Objective
To carry out modal analysis of a plate.
Problem statement
To determine the natural frequencies and mode shapes of a simply supported plate.
Design model
A simply supported rectangular plate with material density ρ.
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Analytical solution
M.V. Barton, "Vibration of rectangular and skew cantilever plates", Journal of Applied Mechanics, vol. 18, 1951, P. 129-134.
Geometry
Length a = 1,5 m
Width b = 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
The plate is restrained along its perimeter against displacement in the w (Z) direction.
Loads
Distributed mass weight for modal analysis (t*ρ*g).
Тщеуы
The problem is solved in 2D formulation (model type 3 – XOY-plane).
The model is generated with FE type 19 – quadrilateral FE of slab.
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: 0,075 × 0,05 m.
Nodes: 441. Elements: 400.
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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Comparison of calculation results
| Parameters | Mode shape | Analytical solution | LIRA-FEM | Error, % |
| Frequency, Hz | 1 | 35,63 | 35,587 | 0,1207 |
| 2 | 68,51 | 68,356 | 0,2248 | |
| 3 | 109,62 | 109,434 | 0,1697 | |
| 4 | 123,32 | 122,986 | 0,2708 | |
| 5 | 142,51 | 141,733 | 0,5452 | |
| 6 | 197,32 | 195,676 | 0,8332 |
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