MAE 4263 Spaceflight Mechanics Dr. Brian Kaplinger

UNIT 1 : Vector and Dynamics “Review”

What is a Vector?

What is a Vector? • A vector has magnitude and direction.

What is a Vector? • A vector has magnitude and direction.

• Consider the following vector:

What is a Vector? • Clearly, this fits the

definition, having both magnitude and direction.

What is a Vector?

What is a Vector?

What is a Vector? • What if we rotated the

unit axes?

What is a Vector?

What is a Vector?

What is a Vector? • Clearly, we have some

counterintuitive behavior.

• We need to reevaluate our definition of a vector, so let’s try again:

• A vector is a mathematical object to which

we can attribute magnitude and direction, which is independent of coordinates.

How do we treat vectors? • We can evaluate the components of a vector

relative to a coordinate system.

• This generates a set of numbers related to the vector.

• This set is often arranged in a column array, e.g.:

How do we treat vectors? • We can evaluate the components of a vector

relative to a coordinate system.

• This generates a set of numbers related to the vector.

• This set is often arranged in a column array, e.g.:

WARNING:

is not generally true and can wreck havoc if the coordinate system is not implied.

What is a Coordinate System?

• We will be dealing with multiple coordinate systems.

• A coordinate system is a set of vectors that is ordered, orthonormal, right-handed, and forms a basis for a space.

What is a Coordinate System?

What is a Coordinate System?

What is a Coordinate System?

What is a Coordinate System?

What is a Coordinate System?

• Defining right-handed is a bit trickier.

• How many orthogonal vectors can we have in a 3D space?

What is a Coordinate System?

• Defining right-handed is a bit trickier.

• How many orthogonal vectors can we have in a 3D space?

• Clearly, in an ND space we can have at most N orthogonal vectors.

What is a Coordinate System?

What is a Coordinate System?

What is a Coordinate System?

• Now we have two orthogonal vectors, how do we add the final vector to our set?

What is a Coordinate System?

What is a Coordinate System?

• The outcome of the cross-product is defined to be right-handed precisely because this choice between two candidates is involved where both satisfy the orthogonal requirement.

What is a Coordinate System?

What is a Coordinate System?

What is a Coordinate System?

• This is called cyclic behavior mathematically, and is where being ordered becomes important

What is a Coordinate System?

What is a Coordinate System?

What is a Coordinate System?

What is a Coordinate System?

• Orthogonal will always mean linear independent, so the only other requirement is that we have the right number of vectors.

• An orthogonal set with the same number of elements as the dimensionality of the space will be a basis for that space.

A Note on Notation

• A coordinate system is composed of vectors.

• A vector is not dependent on a coordinate system.

• The paradox described earlier must be a function of mixing the two concepts.

A Note on Notation • The process of extracting

numerical values from a vector and presenting them as an array is evaluating a vector relative to a coordinate system.

• The numbers we attribute to the vector (elements of the array) are the coordinates of the vector.

• Coordinates are a function of both the vector and a coordinate system, and have no meaning outside of this relationship.

A Note on Notation

• To demonstrate

A Note on Notation

A Note on Notation

Coordinate System Motion

• This concept is important dynamically because not all coordinate systems are stationary!

• Example: A coordinate system fixed to the surface of a rotating Earth.

Coordinate System Motion

Coordinate System Motion

Coordinate System Motion

Coordinate System Motion

Coordinate System Motion

• So the angular velocity vector is

Coordinate System Motion

• For an inertial coordinate system (non-accelerating) the derivatives of the coordinate vectors are all zero.

• It should be clear that this is not the case for the rotating example:

Coordinate System Motion

• For an inertial coordinate system (non-accelerating) the derivatives of the coordinate vectors are all zero.

• It should be clear that this is not the case for the rotating example:

Actually we have

Coordinate System Motion

Coordinate System Motion

Because

Coordinate System Motion

Because

Coordinate System Motion

Because

Coordinate System Motion

Because

Coordinate System Motion

Because

Coordinate System Motion

Which gives us the rule applying to all vector derivatives:

Dynamics Review

• Newton’s Laws (wording from Wikipedia)

• When measured in an inertial reference frame:

• 1: an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.

• 2: the net force on an object is equal to the rate of change of its linear momentum.

• 3: when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

Dynamics Review

Dynamics Review

Dynamics Review

Dynamics Review

So

Dynamics Review

Also

Dynamics Review

• Newton’s Law of Gravitation

• One special case for an extended (rigid) body is gravity of a sphere, which acts like a point mass with all of the sphere’s mass concentrated at a single point.

• Important proof; Shown in book appendix.

Dynamics Review

• Impulse is the integral of force with respect to time:

Dynamics Review

• Impulse is the integral of force with respect to time:

Dynamics Review

• Impulse is the integral of force with respect to time:

Dynamics Review

• Impulse is the integral of force with respect to time:

Dynamics Review

• Impulse is the integral of force with respect to time:

Or

Impulsive Spaceflight

• During this course we will often assume propulsive events are impulsive. Thus, the window of time over which an event occurs is small compared to the overall time we are interested in (say, an orbit period).

Impulsive Spaceflight

• During this course we will often assume propulsive events are impulsive. Thus, the window of time over which an event occurs is small compared to the overall time we are interested in (say, an orbit period).

Impulsive Spaceflight

• During this course we will often assume propulsive events are impulsive. Thus, the window of time over which an event occurs is small compared to the overall time we are interested in (say, an orbit period).

We model as

Impulsive Spaceflight

• During this course we will often assume propulsive events are impulsive. Thus, the window of time over which an event occurs is small compared to the overall time we are interested in (say, an orbit period).

We model as

Impulsive Spaceflight

• During this course we will often assume propulsive events are impulsive. Thus, the window of time over which an event occurs is small compared to the overall time we are interested in (say, an orbit period).

Velocity looks like

Impulsive Spaceflight

• During this course we will often assume propulsive events are impulsive. Thus, the window of time over which an event occurs is small compared to the overall time we are interested in (say, an orbit period).

Velocity looks like We model as

Impulsive Spaceflight

• During this course we will often assume propulsive events are impulsive. Thus, the window of time over which an event occurs is small compared to the overall time we are interested in (say, an orbit period).

Velocity looks like We model as

Impulsive Spaceflight

• As time interval approaches zero, all forces with a net effective change on momentum must be modeled as infinite in magnitude, since a change happens in infinitely small time.

• Forces modeled as “delta functions”.

• Velocity is integral of delta function, or a step function.

• What about position?

Consequences