Lectures d25-D26 3D Rigid body dynamics 12 November 2004
Lectures D25-D26 : 3D Rigid Body Dynamics 12 November 2004
Outline o Review of Equations of motion ● Rotational| Motion Equations of Motion in Rotating Coordinates ● Euler Equations o EXample: Stability of Torque Free Motion ● Gyroscopic Motion Euler Angles Steady Precession e Steady precession with M=0 AERO Dynamics 16.07 Dynamics D25-D26 1
Outline Dynamics 16.07 Dynamics D25-D26 1 • Review of Equations of Motion • Rotational Motion • Equations of Motion in Rotating Coordinates • Euler Equations • Example: Stability of Torque Free Motion • Gyroscopic Motion - Euler Angles - Steady Precession • Steady Precession with M = 0
Equations of Motion Conservation of linear momentum L=F L= mVG Conservation of Angular Momentum HG= MG, HG =IGw or O=M O Ho= low AERO Dynamics 16.07 Dynamics D25-D26 2
Equations of Motion Dynamics 16.07 Dynamics D25-D26 2 Conservation of Linear Momentum L = F ˙ , L = mvG Conservation of Angular Momentum H˙ G = MG, HG = IGω or H˙ O = MO, HO = IOω
Equations of Motion in Rotating Coordinates Angular Momentum (or Ho =low) Time variation Non-rotating axes xYZ(I changes H=I心+D…. I big problen! Rotating axes yz (I constant) H=(H)agz+g×H Cyz +Ω×H AERO Dynamics 16.07 Dynamics D25-D26 3
Equations of Motion in Rotating Coordinates Dynamics 16.07 Dynamics D25-D26 3 Angular Momentum HG = IGω (or HO = IOω) Time variation - Non-rotating axes XY Z (I changes) H =˙ I˙ω + Iω˙ . . . I˙ big problem! - Rotating axes xyz (I constant) H˙ = (H) ˙ xyz + Ω × H = I( ˙ω)xyz + Ω × H
Equations of Motion in Rotating Coordinates (H)eyz+×H=M x -h,nz+h, nn=m H,-H2 Q+Hn2=My H2-H2nytHyn2x 之 yz axis can be any right-handed set of axis but will choose yz(@ to simplify analysis(e.g. I constant) AERO Dynamics 16.07 Dynamics D25-D26 4
Equations of Motion in Rotating Coordinates Dynamics 16.07 Dynamics D25-D26 4 (H) ˙ xyz + Ω × H = M or, H˙ x − HyΩz + HzΩy = Mx H˙ y − HzΩx + HxΩz = My H˙ z − HxΩy + HyΩx = Mz xyz axis can be any right-handed set of axis, but . . . will choose xyz (Ω) to simplify analysis (e.g. I constant)