Flight dynamics

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FLIGTH DYNAMICS - TOPIC 3 EQUATION OF MOTION AND DYNAMIC STABILITY 3.1 Dynamic equations The dynamic equations can be derived by integrating the Newton law for the force and for the moment on the body domain. The comprensive acceleration is expressed as: ! where ! is the inertial frame and ! is the body frame. Integrating with respect to the body mass, one can find: ! dove ! . As far as the moment concerns, it is possible to write: ! Applying the Jacobi identity and defining the inertial tensor as ! , yields: ! If the point ! is the center of gravity !, the equations of motion becomes the Euler equations, since the tensor ! is null: ! If the dynamic of the aircraft is analyzed, some simplifications may be applied to the Euler equation, since some of the terms of the inertial tensor are negligible. In particular: ! aQ/I=dVP/Idt+ω×VP/I+dωdtB×rP/Q+ω×ω×rP/QIQf=mdVP/Idt+ω×mVP/I+SPTdωdt+ω×SPTωSPT=−rP/Q()×dmB∫m=rPQ×aQ/I=mSPTdVPdt+rPQ×ω×VPdmB∫−rPQ×rPQ×dωdtB∫dm+rPQ×ω×ω×rPQdmB∫B∫JP=−rPQ()×rPQ()×dmB∫m=SPdVPdt+JPdωdt+ω×SPVP+JPω()+VP×SPTω+mVP()PGSPm!VG+ω×mVG=fJG!ω+ω×JGω=m⎧⎨⎪⎩⎪JG=Ix−Ixy−Ixz−IxyIy−Iyz−Ixz−IyzIz⎡⎣⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥!Ix0−Ixz0Iy0−Ixz0Iz⎡⎣⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥ 1 being ! the roll axis, ! the pitch axis and ! the yaw axis. If ! and ! , yields: ! and ! A relation for ! and the Euler angle should be added to the system in order to close the equations- unknowns balance. It should be noted that the Euler equations are not linear. 3.2 Derivation of equations of motion based on momentum and angular momentum The equations of motion can be derived by applying the Newton law considering the overall acceleration of a point which belongs to a body which is attached to a moving reference frame. If the motion of the center of gravity is considered, the same equations can be derived by applying the…