orbital mechanics for -...

Orbital Mechanics for Engineering Students

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Orbital Mechanics for Engineering Students

Second Edition

Howard D. Curtis Professor of Aerospace Engineering

Embry -Riddle Aeronautical University Daytona Beach, Florida

AMSTERDAM BOSTON HEIDELBERG LONDONNEW YORK OXFORD PARIS SAN DIEGO

SAN FRANCISCO SINGAPORE SYDNEY TOKYO

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To my parents, Rondo and Geraldine.

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

CHAPTER 1 Dynamics of point masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Mass, force and Newtons law of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Newtons law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.6 Time derivatives of moving vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.7 Relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.8 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.8.1 Runge-Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.8.2 Heuns Predictor-Corrector method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.8.3 Runge-Kutta with variable step size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54List of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

CHAPTER 2 The two-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2 Equations of motion in an inertial frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.3 Equations of relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.4 Angular momentum and the orbit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.5 The energy law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.6 Circular orbits (e 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.7 Elliptical orbits (0 < e < 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.8 Parabolic trajectories (e 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002.9 Hyperbolic trajectories (e > 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042.10 Perifocal frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.11 The lagrange coeffi cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.12 Restricted three-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

2.12.1 Lagrange points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332.12.2 Jacobi constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146List of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Contents

viii Contents

CHAPTER 3 Orbital position as a function of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.2 Time since periapsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.3 Circular orbits (e 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.4 Elliptical orbits (e < 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.5 Parabolic trajectories (e 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1723.6 Hyperbolic trajectories (e < 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1743.7 Universal variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194List of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

CHAPTER 4 Orbits in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1994.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1994.2 Geocentric right ascension-declination frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.3 State vector and the geocentric equatorial frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.4 Orbital elements and the state vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.5 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.6 Transformation between geocentric equatorial and perifocal frames . . . . . . . . . . . . . . . 2294.7 Effects of the Earths oblateness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2334.8 Ground tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244


CHAPTER 5 Preliminary orbit determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.2 Gibbs method of orbit determination from three position vectors . . . . . . . . . . . . . . . . . . 2565.3 Lamberts problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2635.4 Sidereal time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2755.5 Topocentric coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2805.6 Topocentric equatorial coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2835.7 Topocentric horizon coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2845.8 Orbit determination from angle and range measurements . . . . . . . . . . . . . . . . . . . . . . . . 2895.9 Angles only preliminary orbit determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2975.10 Gauss method of preliminary orbit determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297


CHAPTER 6 Orbital maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.2 Impulsive maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3206.3 Hohmann transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.4 Bi-elliptic Hohmann transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3286.5 Phasing maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3326.6 Non-Hohmann transfers with a common apse line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3386.7 Apse line rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3436.8 Chase maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

6.9 Plane change maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3556.10 Nonimpulsive orbital maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368


CHAPTER 7 Relative motion and rendezvous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.2 Relative motion in orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3927.3 Linearization of the equations of relative motion in orbit . . . . . . . . . . . . . . . . . . . . . . . . 4007.4 Clohessy-Wiltshire equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4077.5 Two-impulse rendezvous maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4117.6 Relative motion in close-proximity circular orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419


CHAPTER 8 Interplanetary trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4298.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4298.2 Interplanetary Hohmann transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4308.3 Rendezvous Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4328.4 Sphere of infl uence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.5 Method of patched conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.6 Planetary departure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4428.7 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4488.8 Planetary rendezvous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.9 Planetary fl yby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4588.10 Planetary ephemeris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4708.11 Non-Hohmann interplanetary trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475


CHAPTER 9 Rigid-body dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4859.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4859.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4869.3 Equations of translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4959.4 Equations of rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4979.5 Moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

9.5.1 Parallel axis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5179.6 Eulers equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5249.7 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5309.8 The spinning top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5339.9 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5389.10 Yaw, pitch and roll angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5499.11 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552


Contents ix

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CHAPTER 10 Satellite attitude dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57310.2 Torque-free motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57410.3 Stability of torque-free motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58410.4 Dual-spin spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58910.5 Nutation damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59310.6 Coning maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60110.7 Attitude control thrusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60510.8 Yo-yo despin mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

10.8.1 Radial release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61310.9 Gyroscopic attitude control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61510.10 Gravity gradient stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644List of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

CHAPTER 11 Rocket vehicle dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65511.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65611.3 The thrust equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65811.4 Rocket performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66011.5 Restricted staging in fi eld-free space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66711.6 Optimal staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

11.6.1 Lagrange multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686List of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

Appendix A Physical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

Appendix B A road map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

Appendix C Numerical intergration of the n-body equations of motion . . . . . . . . . . . . 693

Appendix D MATLAB algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

Appendix E Gravitational potential energy of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

The purpose of this book, like the fi rst edition, is to provide an introduction to space mechanics for under-graduate engineering students. It is not directed towards graduate students, researchers and experienced practitioners, who may nevertheless fi nd useful review material within the books contents. The intended readers are those who are studying the subject for the fi rst time and have completed courses in physics, dynamics and mathematics through differential equations and applied linear algebra. I have tried my best to make the text readable and understandable to that audience. In pursuit of that objective I have included a large number of example problems that are explained and solved in detail. Their purpose is not to over-whelm but to elucidate. I fi nd that students like the teach by example method. I always assume that the material is being seen for the fi rst time and, wherever possible, I provide solution details so as to leave little to the readers imagination. The numerous fi gures throughout the book are also intended to aid comprehen-sion. All of the more labor-intensive computational procedures are implemented in MATLAB code.

CHANGES TO THE SECOND EDITION Most of the content and style of the fi rst edition has been retained. Some topics have been revised, rearranged or relocated. I have corrected all of the errors that I discovered or that were reported to me by students, teach-ers, reviewers and other readers. Key terms are now listed at the end of each chapter. The answers in the example problems are boxed instead of underlined. The homework problems at the end of each chapter have been grouped by applicable section. There are many new example problems and homework problems.

Chapter 1, which is a review of particle dynamics, begins with a new section on vectors, which are used throughout the book. Therefore, I thought a brief review of basic vector concepts and operations was appro-priate. The chapter concludes with a new section on the numerical integration of ordinary differential equa-tions (ODEs). These Runge-Kutta and predictor-corrector methods, which I implemented in the MATLAB codes rk1_4.m , rkf45.m and heun.m , facilitate the investigation and simulation of space mechanics problems for which analytical, closed-form solutions are not available. Many of the books new example problems illustrate applications of this kind. Throughout the text I mostly use the ODE solvers heun.m (fi xed time step) and rkf45.m (variable time step) because they work well and the scripts (see Appendix D) are short and easy to read. In every case I checked their results against two of MATLABs own suite of ODE solvers, primarily ode23.m and ode45.m . These general-purpose codes are far more elegant (and lengthy) than the ones mentioned above. They may be listed by issuing the MATLAB type command.

