Building Blocks of Mathematical Modeling

With the general principles of classical mechanics under our belt, we employ a mathematical language that allows us to formulate dynamic problems concisely and to solve them readily with computers. We make use of two fundamental mathematical notions: Points are mathematical models of a physical object whose spatial extension is irrelevant. Frames are unbounded continuous sets of points over the Euclidean three-space whose distances are time invariant and which possess a subset of at least three noncollinear points.

Points and frames, although mathematical concepts, are regarded as idealized physical objects that exist independently of observers and coordinate systems. A point designates the location of a particle, but it is not a particle in itself. It does not have any mass or volume associated with it. For instance, a point marks the c. m. of a satellite; but for modeling the dynamics of the trajectory, the satellite’s mass has to join the point to become a particle. Only then can Newton’s second law be applied.

Combining at least three noncollinear points, mutually at rest, creates a frame. The best known frames are the frames of reference. Any frame can serve as a frame of reference. We will encounter inertial frames, Earth frames, body frames, and others. A frame can fix the position of a rigid body, but it is not a rigid body in itself. Only a collection of particles, mutually at rest, forms a rigid body. It is essential for you to remember that both, points and frames, are physical objects, albeit idealized.

Points and frames are the building blocks for modeling aerospace vehicle dy­namics. I will show by example that they are the only two concepts needed to formulate any problem in flight dynamics. Surprised? Follow me and you be the judge and jury.

We need a mathematical shorthand notation to describe points and frames and their interactions in space and time. Tensors in their simple Cartesian form will serve us splendidly. They exist independently of observers and coordinate systems, and their physical content is invariant under coordinate transformations.

Coordinate systems are required for measurements and numerical problem solv­ing. They establish the relationship between tensors and algebraic numbers and are a purely mathematical concept. Be careful however! Truesdell3 warns, “In particular, frame of reference should not be regarded as a synonym for coordinate system.” They are two different entities. Frames model physical objects, while coordinate systems embed numbers, called coordinates.

These coordinates are ordered numbers, arranged as matrices. Matrices are algebraic arrays that present the coordinates of tensors in a form that is convenient for algebraic manipulations. You will build simulations mostly from matrices. Computers love to chew on these arrays.

The modeling chain is now complete. The mathematical modeling of aerospace vehicles is a three-step process: 1) formulation of vehicle dynamics in invariant tensor form, 2) introduction of coordinate systems for component presentation, and 3) formulation of problems in matrices for computer programming and numerical solutions.

First, you have to think about the physics of the problem. What laws govern the motions of the vehicle? What are the parameters and variables that interact with each other? Which elements are modeled by points and which by frames? Then introduce tensors for the physical quantities and model the dynamics in an invariant form, independent of coordinate systems. Manipulate the equations until they divulge the variables that you want to simulate.

As a physicist you would be finished, but as an engineer your toil has just begun. You have to select the proper coordinate systems for numerical examination. What coordinate systems underlie the aerodynamic and thrust data? In what coordinates are the moments of inertia given? Does the customer want the trajectory output in inertial coordinates or in longitude and latitude? There are many questions that you have to address and translate into the mathematical framework of coordinate systems.

Eventually all equations are coordinated and linked by coordinate transforma­tions. The tensors have become matrices and are ready for programming. Any of the modem computer languages enable programming of matrices directly or at least permit you to create appropriate objects or subroutines. Finally, building the simulation should be straightforward, although very time consuming.