Estimating and Tp
Having measured the frequency response of the aircraft to control excitation, the remaining task is to estimate the bandwidth and phase delay from graphical representations of the amplitude and phase of the response as shown in Fig. 6.25. But how do we ensure that the estimated frequency response functions are as accurate as possible or even valid? The frequency response analysis, whereby the time response data are converted into the frequency domain using a Fast Fourier Transform (FFT) technique (Ref. 6.50), assumes that the input-output relationship is approximately linear and that any ‘noise’ on the signals is random and uncorrelated with the response. Both of these assumptions break down to some degree in practice and it is important to process the time histories systematically to calibrate the data quality. The linear FFT converts a sweep time history of, say, roll rate of duration T, into a complex function of frequency (with in-phase and quadrature components) given by the relation
j p(t) e-jwt dt 0
The minimum frequency in the transformed function is related to the time duration of the sweep by the simple function
In practice, with digitized data, the transformation is conducted discretely, over the time response samples pn, measured every At, in the form
____ v— / Kfl
Pk(®k) = At 2^ pnexp( – j2nn
The frequency response functions H for all required input-output pairs (e. g., mc, p) can be assembled from the spectral density functions G (Ref. 6.50) as
A measure of accuracy of the derived frequency response function in terms of the linear correlation between output and input is given by the coherence function
Any coherence less than unity signifies the presence of nonlinearities or correlated noise on the response. In close-to-ideal conditions, the computations given by eqns 6.15 and 6.16 will generate frequency response data from which good estimates of bandwidth and phase delay can be derived. In practice, further and more detailed processing is often required to ensure that the handling qualities parameter estimates are the best obtainable. In Ref. 6.51, Tischler and Cauffman discuss the details as implemented in the US Army’s CIPHER analysis software, involving concatenation of multiple sweeps in the time domain and windowing to derive the best estimates of the individual power spectra. A second stage involves the derivation of the conditional frequency responses to take account of the effects of corrective control inputs in secondary axes. The associated partial coherence functions serve as a guide to the accuracy of the results and the linearity of the input-output relationships. The third stage in the data quality improvement ensures that the degrading effects of noise on the data are minimized. Effectively, composite frequency responses are derived from averaging with different data ‘window’ sizes in the frequency domain – small for the high-frequency range and large for the lower frequencies. A rough rule of thumb for data validity is given when the coherence function exceeds 0.8.
The calculation of bandwidth and phase delay follows according to the procedure given in Fig. 6.25. Most of the data improvement process described above is actually aimed at raising the coherence in the critical frequency range between «180 and 2«i80, where the phase delay is computed. An accurate estimate of phase delay is clearly important to define the handling qualities, but measuring the slope from the phase roll-off is not always straightforward. Reference 6.41 describes how the least-squares fit of the phase line had to be restricted to avoid being distorted by a high-frequency phase drop due to a rotor structural mode.
Bandwidth andphase delay have emerged as two key parameters reflecting attitude handling qualities in the small amplitude regime. The supporting test and analysis methodologies have received considerable attention since the initial debate on the merits of time and frequency domain methods, and the extensive, and more general, coverage given to the topic in this roll control section reflects the level of effort and importance given to the bandwidth concept by the rotorcraft community.
Table 6.3 gives the roll axis bandwidth and phase delay estimates for a number of current operational helicopters in hover, together with the relevant data sources.
In the characterization of helicopter response portrayed by the framework diagram, Fig. 6.5, there is no reference to a handling quality that enjoyed centre stage prior to the publication of ADS-33 – the control sensitivity, and before leaving
Table 6.3 Roll attitude bandwidth results for current helicopters
small amplitude dynamics, it is important to discuss the apparent demise of this parameter.