Phugoid Approximation for Conventional Airplanes

Abstract

An improved closed-form approximation for phugoid motion in conventional airplanes is presented. Although several closed-form approximations for phugoid motion are currently available and widely used, none of these approximations accurately predict all of the fundamental characteristics of phugoid motion. The new approximation accounts for changes in angle of attack as well as the effects of pitch stability and pitch damping. The total phugoid damping is shown to depend on pitch damping aswell as aircraftdrag. In addition, thissolution pointsout another important contribution to phugoid damping called phase damping. It is shown that the phase-damping contribution to the real component of the phugoid eigenvalue is always positive and tends to reduce the total phugoid damping. Under certain conditions this phase damping can cause the phugoid mode to become divergent. Nomenclature Aw = planform area of the wing CD = total drag coefe cient CDp = parasitic drag coefe cient CD,a = change in drag coefe cient with angle of attack CL = lift coefe cient CL,a = change in lift coefe cient with angle of attack CM = pitching moment coefe cient CM,a = change in pitching moment coefe cient with angle of attack CM,$ = change in pitching moment coefe cient with dimensionless pitching rate ¯ c = mean chord length e = Oswald efe ciency factor FT = thrust force g = acceleration of gravity Iyy = pitching moment of inertia in body-e xed coordinates m = aircraft mass RA = aspect ratio Rd = phugoid pitch-damping ratio Rg = dimensionless gravitational acceleration RM = dimensionless change in pitching moment with axial velocity RM,a = dimensionless change in pitching moment with angle of attack RM,$ = dimensionless change in pitching moment with pitching rate Rp = phugoid phase-divergence ratio Rs = phugoid stability ratio Rx = dimensionless change in axial force with axial velocity Rxa = complex amplitude Rxc = complex coefe cient Rxp = complex phase Rx,a = dimensionless change in axial force with angle of attack Rz = dimensionless change in normal force with axial velocity Rza = complex amplitude Rzc = complex coefe cient Rzp = complex phase Rz,a = dimensionless change in normal force with angle of attack