Chladni Plate Frequency Calculator

Frequency formula for a thin square plate

For a thin, flat, square plate of side length L, thickness h, density ρ, Young’s modulus E, and Poisson’s ratio ν, the flexural (bending) vibration frequencies can be approximated by plate theory. For a mode identified by two positive integers, m and n, an often used relationship is:

Here, D is the flexural rigidity of the plate:

In these expressions:

• f is the resonance frequency (Hz).
• L is the plate side length (m).
• h is the plate thickness (m).
• ρ is the material density (kg/m³).
• E is Young’s modulus (Pa). In the form you enter here, it is in GPa and converted internally to Pa.
• ν (nu) is Poisson’s ratio (dimensionless).
• m and n are positive integers representing the number of half-wavelengths along each edge direction.

The calculator evaluates these formulas numerically using the SI units you provide.

How to use this Chladni plate frequency calculator

All inputs are in SI units. To estimate a resonance frequency:

1. Plate side length (m): Enter the length of one side of your square plate, measured in meters (e.g., 0.3 for a 30 cm plate).
2. Plate thickness (m): Enter the plate thickness in meters (e.g., 0.002 for a 2 mm metal sheet).
3. Density (kg/m³): Enter the material density. You can start from the typical values in the table below.
4. Young’s modulus (GPa): Enter the material’s Young’s modulus in gigapascals. The script converts this to pascals internally.
5. Poisson ratio: Use a value between about 0.2 and 0.4 for most structural metals and wood-based materials. The default 0.3 is reasonable for many alloys.
6. Mode number m and mode number n: Enter positive integers (1, 2, 3, …). Non-integer values are not physically meaningful for this simple plate model.

After filling out the fields, press the Compute Frequency button. The calculator will output the estimated resonance frequency for the selected mode.

Interpreting the calculated frequency

The result is the approximate natural frequency at which your plate will strongly resonate in the specified (m, n) mode. In practice, this means:

• If you drive the plate at this frequency (for example, with a loudspeaker, shaker, or violin bow), the corresponding Chladni pattern is likely to appear.
• Modes with small (m, n) values tend to have lower frequencies and simpler nodal patterns.
• As m and n increase, the number of nodal lines and the geometric complexity of the Chladni figure increase, and so does the frequency.

The formula shows some key trends:

• Stiffer plates resonate higher: Increasing Young’s modulus E (for example, switching from plywood to steel) increases the flexural rigidity D and therefore the frequency.
• Thicker plates resonate higher: Flexural rigidity scales as h³, so doubling the thickness increases D by a factor of 8. The frequency scales with the square root of D, so frequency roughly multiplies by √8 ≈ 2.8.
• Heavier plates resonate lower: Increasing density ρ makes the plate more massive per unit area, which lowers the frequency.
• Larger plates resonate lower: Frequency varies approximately with 1/L². Doubling the side length reduces frequencies by a factor of about 4.

Worked example: steel Chladni plate

Consider a square steel plate you plan to use in a classroom demonstration. Suppose:

• Side length L = 0.3 m
• Thickness h = 0.002 m (2 mm)
• Density ρ ≈ 7850 kg/m³
• Young’s modulus E ≈ 200 GPa
• Poisson’s ratio ν ≈ 0.3
• Mode numbers m = 1 and n = 2

Using the flexural rigidity expression:

and then substituting D, L, ρ, h, m, and n into the frequency formula, the calculator will return an approximate value for f in hertz. In a lab, you would then sweep an audio signal around that frequency and watch for the appearance of a pattern with one nodal line in one direction and two in the other.

You can repeat this process with different thicknesses, materials, or mode numbers to see how the predicted frequencies shift and to plan which modes are practical to excite with your available equipment.

Typical material properties and their effect

The table below lists approximate densities and Young’s moduli for common plate materials. You can enter these values into the calculator as starting points.

Interpreting this table:

• Aluminum is relatively light and moderately stiff, leading to midrange frequencies for a given size and thickness.
• Steel is heavier but also very stiff. Depending on geometry, its higher stiffness often dominates, giving higher resonances than aluminum plates of the same dimensions.
• Brass is both dense and reasonably stiff, typically producing lower frequencies than aluminum but sometimes comparable to steel, depending on the mode.
• Plywood is light and flexible, so plates made from it vibrate at much lower frequencies and are easier to excite at low audio tones.

Comparison: how geometry and material change frequency

The table below summarizes general trends for how changing one parameter while keeping the others fixed affects the predicted resonance frequencies.

Assumptions and limitations of this model

The calculator is based on idealized plate theory and is intended for educational, exploratory, and preliminary design use. Real plates often deviate from these assumptions. Important points to keep in mind:

• Square geometry: The formula assumes a perfectly square plate with equal side lengths. Rectangular or irregular shapes have different frequency relationships.
• Thin-plate approximation: The derivation uses classical thin-plate theory, which is valid when the thickness is small compared to the in-plane dimensions. Very thick plates require more advanced models.
• Uniform, isotropic material: The material is treated as homogeneous and isotropic. Layered, anisotropic, or strongly direction-dependent materials (such as some composites or carefully carved tonewood) will not be captured accurately.
• Clamped or strongly constrained edges: The underlying formula is most appropriate for plates whose edges are effectively fixed (clamped). Simply supported or free edges change the mode shapes and numeric constants, so real frequencies may differ.
• Small deflections and linear behavior: The analysis assumes small-amplitude vibrations where stress is proportional to strain. At very large amplitudes, nonlinear effects can shift frequencies.
• No damping or losses: Damping from material internal friction, air, or mounting hardware is neglected. In reality, damping broadens resonances and may slightly shift peak frequencies.
• Approximate constants: The exact numerical factors in plate-frequency formulas can vary with edge conditions and modeling choices. The implemented relationship captures main trends rather than delivering metrology-grade predictions.

Because of these limitations, measured frequencies on a physical setup will usually differ somewhat from the calculator’s outputs. You can treat the results as a guide for where to search for resonances and how design changes will move them, rather than as exact predictions.

Practical tips for experiments

To make the most of this tool in a lab or workshop setting:

• Start with lower modes (small m and n) because they are easier to excite and observe.
• Use the computed frequency as a starting point, then sweep a range of ±10–20% around it to find the actual resonance.
• Ensure that the plate is supported in a way that approximates the assumed boundary condition (e.g., clamped or firmly held near its edges).
• Record your measured frequencies and compare them to the predictions to build intuition about how your specific mounting and material behave.