Product: ABAQUS/Standard
This example illustrates and verifies the random response analysis capability in ABAQUS with a simple beam example that was originally studied by Olson (1972). The problem is a five-span continuous beam exposed to jet noise. The example is solved using the built-in moving noise loading option and, as an illustration, with user subroutines UPSD and UCORR.
Except for the assumption that time is measured in seconds (so that frequencies are expressed in Hz), no specific set of units is used in this example. The units are assumed to be consistent.
The structure is a five-span straight beam, simply supported at its ends and at the four intermediate supports (Figure 1.4.101). Each span has unit length. The beam is excited in bending. It has unit bending stiffness and mass of 1 × 104 per unit length.
Each span is modeled with four elements of type B23 (cubic beam in a plane), as shown in Figure 1.4.101. No mesh convergence studies have been performed; however, the first 15 natural frequencies agree quite well with the exact values given by Olson, so we assume that the mesh is reasonable. The response analysis is based on 1% of critical damping in each mode, as used by Olson.
Jet noise is an acoustic excitation that applies random pressure loading to the surface of a structure. The pressure at a point is assumed to have a power spectral density , where is frequency, measured in cycles per time. For this case, following Olson, we assume that the excitation is white noise (1.0 at all frequencies) and that the acoustic waves are traveling along the structure with a velocity (where is taken to be 6.0 in this case). The cross-spectral density of the pressure loading between any two points can then be written as
For purposes of illustration we also show input data for the case where we apply the loading via user subroutines UPSD and UCORR. These subroutines allow the user to define a different frequency dependence and magnitude for each entry in the cross spectral density matrix. Any number of frequency functions can be used to define the cross spectral density of the loading as
The first 15 natural frequencies agree closely with the exact values given by Olson, suggesting that the mesh is suitable for frequencies up to at least 110 Hz.
The random response results obtained with the two approaches are identical within numerical accuracy. Figure 1.4.102 illustrates the power spectral density of the transverse displacement at node 2. These results, and similar plots for other nodes and for rotations, are in good agreement with those obtained by Olson (1972).
Figure 1.4.103 shows the root mean square (RMS) value of the transverse displacement at node 2. Since the higher modes tend to contribute less and less to the response, we expect the RMS values to level off as the frequency increases. As shown in Table 1.4.101, the RMS values of rotation and transverse displacement at all nodes along the beam are seen to be in good agreement with Olson's results.
Eigenvalue extraction step.
Restart run for the random response analysis.
Restart run for the random response analysis.
Olson, M. D., A Consistent Finite Element Method for Random Response Problems, Computers and Structures, vol. 2, 1972.
Table 1.4.101 Root mean square displacements and rotations.
Node | Displacement | Rotation | ||
---|---|---|---|---|
Olson | ABAQUS | Olson | ABAQUS | |
1 | 0. | 0. | 0.7988 | 0.8679 |
2 | 0.1719 | 0.1820 | 0.5101 | 0.5289 |
3 | 0.2274 | 0.2349 | 0.2775 | 0.3867 |
4 | 0.1557 | 0.1656 | 0.5319 | 0.5537 |
5 | 0. | 0. | 0.6308 | 0.6811 |
6 | 0.1225 | 0.1301 | 0.3436 | 0.3619 |
7 | 0.1534 | 0.1589 | 0.2421 | 0.3230 |
8 | 0.1040 | 0.1123 | 0.3662 | 0.3840 |
9 | 0. | 0. | 0.4378 | 0.4921 |
10 | 0.0904 | 0.0998 | 0.2819 | 0.3044 |
11 | 0.1176 | 0.1245 | 0.2253 | 0.3050 |
12 | 0.0841 | 0.0932 | 0.2902 | 0.3139 |
13 | 0. | 0. | 0.3801 | 0.4315 |
14 | 0.0889 | 0.0954 | 0.3308 | 0.3469 |
15 | 0.1360 | 0.1400 | 0.2216 | 0.2877 |
16 | 0.1129 | 0.1188 | 0.3005 | 0.3185 |
17 | 0. | 0. | 0.6113 | 0.6539 |
18 | 0.1585 | 0.1670 | 0.5652 | 0.5911 |
19 | 0.2391 | 0.2478 | 0.2198 | 0.3042 |
20 | 0.1793 | 0.1884 | 0.5378 | 0.5615 |
21 | 0. | 0. | 0.8235 | 0.8821 |