I have added two algorithms to Chapter 2 for numerically integrating the two-body equations of motion: an algorithm for propagating a state vector as a function of true anomaly, and an algorithm for fi nding the roots of a function by the bisection method. The last one is useful for determining the Lagrange points in the restricted three-body problem.

Chapter 4 now includes the material on coordinate transformations previously found in this and other chapters. Section 4.5 includes a more general treatment of the Euler elementary rotation sequences, with emphasis on the classical (3-1-3) Euler sequence and the yaw-pitch-roll (3-2-1) sequence. Algorithms were added to calculate the right ascension and declination from the position vector and to calculate the classical Euler angles and the yaw, pitch and roll angles from the direction cosine matrix. I also moved all discussion

Preface

xii Preface

of ground tracks into Chapter 4 and offer an algorithm for obtaining the ground track of a satellite from its orbital elements.

Chapter 6 concludes with a new section on nonimpulsive (fi nite burn time) orbital change maneuvers, including MATLAB simulations.

Chapter 7 now includes an algorithm to fi nd the position, velocity and acceleration of a spacecraft rela-tive to an LVLH frame. Also new to this chapter is the derivation of the linearized equations of relative motion for an elliptical (not necessarily circular) reference orbit.

New to Chapter 9 is a discussion of quaternions and associated algorithms for use in numerically solv-ing Eulers equations of rigid body motion to obtain the evolution of spacecraft attitude. Quaternions can be used with MATLABs rotate command to produce simple animations of spacecraft motion.

Appendices C and D have changed. The MATLAB script in Appendix C was revised. Appendix D no longer contains the listings of MATLAB codes. Instead, the algorithms are listed along with the world wide web addresses from which they may be downloaded. This edition contains over twice the number of MATLAB M-fi les as did the fi rst.

ORGANIZATION The organization of the book remains the same as that of the fi rst edition. Chapter 1 is a review of vector kinematics in three dimensions and of Newtons laws of motion and gravitation. It also focuses on the issue of relative motion, crucial to the topics of rendezvous and satellite attitude dynamics. The new material on ordinary differential equation solvers will be useful for students who are expected to code numerical simula-tions in MATLAB or other programming languages. Chapter 2 presents the vector-based solution of the clas-sical two-body problem, resulting in a host of practical formulas for the analysis of orbits and trajectories of elliptical, parabolic and hyperbolic shape. The restricted three-body problem is covered in order to introduce the notion of Lagrange points and to present the numerical solution of a lunar trajectory problem. Chapter 3 derives Keplers equations, which relate position to time for the different kinds of orbits. The universal vari-able formulation is also presented. Chapter 4 is devoted to describing orbits in three dimensions. Coordinate transformations and the Euler elementary rotation sequences are defi ned. Procedures for transforming back and forth between the state vector and the classical orbital elements are addressed. The effect of the earths oblateness on the motion of an orbits ascending node and eccentricity vector is examined. Chapter 5 is an introduction to preliminary orbit determination, including Gibbss and Gausss methods and the solution of Lamberts problem. Auxiliary topics include topocentric coordinate systems, Julian day numbering and sidereal time. Chapter 6 presents the common means of transferring from one orbit to another by impulsive delta-v maneuvers, including Hohmann transfers, phasing orbits and plane changes. Chapter 7 is a brief intro-duction to relative motion in general and to the two-impulse rendezvous problem in particular. The latter is analyzed using the Clohessy-Wiltshire equations, which are derived in this chapter. Chapter 8 is an introduc-tion to interplanetary mission design using patched conics. Chapter 9 presents those elements of rigid-body dynamics required to characterize the attitude of a space vehicle. Eulers equations of rotational motion are derived and applied in a number of example problems. Euler angles, yaw-pitch-roll angles and quaternions are presented as ways to describe the attitude of rigid body. Chapter 10 describes the methods of controlling, changing and stabilizing the attitude of spacecraft by means of thrusters, gyros and other devices. Finally, Chapter 11 is a brief introduction to the characteristics and design of multi-stage launch vehicles.

Chapters 1 through 4 form the core of a fi rst orbital mechanics course. The time devoted to Chapter 1 depends on the background of the student. It might be surveyed briefl y and used thereafter simply as a ref-erence. What follows Chapter 4 depends on the objectives of the course.

Chapters 5 through 8 carry on with the subject of orbital mechanics. Chapter 6 on orbital maneuvers should be included in any case. Coverage of Chapters 5, 7 and 8 is optional. However, if all of Chapter 8 on

interplanetary missions is to form a part of the course, then the solution of Lamberts problem (Section 5.3) must be studied beforehand.

Chapters 9 and 10 must be covered if the course objectives include an introduction to spacecraft dynam-ics. In that case Chapters 5, 7 and 8 would probably not be covered in depth.

Chapter 11 is optional if the engineering curriculum requires a separate course in propulsion, including rocket dynamics.

The important topic of spacecraft control systems is omitted. However, the material in this book and a course in control theory provide the basis for the study of spacecraft attitude control.

To understand the material and to solve problems requires using a lot of undergraduate mathematics. Mathematics, of course, is the language of engineering. Students must not forget that Sir Isaac Newton had to invent calculus so he could solve orbital mechanics problems in more than just a heuristic way. Newton (1642 1727) was an English physicist and mathematician, whose 1687 publication Mathematical Principles of Natural Philosophy ( the Principia ) is one of the most infl uential scientifi c works of all time. It must be noted that the German mathematician Gottfried Wilhelm von Leibnitz (1646 1716), is credited with invent-ing infi nitesimal calculus independently of Newton in the 1670s.

In addition to honing their math skills, students are urged to take advantage of computers (which, inci-dentally, use the binary numeral system developed by Leibnitz). There are many commercially available mathematics software packages for personal computers. Wherever possible they should be used to relieve the burden of repetitive and tedious calculations. Computer programming skills can and should be put to good use in the study of orbital mechanics. The elementary MATLAB programs referred to in Appendix D of this book illustrate how many of the procedures developed in the text can be implemented in software. All of the scripts were developed and tested using MATLAB version 7.7. Information about MATLAB, which is a registered trademark of The MathWorks, Inc., may be obtained from

The MathWorks, Inc. 3 Apple Hill Drive Natick , MA, 01760-2089, USA www.mathworks.com

Appendix A presents some tables of physical data and conversion factors. Appendix B is a road map through the fi rst three chapters, showing how the most fundamental equations of orbital mechanics are related. Appendix C shows how to set up the n -body equations of motion and program them in MATLAB. Appendix D contains the web locations of the M-fi les of all of the MATLAB-implemented algorithms and example problems presented in the text. Appendix E shows that the gravitational fi eld of a spherically sym-metric body is the same as if the mass were concentrated at its center.

The fi eld of astronautics is rich and vast. References cited throughout this text are listed at the end of the book. Also listed are other books on the subject that might be of interest to those seeking additional insights.

SUPPLEMENTS TO THE TEXT For purchasers of this book:

Copies of the MATLAB M-fi les listed in Appendix D can be freely downloaded from the companion website accompanying this book. To access these fi les please visit www.elsevierdirect.com/9780123747785 and click on the companion site link.

