclic: tabular / algebraic method with arm fixed then unlock

Degree of freedom (Gruebler/Kutzbach planar): DoF = 3(N−1) − 2J1 − J2

Velocity: v = ωr ; Instantaneous center method

Gear ratio: ω2/ω1 = −T1/T2 (external spur)

Balancing: Σ m r = 0 ; Σ m r cosθ = 0 ; Σ m r sinθ = 0

Critical speed (approx. Jeffcott): ωcr ≈ √(k/m)

Definitions

Link: Rigid body with hinges/contacts — binary, ternary…
Kinematic pair: Connection constraining relative motion (lower/higher)
Inversion: Different link fixed → different mechanism from same chain
Grashof: Condition for at least one link to make full revolution
Pressure angle (cam/gear): Force direction vs motion — affects side thrust
Module m: Pitch diameter / teeth (metric gear sizing)
Balancing: Cancel rotating/reciprocating inertia forces & couples
Governor: Speed-sensitive device varying fuel/steam for steady speed
Flywheel: Energy reservoir smoothing speed fluctuation over cycle
Coriolis component: 2 v ω when slider moves on rotating link
Gyroscopic couple: τ = Iωωp — precession of spinning rotor

Topic-wise short notes

Study these first — one block per syllabus topic. Then read the deep notes below.

Linkages, cams, gears

Four-bar types: crank-rocker, double-crank, double-rocker from Grashof.
Slider-crank: engine mechanism; inversions → Whitworth, rotary engine, hand pump.
Quick return: time ratio forward/return ≠ 1 (shaper).
Cam followers: knife, roller, flat; radial vs offset.
Spur vs helical vs bevel vs worm — axis relation & thrust.
Interference: minimum teeth on pinion for rack/gear to avoid undercutting.

Balancing, governors, gyroscopes

Single rotating mass balance with 180° opposite m r.
Multi-plane: dynamic balance with two correction planes (as in rotors).
Locomotive balancing: partial primary to limit rail hammer vs variation of tractive force.
Porter/Hartnell/Watt governors — spring vs gravity controlled.
Gyroscopic effect on ships/aeroplanes/two-wheelers — sense of reaction couple.

Linkages, DoF, inversions

Structure DoF=0; mechanism DoF=1 typically for constrained motion.
Higher pairs (cam, gear tooth) count differently than revolute/prismatic.

Gears, cams, balancing

Contact ratio = arc of contact / circular pitch — higher → quieter.
Epicyclic used in gearboxes/differentials for compact high ratio.

Flywheels & energy fluctuation

ΔE from turning-moment diagram; size I for given Cs.
Punch presses need large ΔE storage — classic flywheel sizing.

Gear force & efficiency

Tangential force from power/ωr; radial from pressure angle.
Efficiency of worm gear low — heat & self-locking possible.

Exam traps & quick notes

Four-bar: Grashof condition for crank-rocker vs double-crank.
Cams: follower motion laws (SHM, cycloidal) trade jerk vs size.
Project tie: BAJA steering geometry; Aero prop/motor dyno dynamics.
Count DoF carefully — redundant constraints can fool Kutzbach.
Velocity diagrams: close polygon; acceleration needs Coriolis/centripetal terms.
Cam: check undercutting and pressure angle limits.
Gear trains: keep track of sign/direction every mesh.
Dynamic balancing ≠ static — add couples in axial planes.
Governors: stability from controlling force curve slope.

Comprehensive notes

Linkages, cams, gears

Velocity/acceleration diagrams, Coriolis component in sliding links, epicyclic gear trains, cam profiles.

External: NPTEL Theory of Machines ↗ Kinematics applet notes ↗

Balancing, governors, gyroscopes

Static/dynamic balancing of rotors; Hartnell/Porter governors; gyroscopic effect on ships/aircraft/vehicles.

External: Roymech balancing ↗

Linkages, DoF, inversions

Kutzbach/Gruebler for planar DoF. Four-bar Grashof condition. Slider-crank inversions (engine, Whitworth, etc.). Velocity/acceleration diagrams; Coriolis when a point slides on a rotating link.

External: NPTEL ↗ Engineering Toolbox ↗

Gears, cams, balancing

Spur gear ratio and contact ratio basics; epicyclic trains. Cam follower laws (SHM, cycloidal) trade peak jerk vs size. Static/dynamic balancing of rotating masses; governors sense speed.

External: NPTEL ↗ Engineering Toolbox ↗

Interview Q&A for this subject

Q: What is degree of freedom of a mechanism?
A: Independent inputs needed to define configuration — use Kutzbach with care about redundant constraints.

External: ASQ / quality ↗ MIT OCW ↗