Example: Parallel Plane Motion 0. ≠0 awag y 02z 之 Body fixed axis =w(and z= Z +I2 y Iy2Wx-Iazw2= My (3) Solve (3)for wx, and then, (1)and(2)for Mr and My AERO Dynamics 16.07 Dynamics D25-D26 5
Example: Parallel Plane Motion Dynamics 16.07 Dynamics D25-D26 5 ωx = ωy = 0, ωz 6= 0 Hx = −Ixzωz, Hy = −Iyzωz, Hz = Izωz Body fixed axis Ω = ω (and z ≡ Z) − Ixzω˙ z + Iyzω 2 z = Mx (1) −Iyzω˙ z − Ixzω 2 z = My (2) Izω˙ z = Mz (3) Solve (3) for ωz, and then, (1) and (2) for Mx and My
Euler's Equations If yz are principal axes of inertia h,=Lwh 之 之0z I2wx-(Iy-I2wyWx=M2 Iywy-(Ix-Irwxwr= M. Iz心2-(L a一1)0:t=xvz AERO Dynamics 16.07 Dynamics D25-D26 6
Euler’s Equations Dynamics 16.07 Dynamics D25-D26 6 If xyz are principal axes of inertia • Hx = Ixωx, Hy = Iyωy, Hz = Izωz • Ω = ω Ixω˙ x − (Iy − Iz)ωyωz = Mx Iyω˙ y − (Iz − Ix)ωzωx = My Izω˙ z − (Ix − Iy)ωxωy = Mz
Euler's Equations o body fixed principal axes o Right-handed coordinate frame Origin at Center of mass G(possibly accelerated Fixed point O o non-linear equations. . hard to solve Solution gives angular velocity components unknown directions(need to integrate w to determine orientation) AERO Dynamics 16.07 Dynamics D25-D26 7
Euler’s Equations Dynamics 16.07 Dynamics D25-D26 7 • Body fixed principal axes • Right-handed coordinate frame • Origin at: – Center of mass G (possibly accelerated) – Fixed point O • Non-linear equations . . . hard to solve • Solution gives angular velocity components . . . in unknown directions (need to integrate ω to determine orientation)