This course is part 2 of the specialization Advanced Spacecraft Dynamics and Control. It assumes you have a strong foundation in spacecraft dynamics and control, including particle dynamics, rotating frame, rigid body kinematics and kinetics. The focus of the course is to understand key analytical mechanics methodologies to develop equations of motion in an algebraically efficient manner.
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The course starts by first developing D’Alembert’s principle and how the associated virtual work and virtual displacement concepts allows us to ignore non-working force terms. Unconstrained systems and holonomic constrains are investigated. Next Kane's equations and the virtual power form of D'Alembert's equations are briefly reviewed for particles.
Next Lagrange’s equations are developed which still assume a finite set of generalized coordinates, but can be applied to multiple rigid bodies as well. Lagrange multipliers are employed to apply Pfaffian constraints.
Finally, Hamilton’s extended principle is developed to allow us to consider a dynamical system with flexible components. Here there are an infinite number of degrees of freedom. The course focuses on how to develop spacecraft related partial differential equations, but does not study numerically solving them. The course ends comparing the presented assumed mode methods to classical final element solutions.
What You Will Learn
- Use virtual work methods to develop equations of motion of mechanical systems.
- Understand how to use Lagrange multipliers to study constrained dynamical systems.
- Be able to derive the equations of motion of a spacecraft with flexible sub-components.
Syllabus
WEEK 1
Generalized Methods of Analytical Mechanics
Learn the methodology of developing equations of motion using D'Alembert's principle, virtual power forms, Lagrange's equations as well as the Boltzmann-Hamel equations. These methods allow for more efficient equations of motion development where state based (holonomic) and rate based (Pfaffian constraints) are considered.
WEEK 2
Energy Based Equations of Motion
Derive methods to develop the equations of motion of a dynamical system with finite degrees of freedom based on energy expressions.
WEEK 3
Variational Methods in Analytical Dynamics
Learn to develop the equations of motion for a dynamical system with deformable shapes. Such systems have infinite degrees of freedom and lead to partial differential equations.
