Theory of Machines and Mechanical Vibrations: Mechanisms, Gears and Resonance
Kinematics and mechanisms
Degrees of freedom by Grübler's criterion: F = 3(n − 1) − 2j − h, where n includes the frame, j is the number of lower pairs and h the number of higher pairs. A four-bar linkage has one DOF; a slider-crank also has one.
Gears, trains and governors
- Module m = d/T; circular pitch p = πm; involute gears mesh only with equal module and pressure angle (usually 20°).
- Interference is avoided by undercutting, raising the pressure angle or adding teeth; the minimum pinion tooth count at 20° is about 18.
- A simple gear train's velocity ratio is the product of driven teeth over driving teeth; an epicyclic train needs the tabular relative-velocity method.
- A flywheel controls cycle-to-cycle speed fluctuation; a governor controls mean speed under varying load. They are not interchangeable.
Vibration
- Undamped natural frequency ω_n = √(k/m) rad/s; f_n = ω_n/2π.
- Damping ratio ζ = c/(2√(km)); logarithmic decrement δ = (1/n)ln(x₀/x_n) = 2πζ/√(1 − ζ²).
- Resonance occurs near ω = ω_n; isolation requires ω/ω_n > √2.
Worked example
A 2 kg mass on a spring of stiffness 800 N/m with a dashpot of 8 N·s/m. ω_n = √(800/2) = 20 rad/s; f_n = 20/6.283 = 3.18 Hz. Critical damping c_c = 2√(km) = 2√1600 = 80 N·s/m, so ζ = 8/80 = 0.1 - underdamped, oscillating and decaying.
Traps
- Omitting the frame from n in Grübler's equation.
- Using static deflection in mm in ω_n = √(g/δ) without converting to metres.
- Confusing governor with flywheel, and interference with undercutting.
- Reporting ω_n in Hz without dividing by 2π.