A forced frequency response analysis was performed on each model using Abaqus. A sinusoidal excitation of a one pound force was applied to the center of the plate through a frequency range of 1 to 1000 Hz at 1 Hz increments. Figure shows the baseline model being driven at a node in the center. The vibrational displacement was saved along the line of nodes also shown in Figure . Using the same method as was used for the boundary conditions, the drive and response nodes were selected along the midplane for the solid element models.
Figure Drive Location and Response Nodes
A wavenumber analysis breaks down the response of the plate into the wavelength components and allowing characterization of the contributions of the waves. A spatial Fourier transform (or wavenumber transform) is used to convert the response from the spatial domain to the wavenumber domain using Equation 7 as described in Fahy, [14].
[7]
Here f(x) is the spatial response and F(k_{s}) is the wavenumber response. The spatial Fourier transform can be found using a Fast Fourier Transform (FFT).
There were 67 response nodes across the plate. For each model, the vertical responses across these nodes were interpolated to 100 response points across the plate for spacing of 1 inch. Since the plate length was 100 inches, the sample rate was 100 response points/100 inches or 1 1/in. The sample rate divided by 2 is the maximum wavenumber supported. The wavenumber increment was determined to be the sample rate divided by the number of response points. The interpolated responses were transformed to the wavenumber domain using the builtin Matlab FFT function. The full Matlab code used for processing and plotting the results is provided in Appendix C. This process was derived from the MathWorks Fast Fourier Transform documentation and examples, [15].
3.Results/Discussion
The models were run using Abaqus and analyzed using Matlab. The results are compared in the following sections. Appendix B contains sample Abaqus input files for the analysis inputs. Appendix C contains the Matlab scripts.
An eigen analysis was performed on the baseline model to compare the mode shapes and frequencies of the plate against analytical values. Appendix B includes a sample Abaqus input file for the eigen analysis.
The side profile of the first mode shape of the plate is shown in Figure . This sort of profile is expected for a plate, simplysupported on all sides [10].
Figure Baseline First Mode Shape Profile
The first four mode frequencies were calculated analytically as well as by the model. Equation 8, [10], was used to find the angular frequency, ω, for the plate.
[8]
Here D is the bending stiffness calculated previously using Equation 1, ρ is the density of the plate material, a and b are the length and width of the plate, and m and n are the order of the waves traveling along each dimension. Dividing the angular frequency, ω, by 2π gives the frequency, f, in cycles per second (Hz). Table provides a comparison of the first four mode frequencies calculated by the model versus the frequencies calculated analytically. The full calculation of the analytical solution can be found in Appendix A.
Table Frequency Comparison of First Four Modes
Mode

Abaqus Solution (Hz)

Analytical Solution (Hz)

Wave Order

1

19.192

19.199

m = 1, n = 1

2

47.982

47.998

m = 2, n = 1

3

47.984

47.998

m = 1, n = 2

4

76.728

76.797

m = 2, n = 2

Figure shows the first four modes of the plate.
Figure Baseline Modes:
(a) First Mode at 19.192 Hz, (b) Second Mode at 47.982 Hz, (c) Third Mode at 47.987 Hz, (d) Fourth Mode at 76.728 Hz
The excitation of the plate results in a high response at the natural frequencies of the plate. Any differences in response due to the differing element formulations will be most apparent at these peak amplitudes. Figure plots the response at the location of the driving force for each frequency.
Figure Resonant Peaks in Response at Drive Location
Seven significant resonant peaks were selected from this figure for comparison. These are natural frequencies of the plate. It was observed that the element types shifted the frequencies of the resonances with significant shifting occurring in peaks above 400 Hz. Table shows the frequencies for each case.
Table Frequencies for Each Peak (Hz)
Peak

Case 1

Case 2

Case 3a

Case 3b

1

37

37

38

37

2

96

96

96

96

3

227

227

223

228

4

255

255

259

262

5

279

279

288

280

6

900

900

884

881

7

966

966

945

952

All the cases share the same frequencies for the first two peaks. Some frequency shifting between cases occurs for the next three peaks. After 400 Hz, it becomes more difficult to determine which the frequencies are part of the same resonance. The last two peaks occur in Cases 3a and 3b but it is more unclear where they occur in Cases 1 and 2. A higher frequency resolution might make these resonances more clear and would ensure that no resonances were missed.