For instructors using this book as text for their course: Please visit www.textbooks.elsevier.com to register for access to the solutions manual, PowerPoint lec-

ture slides and other resources.

Preface xiii

Since the publication of the fi rst edition and during the preparation of this one, I have received helpful criticism, suggestions and advice from many sources locally and worldwide. I thank them all and regret that time and space limitations prohibited the inclusion of some recommended additional topics that would have enhanced the book. I am especially indebted to those who reviewed the proposed revision plan and second edition manuscript for the publisher for their many suggestions on how the book could be improved. Thanks to:

Rodney Anderson University of Colorado at Boulder

Dale Chimenti Iowa State University

David Cicci Auburn University

Michael Freeman University of Alabama

William Garrard University of Minnesota

Peter Ganatos City College of New York

Liam Healy University of Maryland

Sanjay Jayaram St. Louis University

Colin McInnes University of Strathclyde

Eric Mehiel Cal Poly , San Luis Obispo

Daniele Mortari Texas A & M University

Roy Myose Wichita State University

Steven Nerem University of Colorado

Gianmarco Radice University of Glasgow

Alistair Revell University of Manchester

Trevor Sorensen University of Kansas

David Spencer Penn State University

Rama K. Yedavalli Ohio State University

It has been a pleasure to work with the people at Elsevier, in particular Joseph P. Hayton, Publisher, Maria Alonso, Assistant Editor, and Anne B. McGee, Project Manager. I appreciate their enthusiasm for the book, their confi dence in me, and all the work they did to move this project to completion.

Acknowledgments

xvi Acknowledgements

Finally and most importantly, I must acknowledge the patience and support of my wife, Mary, who was a continuous source of optimism and encouragement throughout the yearlong revision effort.

Howard D. Curtis Embry -Riddle Aeronautical University

Daytona Beach, Florida

2010 Elsevier Ltd. All rights reserved.2010

1.1 INTRODUCTION This chapter serves as a self-contained reference on the kinematics and dynamics of point masses as well as some basic vector operations and numerical integration methods. The notation and concepts summarized here will be used in the following chapters. Those familiar with the vector-based dynamics of particles can simply page through the chapter and then refer back to it later as necessary. Those who need a bit more in the way of review will fi nd the chapter contains all of the material they need in order to follow the develop-ment of orbital mechanics topics in the upcoming chapters.

We begin with a review of vectors and some vector operations after which we proceed to the problem of describing the curvilinear motion of particles in three dimensions. The concepts of force and mass are considered next, along with Newtons inverse-square law of gravitation. This is followed by a presentation of Newtons second law of motion ( force equals mass times acceleration ) and the important concept of angular momentum.

As a prelude to describing motion relative to moving frames of reference, we develop formulas for cal-culating the time derivatives of moving vectors. These are applied to the computation of relative velocity and acceleration. Example problems illustrate the use of these results, as does a detailed consideration of how the earths rotation and curvature infl uence our measurements of velocity and acceleration. This brings in the curious concept of Coriolis force. Embedded in exercises at the end of the chapter is practice in veri-fying several fundamental vector identities that will be employed frequently throughout the book.

The chapter concludes with an introduction to numerical integration methods, which can be called upon to solve the equations of motion when an analytical solution is not possible.

Dynamics of point masses 1 CHAPTER

Chapter outline

1.1 Introduction 11.2 Vectors 21.3 Kinematics 101.4 Mass, force and Newtons law of gravitation 151.5 Newtons law of motion 191.6 Time derivatives of moving vectors 241.7 Relative motion 291.8 Numerical integration 38

2 CHAPTER 1 Dynamics of point masses

1.2 VECTORS A vector is an object that is specifi ed by both a magnitude and a direction. We represent a vector graphi-cally by a directed line segment, that is, an arrow pointing in the direction of the vector. The end opposite the arrow is called the tail. The length of the arrow is proportional to the magnitude of the vector. Velocity is a good example of a vector. We say that a car is traveling east at eighty kilometers per hour. The direction is east and the magnitude, or speed, is 80 km/h. We will use boldface type to represent vector quantities and plain type to denote scalars. Thus, whereas B is a scalar, B is a vector.

Observe that a vector is specifi ed solely by its magnitude and direction. If A is a vector, then all vectors having the same physical dimensions, the same length and pointing in the same direction as A are denoted A , regardless of their line of action, as illustrated in Figure 1.1 . Shifting a vector parallel to itself does not mathematically change the vector. However, parallel shift of a vector might produce a different physical effect. For example, an upward 5 kN load (force vector) applied to the tip of an airplane wing gives rise to quite a different stress and defl ection pattern in the wing than the same load acting at the wings mid-span.

The magnitude of a vector A is denoted A , or, simply A . Multiplying a vector B by the reciprocal of its magnitude produces a vector which points in the direction

of B , but it is dimensionless and has a magnitude of one. Vectors having unit dimensionless magnitude are called unit vector s . We put a hat ( )^ over the letter representing a unit vector. Then we can tell simply by inspection that, for example, u is a unit vector, as are B and e .

It is convenient to denote the unit vector in the direction of the vector A as uA . As pointed out above, we obtain this vector from A as follows

u

AA A

(1.1)

Likewise , u CC C / , u FF F / , etc. The sum or resultant of two vectors is defi ned by the parallelogram rule ( Figure 1.2 ). Let C be the sum

of the two vectors A and B . To form that sum using the parallelogram rule, the vectors A and B are shifted parallel to themselves (leaving them unaltered) until the tail of A touches the tail of B . Drawing dotted lines through the head of each vector parallel to the other completes a parallelogram. The diagonal from the tails of A and B to the opposite corner is the resultant C . By construction, vector addition is commutative, that is,

A B B A (1.2)

A Cartesian coordinate system in three dimensions consists of three axes, labeled x , y and z , which inter-sect at the origin O . We will always use a right-handed Cartesian coordinate system, which means if you wrap the fi ngers of your right hand around the z axis, with the thumb pointing in the positive z direction,

A

FIGURE 1.1 All of these vectors may be denoted A , since their magnitudes and directions are the same.

your fi ngers will be directed from the x axis towards the y axis. Figure 1.3 illustrates such a system. Note that the unit vectors along the x , y and z -axes are, respectively, i , j and k .

In terms of its Cartesian components, and in accordance with the above summation rule, a vector A is written in terms of its components Ax , Ay and Az as

A A A Ax y z i j k (1.3)

The projection of A on the xy plane is denoted Axy . It follows that

A i jxy x yA A

According to the Pythagorean theorem, the magnitude of A in terms of its Cartesian components is

A A A Ax y z

2 2 2 (1.4)

From Equations 1.1 and 1.3, the unit vector in the direction of A is

cos cos cos u i j kA x y z (1.5)

1.2 Vectors 3

A

B

C

FIGURE 1.2 Parallelogram rule of vector addition.

x

y

z

i

j

k

AxAy

AzO

Axy

FIGURE 1.3 Three-dimensional, right-handed Cartesian coordinate system.


where

cos cos cos x

xy

yz

zA

A

A

A

A

A (1.6)

The direction angles x , y and z are illustrated in Figure 1.4 , and are measured between the vector and the positive coordinate axes. Note carefully that the sum of x , y and z is not in general known a priori and cannot be assumed to be, say, 180 degrees.

