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M N ] ⋅ [ M 1 + ..... + M k + ..... 115) 0 As can be seen, the transition from a single summation to a double summation is necessary to capture all the cross products. Introducing Eqs. 116) where ρq y ( ∆x ) is the covariance coefficient to the distributed load, and where ∆x = x m − x n . The expression in Eq. 116 is equivalent to that which was obtained in Eq. 103. 4: Considering the cantilevered beam shown in Fig. 10 SPATIAL AVERAGING IN STRUCTURAL RESPONSE CALCULATIONS  σ  M  σ q L2  y 2   = 1  N2  N N ∑ ∑ Gˆ M ( xˆn ) ⋅ Gˆ M ( xˆm ) ⋅ ρq y ( ∆xˆ ) n =1 m =1 where Gˆ M ( xˆ ) = G M L = x L = xˆ , ∆x = x m − x n ( covariance coefficient ρq y ( ∆x ) = exp − ∆x ρq y ( ∆x ) = exp ( −∆xˆ ) .

Thus, a symmetric double-sided auto spectrum associated with −ωk as well as +ωk may be defined with a value that is half the corresponding value of the single sided autospectrum. Extending the frequency axis from minus infinity to plus infinity and using the complex Fourier components X k given in Eq. 63) −∞ Thus, the connection between double- and single-sided spectra is simply that S x (ω ) = 2 ⋅ S x ( ±ω ) . 67) where it is taken for granted that N is sufficiently large. The auto-spectral density S x (ω ) is defined by use of circular frequency ω as shown above.

Thus, a symmetric double-sided auto spectrum associated with −ωk as well as +ωk may be defined with a value that is half the corresponding value of the single sided autospectrum. Extending the frequency axis from minus infinity to plus infinity and using the complex Fourier components X k given in Eq. 63) −∞ Thus, the connection between double- and single-sided spectra is simply that S x (ω ) = 2 ⋅ S x ( ±ω ) . 67) where it is taken for granted that N is sufficiently large. The auto-spectral density S x (ω ) is defined by use of circular frequency ω as shown above.

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