Dynamics of Machines

A rotating unbalance produces centrifugal force F_c = m·r·ω². Balancing sets the resultant of all m·r vectors (and their moments) to zero; single-plane balance needs Σm·r = 0, dynamic balance needs Σm·r = 0 and Σm·r·l = 0, as SS Rattan shows.

The reciprocating mass produces a primary force F_p = m·r·ω²·cosθ and a secondary force F_s = (m·r·ω²/n)·cos2θ, with n = L/r. Partial primary balance is by a rotating mass; secondary forces are handled by multi-cylinder phasing, per SS Rattan.

A flywheel stores kinetic energy $E=\tfrac{1}{2}I\omega^{2}$ and smooths cyclic speed fluctuation. Required inertia is $I=\dfrac{\Delta E}{\omega^{2}C_{s}}$, where $C_{s}=\dfrac{\omega_{\max}-\omega_{\min}}{\omega_{\mathrm{mean}}}$ is the coefficient of fluctuation of speed (SS Rattan).

An SDOF system oscillates at natural frequency ω_n = √(k/m); damping ratio ζ = c/(2√(km)) decides whether the response is under-, critically, or over-damped. The logarithmic decrement δ = 2πζ/√(1−ζ²) extracts ζ from a decay trace, per SS Rattan.

A rotating shaft whirls violently when its speed equals its lateral natural frequency: the critical speed ω_cr = √(g/δ_st) = √(k/m). Operating well below (rigid) or above (flexible) ω_cr keeps deflection bounded, as SS Rattan explains.