Example 1.1 Calculate the direction angles of the vector A i j k 4 8 .

Solution First , compute the magnitude of A by means of Equation 1.4:

A 1 4 8 92 2 2( )

Then Equations 1.6 yield

x

xx

A

A cos cos .1 1

1

983 62

yy

y

A

A

cos cos .1 1

4

9116 4

z

zz

A

A cos cos .1 1

8

927 27

Observe that x y z 227.3 .

x

y

z

i

j

k

A

x y

z

Axy

Az

FIGURE 1.4

Direction angles in three dimensions.

Multiplication and division of two vectors are undefi ned operations. There are no rules for computing the product AB and the ratio A/B . However, there are two well-known binary operations on vectors: the dot product and the cross product. The dot product of two vectors is a scalar defi ned as follows,

A B ABcos (1.7)

where is the angle between the heads of the two vectors, as shown in Figure 1.5 . Clearly,

A B B A (1.8)

If two vectors are perpendicular to each other, then the angle between them is 90 . It follows from Equation 1.7 that their dot product is zero. Since the unit vectors i , j and k of a Cartesian coordinate sys-tem are mutually orthogonal and of magnitude one, Equation 1.7 implies that

i i j j k ki j i k j k

10 (1.9)

Using these properties it is easy to show that the dot product of the vectors A and B may be found in terms of their Cartesian components as

A B A B A B A Bx x y y z z (1.10)

If we set B A , then it follows from Equations 1.4 and 1.10 that

A A A (1.11)

The dot product operation is used to project one vector onto the line of action of another. We can imag-ine bringing the vectors tail to tail for this operation, as illustrated in Figure 1.6 . If we drop a perpendicular

1.2 Vectors 5

B

A

FIGURE 1.5 The angle between two vectors brought tail to tail by parallel shift.

uA

B

A

BA

FIGURE 1.6 Projecting the vector B onto the direction of A .


line from the tip of B onto the direction of A , then the line segment B A is the orthogonal projection of B onto line of action of A . B A stands for the scalar projection of B onto A . From trigonometry, it is obvious from the fi gure that

B BA cos

Let uA be the unit vector in the direction of A . Then

B u B u cos cosA A B

1

Comparing this expression with the preceding one leads to the conclusion that

B

AA A B u B

A (1.12)

where uA is given by Equation 1.1. Likewise, the projection of A onto B is given by

A

BB A

B

Observe that A B B A only if A and B have the same magnitude.

Example 1.2 Let A i k 6 18j and B i j k 42 69 98 . Calculate

(a) The angle between A and B ; (b) The projection of B in the direction of A ; (c) The projection of A in the direction of B .

Solution First we make the following individual calculations.

A B ( )( ) ( )( ) ( )( )1 42 6 69 18 98 1392 (a)

A ( ) ( ) ( )1 6 18 192 2 2 (b)

B ( ) ( ) ( )42 69 98 1272 2 2 (c)

(a) According to Equation 1.7, the angle between A and B is

cos 1A BAB

Substituting (a), (b) and (c) yields

cos .1

1392

19 12754 77

(b) From Equation 1.12 we fi nd the projection of B onto A :

B

A AA

B

A A B

Substituting (a) and (b) we get

BA

1392

19= 73.26

(c) The projection of A onto B is

A

B BB

A

B A B

Substituting (a) and (c) we obtain

AB

1392

12710 96.

The cross product of two vectors yields another vector, which is computed as follows,

A B n ( sin )AB AB (1.13)

where is the angle between the heads of A and B, and nAB is the unit vector normal to the plane defined by the two vectors. The direction of nAB is determined by the right hand rule. That is, curl the fingers of the right hand from the first vector ( A ) towards the second vector ( B ), and the thumb shows the direction of nAB . See Figure 1.7 . If we use Equation 1.13 to compute B A , then nAB points in the opposite direction, which means

B A A B ( ) (1.14)

Therefore , unlike the dot product, the cross product is not commutative. The cross product is obtained analytically by resolving the vectors into Cartesian components.

A B i j k i j k ( ) ( )A A A B B Bx y z x y z (1.15)

Since the set i jk is a mutually perpendicular triad of unit vectors, Equation 1.13 implies that

i i 0 j j 0 k k 0i j k j k i

k i j (1.16)

1.2 Vectors 7

B

A

nAB

FIGURE 1.7 nAB is normal to both A and B and defi nes the direction of the cross product A B .


Expanding the right side of Equation 1.15, substituting Equation 1.16 and making use of Equation 1.14 leads to

A B i j k ( ) ( ) ( )A B A B A B A B A B A By z z y x z z x x y y x (1.17)

It may be seen that the right-hand side is the determinant of the matrix

i j k

A A A

B B B

x y z

x y z

Thus , Equation 1.17 can be written

A B

i j k

A A A

B B B

x y z

x y z

(1.18)

where the two vertical bars stand for determinant. Obviously the rule for computing the cross product, though straightforward, is a bit lengthier than that for the dot product. Remember that the dot product yields a scalar whereas the cross product yields a vector.

The cross product provides an easy way to compute the normal to a plane. Let A and B be any two vec-tors lying in the plane, or, let any two vectors be brought tail-to-tail to defi ne a plane, as shown in Figure 1.7 . The vector C A B is normal to the plane of A and B . Therefore, n CAB C / , or

n

A BA BAB

(1.19)

Example 1.3 Let A i j k 3 7 9 and B i j k 6 5 8 . Find a unit vector that lies in the plane of A and B and is perpendicular to A .

Solution The plane of the vectors A and B is determined by parallel shifting the vectors so that they meet tail to tail. Calculate the vector D A B .

D

i j k

i j k

3 7 9

6 5 8

101 78 27

Note that A and B are both normal to D . We next calculate the vector C D A .

C

i j k

i j k

101 78 27

3 7 9

891 828 941

C is normal to D as well as to A. A , B and C are all perpendicular to D . Therefore they are coplanar. Thus C is not only perpendicular to A , it lies in the plane of A and B . Therefore, the unit vector we are seeking is the unit vector in the direction of C , namely,

( )u

C i j kC C

891 828 941

891 828 9412 2 2

. . . u i j kC 0 5794 0 5384 0 6119

In the chapters to follow we will often encounter the vector triple product, A ( B C ). By resolving A , B and C into their Cartesian components, it can easily be shown (see Problem 1.1c) that the vector triple product can be expressed in terms of just the dot products of these vectors as follows:

A B C B A C C A B ( ) ( ) ( ) (1.20)

Because of the appearance of the letters on the right-hand side, this is often referred to as the bac-cab rule .

Example 1.4 If F E { D [ A ( B C )] } , use the bac-cab rule to reduce this expression to one involving only dot products.

Solution First we invoke the bac-cab rule to obtain

F E D B A C C A B

[ ( ) ( )]bac cab rule

Expanding and collecting terms leads to

F A C E D B A B E D C ( )[ ( )] ( )[ ( )]

We next apply the bac-cab rule twice on the right-hand side.

