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Natural Frequency Calculator

Calculate resonant frequency of mass-spring systems

Calculation Parameters

Based on ISO 2041:2018 and vibration theory

Metric System

Imperial System

System Type

System Mass

Second Mass

Spring Stiffness

Damping Ratio (ζ)

0 = undamped, 1 = critically damped

Moment of Inertia

Mounting Configuration

Inclination Angle

degrees

Calculation Results

Natural Frequency (fn):
—

Natural Angular Frequency (ωn):
—

Natural Period (T):
—

Static Deflection:
—

Damped Natural Frequency (fd):
—

Frequency Range Assessment:

< 1 Hz: Very low frequency – seismic isolation

1-10 Hz: Low frequency – building vibration range

10-100 Hz: Medium frequency – machine vibration

> 100 Hz: High frequency – precision equipment

How the Calculator Works

Undamped Natural Frequency

For a simple mass-spring system:

fn = (1/2π) × √(k/m)

where:

fn — natural frequency (Hz)
k — spring stiffness (N/m)
m — mass (kg)

Damped Natural Frequency

When damping is present:

fd = fn × √(1 – ζ²)

where ζ is the damping ratio (dimensionless)

Static Deflection Method

Natural frequency from static deflection:

fn = (1/2π) × √(g/δst) ≈ 15.76/√δst

where δst is static deflection in mm

Torsional Systems

For rotational vibration:

fn = (1/2π) × √(kt/J)

where kt is torsional stiffness and J is moment of inertia

Two-Mass Systems

Systems with two masses have two natural frequencies:

First mode: masses move together
Second mode: masses move opposite

Important Considerations

Avoid operating near natural frequency (resonance)
Stay below 0.7×fn or above 1.4×fn for isolation
Added mass lowers natural frequency
Stiffer springs increase natural frequency
Damping reduces amplitude but not natural frequency significantly

Applications

Vibration Isolation: Design for fn < forcing frequency/√2
Seismic Protection: Very low fn (0.5-2 Hz)
Machine Mounts: Typically 5-15 Hz
Precision Equipment: High fn to avoid building vibrations

📘 Complete Guide: Natural Frequency Calculator

🎯 What This Calculator Does

Calculates natural frequency of mass-spring systems. Critical for preventing resonance and designing vibration isolation.
Formula: fn = (1/2π) × √(k/m)

💼 Key Applications

Vibration Isolation: Compressor 1200 kg, 1500 RPM (25 Hz). For isolation: fn < 25/3 ≈ 8 Hz. Required spring stiffness: k < 30000 N/m.
Resonance Prevention: Turbine on foundation, fn = 4.2 Hz. Rotation: 3000 RPM = 50 Hz. Ratio 50/4.2 = 12 → No resonance danger.
Dynamic Absorber: Shaft vibrates at 180 Hz. Install absorber with fn = 180 Hz to suppress vibration.

Isolation Principle:

For effective isolation from vibration at frequency f:

Good isolation: fn < f/√2 (transmissibility TR < 1)
Effective: fn < f/3 (TR < 0.1, 90% reduction)
Excellent: fn < f/5 (TR < 0.04, 96% reduction)

📖 Quick Reference

Resonance: Amplification occurs when external frequency = natural frequency (can increase 10-50×)
Static Deflection: δst = mg/k. Relation: fn ≈ 5/√δst (δst in mm)
Damping (ζ): Steel springs: 0.01-0.03, Rubber: 0.05-0.15, Critical: 1.0

Natural Frequency Calculator – Mass-Spring System

Published by admin on October 20, 2025 October 20, 2025