Dynamics of Systems of Rigid Bodies

1 Mathematical Notation.- 2 Rigid Body Kinematics.- 2.1 Generalized coordinates for the angular orientation of a rigid body.- 2.1.1 Euler angles.- 2.1.2 Bryant angles.- 2.1.3 Euler parameters.- 2.2 The notion of angular velocity.- 2.3 Relationships between the angular velocity of a body and generalized coordinates describing the angular orientation of the body.- 2.3.1 Direction cosines.- 2.3.2 Euler angles.- 2.3.3 Bryant angles.- 2.3.4 Euler parameters.- 3 Basic Principles of Rigid Body Dynamics.- 3.1 Kinetic energy.- 3.2 Angular momentum.- 3.3 Properties of moments and products of inertia.- 3.3.1 Transition to another reference point without change of the reference base.- 3.3.2 Transition to another reference base without change of the reference point.- 3.3.3 Principal axes and principal moments of inertia.- 3.3.4 Invariants and inequalities for moments and products of inertia.- 3.4 The law of moment of momentum.- 3.5 D'Alembert's principle applied to a rigid body.- 4 Classical Problems of Rigid Body Mechanics.- 4.1 The unsymmetric torque-free rigid body.- 4.1.1 Polhodes and permanent rotations.- 4.1.2 Poinsot's geometric interpretation of the motion.- 4.1.3 The solution of Euler's equations of motion.- 4.1.4 The solution of the kinematic differential equations.- 4.2 The symmetric torque-free rigid body.- 4.3 The self-excited, symmetric rigid body.- 4.4 The symmetric heavy top.- 4.5 The symmetric heavy body in a cardan suspension.- 4.6 The gyrostat. General considerations.- 4.7 The torque-free gyrostat.- 4.7.1 Polhodes and permanent rotations.- 4.7.2 The solution of the dynamic equations of motion.- 5 General Multi-Body Systems.- 5.1 Introductory remarks.- 5.2 Equations of motion for systems with tree structure.- 5.2.1 The mathematical description of the interconnection structure.- 5.2.2 Systems with ball-and-socket joints. One body is coupled to an external body whose motion is prescribed.- 5.2.3 The special case of plane motions.- 5.2.4 Systems with ball-and-socket joints without coupling to an external body whose motion is prescribed.- 5.2.5 The special case of a multi-body satellite in a circular orbit.- 5.2.6 Systems with ball-and-socket, universal and pin joints.- 5.2.7 Programming instructions.- 5.2.8 Systems with arbitrary holonomic constraints in the hinges.- 5.2.9 Internal forces and torques in the hinges of a system with arbitrary holonomic constraints.- 5.3 Multi-body systems with closed chains and with arbitrary constraints.- 5.3.1 The mathematical description of the interconnection structure. A generalization of Section 5.2.1.- 5.3.2 Equations of motion.- 5.4 Concluding remarks.- 6 Impact Problems in Holonomic Multi-Body Systems.- 6.1 Basic assumptions.- 6.2 Instantaneous velocity increments.- 6.3 An analogy to the law of Maxwell and Betti.- 6.4 Internal impulses and impulse couples in hinges.- Answers to Problems.- Literature References.