F A C D E B B E D A B D

( ) ( ) ( ) ( )bac cab rule

(( ) ( )E C C E D bac cab rule

Expanding and collecting terms yields the sought-for result.

F A C E B A B E C D A C E D B A B E D C [( )( ) ( )( )] ( )( ) ( )( )

1.2 Vectors 9


Another useful vector identity is the interchange of the dot and cross :

A B C A B C ( ) ( ) (1.21)

It is so-named because interchanging the operations in the expression A B C yields A B C . The parentheses in Equation 1.21 are required to show which operation must be carried out fi rst, according to the rules of vector algebra. (For example, ( A B ) C , the cross product of a scalar and a vector, is unde-fi ned.) It is easy to verify Equation 1.21 by substituting A i j k A A Ax y z , B i j k B B Bx y z and C i j k C C Cx y z and observing that both sides of the equal sign reduce to the same expression (Problem 1.1b).

1.3 KINEMATICS To track the motion of a particle P through Euclidean space we need a frame of reference, consisting of a clock and a Cartesian coordinate system. The clock keeps track of time t and the xyz axes of the Cartesian coordinate system are used to locate the spatial position of the particle. In nonrelativistic mechanics, a sin-gle universal clock serves for all possible Cartesian coordinate systems. So when we refer to a frame of reference we need think only of the mutually orthogonal axes themselves.

The unit of time used throughout this book is the second (s). The unit of length is the meter (m), but the kilometer (km) will be the length unit of choice when large distances and velocities are involved. Conversion factors between kilometers, miles and nautical miles are listed in Table A.3.

Given a frame of reference, the position of the particle P at a time t is defi ned by the position vector r ( t ) extending from the origin O of the frame out to P itself, as illustrated in Figure 1.8 . The components of r ( t ) are just the x , y and z coordinates,

r i j k( ) ( ) ( ) ( )t x t y t z t

The distance of P from the origin is the magnitude or length of r , denoted r or just r ,

r r x y z2 2 2

x

y

z

O

r

v

a P

s

o

path

FIGURE 1.8 Position, velocity and acceleration vectors.

As in Equation 1.11, the magnitude of r can also be computed by means of the dot product operation,

r r r

The velocity v and acceleration a of the particle are the fi rst and second time derivatives of the position vector,

v i j k i j k( )

( ) ( ) ( )( ) ( ) ( )t

dx t

dt

dy t

dt

dy t

dtv t v t v tx y z

a i j k i j( )

( ) ( ) ( )( ) ( ) (t

dv t

dt

dv t

dt

dv t

dta t a t a tx

y zx y z

))k

It is convenient to represent the time derivative by means of an overhead dot. In this shorthand overhead dot notation , if ( ) is any quantity, then

( )

( )( )

( )( )

( )

d

dt

d

dt

d

dt

2

2

3

3, etc.

Thus , for example,

v ra v r

v x v y v z

a v x a v y a v zx y z

x x y y z z

The locus of points that a particle occupies as it moves through space is called its path or trajectory. If the path is a straight line, then the motion is rectilinear. Otherwise, the path is curved, and the motion is called curvilinear. The velocity vector v is tangent to the path. If ut is the unit vector tangent to the trajec-tory, then

v u v t (1.22)

where the speed v is the magnitude of the velocity v . The distance ds that P travels along its path in the time interval dt is obtained from the speed by

ds vdt

In other words,

v s

The distance s , measured along the path from some starting point, is what the odometers in our automo-biles record. Of course, s , our speed along the road, is indicated by the dial of the speedometer.

Note carefully that v r , that is, the magnitude of the derivative of r does not equal the derivative of the magnitude of r .

1.3 Kinematics 11


Example 1.5 The position vector in meters is given as a function of time in seconds as

r i j k ( ) ( ) ( ) (m)8 7 6 5 4 0 3 2 12 3 4 2t t t t t . (a)

At t 10 seconds, calculate (a) v (the magnitude of the derivative of r ) and (b) r (the derivative of the magnitude of r ).

Solution (a) The velocity v is found by differentiating the given position vector with respect to time,

v

ri j k

d

dtt t t t( ) ( )16 7 15 1 2 42 3 .

The magnitude of this vector is the square root of the sum of the squares of its components,

v t t t t ( . . )1 44 234 6 272 224 496 4 2

1

2

Evaluating this at t 10 s, we get

v 1953 3. m/s

(b) Calculating the magnitude of r in (a) leads to

r t t t t t t ( . . . )0 09 26 2 68 6 152 149 84 538 6 4 3 2

1

2

The time derivative of this expression is:

rdr

dt

t t t t t

t t

0 36 78 6 137 2 228 149 42

0 09 26 2

7 5 3 2

8 6

. . .

( . . 668 6 152 149 84 534 3 21

2. )t t t t

Substituting t 10 s yields

r 1935 5. m/s

If v is given, then we can fi nd the components of the unit tangent ut in the Cartesian coordinate frame of reference by means of Equation 1.22:

u

vi j kt

x y zx y zv

v

v

v

v

v

vv v v v 2 2 2( ) (1.23)

The acceleration may be written

a u u a at t n n (1.24)

where a t and a n are the tangential and normal components of acceleration, given by

a v s av

t n ( )2

(1.25)

is the radius of curvature, which is the distance from the particle P to the center of curvature of the path at that point. The unit principal normal un is perpendicular to ut and points towards the center of curvature C , as shown in Figure 1.9 . Therefore, the position of C relative to P , denoted r C / P , is

r uC P n (1.26)

The orthogonal unit vectors ut and un form a plane called the osculating plane. The unit normal to the osculating plane is ub , the binormal, and it is obtained from ut and un by taking their cross product:

u u ub t n (1.27)

From Equations 1.22, 1.24 and 1.27 we have

v a u u u u u u v a u v a a va vat t t n n n t n n b b ( ) ( )

That is, an alternative to Equation 1.27 for calculating the binormal vector is

uv av ab

(1.28)

Note that ut , un and ub form a right-handed triad of orthogonal unit vectors. That is,

u u u u u u u u ub t n t n b n b t (1.29)

The center of curvature lies in the osculating plane. When the particle P moves an incremental distance ds the radial from the center of curvature to the path sweeps out a small angle d , measured in the osculat-ing plane. The relationship between this angle and ds is

ds d so that s , or

v

(1.30)

1.3 Kinematics 13

x

y

z

O

P

C d

ds

ut

un

ub

Osculating plane

FIGURE 1.9 Orthogonal triad of unit vectors associated with the moving point P .


Example 1.6 Relative to a Cartesian coordinate system, the position, velocity and acceleration of a particle P at a given instant are

r i j k 250 630 430 ( )m (a)

v i j k 90 125 170 ( )m/s (b)

a i j k 16 125 302 ( )m/s (c)

Find the coordinates of the center of curvature at that instant.

Solution The coordinates of the center of curvature C are the components of its position vector r C . Consulting Figure 1.9 , we observe that

r r uC n (d)

where r is the position vector of the point P , is the radius of curvature and un is the unit principal nor-mal vector. The position vector r is given in (a), but and un are unknowns at this point. We must use the geometry of Figure 1.9 to fi nd them.

