spectral density and noted that, as far as displacement and stress responses are concerned, the spectral density approximately equals the one that would be excited in a system by a white noise excitation with spectral density equal to the actual input spectral density at the natural frequency of the SDF system. Specifically, if the excitation random process is zero-mean and stationary, with spectral density ( ) −∞< <∞ ω ω , SQQ , and if the SDF system has FRF ( ) −∞< <∞ ω ω , H , then the response spectral density is given by Eq. (33). Miles’ approximation states that the response spectral density can be approximated by ( ) ( ) ( ) −∞< <∞ = ω ω ω ω QQ n XX H S S 2 (37) where nω is the natural frequency of the SDF system. For example, when an SDF system with natural frequency rad / sec nω π 2= , and damping factor 0 05 . =ζ , is excited by an input with the spectral density shown in Figure 13, its displacement response is a random process with the spectral density shown in Figure 13. The response spectral density is the product of the modulus squared of the FRF (also shown) and the input spectral density. Miles’ approximation to the response spectral density is the product of the white noise spectral density with magnitude ( ) QQ n S ω and the modulus squared of the FRF, both shown in Figure 13. The RMS value of the response is the square root of the area under the response spectral density curve. In this case the exact value is 0.312 in, and the value that comes from the approximation is 0.297 in. For this example, the approximation yields accuracy of about five percent. (Of course, in individual cases, the approximation may be better or worse.) Miles’ paper may be better known for this intermediate result than for its final result involving fatigue analysis. Figure 13. Graphic showing the meaning of Miles’ approximation to the response spectral density. The response spectral density approximation that Miles defined is so attractive that it has gained widespread acceptance, and is in use in many applications. For example, NASA uses it in combination with finite element models and a load combination scheme to establish an approximation for the overall load on a system. (See, for example, Himelblau, et al., 2001, Ferebee, 2000, and NASA, 2002.) Thomson and Barton (1957) wrote a paper that extended the approximation of Miles. They pointed out that through modal analysis the equations of motion of a complex linear structure can be reduced to a set of equations that have the form of the equation governing motion of an SDF system. These simple equations can be evaluated individually, then the mean square responses in the modes can be synthesized to approximate the mean square response at a point on the system. Their modal assumption is that the response at a location r on a complex system ( ) ∈ −∞< <∞ t , x ,t , r r V , can be expressed as a series 10-1 100 101 10-4 10-2 100 Frequency, Hz SQQ(f) |H(f)|2 SXX(f) SQQ(fn) |H(f)|2SQQ(fn) 124
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