Objective
Modal analysis of a thin plate.
Problem statement
To determine the natural frequencies and mode shapes of a thin plate.
Design model
A square plate with material density ρ.
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Variant 1
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Variant 2
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Reference
M.V. Barton, "Vibration of rectangular and skew cantilever plates", Journal of Applied Mechanics, vol. 18, 1951, P. 129-134.
Geometry
Side length a = 1 m
Thickness t = 0,01 m
Material properties
Modulus of elasticity Е = 2,1 * 108 tf/m2
Poisson's ratio v = 0,3
Material density ρ = 7800 kg/m3
Boundary conditions
Variant 1: All degrees of freedom are restrained along edge DC.
Variant 2: The plate is free of rigid restraints and supported by elastic supports.
Loads
Distributed mass weight for modal analysis (t*ρ*g).
Note
The problem is solved in 3D formulation (model type 5).
To ensure geometric stability in variant 2, elastic spring elements (FE 56) with a stiffness of 0.001 tf/m are assigned to all nodes of the model in the w, u, and v displacement directions (Z, X, and Y).
The model is generated with FE type 44 – arbitrary quadrilateral FE of shell.
The mass weights are defined using the «Weight of distributed dynamic mass» load.
A dynamic analysis (modal analysis) is performed.
Number of mode shapes considered: 10.
Finite element size: 0,05 × 0,05 m.
Variant 1: Nodes: 441. Elements: 400.
Variant 2: Nodes: 441. Elements: 841.
Output data
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1st mode shape
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2nd mode shape
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3rd mode shape
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4th mode shape
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5th mode shape
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6th mode shape
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7th mode shape
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8th mode shape
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9th mode shape
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10th mode shape
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11th mode shape
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Comparison of calculation results
| Parameters | Mode shape | Analytical solution | LIRA-FEM | Error, % | |
| Frequency, Hz | Var.1 | 1 | 8,7266 | 8,668 | 0,6715 |
| 2 | 21,3042 | 21,206 | 0,4609 | ||
| 3 | 53,5542 | 52,962 | 1,1058 | ||
| 4 | 68,2984 | 67,487 | 1,1880 | ||
| 5 | 77,7448 | 76,95 | 1,0223 | ||
| 6 | 136,0471 | 134,098 | 1,4327 | ||
| Var.2 | 1 – 3 | 0 | 0,3786 | - | |
| 4 – 6 | 0 | 0,3961 | - | ||
| 7 | 33,7119 | 33,548 | 0,4862 | ||
| 8 | 49,4558 | 48,568 | 1,7951 | ||
| 9 | 61,0513 | 60,263 | 1,2912 | ||
| 10 | 87,516 | 86,229 | 1,4706 | ||
| 11 | 87,516 | 86,229 | 1,4706 | ||
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