Three-Degrees-of-Freedom Simulation

What a journey it has been so far! Provided you have not skipped the first seven chapters, you have reached Part 2 with a tool chest full of gadgets that aspire to be used for challenging simulation tasks. You are trained in coordinate systems, translational and rotational kinematics, and are able to apply Newton’s and Euler’s laws to the dynamics of aerospace vehicles.

If you skimmed over the first part, because of your maturity in such matters, you are also welcome to join us. Make sure, however, that you understand my notation and the invariant formulation of dynamic equations. Then it should be easy for you to follow us. To make the following three chapters self-contained, I will derive the equations of motion from first principles.

Let us ease into the world of simulation with simple three-DoF, point-mass models. They are suitable for trajectory studies of rockets, missiles, and aircraft. All you need is an understanding of Newton’s second law and basic aerodynamic and propulsion data. In no time will you be productive, churning out time histories of key flight parameters. The more sophisticated five – and six-DoF simulations are left for the following chapters.

In preliminary design, vehicle characteristics are often sketchy and aerodynam­ics and propulsion data only known approximately. There may be just enough infor­mation to build simple three-DoF simulations. Fortunately, the trajectory of the c. m. of the vehicle is of greater interest than its attitude motions. Therefore, these three – DoF simulations are very useful for initial performance estimates and trade studies.

Newton’s second law governs the three translational degrees of freedom of three-DoF simulations. Aerodynamic, propulsive, and gravitational forces must be known. In contrast to six-DoF simulations, Euler’s law is not used, and body rates and attitudes are not calculated. Therefore, there is no requirement for aerodynamic and propulsive moments.

In deriving the equations of motion, three different perspectives can be taken according to the state variables selected for integration. Vinh1 takes the direct approach and derives the equations for the following state variables: geographic speed, flight-path angle, heading angle, radial distance, and longitude and latitude. The isolation of the state variables on the left side of the differential equations requires complicated manipulations that are not documented in his book. (Vinh’s equations are implemented by the TEST case, supplied with CADAC-Studio.)

The second method, the so-called Cartesian approach, formulates the equations in Cartesian coordinates. The state variables are the vehicle’s inertial velocity and position components [ujj]7 and [лй/]’, expressed in inertial coordinates.

The third method takes the perspective of the missile’s velocity vector wrt Earth [и|]’/ in velocity coordinates. Its polar components |uf |, /, у are the velocity state variables and the position states, expressed in Earth coordinates. We

refer to it as the polar approach.

The derivations of the Cartesian and polar equations of motion will be pro­vided first as invariant tensor forms and then expressed as matrices for program­ming. Before you can code up the simulation, you need to have some elementary understanding of the atmosphere, gravitational acceleration, aerodynamics, and propulsion. To help you take the first step, I include two example simulations of a three-stage rocket and a hypersonic vehicle. The complete code is provided on the CADAC CD as ROCKET3 and GHAME3.