in which n, and Z, are the frequency and damping ratios in the ¿th mode. The spectral density of the generalized force takes the form r- H r- H

%(n)=p2CDB2Z2(«W / U(zJU(z2)SUlU2(r, n)f,(zf(z2) dz: dz2 (22.38)

i Jo Jo where (r, n) is the cross-spectral density defined in Equation 22.14 with r being the distance between the coordinates z1 and z2. In Equation 22.38, the aerodynamic admittance has been incorporated to account for the distortion caused by the structure to the turbulent velocity. In view of Equation 22.15, Equation 22.38 may be expressed as r- H r- H

Sp.(n) = r2CDB2Z2(n) / / f,(zf (%) U(zi) u2(n) g(r, n) dz 1 dz2 (22.39)

i Jo Jo where g(r, n) is the square root of the coherence given in Equation 22.16 and Su(n) is the spectral density of the turbulent velocity. The variance of the ith normal coordinate is obtained from aX = Sq,(n) dn = —2 |H,(n)|2 Sp.(n) dn (22.40)

The calculation of the above integral is very much simplified by observing the plot of the two components of the integrant shown in Figure 22.6o. The mechanical admittance function is either 1.o or zero for most of the frequency range. However, over a relatively small range of frequencies around the natural frequency of the system, it attains very high values if the damping is small. As a result, the integrant takes the shape shown in Figure 22.6oc. It has a sharp spike around the natural frequency of the system.


FIGURE 22.60 Schematic diagram for computation of response.

FIGURE 22.60 Schematic diagram for computation of response.

The broad hump is governed by the shape of the turbulent velocity spectrum which is modified slightly by the aerodynamic admittance function. The area under the broad hump is the broad band or non-resonant response, whereas the area in the vicinity of the natural frequency gives the narrow band or resonant response. Thus, Equation 22.40 can be rewritten as

0 0

Post a comment