VM-DX-MECH-004

VM-DX-MECH-004
Optimization of Frequency for a Plate With Simple Support at all Vertices

Overview

Reference:	Blevins, Formula for Natural Frequency and Mode Shape, Van Nostrand Reinhold Company Inc., 1979, pg. 269-271
Analysis Type(s):	Goal Driven Optimization
Element Type(s):	3-D Shell

A square plate of side 250 mm and thickness 5 mm is simply supported on all its vertices.

Input Parameters: — Young's modulus, Poisson's ratio and density

Response Parameters: — First natural frequency

Figure 222: Schematic

Material Properties

E = 2e5 MPa

ν = 0.3

ρ = 7.850 e-6 kg/mm³

Geometric Properties

Length a = 250 mm

Width b = 250 mm

Thickness h = 5 mm

Loading

All vertices are simply supported

Parameter	Type	Constraints	Desired Value	Importance
Young's Modulus E	Input	1.8e11 Pa E 2.2e11Pa	No Preference	Low
Poisson's Ratio μ	Input	0.27 μ 0.30	No Preference	Low
Density ρ	Input	7065 kg/m³ ρ 8635 kg/m³	No Preference	Low
First Natural Frequency w	Output	N/a	Minimum Possible	High

First Natural Frequency:

Objective function becomes:

Minimizing ϕ we get dimensions as:

Young's Modulus E = 1.8e11 Pa

Poisson's Ratio μ = 0.27

Density ρ = 8635 kg/m³

First Natural Frequency w = 124.0913 rad/s

Results	Target	DesignXplorer	Error (%)
Young's Modulus E	1.8e11 Pa	1.8e11 Pa	0.00
Poisson's Ratio μ	0.27	0.27	0.00
Density ρ	8635 kg/m³	8635 kg/m³	0.00
First Natural Frequency w	124.0913 rad/s	123.3403 rad/s	-0.6060