Natural Frequency Fundamentals

Natural frequency is the frequency at which a system tends to oscillate when disturbed from equilibrium. Every mechanical system has one or more natural frequencies determined by its mass and stiffness properties.

fn = (1/2pi) * sqrt(k/m)

• fn: Natural frequency in Hz (cycles per second)
• k: System stiffness (N/m or lb/in) - resistance to deformation
• m: System mass (kg or slugs) - inertia of the system
• omega_n: Angular natural frequency = 2*pi*fn (rad/s)

Resonance Avoidance

Resonance occurs when the operating frequency equals the natural frequency, causing amplitude amplification that can lead to catastrophic failure. The amplification factor (Q) at resonance can be 10-100x or more.

Avoid operating within +/- 20% of natural frequency

• Subcritical: Operating below 0.7 * fn - safe but may need to pass through resonance during startup
• Supercritical: Operating above 1.4 * fn - safe if startup is fast enough
• Danger Zone: 0.7 * fn to 1.4 * fn - high vibration, potential damage

Critical Speed of Rotating Shafts

Critical speed is the rotational speed at which shaft whirl occurs. For a simply-supported shaft with center mass:

Nc = (60/2pi) * sqrt(48*E*I / m*L^3)

• E: Elastic modulus (Pa or psi)
• I: Area moment of inertia = pi*d^4/64 for solid shaft
• L: Span length between supports
• Higher modes exist at approximately 4x, 9x, 16x first critical

Damping Ratio (zeta)

Damping ratio describes how oscillations decay over time. It affects both free vibration decay and forced response amplitude:

zeta = c / (2 * sqrt(k * m)) = c / c_critical

• zeta = 0: Undamped - oscillates forever
• 0 < zeta < 1: Underdamped - decaying oscillation (most structures)
• zeta = 1: Critically damped - fastest return without overshoot
• zeta > 1: Overdamped - slow return, no oscillation
• Typical values: Welded steel 0.02-0.05, Bolted joints 0.03-0.10, Rubber mounts 0.05-0.20

Vibration Isolation Efficiency

Isolation mounts reduce force transmission to foundations. Efficiency depends on the frequency ratio (r = f_operating / f_natural):

Transmissibility = 1 / |1 - r^2| (undamped)

Isolation Efficiency = (1 - T) * 100%

• r < 1: No isolation - force may be amplified
• r = sqrt(2): 0% isolation - transition point
• r > sqrt(2): Positive isolation begins
• r > 3: Good isolation (>85%)
• r > 5: Excellent isolation (>95%)

Formulas Quick Reference

• fn = (1/2pi) * sqrt(k/m) - Natural frequency from stiffness and mass
• fn = (1/2pi) * sqrt(g/delta) - Natural frequency from static deflection
• fn = 15.76 / sqrt(delta_mm) - Quick formula (delta in mm, fn in Hz)
• fn = 3.13 / sqrt(delta_in) - Quick formula (delta in inches, fn in Hz)
• omega_n = 2 * pi * fn - Angular natural frequency (rad/s)
• Nc = fn * 60 - Critical speed in RPM