航天动力学与控制Lecture2
航天动力学与控制 Lecture 2
Basic ConceptandDynamicalEquationGeneral Rigid BodyMotionThe concept of Rigid BodyA rigid body can be defined as a system ofparticles whose relative distancesarefixed with time. The internal potential energy of a rigid body is constantOrbital Mechanics: translational motion of spacecraft underthe influence of gravitational and otherforces becomes orbital mechanicsAttitude Mechanics: Rotation about the center of mass underthe influence ofappliedtorquesbecomes attitudemechanics
General Rigid Body Motion − The concept of Rigid Body • A rigid body can be defined as a system of particles whose relative distances are fixed with time. The internal potential energy of a rigid body is constant − Orbital Mechanics: translational motion of spacecraft under the influence of gravitational and other forces becomes orbital mechanics − Attitude Mechanics: Rotation about the center of mass under the influence of applied torques becomes attitude mechanics. Basic Concept and Dynamical Equation
Two-bodyand central forcemotionn-bodyproblemn-body problem has only1oknown integrals ofmotion:3velocitycomponents,3positioncomponents,3angularmomentumcomponents,andkineticenergy.only thetwo-body problem has an unrestricted solution.Special cases of thethree-bodyproblemhavebeentreatedinclosedformAssumption Masses could be two bodies whose minimum distance apart is largecomparedtotheirlargest dimensions orcouldhavespherically symmetricmass distributions andnevertoucheachother.Thisrestriction allows thesemasses to be treated as particles in thefollowing analysis
n-body problem − n-body problem has only 10 known integrals of motion: 3 velocity components, 3 position components, 3 angular momentum components, and kinetic energy. − only the two-body problem has an unrestricted solution. Special cases of the three-body problem have been treated in closed form Two-body and central force motion • Assumption • Masses could be two bodies whose minimum distance apart is large compared to their largest dimensions or could have spherically symmetric mass distributions and never touch each other. This restriction allows these masses to be treated as particles in the following analysis
Two-body and central forcemotion (basic differentialCenter of mass:cm(r-r)+m(r-r)=0Geometry dictates thatr-r=rWhich permits expressions to becomem2-m1-r.=r2-m,+m,m,+m2thenmm_rm,mF=mir=mr+LF, =mi, =mrm,+mzm,+mz
Two-body and central force motion (basic differential equation) Center of mass: c 1 1 2 2 ( ) ( ) 0 m m − + − = c c r r r r Geometry dictates that r r r 2 1 − = Which permits expressions to become 2 1 1 2 c m m m − = + r r r 1 2 1 2 c m m m − − = + r r r then 1 2 1 1 1 1 1 2 c m m m m m m = = + + F r r r 1 2 2 2 2 2 1 2 c m m m m m m = = − + F r r r
Two-bodyand central forcemotion (basic differentialMutual attraction requires that F=-Fmr,=-mrthen r,=0ApplyingthisresultsleadsimmediatelytommrF=-F=m,+m,In terms ofthe gravitational attraction yields thebasiedifferentialequation ofmotionforthetwo-body system.Xd'r从3=0u=G(m+m2)di?
Two-body and central force motion (basic differential equation) Mutual attraction requires that F F 1 2 = − m m 1 2 r r c c = − then 0 r c = Applying this results leads immediately to 1 2 1 2 1 2 m m m m = − = + F F r In terms of the gravitational attraction yields the basic differential equation of motion for the two-body system. 2 2 3 0 d dt r + = r r 1 2 = + G m m ( )