Engineering Tools

Cantilever Beam Natural Frequency Calculator

Natural bending frequencies of a fixed-free beam for the first three transverse vibration modes.

Enter material, length, and cross-section to get the first three natural frequencies, mode by mode.

Why a beam has more than one natural frequency

A cantilever — clamped at one end, free at the other — does not vibrate at a single frequency. It has an infinite family of natural bending modes, each with its own characteristic mode shape and frequency. This calculator returns the first three, which is where nearly all the vibration energy in a real strike or disturbance ends up.

Each mode's frequency comes from the beam's stiffness (Young's modulus times the second moment of area of the cross-section) working against its inertia (density times cross-sectional area), scaled by a mode-specific constant that encodes the fixed-free boundary condition and the mode number. A stiffer or lighter beam rings higher; a longer or heavier one rings lower — and the length term matters more than either, since it enters squared.

The cross-section only shows up through its second moment of area, so a rectangular, solid round, or hollow round section of the same material and length can have very different natural frequencies purely from how that shape resists bending. This calculator does not account for damping, so the numbers describe where resonance sits, not how sharply the beam would actually ring at it.

When to Use This

  • Checking whether a bracket, sensor arm, or probe tip sits too close to a known excitation frequency (motor RPM, gear mesh, fan blade pass) before it is built.
  • Comparing how much a cross-section change (going from solid round to a lighter hollow tube, or thinning a rectangular bar) shifts the fundamental frequency.
  • Getting a first-pass estimate before committing to a full finite-element modal analysis, especially for a simple cantilevered part.

Common Mistakes

  • Treating the fundamental (first mode) frequency as the only one that matters — higher modes can still be excited if the disturbance has energy at those frequencies, and this calculator reports the first three for that reason.
  • Forgetting that this result assumes a genuinely fixed (rigid, non-compliant) support at one end and a genuinely free end. A soft or partially constrained mount lowers the real frequency below this calculation.
  • Applying a cantilever result to a part that is actually supported at both ends, or fixed at both ends — the boundary condition changes the mode-shape constants, not just a scale factor.

FAQ

Where do the default Young's modulus and density values come from?

Selecting a material (steel, stainless, aluminum, copper, or titanium) looks up a fixed pair of typical values from a small built-in table — it is not derived from the entered geometry or from a specific alloy grade. Choose "Custom" to enter your own Young's modulus and density directly if you need a specific material grade.

Why does a longer beam always show a much lower frequency?

Length enters the frequency formula squared and in the denominator, so doubling the length roughly quarters the natural frequency for a given cross-section and material — length change dominates over comparable changes in thickness or material stiffness.

Does this account for damping or for how a real structure is actually mounted?

No. This calculates undamped natural frequencies assuming an ideal rigid clamp at one end and a completely free tip. Real mounts, added mass, and material damping will shift and broaden the actual resonance — treat these numbers as a first-pass estimate, not a substitute for measurement or finite-element modal analysis.