Ashish Tewari - Foundations of Space Dynamics

(2.1)

All the variables representing the motion of a spacecraft are changing with time , . The overdots represent the time derivatives, e.g., , , . The time derivative of a vector , which is changing both in its magnitude and its direction, requires an explanation.

The time derivative of a vector, , which is changing both in magnitude and direction can be resolved in two mutually perpendicular directions – one along the original direction of , and the other normal to it on the plane of the rotation of . The instantaneous angular velocity , , of denotes the vector rate of change in the direction, whereas is the rate of change in its magnitude. By definition, is normal to the direction of the unit vector, , and lies in the instantaneous plane of rotation normal to . The rotation of is indicated by the right‐hand rule , where the thumb points along , and the curled fingers show the instantaneous direction of rotation, 1 . The time derivative of is therefore expressed as follows:

(2.2)

where the term represents a unit vector in the original direction of , and is the change normal to caused by its rotation. Equation ( 2.2) will be referred to as the chain rule of vector differentiation in this book.

Similarly, the second time derivative of is given by the application of the chain rule to differentiate as follows:

(2.3)

Applying Eq. (2.1)to the time derivative of the angular velocity, , we have the following expression for the angular acceleration of :

(2.4)

where is the instantaneous angular velocity at which the vector is changing its direction. Hence, the second time derivative of is expressed as follows:

(2.5)

The bracketed term on the right‐hand side of Eq. (2.5)is parallel to , while the second term on the right‐hand side is perpendicular to both and . The last term on the right‐hand side of Eq. (2.5)denotes the effect of a time‐varying axis of rotation of .

As a special case, consider the motion of a point, P , in a fixed plane described by the radius vector, , which is changing in time. The vector is drawn from a fixed point, o , on the plane, to the moving point, P , and hence denotes the instantaneous radius of the moving point from o . The instantaneous rotation of the vector is described by the angular velocity, , which is fixed in the direction given by the unit vector , normal to the plane of motion. Thus we have the following in Eq. (2.4):