Random response to jet noise excitation

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 $S_0\left(\mathrm{\Omega }\right)$, where $\mathrm{\Omega }$ is frequency, measured in cycles per time. For this case, following Olson, we assume that the excitation is white noise ($S_0=$1.0 at all frequencies) and that the acoustic waves are traveling along the structure with a velocity $c_0$ (where $c_0$ 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 $\overline{x}$ is the distance between the two points for which $S^F$ is being given. This type of loading is specified as a moving noise load in the correlation definition for random response analysis. The correlation definition 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, $\overline{x}$ 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.

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 $P_J\left(\mathrm{\Omega }\right)$ is a complex frequency function defined in the power spectral density definition and referenced in the Jth correlation definition, ${\mathrm{\Psi }}_{\left(N,i\right)\left(M,j\right)}^{IJ}$ is the corresponding Jth correlation matrix for load case I for degrees of freedom i at node N and j at node M, and $F_{\left(N,i\right)}^I$ 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 $S^F\left(\mathrm{\Omega },\overline{x}\right)$ can be accomplished as follows. Concentrated nodal loads $F_{\left(N,i\right)}^1$ 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 $x_N-x_M$, the distance between two nodes N and $M.$$S_0$ is equal to 1.0 (white noise) in this case. These frequency functions are referenced in 21 correlation definitions in the random response step. For each of these options matrices ${\mathrm{\Psi }}_{\left(N,i\right)\left(M,j\right)}^{IJ}$ are defined in user subroutine UCORR, each with unity in the appropriate $\left(N,i\right)\left(M,j\right)$ positions and zero elsewhere.