Flat Plate Natural Frequency Calculator

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This tool provides a calculated natural frequency for flat plates (typically circuit cards) with various boundary support scenarios.

2.0   SCOPE

The tool can be used to quickly identify adequate support methods for a plate of certain size to achieve a desired natural frequency.


Steinberg, D. (2000).  Vibration Analysis for Electronic Equipment, 3rd Edition. New York: John Wiley & Sons Inc. Chapter 6, “Printed-Circuit Boards and Flat Plates”


This tool was developed to provide a means to quickly estimate the natural frequency of circuit cards during the design phase.  The material properties such as density, Poisson’s and Young’s Modulus are preset to typical PWB properties, with the main properties left to adjust being the dimensions.

The tool uses standard flat plate theory from Steinberg, the units are in English.  Metric values could be used if the units involving weght/mass/gravity are minded, however the scaling approximations require English inputs.  The equations are detailed on the References sheet, with both trigometric and polynomial forms of the equations, depending on the origin used.  The References sheet also includes the images with resonances for regular shapes in a multitude of support conditions.

Variables used:
“a” and “b”, the dimensions of the plate
“c” and “d”, the half-dimensions of the plate (for polynomial equations)
“E”, plate material Young’s modulus
“h”, the plate thickness/height
“μ”, plate material Poisson’s Ratio
“γ” or “
ν”, plate material density
“D”, plate stiffness factor (defined on References sheet)
“ρ”, mass per unit area (defined on References sheet)
“Z”, plate deflection at resonance


The natural frequency calculations are detailed for rectangular plates only, however does provide reference values for circular, triangular and hexagonal plates with calculations to scale the results based on the desired plate properties.  As noted on the comment attached to the header, the scaling approximations are more accurate if the proportions are similar and if the plate material properties are similar to the original source as detailed on the References sheet.

The components to build deflection equations are for rectangular plates only.


6.1   Plate Frequencies

Enter the material properties and plate dimensions in the Input sections of the Calculations sheet.  √(D/ρ) will calculate according to the rest of the properties, although can be overridden if this is a known value.  To restore it to the calculated values, set the equation to equal cell C4.  A comment is also attached to this cell with these instructions.

Use the table to find the support condition that best matches your application, or look for a support condition that yields an acceptable natural frequency.  The first fn column is with the supports as defined in the Boundary column.  The “fn (rot 90)” column has swapped the “a” and “b” dimensions; so the “xb” row will first have both “b” edges fixed, then have both “a” edges fixed.  “x” denotes fixed support, “s” denotes simply supported, and free edges are just omitted.  The numerical “a1” or “b2” notation represents a single side per the box on the right, while a lack of number means both sides.

6.2   Scaling Approximations

The scaling equations use the inputs for the 2nd plate in equation 6.77, defined on the References sheet.  The properties for the first plate are the reference values that the images are based on.  Choose the boundary condition per the images on the References sheet, then enter the frequency value into the appropriate f1 cell on the Calculations sheet.

6.3   Deflection Equations


There are no calculations for this portion, merely the pieces to assemble the deflection equation for a particular boundary condition.  Use the trigometric or polynomial (if available) forms according to the coordinate system (they can be mixed).  Select the X variant and the Y variant as shown in the example.

Calculation Reference
Flat Plates
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Plate Bending
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Strength of Materials
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Submitted On:
02 Mar 2017
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#1 JohnDoyle[Admin] 2017-03-02 14:29
Thank you for your debut calculation I have awarded a free 3 month subscription to the site by way of thanks.

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