Natural Frequency Calculator Formula
Understand the math behind the natural frequency calculator. Each variable explained with a worked example.
Formulas Used
Angular Natural Frequency
omega_n = sqrt(spring_k / mass)Natural Frequency
freq_hz = sqrt(spring_k / mass) / (2 * pi)Natural Period
period = 2 * pi / sqrt(spring_k / mass)Variables
| Variable | Description | Default |
|---|---|---|
spring_k | Spring Stiffness (k)(N/m) | 10000 |
mass | Mass (m)(kg) | 5 |
How It Works
Natural Frequency of a Spring-Mass System
Every elastic system has a natural frequency at which it tends to vibrate when disturbed.
Formulas
omega_n = sqrt(k / m) (angular frequency in rad/s)
f_n = omega_n / (2 pi) (frequency in Hz)
T = 1 / f_n = 2 pi / omega_n (period in seconds)
Resonance occurs when the excitation frequency matches the natural frequency, leading to large amplitude vibrations that can cause structural failure.
Worked Example
A 5 kg mass on a spring with stiffness 10,000 N/m.
- 01omega_n = sqrt(10000 / 5) = sqrt(2000) = 44.72 rad/s
- 02f_n = 44.72 / (2 x 3.1416) = 7.12 Hz
- 03T = 1 / 7.12 = 0.1405 s
Frequently Asked Questions
What is resonance and why is it dangerous?
Resonance occurs when an external vibration matches the natural frequency. At resonance, vibration amplitudes grow dramatically (limited only by damping), potentially causing fatigue failure. The Tacoma Narrows Bridge collapse in 1940 is a classic example.
How does damping affect natural frequency?
Damping slightly lowers the actual oscillation frequency. The damped frequency is f_d = f_n x sqrt(1 - zeta^2), where zeta is the damping ratio. For typical structural damping (zeta < 0.1), the difference is negligible.
How do I avoid resonance in machine design?
Design the system so that operating frequencies are at least 20-30% away from any natural frequency. Alternatively, add damping, change mass, change stiffness, or use vibration isolators.
Learn More
Guide
Understanding Vibration Analysis: Natural Frequency, Damping, and Resonance
Learn the fundamentals of mechanical vibration analysis. Covers natural frequency, damping ratio, resonance, forced vibration, vibration isolation, and practical measurement techniques.
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