Natural Frequency Calculator
Calculate natural frequencies for springs, beams, cantilevers, and structural systems
System Configuration
Spring-Mass System: Mass connected to a spring with elastic constant k
Formula: f = (1/2π) × √(k/M)
System Parameters
Output Units
Natural Frequency Results
Example Calculation
Spring-Mass System
Given: Spring constant k = 100 N/m, Mass M = 2 kg
Angular frequency: ω = √(k/M) = √(100/2) = 7.07 rad/s
Natural frequency: f = ω/(2π) = 7.07/(2π) = 1.125 Hz
Period: T = 1/f = 0.889 s
Simply Supported Beam
Given: Steel beam, L = 10 m, E = 200 GPa, I = 2.14×10⁻⁵ m⁴, M = 500 kg
f = (1/2π) × √(48EI/(ML³))
f = (1/2π) × √(48×200×10⁹×2.14×10⁻⁵/(500×10³)) = 3.04 Hz
System Types
Typical Elastic Moduli
Frequency Ranges
Vibration Tips
Natural frequency is intrinsic property
Avoid resonance for structural safety
Higher stiffness = higher frequency
Higher mass = lower frequency
Period = 1/frequency
Understanding Natural Frequency
What is Natural Frequency?
Natural frequency is the frequency at which a system oscillates when not subjected to a continuous or repeated external force. Every object has at least one natural frequency, and some complex objects have multiple natural frequencies. When an external force matches the natural frequency, resonance occurs, which can lead to dramatic amplitude increases.
vs. Resonant Frequency
- •Natural Frequency: Frequency of free vibration (no external force)
- •Resonant Frequency: Frequency at which maximum response occurs under forced vibration
- •Relationship: Nearly identical in lightly damped systems
System Formulas
Spring-Mass System
f = (1/2π) × √(k/M)
ω = √(k/M)
k = spring constant, M = mass
Supported Beam
f = (1/2π) × √(48EI/(ML³))
E = elastic modulus, I = moment of inertia, L = length
Cantilever Beam
f = (1/2π) × √(3EI/(FL³))
F = force at free end
Engineering Applications
Structural Design
Avoid resonance with operational frequencies (machinery, wind, seismic)
Vibration Control
Design dampers and isolators based on natural frequencies
Modal Analysis
Identify critical modes and frequencies for complex structures
Design Considerations
Avoid Resonance
Keep natural frequency away from excitation frequencies
Stiffness vs. Mass
Increase stiffness or reduce mass to raise natural frequency
Damping
Add damping to reduce resonance amplification