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4.Conclusions


The results of the wavenumber analysis show that there is little difference in wavenumber content for all modeled cases for excitation frequencies below 300 Hz. By 800 Hz, the wavenumber content varies greatly between the modeled cases. Different wavenumber content means that the actual shape of vibration is different between models. Since the models compared have identical geometry and mesh resolution, the changes in wavenumber content are due to the element formulations.

The Case 1 model (shell element plate with beam element stiffening) and the Case 2 model (shell element plate with shell element stiffening) provided the same response throughout the entire frequency range. This suggests that refining the stiffeners by switching from beams to plates did not affect the modal response of the plate. However, at frequencies above 800 Hz, there appeared to be little to no resonant character present in the Case 1 and Case 2 models, whereas the Case 3a (solid element, reduced integration) and Case 3b (solid element, incompatible mode formulation) continued to predict resonant character.

It should be noted that the small amplitude differences seen at low frequencies between model wavenumber responses that otherwise followed the same trend is due to the frequency resolution of the analysis. An attempt was made to plot the wavenumber response of each model at its resonant frequency. However, the 1 Hz frequency resolution was not sufficient to fully align all resonant frequencies. For a future study, it is recommended that an Eigen study be performed throughout the frequency range to establish the exact natural frequencies of vibration for each model. Performing the Fourier analysis at the exact resonant frequency of each model should result in better amplitude correlation.

In addition to isolating the exact natural frequencies, future work should include evaluation of different plate geometries and stiffening arrangements, such as an isogrid stiffening arrangement, curved plates, and thick plate structures (plate thickness less than 1/10 the length). Higher order elements, such as the 20-node brick element, should also be evaluated. Finally, a two-dimensional wavenumber analysis technique could be used to characterize the entire surface of the plate.


5.References


[1] Wang, Wenchao, Mohamad S. Qatu, and Shantia Yarahmadian. “Accuracy of shell and solid elements in vibration analyses of thin–and thick–walled isotropic cylinders”. International Journal of Vehicle Noise and Vibration 8.3 (2012): 221-236.
[2] Brown, Jeff. "Characterization of MSC/NASTRAN & MSC/ABAQUS elements for turbine engine blade frequency analysis." Proc. MSC Aerospace Users' Conference. 1997.
[3] Dassault Systèmes Simulia. 2012. Abaqus Software. Abaqus/CAE 6.12-1
[4] Benzley, Steven E., et al. "A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis." Proceedings, 4th International Meshing Roundtable. Vol. 17. Albuquerque, NM: Sandia National Laboratories, 1995.
[5] Thompson, Lony L., and Peter M. Pinsky. "Complex wavenumber Fourier analysis of the p-version finite element method." Computational Mechanics 13.4 (1994): 255-275.
[6] Mathworks Software, R2012a, MATLAB Student Version 7.14.0.739.
[7] Cook, R., Malkus, D. Plesha, M., and Witt, J. 2002. Concepts and Applications of Finite Element Analysis, 4th Ed. John Wiley & Sons, Inc.
[8] Altair Engineering Inc., Altair HyperWorks, Version 12.0, HyperMesh v12.0
[9] MatWeb. HY-80 Steel Material Properties. [Online] [Cited: October 15, 2014.]
[10] Leissa, Arthur. 1993. Vibration of Plates. Acoustical Society of America
[11] Dassault Systèmes Simulia, 2013. Abaqus 6.13. Abaqus/CAE User’s Guide
[12] Junger, Miguel C., Feit, David. 1993. Sound, Structures, and Their Interaction. Acoustical Society of America.
[13] Sun, Eric Qiuli. “Shear Locking and Hourglassing in MSC Nastran, ABAQUS, and ANSYS”. Msc software users meeting. 2006.
[14] Fahy, Frank.1985. Sound and Structural Vibration: Radiation, Transmission and Response. Academic Press Inc. (London) Ltd.
[15] MathWorks. R2014b Documentation of Fast Fourier Transform (FFT). [Online] [Cited December 11, 2014]

6.Appendix A – Mesh Refinement Calculation

Note: All these calculations were performed in Excel Spreadsheet



Table 6-1 Geometry Assumptions

Plate Dimensions

Dimension

Units

Value

Length, a

in

100

Width, b

in

100

Thickness, t

in

1

Stiffener Dimensions

Height, h

in

5

Thickness, ts

in

0.5

Table 6-2 Material Properties

Property

Units

Value

Elastic Modulus, E

psi

3.00E+07

Poisson’s Ratio, ν




0.3

Mass Density, ρ

Lbf*s2/in4

(0.284lbm/in3)/(32.174ft/s2)*(12in/ft) = 0.000736

Table 6-3 Wavelength Calculations for Infinite, Unstiffened Plate (Reference 10)

Description

Equation

Value

1. Bending Stiffness

D = E*t^3/(12*(1-ν^2))




(Equ. Around 7.59)

3.00E+07*(1^3)/(12*(1-(0.3^2))) =

2747253

2. Plate Flexural Wavenumber

kf = [ρ*t*(2*π*f)^2/D]^1/4




(Equ. 7.62)

(0.000736*1*((2*PI()*1000)^2)/ 2747253)^(1/4) =

0.320644127

3. Flexural Wavelength

λ = 2*π/kf




(Equ. 2.21)

2*PI()/0.320644127 =

19.59551036

Element Length for Bending (Shell, Brick)

19.59551036/12 = 1.633

4. Compressional Wave Velocity

cp = [E/((1-ν^2)*ρ)]^1/2




(Equ. 2.53)

(3.00E+07/((1-(0.3^2))* 0.000736))^(1/2) =

211701

5. Compressional Wavenumber

c = 2*π*f/k -> kp = 2*pi*f/cp




(Equ. Around 2.21)

2*PI()*1000/211701 =

0.029679457

6. Compressional Wavelength

λ = 2*π/kp




(Equ. 2.21)

2*PI()/0.029679457 =

211.7014891

Element Length for Compression (Brick)

211.7014891/12 = 17.642


Table 6-4 Eigen Modes of SSSS Baseline Plate

m, n

ω = SQRT(D/ρ)*

[(m*π/a)^2 + (n*π/b)^2]

Value

f = ω/2*π

Value

m = 1

n = 1


SQRT(2747253/0.000736)* (((1*PI()/100)^2)+((1*PI()/100)^2))

120.6321

120.6321/(2*PI())

19.1992

m = 2

n = 1


SQRT(2747253/0.000736)*

(((2*PI()/100)^2)+((1*PI()/100)^2))



301.5803

301.5803/(2*PI())

47.9980

m = 1

n = 2


SQRT(2747253/0.000736)* (((1*PI()/100)^2)+((2*PI()/100)^2))

301.5803

301.5803/(2*PI())

47.9980

m = 2

n = 2


SQRT(2747253/0.000736)*

(((2*PI()/100)^2)+((2*PI()/100)^2))



482.5286

482.5286/(2*PI())

76.7968



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