Rotational Time Derivative

The challenge is to find the time operator that preserves the form on both sides of Eq. (4.35). We venture to define the terms in parentheses as that operator and see what happens to the left side.

Definition: The rotational time derivative of a first-order tensor x wrt any

frame A, DAx, and expressed in any allowable coordinate system ]B is defined by

Notice in [Dax]b the appearance of frame A and coordinate system ]B. Both are arbitrary, but the coordinate system ]4 of [T]BA on the right-hand side has to be associated with frame A.

Let us break the suspense and apply the rotational derivative to the left side of Eq. (4.35). Instead of ]B the coordinate system is now ]4:

ds	A	l-dTl	-AA	ds
	+ [T]AA	– dt _	=
_d t _	_dt _

[Das]a =

which is, with the time derivative of the unit matrix being zero, exactly in the desired form. Therefore, we can write Eq. (4.35) as

[Das]a = [T]ab[Das]b

That looks to me like a tensor transformation. Although the ]B coordinate system is arbitrary, the ]A system is still unique by its association with frame A. For a true tensor form any allowable coordinate system must be admitted. Therefore, consider an arbitrary allowable coordinate system ]Dto replace ]Aon the left side of Eq. (4.35):

By definition of the rotational derivative, we obtain the true tensor transformation

[Das]d = [T]db[Das]b

Therefore, the rotational derivative of a vector transforms like a tensor, and DAs

is a tensor.

A comparable rationale defines the rotational derivative for second-order ten­sors. I will just state the results. The details can be found in Appendix D.

Definition: The rotational time derivative of a second-order tensor X wrt any

frame A, DAX, and expressed in any allowable coordinate system ]B, is defined as

We have two terms pre – or postmultiplying the tensor with the term [T]BA [dr /dt] and its transpose.

The rotational derivative of tensors transforms like a second-order tensor. For any two allowable coordinate systems ]B and ]D,

[.DaX]d = [T]db[DaX]b[T]db

Therefore if X is a second-order tensor, so is DAX.

The rotational time derivative has some nice properties that will make it a joy to work with.

Property 1

The rotational derivative wrt any frame A is a linear operator. 1) For any constant scalar к and vector*

DA(kx) = kDAx	(4.38)
2) For any two vectors x and у
DA(x+y) = DAx + DAy	(4.39)
Property 2

The rotational time derivative abides by the chain rule. For any tensor Y and vector x

Da(Yx) = (DaY)x + YDax

Note that you must maintain the order of the tensor multiplication. The parenthe­ses on the right side are redundant because, by convention, the derivative operates only on the adjacent variable.

The rotational derivative strengthens the modeling process of aerospace vehicles. It enables the formulation of Newton’s and Euler’s laws as invariants under time – dependent coordinate transformations. As you will see in Chapters 5 and 6, we remain true to our principle from tensor formulation to matrix coding. With the instrument of an invariant time operator in hand, we can create linear and angular velocities and their accelerations.