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        -  L-dDap                    (4.566)
   - pUo2[ao(y)ry+[ao(y)+CD.,](7jY)l,(y)dy (4.567)
                = -;pl
The rolling moment due to the elemental strip RT is given by
 dRM=-l2pUo2[ao(y)a,+[ao(y)+CD,l](ZrY)].(y)ydy (4.568)
The total rolling moment due to the right wing is given by
       ,      1
      RMR = _~p Uo2 l,b/2 [ao(y)a- + [ao(Y) + CD.d (7jY) ].(y)y dy     (4.569)
Similarly, the rolling moment due to the left wing (change +y to - y) is given by
 RM,=;pUo2l,'/2[ao(y)u-[ao(y)+CD.,]7jY],(y)ydy (4.570)
The total rolling moment
   RM=-pUo2(3)l,%[ao(y)+CD.tlc(y)y2dy (4.571)
We have
            CL= pRUMS7         (4.572)
            Ctp= lCt         (4.573)
      a~)
Then, '
                         (C,p)W = Sb4  f.g[ao(Y) + CD,tlc(y)y2dy               (4.574)
For an untwisted rectangular wing with a constant chord and an identical airfoil
section along the span, Eq. (4.574) reduces to the following form:
(CLp)W = -61(0o + CD)                                 (4.575)
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410                PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Usually, lao(Y)I > CD,I SO that the sign of Ciy depends on the sign ofao(Y). There-
fore, for angles of attack below stall where the lift-curve slope ao is positive,
(Ctp)W < O, i.e., the wing provides a positive damping or simply damping effect.
For angles of attack above the stall angle, the lift coefficient decreases with an
increase in the angle of attack so that ao becomes negatrve and (Ctp)W > 0. This
loss of damping in roll is one of the main causes of autorotati,on of the wings in
poststall region and aircraft spin entry, which we will discuss in Chapter 7.
     The above result based on strip theory is very approximate because the str:ip the-

　
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