Steady Pressure Term

Now consider the work due to the steady pressure for the same rigid body mode shapes. It can be shown that for the forward traveling wave.

Wps = —nf [Ps (a • ns – as • nc)] dA =

A

—n I (ashc, t — a, c hs,£) J Ps cos (On (A)) dA + (ac hs, n — as hc, rj) J Ps sin (On (A)) dA

AA

The term 0N is the local angle between the surface normal and the chord line. Therefore the first integration term is the steady lift and the second is the steady drag. In general the work associated with the steady pressure is nonzero for airfoils producing lift and drag, and having both translational and rotational motion.

3. Backward Traveling Waves

For backward traveling waves the interblade phase angle changes sign and the mode shape is

ф = ac + ias

The work associated with the unsteady pressure then becomes Wup (backward traveling waves) = Wcc + Wss + Wcs — Wsc

Thus, the sign of the interaction terms is negative of that for the forward trav­eling waves. Note that all of these individual work terms are different from those of the forward traveling waves because the interblade phase angle is also switched in sign. The work associated with the unsteady pressure, W Ps, is the negative of that shown above for the forward traveling wave.