We fi rst seek the value of un , starting with Equation 1.29 1 ,

u u un b t (e)

The unit tangent vector ut is found at once from the velocity vector in (b) by means of Equation 1.23:

u

vt v

where

v 90 125 170 229 42 2 2 . (f)

Thus

.. . . u

i j ki j kt

90 125 170

229 40 39233 0 5449 0 74106 (g)

To fi nd the binormal ub we insert the given velocity and acceleration vectors into Equation 1.28:

uv av a

i j k

v a

i j k

b

90 125 170

16 125 30

17500 20 9250

177500 20 9250

0 88409 0 0010104 0 46731

2 2 2( ) . . . i j k (h)

Substituting (g) and (h) back into (e) fi nally yields the unit principal normal:

. . .

. . .

.u

i j k

n 0 88409 0 0010104 0 46731

0 39233 0 5449 0 74106

0 225389 0 8385 0 48214 . . i j k (i)

The only unknown remaining in (d) is , for which we appeal to Equation 1.25:

v

an

2

(j)

The normal acceleration a n is calculated by projecting the acceleration vector a onto the direction of the unit normal un ,

an n a u i j k i j k ( ) ( . . . )16 125 30 0 25389 0 8385 0 48214 86 287. m/s (k)

Putting the values of v and a n in Equations (f) and (k) into (j) yields the radius of curvature,

229 4

86 287609 89

2.

.. m (l)

Upon substituting (a), (i) and (l) into (d) we obtain the position vector of the center of curvature C :

r i j k i jC ( ) (250 630 430 609 89 0 25389 0 8385 0 48214 . . . . kki j k

) (km) 95 159 1141 4 135 95. . .

Therefore , the coordinates of C are

x y z 95 16 1141 136 0. .m m m

1.4 MASS, FORCE AND NEWTONS LAW OF GRAVITATION Mass , like length and time, is a primitive physical concept: it cannot be defi ned in terms of any other physical concept. Mass is simply the quantity of matter. More practically, mass is a measure of the inertia of a body. Inertia is an objects resistance to changing its state of motion. The larger its inertia (the greater its mass), the more diffi cult it is to set a body into motion or bring it to rest. The unit of mass is the kilogram (kg).

Force is the action of one physical body on another, either through direct contact or through a distance. Gravity is an example of force acting through a distance, as are magnetism and the force between charged particles. The gravitational force between two masses m1 and m2 having a distance r between their centers is

F G

m m

rg

1 22

(1.31)

This is Newtons law of gravity , in which G , the universal gravitational constant , has the value G 6.6742 10 11 m 3 /(kg s 2 ). Due to the inverse-square dependence on distance, the force of gravity

1.4 Mass, force and Newtons law of gravitation 15


rapidly diminishes with the amount of separation between the two masses. In any case, the force of gravity is minuscule unless at least one of the masses is extremely big.

The force of a large mass (such as the earth) on a mass many orders of magnitude smaller (such as a person) is called weight, W . If the mass of the large object is M and that of the relatively tiny one is m , then the weight of the small body is

W G

Mm

rm

GM

r

2 2

or

W mg (1.32)

where

g

GM

r

2 (1.33)

g has units of acceleration (m/s 2 ) and is called the acceleration of gravity. If planetary gravity is the only force acting on a body, then the body is said to be in free fall. The force of gravity draws a freely falling object towards the center of attraction (e.g., center of the earth) with an acceleration g. Under ordinary con-ditions, we sense our own weight by feeling contact forces acting on us in opposition to the force of gravity. In free fall there are, by definition, no contact forces, so there can be no sense of weight. Even though the weight is not zero, a person in free fall experiences weightlessness, or the absence of gravity.

Let us evaluate Equation 1.33 at the surface of the earth, whose radius according to Table A.1 is 6378 km. Letting g 0 represent the standard sea-level value of g , we get

gGM

RE0 2

(1.34)

In SI units,

g029 807 . m/s (1.35)

Substituting Equation 1.34 into Equation 1.33 and letting z represent the distance above the earths surface, so that r R E z , we obtain

g gR

R z

g

RE

E E

0

2

20

21( ) ( )z/ (1.36)

Commercial airliners cruise at altitudes on the order of ten kilometers (six miles). At that height, Equation 1.36 reveals that g (and hence weight) is only three-tenths of a percent less than its sea level value. Thus, under ordinary conditions, we ignore the variation of g with altitude. A plot of Equation 1.36 out to a height of 1000 km (the upper limit of low-earth orbit operations) is shown in Figure 1.10 . The variation of g over that range is signifi cant. Even so, at space station altitude (300 km), weight is only about 10 percent less that it is on the earths surface. The astronauts experience weightlessness, but they clearly are not weightless.

Example 1.7 Show that in the absence of an atmosphere, the shape of a low altitude ballistic trajectory is a parabola. Assume the acceleration of gravity g is constant and neglect the earths curvature.

Solution Figure 1.11 shows a projectile launched at t 0 with a speed v 0 at a fl ight path angle 0 from the point with coordinates ( x 0 , y 0 ). Since the projectile is in free fall after launch, its only acceleration is that of gravity in the negative y -direction:

xy g

0

Integrating with respect to time and applying the initial conditions leads to

x x v 0 0 0( cos ) (a)

y y v t gt 0 0 0

21

2( sin ) (b)

Solving (a) for t and substituting the result into (b) yields

y y x xg

vx x

o

0 0 00

021

2( ) tan

cos( )

(c)

This is the equation of a second-degree curve, a parabola, as sketched in Figure 1.11 .

1.4 Mass, force and Newtons law of gravitation 17

200 400 600 800

0.7

0.8

0.9

1.0

1000z, km

g

g0

00

FIGURE 1.10 Variation of the acceleration of gravity with altitude.


Example 1.8 An airplane fl ies a parabolic trajectory like that in Figure 1.11 so that the passengers will experience free fall (weightlessness). What is the required variation of the fl ight path angle with speed v ? Ignore the curvature of the earth.

Solution Figure 1.12 reveals that for a fl at earth, d d , i.e.,

It follows from Equation 1.30 that

v (1.37)

The normal acceleration a n is just the component of the gravitational acceleration g in the direction of the unit principal normal to the curve (from P towards C ). From Figure 1.12 , then,

a gn cos (a)

x

y

(xo, yo)

vo

P

g

o

FIGURE 1.11 Flight of a low altitude projectile in free fall (no atmosphere).

d

d

C

y

x

g

P

FIGURE 1.12 Relationship between d and d for a fl at earth.

Substituting Equation 1.25 2 into (a) and solving for the radius of curvature yields

v

g

2

cos (b)

Combining Equations 1.37 and (b), we fi nd the time rate of change of the fl ight path angle,

g

v

cos

1.5 NEWTONS LAW OF MOTION Force is not a primitive concept like mass because it is intimately connected with the concepts of motion and inertia. In fact, the only way to alter the motion of a body is to exert a force on it. The degree to which the motion is altered is a measure of the force. Newtons second law of motion quantifi es this. If the resul-tant or net force on a body of mass m is F net , then

F anet m (1.38)

In this equation, a is the absolute acceleration of the center of mass. The absolute acceleration is measured in a frame of reference which itself has neither translational nor rotational acceleration relative to the fi xed stars. Such a reference is called an absolute or inertial frame of reference .

