Pick a structure (cantilever, simply supported or fixed-fixed beam, circular plate, rectangular membrane), set geometry and material, and get the first five natural frequencies plus an animated mode shape, all in real time.
What does "natural frequency" really mean? What can I learn from this simulator?
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Roughly, it's the rhythm a structure wobbles at on its own, like a guitar string going "twang". Try moving the Length L slider — see how the first natural frequency drops fast as L grows.
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Doubling the length cut the frequency to about a quarter — why so dramatic?
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Beam fn ∝ 1/L², so doubling L gives 1/4 the frequency. Long beams are floppy; short ones are stiff. Now switch the structure type from cantilever to fixed-fixed and watch the frequency jump — even at the same length.
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The animation shape is different for each "mode". What does mode 2 mean?
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Mode 1 is the simplest hump, mode 2 has one node (a point that doesn't move), mode 3 has two, and so on. Each mode has its own frequency. In real machines, if any external excitation lands on one of these, you get resonance — designers detune by tweaking material (E, ρ) or adding ribs.
FAQ
Cantilever beams have the lowest first frequency, fixed-fixed the highest, simply supported in between. The eigenvalue β1L drops into the formula directly, so design first matches the actual constraint.
Material datasheets or any engineering handbook (steel ≈ 200 GPa / 7850 kg/m³, aluminum ≈ 70 GPa / 2700 kg/m³, etc.). I (second moment of area) is computed for you from b and h.
For higher than the 5th mode, use a general-purpose FEM solver such as ANSYS or Abaqus. In most practical designs the lowest few modes dominate.
Common causes: wrong E or ρ, idealized boundary condition (real fixings are never perfectly clamped or simply supported), neglected damping, added mass, or measurement error in length and thickness.
Real-world applications
Skyscrapers: Behave like vertical cantilevers; designers detune the first mode away from dominant wind and earthquake periods.
Turbine blades: Spinning cantilevers whose natural frequencies must avoid integer multiples of the rotation × stator-blade count to escape high-cycle fatigue.
Hard-disk and brake-disc design: Circular plates with clamped boundary conditions; the simulator's plate option gives a quick first estimate.
Musical instruments: Guitar strings (membrane analog) and piano sound boards (plates) — pitch and tone come from the very same equations.
Common misconceptions
The calculated natural frequency is an estimate, not a guaranteed safety value: real structures usually have stiffer-than-ideal joints, so measured frequencies tend to come in higher. Watch your units — entering E in MPa instead of GPa makes the answer 1000× wrong. Don't stop at the 1st mode either: if the excitation lies above f1, mode 2 or 3 may resonate, and the animation shows you exactly where the nodes are so you know where damping treatment will be most effective.
Select structure type (cantilever beam, simply supported beam, fixed-fixed beam, circular plate, or rectangular membrane) from the dropdown.
Input geometry: length (valLNum in mm), width (valB2Num in mm), height (valHNum in mm). Use sliders for fine adjustment.
Set elastic modulus (valENum in GPa) for your material—steel 200 GPa, aluminum 69 GPa, carbon fiber 120 GPa.
Click Calculate to compute fundamental natural frequency in Hz and display mode shape visualization.
Worked Example
A cantilever steel beam: length L=500 mm, rectangular cross-section 20×40 mm (width×height), E=200 GPa, density ρ=7850 kg/m³. The calculator derives second moment of inertia I=10,667 mm⁴, then applies f₁=(λ₁²/2π)√(EI/μL⁴) where λ₁=1.875 for cantilever and linear mass μ=0.628 kg/m. Result: f₁≈18.4 Hz. For comparison, a simply supported beam with identical properties yields f₁≈12.2 Hz due to boundary condition differences.
Practical Notes
Cantilever beams resonate lower than simply supported spans of equal length; fixed-fixed beams raise frequency by ~60% due to constraint stiffness.
Rectangular plate natural frequencies scale with thickness⁰·⁵ and inversely with length²—doubling height increases f₁ by √2≈1.41.
Density is material-dependent: titanium (4500 kg/m³) yields higher frequencies than aluminum (2700 kg/m³) at same E and geometry.
Verification: compare simulator results against published tables (e.g., Roark's Formulas for Stress and Strain) for validation before production design.