Flat Plate Natural Frequency Calculator

Rating:
4

Description


“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
“?”, plate material Poisson’s Ratio
“?” or “ “D”, plate stiffness factor (defined on References sheet)
“Z”, plate deflection at resonance

Calculation Reference
Flat Plates
Plate Bending
Strength of Materials

The natural frequency of a flat plate is an important parameter when designing structures or components subjected to dynamic loading or vibrations. To calculate the natural frequency of a flat plate, you can use the following steps based on classical plate theory and the Rayleigh-Ritz method:

  1. Define plate geometry and properties: Determine the dimensions (length 'a' and width 'b') and thickness 'h' of the flat plate. Identify the material properties, such as Young's modulus 'E' and Poisson's ratio 'v', and calculate the mass density 'ρ'.

  2. Boundary conditions: Specify the boundary conditions of the plate, which can be simply supported, clamped, or free on its edges.

  3. Formulate the governing equation: Use the classical plate theory to derive the governing equation for the flat plate, which involves the deflection of the plate (w) and its bending stiffness (D). The bending stiffness (D) can be calculated using the formula:

D = (E * h^3) / (12 * (1 - v^2))

  1. Assume a deflection shape: Assume a sinusoidal deflection shape for the flat plate, which can be expressed as a product of sine functions in both the x and y directions:

w(x, y) = A * sin(m * π * x / a) * sin(n * π * y / b)

where 'A' is the amplitude, and 'm' and 'n' are the mode numbers in the x and y directions, respectively.

  1. Apply the Rayleigh-Ritz method: Apply the Rayleigh-Ritz method by substituting the assumed deflection shape into the governing equation and calculating the strain energy (U) and kinetic energy (T) of the plate.

  2. Calculate the natural frequency: The natural frequency (ω) can be obtained by minimizing the Rayleigh quotient, which is the ratio of the strain energy (U) to the kinetic energy (T):

ω² = U / T

Calculate the natural frequency (ω) by taking the square root of the obtained value for ω².

  1. Convert to frequency: Convert the natural frequency (ω) to frequency (f) using the following formula:

f = ω / (2 * π)

These steps provide an approximate method for calculating the natural frequency of a flat plate based on classical plate theory and the Rayleigh-Ritz method. The accuracy of this method depends on the complexity of the plate geometry, boundary conditions, and material properties. For more accurate results, especially for complex or irregular plates, consider using numerical methods, such as the Finite Element Method (FEM), which can provide detailed solutions for the natural frequencies and mode shapes of the plate.

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Uploaded
01 Mar 2017
Last Modified
24 Apr 2023
File Size:
171.87 Kb
Downloads:
78
File Version:
1.0
File Author:
Timothy
Rating:
4

 
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Comments: 2
ghazard@bakerco.com 4 years ago
Dear Author of ExcelCalcs,You have a very nice program which I am using for determining the natural frequency of flat plates. Questions 1 - 4 have been answered by John Doyle, however I'm curious about responses to questions 5, 6 and 7 below and would appreciate guidance from you on these points. 5. What is the meaning of a1, a2, b1, b2, c and d? Do a1, a2, b1 and b2 signify the dimensions of a generalized polygon with different dimensions a1, a2, b1 and b2? >>> John Doyle writes, 'I am also unclear on this please raise a comment with the author on the download page.'6. What is the meaning of “c” and “d”? They appear to the be midpoints in both dimensions of the plate, but I’d like to verify this interpretation.>>> John Doyle writes, 'I am also unclear on this please raise a comment with the author on the download page.'7. If a1, a2, b1 and b2 are dimensions of a polygon having different dimensions for all 4 sides, then how are these 4 values inputted into the Plate Frequency Calculator in cells B5:6?>>> John Doyle writes, 'I am also unclear on this please raise a comment with the author on the download page.'Thank you in advance,Gary Hazard
JohnDoyle[Admin] 7 years ago
Thank you for your debut calculation I have awarded a free 3 month subscription to the site by way of thanks.