Theory of Machines
Theory of Machines
This domain tests mobility, velocity and acceleration analysis, cams, gears, governors, flywheels, balancing, and vibration. Mechanism counting and gear ratios are high-yield.
Core rules
- Grübler's criterion: DOF = 3(n − 1) − 2j − h, where n is the number of links, j is the number of lower pairs, and h is the number of higher pairs. A four-bar linkage has n = 4, j = 4, h = 0, so DOF = 1.
- Gear velocity ratio: ω₁/ω₂ = N₂/N₁ = T₁/T₂ for external gears. Involute gears avoid interference; the minimum number of teeth without undercutting depends on the pressure angle (about 17 for 20° full-depth).
- Flywheel: ΔE = Iω_m²C_s, where C_s = (ω_max − ω_min)/ω_m. Use ω in rad/s, not rpm.
- Balancing: static balance requires Σmr = 0; dynamic balance requires Σmr = 0 and Σmrl = 0.
Worked example
A flywheel has I = 10 kg·m² and runs at a mean speed of 300 rpm with C_s = 0.05. Convert ω_m = 2π × 300/60 = 31.42 rad/s. ΔE = Iω_m²C_s = 10 × (31.42)² × 0.05 = 10 × 987.0 × 0.05 = 493.5 J. This is the maximum fluctuation of energy the flywheel must absorb between the mean speed and the speed limits.
Traps
Using rpm directly in ω; inverting the gear ratio; counting a spring as a link in Grübler's formula; confusing static and dynamic balancing; and ignoring the effect of centrifugal force in governor problems.