8.4 Dynamic Stability
8.4.1 Analytical Process
The mathematical treatment of dynamic stability given by Bramwell follows the lines of the standard treatment for fixed-wing aircraft. Wind axes are used, with the X axis parallel to the flight path, and the stability derivatives ultimately are fully non-dimensionalized. The classical format is useful because it is basic in character and displays essential comparisons prominently. The most notable distinction which emerges is that, whereas with a fixed-wing aircraft the stability quartic equation splits into two quadratics, leading to a simple physical interpretation of the motion, with the helicopter this unfortunately is not so and as a consequence the calculation of roots becomes a more complicated process.
Industrial procedures for the helicopter tend to be on rather different lines. The analysis is generally made with reference to body axes, with origin at the CG. In this way the X axis remains forward relative to the airframe, whatever the direction of flight or of relative airflow. The classical linearization of small perturbations is still applicable in principle, the necessary inclusion of initial-condition velocity components along the body axes representing only a minor complication. Force and moment contributions from the main rotor, tail rotor, airframe and fixed tail surface are collected along each body axis, as functions of flow parameters, control angles and flapping coefficients, and are then differentiated with respect to each independent variable in turn. In earlier days, computational techniques provided ready solutions to the polynomials. However, computer hardware and software have improved to an extent where many different techniques for solving the equations are possible. Full non – dimensionalization of the derivatives is less useful than for fixed-wing aircraft and a preferred alternative is to ‘normalize’ the force and moment derivatives in terms of the helicopter weight and moment of inertia respectively. This means that linear and rotational accelerations become the yardstick. These normalized terms are often referred to as concise derivatives.