Natural Frequency Calculator

What Is Natural Frequency and Why Does It Matter?

Natural frequency is the frequency at which a structure vibrates freely when disturbed. If an external excitation frequency matches the natural frequency, resonance occurs — amplitudes grow dramatically, potentially causing catastrophic failure. Every rotating machine, beam, shaft, and structure has natural frequencies that must be identified and avoided.

For beams, natural frequency depends on boundary conditions (cantilever, simply supported, fixed-fixed), material stiffness (Young's modulus E), moment of inertia (I), mass per unit length, and span length. The Euler-Bernoulli beam equation provides mode shapes and frequencies. Higher modes have progressively higher frequencies.

In practice, operating speed should be at least 20% away from any natural frequency (above or below). Critical speed analysis is mandatory for high-speed shafts, turbines, and compressors. Finite element analysis (FEA) is used for complex geometries. Note that this margin is a design guideline for avoiding resonance, distinct from ISO 10816's vibration severity limits, which classify measured vibration on running machinery rather than predicted natural frequencies.

Formula: Spring-Mass: fn = (1/2π) × √(k/m) Cantilever: fn = (λn²/2πL²) × √(EI/ρA), λ₁=1.875 Simply Supported: fn = (nπ/2L²) × √(EI/ρA) Shaft-Disk: fn = (1/2π) × √(GJ/(L×Ip)), Ip=½mr²

Example Calculation

A steel cantilever beam: L = 500 mm, rectangular 20×10 mm cross-section. E = 200 GPa, ρ = 7850 kg/m³. I = 20×10³/12 = 1667 mm⁴. A = 200 mm². fn₁ = (1.875²)/(2π×0.5²) × √(200e9×1.667e-9/(7850×2e-4)) = 32.62 Hz.

When to Use This Calculator

• Designing a shaft for a motor or pump and verifying that the operating RPM is at least 20% away from the critical speed
• Troubleshooting excessive vibration in rotating machinery — checking whether the operating frequency is near a natural frequency
• Sizing a mounting bracket or support structure to ensure its natural frequency is above the excitation frequency of nearby equipment
• Performing a preliminary dynamic analysis before committing to a full FEA modal analysis

Common Mistakes to Avoid

• Using the wrong boundary condition — a beam that appears simply supported may actually be partially fixed, which significantly changes the natural frequency
• Ignoring higher modes — the fundamental frequency is important, but 2nd and 3rd modes can also cause resonance at 2× and 3× the operating speed
• Neglecting the mass of the beam/shaft itself — for lightweight shafts with heavy mounted components the beam mass may be negligible, but for uniform beams it is the dominant mass
• Assuming that operating below the first natural frequency is always safe — subharmonics, misalignment, and bearing defect frequencies can excite resonance at fractions of the natural frequency

How to Interpret Results

• If the operating speed is within 20% of any natural frequency, redesign is required — either stiffen the structure (increase frequency) or add mass (decrease frequency) to move away from resonance
• A frequency ratio (operating/natural) between 0.7 and 1.3 is the danger zone where amplification factors exceed 3×
• Multiple modes close together indicate a risk of coupled vibration — consider a full FEA modal analysis in this case
• Very low natural frequency (<5 Hz) suggests the structure may be too flexible for precision machinery applications

Related Standards & References

• ISO 10816 — Mechanical vibration — Evaluation of machine vibration by measurements on non-rotating parts (severity limits for measured vibration; this calculator predicts natural frequencies, a different quantity)
• ISO 1940-1 — Mechanical vibration — Balance quality requirements for rotors
• API 610 — Centrifugal pumps (includes critical speed requirements)
• API 617 — Axial and centrifugal compressors (lateral and torsional vibration analysis requirements)

Frequently Asked Questions

How do I avoid resonance in rotating machinery?

First, calculate natural frequencies of the shaft, bearings, and support structure. Ensure operating speed is at least ±20% from any natural frequency. If resonance cannot be avoided during run-up/coast-down, accelerate through it quickly. Add damping (rubber mounts, tuned mass dampers), stiffen the structure (increase cross-section, shorten spans), or change the mass distribution.

What is the difference between natural frequency and critical speed?

Natural frequency is a general vibration property of any structure. Critical speed specifically refers to a rotating shaft's natural frequency — the RPM at which shaft deflection becomes theoretically infinite due to resonance. A shaft operating above its first critical speed is called 'supercritical' and must pass through the critical speed during startup. Supercritical operation is common in turbomachinery.

How does damping affect the resonance response?

Damping reduces the amplitude at resonance from theoretically infinite (zero damping) to a finite value. The damping ratio ζ determines the amplification factor: at ζ = 0.01 (steel structure), amplification is ~50×; at ζ = 0.05, it is ~10×; at ζ = 0.1, it is ~5×. Adding damping (rubber mounts, constrained layer damping, tuned mass dampers) is often the most practical solution when redesigning for frequency avoidance is not feasible.