Vibrations of  a Simply Supported Plate Because of the Middle Point Oscillation

Let us consider a simply supported plate, the middle section of which harmonically oscillates with an amplitude of 0.5 mm. (see figure)

Vibrations of a Spring-Mass System Because of Oscillating Foundation

Our aim is to find oscillation amplitudes of a point with coordinate x using the specified frequencies.

Let us use the following initial data: length of the plate L = 850 mm, the cross section is a rectangle with width b = 75 mm, height h = 5 mm. Harmonic force is applied at a point with x=0.5L=212.5 mm.

Parameters of the material: modulus of elasticity E=2.1E+011Pa, Poisson's ratio ν=0.28, density γ=7800kg/m3.

 

Classic analytical solving

Natural (resonant) frequencies of the system are:

Where Jx=bh3/12 - the moment of inertia of cross-section;

f1,2,3,4= 16.2826; 65.1304; 146.5434; 260.5216. Thus, first natural frequency falls in range 4Hz to 32Hz.

Dynamic deflection at point x is calculated by formula:

.

Maximum deformation is attained at ωt=π. Deflection under the dynamic load at 4Hz to 32Hz:  ΔZdyn = 0.344039; 0.344910; 0.346381; 0.348479; 0.351244; 0.354734; 0.359022; 0.364205 mm.

Numerical solution

Let us solve this study by AutoFEM Analysis package. Both ends are restrained to simulate simple support: displacements of the left end along XYZ-axis are forbidden and rotation only around Y-axis is allowed; displacements of the right end along YZ-axis are forbidden and rotation only around Y-axis is allowed.

Forced Oscillation of a Spring-Mass System, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

First eigenfrequency is equal to f(1)n =16.286 Hz (the result "Mode 01 (16.286 Hz)" of the study "Study 2 (eigenfrequencies)").

Vibrational amplitudes have the following values: Z*dyn= see table 2 (results "4.000 Hz-Form, Amplitude Z ... 32.000 Hz-Form, Amplitude Z" of the study "Study 3  (forced oscillations)").

Let us compare the results of calculation:

Table 1. Parameters of the finite element mesh

Finite element type

Number of nodes

Number of finite elements

linear triangle

585

352

Table 2. The results

Frequency ff , Hz

Analytical solution
R

Numerical solution
R*

Error δ = 100* | R* - R | / | R |, %

4

0.344039

0.3460

0.57

8

0.344910

0.3478

0.84

12

0.346381

0.3507

1.25

16

0.348479

0.3548

1.81

20

0.351244

0.3603

2.58

24

0.354734

0.3672

3.51

28

0.359022

0.3756

4.62

32

0.364205

0.3858

5.93

 

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

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