Notation

Now we come to a nettlesome issue. What notation is best suited for modeling of aerospace vehicle dynamics? It should be concise, self-defining, and adaptable to tensors and matrices. By “self-defining” I mean that the symbol expresses all char­acteristics of the physical quantity. For intricate quantities it may require several sub- and superscripts.

Surveying the field, I go back to my vector mechanics book. There, as an ex­ample, velocity vectors are portrayed by symbols like v, v, v, or v. An advanced physics book will most likely use the subscripted tensor notation, emphasizing the transformation properties of tensors. The velocity vector is written as u,; і =

1,2,3 over the Euclidean three-space, and the transformation between coordinates is

vi=tijVj j — 1,2,3; і = 1,2,3

with the summation convention over the dummy index j implied, meaning

Draper Laboratory at the Massachusetts Institute of Technology has modified this convention, favoring the form

Vі =tljvj; j = 1,2,3; і = 1,2,3

as a vector transformation.

Our need is driven by our modeling approach, i. e., from invariant tensors to programmable matrices. Vector mechanics emphasizes the symbolic, coordinate – independent notation, whereas the tensor notation focuses on the components. We adopt the best of both worlds. Bolded lower-case letters are used for vec­tors (first-order tensors) and bolded upper-case letters for tensors (second-order tensors). For scalars (zeroth-order tensor) we use regular fonts. These are the only three types of variables that occur in the Euclidean space of Newtonian mechanics.

The sub- and superscript positions immediately after the main symbol are re­served for further specification of the physical quantity. Here we make use of our postulate that points, and frames suffice to describe any physical phenomena in flight dynamics. We fix indelibly the following convention: subscripts for points and superscripts for frames. For both we use capital letters. Some examples should crystallize this practice.

The displacement vector of point A with respect to point В is the vector sAb the velocity vector of point В with respect to the inertial frame I is modeled by v ; and the angular velocity vector of frame В with respect to frame / is annotated by u>BI. All three are first-order tensors. The moment of inertia tensor IB of body (frame) В referred to the reference point C is a second-order tensor. If there are two sub – or two superscripts, they are always read from left to right, joined by the phrase “with respect to” (wrt).

For expressing the tensors in coordinate systems, we could use the subscript notation of tensor algebra or the sub/superscript formulation of the Massachusetts Institute of Technology. However, our sub – and superscript locations would become overloaded. I prefer to emphasize the fact that the tensor has become a matrix (through coordination) by using square brackets with the particular coordinate system identified by a raised capital letter. Let us expand on the four examples.

To express the displacement vector sAB in Earth coordinates E, we write [л’дд]£; the velocity vector Vg becomes [v’H |£; and the angular velocity vector u>BI, stated in inertial coordinates, is [ ooBI]1. All three are 3×1 column matrices. The moment of inertia tensor IB, expressed in body coordinates B, is the 3 x 3 matrix [IB]B■ Usually the bolding of the symbols will be omitted once the variable is enclosed in brackets, and we will write plainly [sAB]E, [v’BE, [coBI]1, and [IB]B.

The nomenclature at the front of this volume summarizes most of the variables that you will encounter throughout the book. I will adhere to these symbols closely, only changing the sub – and superscripts. Let me just point out a few things. All variables are considered tensors either of zeroth-, first-, or second order, but I will use mostly the term vector for the first-order tensor. The transpose is indicated by an overbar. We will distinguish carefully between an ordinary and rotational time derivative.

The advantage of the nomenclature lies in the clear distinction between coordinate-independent (invariant) tensor notation and the coordinate-dependent bracketed matrix formulation. General tensor algebra, with its sub – and superscript notation, emphasizes many types of tensors, e. g., covariant, contravariant tensors, Kronecker delta, and permutation symbol. The dummy indices and contraction (summation) play an important part. This mathematical language was created for the sophisticated world of general relativity embedded in Riemannian space. Our world is still Newtonian and Euclidean. Simple Cartesian tensors are completely adequate. Therefore, I forego the tensorial sub – and superscript notation in favor of the matrix brackets and am able to readily distinguish between the many coordinate systems of flight mechanics.