Force , then, is related to the primitive concepts of mass, length and time by Newtons second law. The unit of force, appropriately, is the newton, which is the force required to impart an acceleration of 1 m/s 2 to a mass of 1 kg. A mass of one kilogram therefore weighs 9.81 newtons at the earths surface. The kilogram is not a unit of force.

Confusion can arise when mass is expressed in units of force, as frequently occurs in U.S. engineer-ing practice. In common parlance either the pound or the ton (2000 pounds) is more likely to be used to express the mass. The pound of mass is offi cially defi ned precisely in terms of the kilogram as shown in Table A.3. Since one pound of mass weighs one pound of force where the standard sea-level acceleration of gravity ( g 0 9.80665 m/s

2 ) exists, we can use Newtons second law to relate the pound of force to the newton:

1 0 4536 9 807 4 4482lb force kg m/s N( ) . . .

The slug is the quantity of matter accelerated at one foot per second 2 by a force of one pound. We can again use Newtons second law to relate the slug to the kilogram. Noting the relationship between feet and meters in Table A.3, we fi nd

11

1

4 448

0 304814 59 14 5

2 2

2

2 slug

lb

ft/s

N

m/s

kg m/s

m/s

.

.. . 99 kg

1.5 Newtons law of motion 19


Example 1.9 On a NASA mission the space shuttle Atlantis orbiter was reported to weigh 239,255 lb just prior to lift off. On orbit 18 at an altitude of about 350 km, the orbiters weight was reported to be 236,900 lb. (a) What was the mass, in kilograms, of Atlantis on the launch pad and in orbit? (b) If no mass were lost between launch and orbit 18, what would have been the weight of Atlantis, in pounds?

Solution (a) The given data illustrates the common use of weight in pounds as a measure of mass. The weights given are actually the mass in pounds of mass. Therefore, prior to launch

mlaunch pad lb mass

kg

lb masskg 239 255

0 4536

1108 500, ( )

.

( ),

In orbit,

morbit lb mass

kg

lb masskg18 236 900

0 4536

1107 500 , ( )

.

( ),

The decrease in mass is the propellant expended by the orbital maneuvering and reaction control rockets on the orbiter.

(b) Since the space shuttle launch pad at Kennedy Space Center is essentially at sea level, the launch-pad weight of Atlantis in lb (force) is numerically equal to its mass in lb (mass). With no change in mass, the force of gravity at 350 km would be, according to Equation 1.36,

W

239 2551

13506378

215 000

2

, ( ) ,lb force

llb force( )

The integral of a force F over a time interval is called the impulse of the force ,

(1.39)

Impulse is a vector quantity. From Equation 1.38 it is apparent that if the mass is constant, then

(1.40)

That is, the net impulse on a body yields a change m v in its linear momentum, so that

(1.41)

If F net is constant, then net F net t , in which case Equation 1.41 becomes

v

FF net netif is constantm

t ( ) (1.42)

Let us conclude this section by introducing the concept of angular momentum. The moment of the net force about O in Figure 1.13 is

M r FOnet net

Substituting Equation 1.38 yields

M r a r

vO m m

d

dtnet (1.43)

But , keeping in mind that the mass is constant,

r

vr v

rr v v v m

d

dt

d

dtm

d

dtm

d

dtm m( ) ( ) ( )v

Since v m v m ( v v ) 0 , it follows that Equation 1.43 can be written

M

HO

Od

dtnet (1.44)

where H O is the angular momentum about O ,

H r vO m (1.45)


x

y

z

r

Inertial frame

O

m

Fnet

a

vk^

i

j^

FIGURE 1.13 The absolute acceleration of a particle is in the direction of the net force.


Thus , just as the net force on a particle changes its linear momentum m v , the moment of that force about a fi xed point changes the moment of its linear momentum about that point. Integrating Equation 1.44 with respect to time yields

M H HOt

t

O Odtnet1

2

2 1 (1.46)

The integral on the left is the net angular impulse . This angular impulse momentum equation is the rota-tional analog of the linear impulse momentum relation given in Equation 1.40.

Example 1.10 A particle of mass m is attached to point O by an inextensible string of length l ( Figure 1.14 ). Initially the string is slack when m is moving to the left with a speed v 0 in the position shown. Calculate (a) the speed of m just after the string becomes taut and (b) the average force in the string over the small time interval t required to change the direction of the particles motion.

Solution (a) Initially, the position and velocity of the particle are

r1 1 c d vo i j v i

The angular momentum about O is

H r v

i j k

k1 1 1 0

0 0

m c d

mv

mv d

o

o

(a)

Just after the string becomes taut

r i j v i j2

2 22 l d d v vx y

(b)

x

y

d

v0

v

l

m

O

c

FIGURE 1.14 Particle attached to O by an inextensible string.

and the angular momentum is

H r v

i j k

k2 2 22 2 2 20

0

m l d d

v v

mv d mv l d

x y

x y

( ) (c)

Initially the force exerted on m by the slack string is zero. When the string becomes taut, the force exerted on m passes through O . Therefore, the moment of the net force on m about O remains zero. According to Equation 1.46,

H H2 1

Substituting (a) and (c), yields

v d l d v v dx y o

2 2 (d)

The string is inextensible, so the component of the velocity of m along the string must be zero:

v r2 2 0

Substituting v 2 and r 2 from (b) and solving for v y , we get

v v

l

dy x

2

21 (e)

Solving (d) and (e) for v x and v y leads to

v

d

lv v

d

l

d

lvx o y o

2

2

2

21 (f)

Thus , the speed, v v vx y2 2 , after the string becomes taut is

v

d

lvo

(b) From Equation 1.40, the impulse on m during the time it takes the string to become taut is

The magnitude of this impulse, which is directed along the string, is



Hence , the average force in the string during the small time interval t required to change the direction of the velocity vector turns out to be

1.6 TIME DERIVATIVES OF MOVING VECTORS Figure 1.15(a) shows a vector A inscribed in a rigid body B that is in motion relative to an inertial frame of reference (a rigid, Cartesian coordinate system which is fi xed relative to the fi xed stars). The magnitude of A is fi xed. The body B is shown at two times, separated by the differential time interval dt . At time t dt the orientation of vector A differs slightly from that at time t , but its magnitude is the same. According to one of the many theorems of the prolifi c eighteenth century Swiss mathematician Leonhard Euler (1707 1783), there is a unique axis of rotation about which B , and therefore, A rotates during the differential time interval. If we shift the two vectors A ( t ) and A ( t dt ) to the same point on the axis of rotation, so that they are tail-to-tail as shown in Figure 1.15(b) , we can assess the difference d A between them caused by the infi nitesimal rotation. Remember that shifting a vector to a parallel line does not change the vector. The rotation of the body B is measured in the plane perpendicular to the instantaneous axis of rotation. The amount of rotation is the angle d through which a line element normal to the rotation axis turns in the time interval dt . In Figure 1.15(b) that line element is the component of A normal to the axis of rotation. We can express the difference d A between A ( t ) and A ( t dt ) as

d d

d

A A n

A

magnitude of

sin ( )

