Simplification of Equations of Motion and Transfer-Function Analysis

5.1 INTRODUCTION

Although simulation of nonlinear flight dynamics, in general, requires full six – degrees-of-freedom (6DOF) equations of motion (EOMs), it is not always necessary to use very complex and coupled models in the design of flight control laws, dynamic analysis, and handling qualities analysis procedures. Simple models with essential derivatives that capture the basic characteristics of the vehicle normally yield adequate results. However, caution must be exercised in simplifying the model too much as this may not be able to represent the system dynamics properly. Some of the reasons why flight analysts prefer simplified models are (1) full complex models are difficult to interpret and analyze, (2) it is possible to separate the model equations into independent subsets without much loss of accuracy, (3) ease of linearization, (4) unnecessary details that have little effect on system dynamics are avoided, (5) excursions during flight test maneuvers can be restricted to apply the assumption of linearity, and (6) easy to implement software codes are available for handling qualities analysis and parameter estimation. The simplification of nonlinear EOMs into workable and easily usable linear models can bring about connections between various aircraft configurations via the transfer-function (TF) analysis and frequency responses. The aircraft may be of different sizes and wing-body config­urations, but the dynamic characteristics may be the same. The TF analysis and zeros/poles disposition can be useful in assessing the dynamic characteristics of these aircraft, rotorcraft, missiles, UAVs (unmanned/uninhibited aerial vehicles), MAVs (micro-air vehicles), and airships. Simplified models are used in the design of autopilot control laws and in evaluation of handling qualities of the vehicle. The detailed effects of particular derivatives can be observed. Also, it becomes easy to obtain the aerodynamic derivatives of the vehicle from the flight data, using linear mathematical models in a parameter-estimation algorithm.

The TF analysis of the aircraft mathematical models gives a deeper insight into the effect of aircraft configurations on its dynamics. Aircraft configuration refers to the external shape of the aircraft and is related to the use of stores, center of gravity (CG) movement (aft, mid, and fore), flap settings, autopilot on or off condition, slat effect, airbrake deployment, etc. Aircraft dynamics for a particular configuration, at a
particular flight condition, are characterized by stability derivatives (Chapter 4). Understanding flight mechanics models via TF helps to properly interpret the effects of the configuration on flight responses as well as on aerodynamic derivatives.

The coupled nonlinear 6DOF EOMs discussed in Chapter 3 include nonlinea­rities because of the gravitational and rotation-related terms in the force equations and the appearance of products of angular rates in the moment equations. The dynamic pressure q also contributes to nonlinearity because it varies with the square of the velocity (q = 1 pV2). Apart from the nonlinearities contributed by the kin­ematic terms, the aerodynamic coefficients Cx, Cy, Cz, Q, Cm, and Cn may contain additional nonlinearities; for example, the lift coefficient due to horizontal tail, when expressed in Taylor’s series, may have the following form:

CLaH aH + CL8ede + CLS3 d3 + CLseaH deaH

The above model form may be useful in explaining nonlinearities arising from control surface deflections and flow effects at higher angles of attack. Analysis of a nonlinear model would generally require implementation of nonlinear programs. However, it would be worthwhile to determine simplified models that can be used in flight regimes where nonlinear effects are not important. In this chapter, we look at the various strategies adopted to simplify aircraft EOMs to obtain model forms that have practical utility and computation efficiency.