Aerospace Dynamics and Control Lecture 2 - Basic Concepts and Dynamical Equations

1. 航天动力学与控制 Lecture 2 2007年2月

2. Basic Concept and Dynamical Equation • 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.

3. Two-body and central force motion • 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 • 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.

4. Two-body and central force motion(basic differential equation) Center of mass: c Geometry dictates that Which permits expressions to become then

5. Two-body and central force motion(basic differential equation) Mutual attraction requires that then Applying this results leads immediately to In terms of the gravitational attraction yields the basic differential equation of motion for the two-body system.

6. Two-body and central force motion(Solution of the differential equation) Using a straightforward vector approach. (1) . Defining angular momentum per unit mass as and Leads to the conclusion that Thus, angular momentum is conserved and three integrals of motion are and this plane must be inertially fixed

7. Two-body and central force motion(Solution of the differential equation) Cross equation (1) with h Applying the standard identity for triple vector products and noting that Leads to (2) Since h is constant, equation (2.) may be integrated directly (3) e is a constant of integration and is called the eccentricity vector. The orientation of e in this plane is taken as a reference direction

8. Two-body and central force motion(Solution of the differential equation) Dot expression (3) with r and using the triple scalar product identity to obtain (4) Vector e is parallel to the direction of minimum r, and angular is measured from this point to the position in the orbit. This angle is known as the true anomaly Kinetic Energy The first term on the right side is just translational kinetic energy of two-body system. The remaining term is rotational kinetic energy about the center of mass.

9. Central Force Motion then the motion of m2 about m1 is essentially the motion of a particle in an inertially fixed field. This type of motion is known as central force motion. If Notice that the two-body problem may be treated as central force motion of a particle of mass The force which is a function of r can be written as the gradient of gravitational potential Kinetic energy per unit mass is

10. Central Force Motion Which permits writing kinetic energy in a more convenient form The energy of a unit mass in a central fields is given introducing a new variable as thus

11. Central Force Motion Consider the inverse-square force law characterized by Define a parameter p as Is replaced by Rearranging gives (5)

12. Eccentricity (4) is identical to form (5) Plus sign in (5) minus sign in (5) e=0, Circle 0<e<1 ellipse e=1 parabola e>1 Hyperbola

13. Kepler’s Time Equation The relationship between time and position in orbit is considered Noting that yields A new variable is introduced Since

14. Kepler’s Time Equation Taking the time derivative of this gives radial speed Evaluation of energy at periapsis Rearranging this leaves Define Then This is known as Kepler’s equation for relating time position in orbit when

15. Kepler’s Time Equation When eccentricity is less than 1.0, the orbit is closed and periodic. The associated period can be determined by considering the rate of area swept out by the radius vector which is the areal velocity. Kepler’s second law. Integrating over the entire orbit gives the period as For an ellipse From geometry, Kepler’s third law.

16. 第一章 基本的物理定律 • 相对运动(加速度) The second term of the expression becomes is the acceleration of the origin of the moving frame is the apparent acceleration of particle in the moving frame is the Coriolis acceleration due to the motion of p in x, y, z is the acceleration of p due to the changes of the angular velocity is the centrifugal acceleration

17. 第一章 基本的物理定律 • Motion of the Earth’s Surface the two centrifugal acceleration terms are negligible compared to the Coriolis term for low altitude orbits.

18. 第一章 基本的物理定律 • 哥氏力的影响 To illustrate the effect of this Coriolis acceleration, consider a satellite traveling due east over the ground station, Thisindicates that an observer would notice the satellite turing to the south or to the right of its initial flight path.