ASSPIN Users' Manual - CAFE Foundation

where A1 and A2(r) are the 3 x 3 rotation matricesA 1 =0 1 0cosc_sinc_ 0-sino_)coso_andA2(r) = sincar coscar 0 .cos0car -sincar00)1The velocity, I_(r), of a blade point with respect to an observer in the medium fixedframe is obtained by differentiating (6) with respect to source time. Thus,0r - A1 0-----7--'7 + \(s)where(sin-cos 0) .0r -- co cos car -sincar 0 .0 0 0In order for the integrands of (2) and (3) to be meaningful, all of the terms mustbe written in the same coordinate system. Most of the terms contain only bladegeometric quantities. These quantities are easily computed in terms of blade fixedcoordinates by evaluating the spline functions that were described in the previoussubsection. Therefore, to facilitate the evaluation of the integrands, all terms thatinvolve vector operations (e.g., dot products) are expressed in blade fixed coordinates.It is natural to write the Mach vector, Jl, of a blade point and the moving observerpoint, :ie, in medium fixed coordinates. Thus, .._I and .Y must be expressed in blade fixedcoordinates in order to maintain consistency.The Mach vector, expressed in terms of medium fixed coordinates, is obtained from(8). Since 2t'I is viewed as a free vector, it follows that its representation with respect tothe aircraft fixed frmne is the same as that of the medium fixed frame. Consequently, attime r, the Math vector of a moving blade point, 77, expressed in blade fixed coordinatesis calculated by applying the inverse transformation of (7) to equation (8) which yieldsc o Or -- co All(r) c)_ 7-[+A21(r)A11 , (9)2ll r12