Analysis of Response to Controls
Previous sections have focused on trim and stability analysis. The analysis of flight behaviour following control or disturbance inputs is characterized under the general heading ‘Response’, and is the last topic of this series of modelling sections. Along with the trim and stability analysis, response forms a bridge between the model building activities in Chapter 3 and the flying qualities analysis of Chapters 6 and 7. In the following sections, results will be presented from so-called system identification techniques, and readers unfamiliar with these methods are strongly encouraged to devote some time to familiarizing themselves with the different tools (Ref. 5.14). Making sense of helicopter dynamic flight test data in the validation context requires a combination of experience (e. g., knowing what to expect) and analysis tools that help to isolate cause and effect, and hence provide understanding. System identification methods provide a rational and systematic approach to this process of gaining better understanding.
Before proceeding with a study of four different response topics, we need to recall the basic equations for helicopter response, given in earlier chapters and Appendix 4A. The nonlinear equations for the motion of the fuselage, rotor and other dynamic elements, combined into the state vector x(t), in terms of the applied controls u(t) and disturbances f(t), can be written as
dt = F(x(t), u(t), f(t); t)
with solution as a function of time given by
x(t) = x(0) + y F(x(r), u(t), f(r); т)dr (5.48)
In a forward simulation, eqn 5.48 is solved numerically by prescribing the form of the evolution of F(t) over each time interval and integrating. For small enough time intervals, a linear or low-order polynomial form for F(t) generally gives rapid convergence. Alternatively, a process of prediction and correction can be devised by iterating on the solution at each time step. The selection of which technique to use will usually not be critical. Exceptions occur for systems with particular characteristics (Ref. 5.15), leading to premature numerical instabilities, largely determined by the distribution of eigenvalues; inclusion of rotor and other higher order dynamic modes in eqn 5.47 can sometimes lead to such problems and care needs to be taken to establish a sufficiently robust integration method, particularly for real-time simulation, when a constraint will be to achieve the maximum integration cycle time. We will not dwell on these clearly important issues here, but refer the reader to any one of the numerous texts on numerical analysis.
The solution of the linearized form of eqn 5.47 can be written in either of two forms (see Appendix 4A)
x(t) = Y(t)x0 + f Y(t – r )(Bu(t ) + f(r)) dr (5.49)
n t –
(t) = ^2 (vTX0) exp(V) + f (vt(Bu(t) + f(r)) exp[X;(t – r)]) dr
i=1 L "
where the principal matrix solution Y(t) is given by
Y(t) = 0, t < 0, Y(t) = W diag[exp(^t)]VT, t > 0 (5.51)
W is the matrix of right-hand eigenvectors of the system matrix A, VT = W-1 is the matrix of eigenvectors of AT, and X( are the corresponding eigenvalues; В is the control matrix. The utility of the linearized response solutions depends on the degree of nonlinearity and the input and response amplitude. In general terms, the linear formulation is considerably more amenable to analysis, and we shall regularly use linear approximations in the following sections to gain improved understanding. In particular, the ability to estimate trends through closed-form analytic solutions, exploited fully in the analysis of stability, highlights the power of linear analysis and, unless a nonlinearity is obviously playing a significant role, equivalent linear systems analysis is always preferred in the first instance.
It is inevitable that the following treatment has to be selective; we shall examine response characteristics in different axes individually, concerning ourselves chiefly with direct response to controls. In several cases, comparisons between flight and He – lisim simulation are shown and reference is made to the AGARD Working Group 18 (Rotorcraft System Identification) flight test databases for the DRA Puma and DLR Bo105 helicopters (Ref. 5.14).