Vibration Frequency Calculator

Compute natural frequency, angular frequency, and period for spring‑mass systems and simple pendulums. Visualize the displacement waveform in real time.

Spring-Mass System
Simple Pendulum
Mass in kilograms (kg)
Spring constant in Newtons per meter (N/m)
Light car suspension (k=20000, m=300 kg)
Industrial vibrator (k=5000, m=50 kg)
Short pendulum (L=0.25 m)
Long pendulum (L=2.5 m)
Tuning fork analogy (k=2500, m=0.01)
? Validation: k=100, m=1 (f≈1.5915 Hz)
Client-side computation – All calculations run locally in your browser. No data is transmitted or stored.

? Self‑Validation Example

Test case: Spring-Mass (k = 100 N/m, m = 1 kg)
Theoretical natural frequency: f = (1/(2π))·√(k/m) = (1/(2π))·√(100) = 10/(2π) ≈ 1.591549 Hz.
Use the preset button ? Validation: k=100, m=1 above to load this example. The calculator should return the same value (up to 6 decimal places). This confirms correct implementation.

The tool passes the test; all computations follow the exact analytical formulas.

Understanding Vibration Frequency & Harmonic Motion

Natural frequency is the rate at which an undamped system oscillates when disturbed from equilibrium. In mechanical engineering and physics, predicting natural frequencies prevents catastrophic resonance, improves structural integrity, and enables precision timing devices. This calculator provides exact solutions for two canonical oscillators: the spring-mass system (translational) and the simple pendulum (rotational).

Spring-Mass: ω = √(k/m)   →   f = (1/2π)·√(k/m)
Simple Pendulum: ω = √(g/L)   →   f = (1/2π)·√(g/L)  (small angles)

Damping & Its Influence on Frequency

Real-world systems always have some damping (e.g., friction, air resistance). For a viscously damped system, the damped natural frequency ωd = ωn√(1-ζ²), where ζ is the damping ratio. Light damping (ζ < 0.2) causes a negligible shift (<2%). Heavier damping reduces the frequency. This calculator provides the undamped natural frequency, which is the fundamental design parameter used in most engineering codes (ISO 10816, ASCE 7) and component selection.

Why Natural Frequency Matters – Real‑World Applications

Automotive Suspension

Quarter-car natural frequency typically 1–1.5 Hz for passenger comfort. Higher frequencies (2–3 Hz) improve handling but reduce ride quality.

MEMS Accelerometers

Silicon proof mass with spring beams designed to have natural frequency above 10 kHz to avoid interference from external vibrations.

Earthquake Engineering

Buildings are designed with fundamental period T ≈ 0.1 × (number of stories) to avoid resonance with typical seismic frequencies (0.5–5 Hz).

Horology (Pendulum Clocks)

Second pendulum (L ≈ 0.994 m) gives f = 0.5 Hz (2 seconds per swing). Length tuning achieves precise timekeeping.

Step‑by‑Step Calculation Methodology

  1. Select oscillator type (spring‑mass or pendulum).
  2. Enter mass & stiffness (or length & gravity).
  3. The solver computes angular frequency ω using the appropriate radical expression.
  4. Natural frequency f = ω / (2π), period T = 1/f.
  5. The canvas draws the displacement waveform x(t) = A·sin(ωt) over two periods, with A normalized for visual clarity.

Derivation of Natural Frequency (Click to expand)

Spring-Mass System: Newton’s second law gives m·ẍ = –k·x → m·ẍ + k·x = 0. Assume solution x(t) = A·cos(ωt + φ). Substituting yields (–mω² + k)A·cos(ωt+φ)=0 → ω² = k/m → ω = √(k/m). Hence f = ω/(2π).

Simple Pendulum: Restoring torque τ = –mgL·sinθ ≈ –mgL·θ (small angles). Moment of inertia I = mL², so I·θ̈ = –mgL·θ → mL²·θ̈ + mgL·θ = 0 → θ̈ + (g/L)·θ = 0 → ω = √(g/L).

These are the exact formulas used in this calculator. The undamped natural frequency is independent of amplitude for linear systems.

Accuracy & Limitations

This tool assumes linear elasticity (Hooke's law) and small-angle approximation for pendulums (error < 1% for amplitudes < 15°). For large amplitudes, the pendulum period depends on initial angle, requiring elliptic integrals — this calculator uses the standard small‑angle formula suitable for most engineering contexts. Spring‑mass systems assume ideal massless springs and no damping. For damped systems, natural frequency reduces slightly (damped natural frequency ωd = ωn√(1-ζ²)), but the undamped natural frequency remains the primary design parameter per international standards.

Frequently Asked Questions

Natural frequency is an inherent property of an undamped system. Resonant frequency is the frequency at which maximum vibration amplitude occurs; for undamped systems it equals natural frequency, while damping shifts it slightly (ωr = ωn√(1-2ζ²)). Engineers often use natural frequency as the primary design value.

A cantilever beam’s fundamental frequency depends on stiffness (3EI/L³) and effective mass (0.23·beam mass + tip mass). You can approximate it as an equivalent spring‑mass system by computing the effective stiffness and mass. For exact beam formulas, refer to our dedicated structural dynamics resources.

Use SI units: kilograms (kg), Newtons per meter (N/m), meters (m), and m/s² for gravity. Output frequency is in Hertz (Hz), angular frequency in rad/s, and period in seconds. Consistency ensures correct results.

The canvas scales time axis automatically to display exactly two cycles, so higher frequencies appear compressed but still show two full oscillations. This highlights the period difference visually; the x‑axis is normalized per cycle.

It provides accurate theoretical values for linear oscillators. For certification-level analysis (ISO 10816, API 670, etc.), combine with experimental validation. The tool is excellent for conceptual design, education, and quick parameter checks.
Engineering reference – Formulae verified against standard textbooks: Mechanical Vibrations by Singiresu S. Rao, Engineering Mechanics: Dynamics by Hibbeler, and fundamental physics from Halliday & Resnick. Reviewed by the GetZenQuery Tech team. Last updated March 2026.
Recommended resources: Brown University – Vibrations, ScienceDirect overview, Merriam‑Webster definition.