3 Simple Harmonic Motion
Simple Harmonic Motion: The foundation of vibration analysis. A particle undergoes SHM when its acceleration is proportional to displacement and directed toward the equilibrium position:
\[a = -\omega^2 x\]
Key relationships — displacement, velocity, acceleration, amplitude, period, and frequency — are all derived from this condition.
3.1 Free Undamped Vibration
3.1.1 Mass-Spring System
The simplest model: a mass \(m\) on a spring of stiffness \(k\):
\[\omega_n = \sqrt{\frac{k}{m}} \qquad f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\]
3.1.2 Simple Pendulum
\[f_n = \frac{1}{2\pi}\sqrt{\frac{g}{L}}\]
3.1.3 Equivalent Stiffness
Springs in series and parallel are reduced to a single equivalent stiffness before applying the standard formulae:
\[k_{\text{series}} = \frac{k_1 k_2}{k_1 + k_2} \qquad k_{\text{parallel}} = k_1 + k_2\]
3.2 Energy Method (Rayleigh’s Method)
An alternative to force methods — equating maximum kinetic energy to maximum potential energy to find the natural frequency. Particularly useful where writing equations of motion is complex:
\[\frac{1}{2}mv_{\max}^2 = \frac{1}{2}kx_{\max}^2\]
3.3 Damped Free Vibration
In practice, energy is dissipated. A viscous damping force proportional to velocity is assumed:
\[F_d = -c\dot{x}\]
The damping ratio \(\zeta\) determines the character of the motion:
\[\zeta = \frac{c}{2\sqrt{mk}}\]
| Condition | Behaviour |
|---|---|
| \(\zeta < 1\) | Underdamped — oscillates with decaying amplitude |
| \(\zeta = 1\) | Critically damped — returns to rest without oscillation |
| \(\zeta > 1\) | Overdamped — sluggish return to rest |
The damped natural frequency is reduced from the undamped value:
\[\omega_d = \omega_n\sqrt{1 - \zeta^2}\]
3.4 Forced Vibration (Undamped)
When a harmonic exciting force \(F = F_0 \sin\omega t\) is applied, the steady-state amplitude is:
\[X = \frac{F_0/k}{1 - (\omega/\omega_n)^2}\]
Resonance occurs when \(\omega \to \omega_n\), causing theoretically infinite amplitude.
3.5 Forced Vibration (Damped)
Damping limits the amplitude at resonance. The dynamic magnification factor (DMF) gives the ratio of dynamic to static deflection:
\[\text{DMF} = \frac{1}{\sqrt{\left[1-\left(\dfrac{\omega}{\omega_n}\right)^2\right]^2 + \left[2\zeta\dfrac{\omega}{\omega_n}\right]^2}}\]
At resonance (\(\omega = \omega_n\)), the DMF reduces to:
\[\text{DMF}_{\text{resonance}} = \frac{1}{2\zeta}\]
3.6 Whirling of Shafts
A rotating shaft has a critical speed at which it deflects violently — directly analogous to resonance. The critical (whirling) speed corresponds to the natural frequency of transverse vibration:
\[N_c = \frac{30}{\pi}\sqrt{\frac{g}{\delta_{st}}}\]
where \(\delta_{st}\) is the static deflection of the shaft under its own (or the rotor’s) weight. Operating speeds should be well clear of \(N_c\).
3.7 Summary of Equations
| Concept | Equation |
|---|---|
| SHM condition | \(a = -\omega^2 x\) |
| Natural frequency (spring-mass) | \(\omega_n = \sqrt{k/m}\) |
| Natural frequency (pendulum) | \(f_n = \frac{1}{2\pi}\sqrt{g/L}\) |
| Damping ratio | \(\zeta = c \, / \, 2\sqrt{mk}\) |
| Damped natural frequency | \(\omega_d = \omega_n\sqrt{1-\zeta^2}\) |
| Resonance condition | \(\omega = \omega_n\) |
| DMF (damped forced) | \(1/\sqrt{[1-r^2]^2+[2\zeta r]^2}\), where \(r=\omega/\omega_n\) |
| Critical shaft speed | \(N_c = (30/\pi)\sqrt{g/\delta_{st}}\) |
3.8 Summary
All vibration problems reduce to understanding the interplay between stiffness (restoring force), mass (inertia), and damping (energy dissipation). Resonance avoidance and the role of damping in controlling amplitudes are the central practical concerns for mechanical design.