Tensor flight dynamics

Tensor flight dynamics elevates the modeling of flight dynamics to a coordinate-invariant form. It is based on Einstein's covariance principle, which states that all physical laws are invariant under all coordinate transformations. While Einstein based his special and general theory of relativity on the covariance principle, tensor flight dynamics, a branch of classical mechanics, applies this principle to Newton's laws of motion.
Newton's Second Law
Newton's second law is the foundation of flight dynamics. It governs the equations of motion, aerodynamics, and propulsion of all flight vehicles. In its vector form it is invariant only if expressed in inertial coordinates. But if, for instance, expressed in airframe coordinates, an additional term appears and, therefore,the invariance of Newton's law is lost.
History
Tensor flight dynamics uses a different time operator, called rotational time derivative, to maintain the invariance of Newton's law. The first allusion is found in an article by Wundheiler in 1932. More recently, Wrede coined the term rotational time derivative. Its analytical derivation from first principles is given by Zipfel, based on his PhD dissertation.
Invariance of Newton's Second Law
While the ordinary time derivative operates on an implied coordinate system, the rotational time derivative operates on a tensor without reference to a coordinate system. Zipfel has shown in his dissertation that the rotational time derivative, operating on a tensor, is a tensor itself. By replacing the ordinary time operator in Newton’s law by the rotational time derivative, the tensor characteristic of the law is maintained; i.e., it has the same form in all coordinate systems, even non-inertial ones. In this formulation, Newton’s law abides by Einstein’s covariance principle.
Achieving covariance has several consequences. A strict distinction has to be made between coordinate systems and reference frames. Reference frames are models of physical entities, like inertial reference frame, airframe, and sensor frame, while coordinate systems are mathematical abstracts that relate the Euclidean three-space to ordered algebraic numbers.
Consequences
This separation of the physical phenomena from the computational process has advantageous consequences for the modeling and simulation process. Faced with building a computer simulation, the engineer first formulates the physics of the aerospace vehicle in tensors, then introduces coordinate systems, and so converts the tensors into matrices for direct programming.
Applications
Zipfel claims that tensor flight dynamics has proven its usefulness in building sophisticated simulations of so-called net-centric engagements, where multiple vehicles engage a variety of targets, supported by several overhead assets. He combines tensor modeling with programming and calls it: “Simulating of aerospace vehicles by object oriented modeling and programming.” His coding framework is CADAC++, which features sample simulations of missiles, UAVs, aircraft, three-stage boosters, and hypersonic vehicles.

Main Menu

See Also

Main Menu