Theory of Machines and Vibrations

Syllabus

• Theory of machines: velocity and acceleration analysis of plane mechanisms, dynamic analysis of linkages, cams, gears and gear trains, flywheels and governors, balancing of reciprocating and rotating masses, gyroscope.
• Vibrations: free and forced vibration of single degree of freedom systems, damping, vibration isolation, resonance, critical speeds of shafts.

Key concepts and formulas

Mobility (Gruebler). For a planar mechanism with n links, j₁ lower pairs and j₂ higher pairs: F = 3(n − 1) − 2j₁ − j₂.

Grashof's law. For a four-bar chain with shortest link s, longest l and others p, q: if s + l ≤ p + q, at least one link can rotate fully. With the shortest link as the frame, it is a double-crank; adjacent to the shortest, a crank-rocker; opposite, a double-rocker.

Instantaneous centres. A mechanism with n links has n(n − 1)/2 instantaneous centres (Kennedy's theorem: any three links' centres lie on a line).

Gears. Module m = D/T; circular pitch πm. Velocity ratio of a simple train = product of driver teeth / product of driven teeth (with sign for direction). Minimum teeth to avoid interference (20° full depth) is about 18. In an epicyclic train, use the tabular method: fix the arm, rotate the sun, add the arm motion.

Flywheel. Fluctuation of energy ΔE = Iω²C_s, where C_s = (ω_max − ω_min)/ω is the coefficient of fluctuation of speed.

Governors. Watt governor height h = g/ω² (h in m, ω in rad/s). A governor is isochronous if its equilibrium speed is the same for all radii; sensitiveness = (N₂ − N₁)/N.

Gyroscope. Gyroscopic couple C = Iωω_p (spin ω, precession ω_p).

Balancing. Rotating masses are balanced if Σ m r = 0 (force) and Σ m r l = 0 (couple), on a single plane or two planes.

Free vibration. ω_n = √(k/m); f_n = ω_n/(2π). With static deflection δ: ω_n = √(g/δ). Damping ratio ζ = c/(2√(km)); damped frequency ω_d = ω_n√(1 − ζ²). Logarithmic decrement δ = 2πζ/√(1 − ζ²) = ln(x_n/x_n+1).

Forced vibration. With frequency ratio r = ω/ω_n, the magnification factor is 1/√((1 − r²)² + (2ζr)²). Transmissibility exceeds 1 for r < √2 and falls below 1 (isolation) for r > √2, whatever the damping. The critical speed of a shaft equals its natural frequency of transverse vibration.

Worked example

A spring-mass system has m = 10 kg, k = 4000 N/m and c = 40 N s/m. Find ω_n, f_n, ζ, ω_d and the logarithmic decrement. Then check the mobility of a four-bar chain and a slider-crank.

import math

m, k, c = 10, 4000, 40
wn = math.sqrt(k / m)
zeta = c / (2 * math.sqrt(k * m))
print(f"wn = {wn:.3f} rad/s, fn = {wn / (2 * math.pi):.4f} Hz")
print(f"zeta = {zeta:.4f}, wd = {wn * math.sqrt(1 - zeta ** 2):.4f} rad/s")
print(f"log decrement = {2 * math.pi * zeta / math.sqrt(1 - zeta ** 2):.4f}")
mobility = lambda n, j1, j2=0: 3 * (n - 1) - 2 * j1 - j2
print("four-bar chain F =", mobility(4, 4), " slider-crank F =", mobility(4, 4), " five-bar F =", mobility(5, 5))

wn = 20.000 rad/s, fn = 3.1831 Hz
zeta = 0.1000, wd = 19.8997 rad/s
log decrement = 0.6315
four-bar chain F = 1  slider-crank F = 1  five-bar F = 2

Practice questions