(1.47)

d

A

dA

A + dA

A(t + dt)

X

Y

Z

t t + dt

A (t)

Rigid body B

(a) (b)

Inertial frame

Instantaneous axis of rotation

v

FIGURE 1.15 Displacement of a rigid body.

where n is the unit normal to the plane defined by A and the axis of rotation, and it points in the direction of the rotation. The angle is the inclination of A to the rotation axis. By definition,

d dt (1.48)

where is the angular velocity vector, which points along the instantaneous axis of rotation and its direc-tion is given by the right-hand rule. That is, wrapping the right hand around the axis of rotation, with the fingers pointing in the direction of d results in the thumbs defining the direction of . This is evident in Figure 1.15(b) . It should be pointed out that the time derivative of is the angular acceleration, usually given the symbol . Thus,

d

dt (1.49)

Substituting Equation 1.48 into Equation 1.47, we get

d dt dtA A n A n sin sin ( ) (1.50)

By defi nition of the cross product, A is the product of the magnitude of , the magnitude of A , the sine of the angle between and A and the unit vector normal to the plane of and A , in the rotation direction. That is,

A A n sin (1.51)

Substituting Equation 1.51 into Equation 1.50 yields

d dtA A Dividing through by dt , we fi nally obtain

d

dt

d

dt

AA A if 0

(1.52)

Equation 1.52 is a formula we can use to compute the time derivative of a rotating vector of constant magnitude .

Example 1.11 Calculate the second time derivative of a vector A of constant magnitude, expressing the result in terms of and its derivatives and A .

Solution Differentiating Equation 1.52 with respect to time, we get

d

dt

d

dt

d

dt

d

dt

d

dt

d

dt

2

2

A AA A

A ( )

Using Equations 1.49 and 1.52, this can be written

d

dt

2

2

AA A ( ) (1.53)

1.6 Time derivatives of moving vectors 25


Example 1.12 Calculate the third derivative of a vector A of constant magnitude, expressing the result in terms of and its derivatives and A .

Solution

d

dt

d

dt

d

dt

d

dtd

dt

d

dtd

3

3

2

2

A AA A

A A

[ ( )]

( ) [ ( )]

ddt

d

dt

d

dt

d

dtd

dt

AA

A A

( ) ( )

A A A AA

( ) ( )

d

dt

d

dt

d

dtd

d

A A A A A( ) { ( ) [ ( )]}

ttd

dt

A A A A A

A A

( ) ( ) [ ( )]

( )

( )

2 ( ) [ ( )]A A

d

dt

d

dt

3

32

AA A A A

( ) [ ( )]

Let XYZ be a rigid inertial frame of reference and xyz a rigid moving frame of reference, as shown in Figure 1.16 . The moving frame can be moving (translating and rotating) freely of its own accord, or it can be attached to a physical object, such as a car, an airplane or a spacecraft. Kinematic quantities measured relative to the fi xed inertial frame will be called absolute (e.g., absolute acceleration), and those measured relative to the moving system will be called relative (e.g., relative acceleration). The unit vectors along the inertial XYZ sys-tem are I , J and K , whereas those of the moving xyz system are i , j and k . The motion of the moving frame is arbitrary, and its absolute angular velocity is . If, however, the moving frame is rigidly attached to an object, so that it not only translates but rotates with it, then the frame is called a body frame and the axes are referred to as body axes. A body frame clearly has the same angular velocity as the body to which it is bound.

Let B be any time-dependent vector. Resolved into components along the inertial frame of reference, it is expressed analytically as

B I J K B B BX Y Z

where B X , B Y and B Z are functions of time. Since I , J and K are fixed, the time derivative of B is simply given by

d

dt

dB

dt

dB

dt

dB

dtX Y ZB I J K

dB X / dt , dB Y / dt and dB Z / dt are the components of the absolute time derivative of B .

B may also be resolved into components along the moving xyz frame, so that, at any instant,

B i j k B B Bx y z (1.54)

Using this expression to calculate the time derivative of B yields

d

dt

dB

dt

dB

dt

dB

dtB

d

dtB

d

dtB

d

dtx y z

x y zB

i j ki j k

(1.55)

The unit vectors i , j and k are not fi xed in space, but are continuously changing direction; therefore, their time derivatives are not zero. They obviously have a constant magnitude (unity) and, being attached to the xyz frame, they all have the angular velocity . It follows from Equation 1.52 that

d

dt

d

dt

d

dt

i i

jj

kk

Substituting these on the right-hand side of Equation 1.55 yields

d

dt

dB

dt

dB

dt

dB

dtB B B

dB

x y zx y z

Bi j k i j k

( ) ( ) ( )

xx y zx y z

x

dt

dB

dt

dB

dtB B B

dB

dt

( ) ( ) ( )

i j k i j k

i

ddB

dt

dB

dtB B B

y zx y z

( )j k i j k

In view of Equation 1.54, this can be written

d

dt

d

dt

B BB

rel (1.56)

1.6 Time derivatives of moving vectors 27

X

Y

Z

x

y

z

Inertial frame

Moving frameO

B

By

Bx

Bz

J

I

K

i

j

k

FIGURE 1.16 Fixed (inertial) and moving rigid frames of reference.


where

d

dt

dB

dt

dB

dt

dB

dtx y zB i j k

rel

(1.57)

d B / dt ) rel is the time derivative of B relative to the moving frame. Equation 1.56 shows how the absolute time derivative is obtained from the relative time derivative . Clearly, d B / dt d B / dt ) rel only when the mov-ing frame is in pure translation ( 0 )).

Equation 1.56 can be used recursively to compute higher order time derivatives. Thus, differentiating Equation 1.56 with respect to t , we get

d

dt

d

dt

d

dt

d

dt

d

dt

2

2

B BB

B

rel

Using Equation 1.56 in the last term yields

d

dt

d

dt

d

dt

d

dt

d

dt

2

2

B BB

BB

rel rel

(1.58)

Equation 1.56 also implies that

d

dt

d

dt

d

dt

d

dt

B B B

rel rel rel

2

2 (1.59)

where

d

dt

d B

dt

d B

dt

d B

dtx y z

2

2

2

2

2

2

2

2

Bi j k

rel

Substituting Equation 1.59 into Equation 1.58 yields

d

dt

d

dt

d

dt

d

dt

2

2

2

2

B B BB

rel rel

d

dt

BB

rel

Collecting terms, this becomes

d

dt

d

dt

d

dt

2

2

2

22

B BB B

B

rel rel

( ) (1.60)

where d dt/ is the absolute angular acceleration of the xyz frame. Formulas for higher order time derivatives are found in a similar fashion.

1.7 RELATIVE MOTION Let P be a particle in arbitrary motion. The absolute position vector of P is r and the position of P relative to the moving frame is r

orbital mechanics for -...

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