As they tumble through space, objects like spacecraft move in dynamical ways. Understanding and predicting the equations that represent that motion is critical to the safety and efficacy of spacecraft mission development. Kinetics: Modeling the Motions of Spacecraft trains your skills in topics like rigid body angular momentum and kinetic energy expression shown in a coordinate frame agnostic manner, single and dual rigid body systems tumbling without the forces of external torque, how differential gravity across a rigid body is approximated to the first order to study disturbances in both the attitude and orbital motion, and how these systems change when general momentum exchange devices are introduced.

After this course, you will be able to…

*Derive from basic angular momentum formulation the rotational equations of motion and predict and determine torque-free motion equilibria and associated stabilities

* Develop equations of motion for a rigid body with multiple spinning components and derive and apply the gravity gradient torque

* Apply the static stability conditions of a dual-spinner configuration and predict changes as momentum exchange devices are introduced

* Derive equations of motion for systems in which various momentum exchange devices are present

Please note: this is an advanced course, best suited for working engineers or students with college-level knowledge in mathematics and physics.

**Who is this class for:** This class is for working engineering professionals looking to add to their skill sets, graduate students in engineering looking to fill gaps in their knowledge base, and enterprising engineering undergraduates looking to expand their horizons.

### Syllabus

**WEEK 1**

Continuous Systems and Rigid Bodies

The dynamical equations of motion are developed using classical Eulerian and Newtonian mechanics. Emphasis is placed on rigid body angular momentum and kinetic energy expression that are shown in a coordinate frame agnostic manner. The development begins with deformable shapes (continuous systems) which are then frozen into rigid objects, and the associated equations are thus simplified.

Graded: Concept Check 1 – Super Particle Theorem

Graded: Concept Check 2 – Kinetic Energy

Graded: Concept Check 3 – Linear Momentum

Graded: Concept Check 4 – Angular Momentum

Graded: Concept Check 5 – Angular Momentum

Graded: Concept Check 6 – Parallel Axis Theorem

Graded: Concept Check 6.1 – Coordinate Transformation

Graded: Concept Check 7 – Kinetic Energy

Graded: Concept Check 8 – Equations of Motion

**WEEK 2**

Torque Free Motion

The motion of a single or dual rigid body system is explored when no external torques are acting on it. Large scale tumbling motions are studied through polhode plots, while analytical rate solutions are explored for axi-symmetric and general spacecraft shapes. Finally, the dual-spinner dynamical system illustrates how the associated gyroscopics can be exploited to stabilize any principal axis spin.

Graded: Concept Check 1 – Rigid Body Polhode Plots

Graded: Concept Check 2 – Torque Free Motion with Axisymmetric Body

Graded: Concept Check 3 – Torque Free Motion General Inertia

Graded: Concept Check 4 – Torque Free Motion Integrals of Motion

Graded: Concept Check 5 – Torque Free Motion Phase Space Plots

Graded: Concept Check 6 – Torque Free Motion Precession

Graded: Concept Check 7 – Dual Spinner Equations of Motion

Graded: Concept Check 8 – Dual Spinner Equilibria

Graded: Concept Check 9 – Dual Spinner Linear Stability

**WEEK 3**

Gravity Gradients

The differential gravity across a rigid body is approximated to the first order to study how it disturbs both the attitude and orbital motion. The gravity gradient relative equilibria conditions are derived, whose stability is analyzed through linearization.

Graded: Concept Check 1 – Gravity Gradient Derivation

Graded: Concept Check 2 – Gravity Gradient Equilibria

Graded: Concept Check 3 – Gravity Gradient Linear Stability

**WEEK 4**

Equations of Motion with Momentum Exchange Devices

The equations of motion of a rigid body are developed with general momentum exchange devices included. The development begins with looking at variable speed control moment gyros (VSCMG), which are then specialized to classical single-gimbal control moment devices (CMGs) and reaction wheels (RW).

Graded: Concept Check 1 – Overview of Momentum Exchange Devices

Graded: Concept Check 2 – VSCMG Equations of Motion

Graded: Concept Check 3 – VSCMG Motor Torque Equations

Graded: Concept Check 4 – VSCMG EOM Variations

Graded: Kinetics Final Assignment

ENROLL IN COURSE