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Accelerated Frames of Reference

A coordinate system used to measure the position and motion of objects is referred to as a “frame of reference”. An inertial frame of reference is one where an object at rest or moving at a constant velocity experiences no net external forces (i.e., Newton’s first law applies). Most aerospace systems, especially during maneuvers, are more conveniently tackled in non-inertial or accelerated frames, however. Such frames come into play when a vehicle is experiencing rapid changes in velocity, direction, or altitude. For instance, when an aircraft is turning, an observer on the plane experiences centrifugal forces that do not exist in the (quasi) inertial frame of a stationary observer on the surface of the Earth. In these cases, so-called fictitious or frame-based forces such as centrifugal force (due to rotation) and the Coriolis force (due to moving within a rotating reference frame) arise.

To navigate between inertial and accelerated frames, one must apply transformations that account for frame-based forces.

Kinematics in Accelerated Frames

A time dependent position vector pointing from the origin to a position in the inertial frame A can be defined as

where is the standard unit basis. For instance, a realization in Cartesian 3D of A could read

The same vector in the accelerated frame B reads

We will assume that the new basis vectors are time dependent and can be expressed in terms of the unit basis vectors such that we can construct a change-of-basis matrix . In Cartesian 3D would read

We will further assume that we can express the vector as

where is the vector connecting the origins of the frames and . To simplfy notation we shall remember that and assume without loss of generatlity that for the remainder of this article.

Accelerated Frame to Inertial Frame

If both origins are identical, i.e., , then the position in frame can be calculated via

The velocity in frame is

and accelerations in frame can be found by differentiating once more with respect to time

Here we have used the product rule, since both and may depend on time. We see that accelerations in the inertial frame are not simply a linear combination of acceleration in the accelerated frame . Instead, the time dependent basis vectors contribure extra terms if .

Inertial Frame to Accelerated Frame

In order to understand what velocity and acceleration vectors in the accelerated frame look like, we can use the fact that positions transform as

Velocities are derived via matrix calculus

Finally, accelerations of our test particle in the accelerated frame read

Heads up!

Note that for velocity and acceleration vectors can be attained through a simple change of basis , and
.

Rotating Frames and the Transport Theorem

Matrices that describe rotating coordinate systems can be considered a special case of time dependent frame transformations. Let be the position vector in the inertial frame, be the position vector in the rotating frame and a rotation matrix from frame B to frame A, then

Realizations of rotation matrices around the x, y and z axis are

respectively. Calculating derivatives and inverting the above matrices is possible to calculate velocity and acceleration in and frames is possible, but not very convenient. The Transport Theorem provides a more efficient way to calculate vector components in accelerated (rotating) frames from inertial-frame quantities.

Transport Theorem

The Transport Theorem states that time derivatives of a vector in inertial frames can be expressed in rotating frame quantities as

How come? The change in time of the unit vectors that make up matrix can be described as

since the rotation has to happen in the plane orthogonal to and vectors . To this we need to add changes of in the frame, and voila.

Using the Transport Theorem we can derive the well known equations for transforming accelerations and velocities in rotating frames. Taking the time derivative of we find

and since (matrix component vector wise)

we have

Differentiating again we find the well known transformation equation for accelerations:

This above expression contains the following acceleration terms:

Euler: ,
Centripetal: ,
Coriolis: , and
In Frame: .