Expert-level mechanical vibrations covering free and forced vibration, damping, resonance, modal analysis, vibration isolation, and rotating machinery.
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Equation of motion: m x-double-dot + c x-dot + k x = F(t). Natural frequency: omega_n = sqrt of k over m. Damping ratio: zeta = c over 2 sqrt of km. Underdamped: oscillatory decay, frequency omega_d = omega_n sqrt of 1 minus zeta squared. Critically damped: fastest return to equilibrium without oscillation.
Steady-state amplitude: X = F0 over k times magnification factor. Resonance: maximum amplitude when excitation frequency equals natural frequency. Phase angle: response lags excitation, 90 degrees at resonance. Frequency response function: complex ratio of output to input versus frequency.
Mass and stiffness matrices: M x-double-dot + K x = F. Natural frequencies: eigenvalues of K inverse M. Mode shapes: eigenvectors, orthogonal with respect to mass matrix. Modal superposition: response is sum of individual mode responses.
Unbalance: mass offset from rotation axis causes rotating force excitation. Critical speed: rotation speed equals natural frequency, resonance in rotor. Balancing: single plane and two plane balancing to reduce unbalance forces. Campbell diagram: plot natural frequencies and excitation orders versus speed.
Transmissibility: ratio of transmitted to applied force. Isolation: below natural frequency amplifies, above attenuates vibration. Design rule: isolator natural frequency below one third excitation frequency. Soft mounts: low stiffness isolators, good isolation but large static deflection.
| Pitfall | Fix |
|---|---|
| Ignoring damping in resonance analysis | Even small damping limits response at resonance |
| Wrong isolation region | Ensure operating frequency is well above isolator natural frequency |
| Missing higher modes in truncated modal analysis | Include sufficient modes for accurate response |
| Unbalance in rotating equipment | Perform field balancing after assembly |
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