1.4.10 Random response to jet noise excitation

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.

Problem description

Loading

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

where is the distance between the two points for which is being given. This type of loading is specified by using the TYPE=MOVING NOISE parameter provided in the *CORRELATION option for random response analysis. The *CORRELATION option acts between loads applied at the nodes of the model. In this case, since the elements are all of equal length, a load of magnitude 0.25 is applied equally to all nodes to simulate the pressure loading. Thus, has only the discrete values of the distance between any combination of two nodes. Olson points out that this approximation reduces the accuracy of the results unless a rather fine mesh is used. However, the mesh used here provides results that agree well with those of Olson up to relatively high frequencies, suggesting that the approximation is not too coarse.

Loading via user subroutines

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

where is a complex frequency function defined in the *PSD-DEFINITION option and referenced in the Jth *CORRELATION option, is the corresponding Jth correlation matrix for load case I for degrees of freedom i at node N and j at node M, and is the load applied to degree of freedom i at node N in load case I. Since there are 21 nodes in our model and the elements are all of equal length, the construction of can be accomplished as follows. Concentrated nodal loads of 0.25 (corresponding to the length of each element) are applied to all of the nodes. In user subroutine UPSD we specify 21 complex frequency functions,

each for a particular value of , the distance between two nodes N and is equal to 1.0 (white noise) in this case. These frequency functions are referenced in 21 *CORRELATION options in the *RANDOM RESPONSE step. For each of these options matrices are defined in user subroutine UCORR, each with unity in the appropriate positions and zero elsewhere.

Results and discussion

Input files

Reference

Table

Table 1.4.10–1 Root mean square displacements and rotations.

NodeDisplacementRotation
OlsonABAQUSOlsonABAQUS
10.0.0.79880.8679
20.17190.18200.51010.5289
30.22740.23490.27750.3867
40.15570.16560.53190.5537
50.0.0.63080.6811
60.12250.13010.34360.3619
70.15340.15890.24210.3230
80.10400.11230.36620.3840
90.0.0.43780.4921
100.09040.09980.28190.3044
110.11760.12450.22530.3050
120.08410.09320.29020.3139
130.0.0.38010.4315
140.08890.09540.33080.3469
150.13600.14000.22160.2877
160.11290.11880.30050.3185
170.0.0.61130.6539
180.15850.16700.56520.5911
190.23910.24780.21980.3042
200.17930.18840.53780.5615
210.0.0.82350.8821


Figures

Figure 1.4.10–1 Beam subjected to jet noise.

Figure 1.4.10–2 Power spectral density of displacement at node 2.

Figure 1.4.10–3 Root mean square of displacement at node 2.