Mechanical Vibrations Calculators

Mechanical Vibrations studies the oscillatory motion of physical systems — from simple spring-mass-damper models to complex multi-degree-of-freedom structures. Understanding vibration is critical for preventing resonance failures, designing isolation systems, and diagnosing machinery health.

The course begins with free vibration of single-degree-of-freedom (SDOF) systems: natural frequency ω_n = √(k/m), damping ratio ζ, and the three response regimes (underdamped, critically damped, overdamped). Forced vibration under harmonic excitation introduces the concepts of resonance, frequency response, and transmissibility — the foundation of vibration isolation design. Multi-degree-of-freedom (MDOF) systems require eigenvalue analysis to find natural frequencies and mode shapes, and modal superposition to compute total response. Vibration isolation uses compliant mounts to reduce transmitted forces or motion at frequencies above √2 times the natural frequency. Dynamic vibration absorbers (tuned mass dampers) cancel vibration at specific frequencies. Critical speed analysis of rotating shafts prevents resonant whirl that damages bearings and seals. FFT (Fast Fourier Transform) spectrum analysis is the primary diagnostic tool for rotating machinery — converting time-domain vibration signals into frequency-domain spectra that reveal imbalance, misalignment, bearing faults, and gear mesh problems.

Vibration analysis is essential in automotive suspension design, aircraft flutter prevention, earthquake-resistant building design, turbomachinery monitoring, precision manufacturing, and any application where unwanted oscillation must be controlled or eliminated.