Estimation using SAFO data

The validity of the linearized form of VVM in Eq.(3.9), for the given SAFO data is checked using the power spectrum density (PSD) maps of CZ(t). This is important as variation of force and moment coefficients becomes nonlinear beyond a certain input amplitude. However, raw wind tunnel test data is not available for this purpose, while this conclusion cannot be drawn reliably from averaged and filtered response data. The processed CZ(t) oscillation cycle shows a significant power only upto first harmonic frequency. This also an indicator of the primarily linear nature of variation of CZ(t).

As shown in chapter 3, linearized form of VVM implied a linear relation between in- phase and out-of-phase derivatives extracted from SAFO data. This linear relationship is found to be true for CZ of GTA, as shown in Fig.(4.5). The separation between

25 30 35 40 45 50 55 60 −3 −2 −1 0 1 CZ α In−phase 25 30 35 40 45 50 55 60 −100 −50 0 Angle−of−attack α CZ α ’ Out−of−phase f=0.5 Hz mod 0.5 Hz f=1 Hz mod 1 Hz f=1.5 Hz mod 1.5 Hz

Fig. 4.6. Comparison of in-phase and out-of-phase derivatives from the estimated model and experimental data forCZof GTA.

the points for different frequencies is distinct only in the angle-of-attack range where unsteady aerodynamic loads are significant, that is α _{∈ (30}o55o). Hence, a(α0), K1(α0) are estimated in this angle-of-attack range by the two-step regression method proposed in [16, 22], and given in APPENDIX A. Accordingly, a(α0) is estimated by linear regression considering Eq.(3.9), and then these values are used in the second step to estimate K1(α0) and CZq(α0) considering Eq.(3.8).

Static wind tunnel test data defines the steady state solutions of the model. The derivative of CZvs. α curve, denoted as CZα,st(α0) is estimated from static wind tunnel

data and it is used as a constraint in the second step of regression. Since the static data for CZ of GTA is available at only 5 deg intervals of angle-of-attack, its derivative has a high uncertainty. This is especially an important issue in the stall angle-of-attack region where CZ,st(α) changes rapidly. Hence, CZ,st(α) is curve-fitted with a 7-th degree polynomial function and then differentiated to get CZα,st(α0). In small amplitude forced oscillation

tests, pitching motion causes a filtering effect on CZ,stvs. α. Also, the oscillation in angle- of-attack is over a small range of α = 6o_{. Hence, the approximation of the value of slope} by curve-fitting is likely to be close to the actual value.

Since the co-variances of in-phase and out-of-phase derivatives are not given, the linear regression is performed by Weighted-Least-Squares method in several iterations. In this process, the estimation error covariance matrix is computed in each iteration, and

20 25 30 35 40 45 50 55 60 −50 −40 −30 −20 −10 0 Time scale a 20 25 30 35 40 45 50 55 60 −6 −4 −2 0 Angle−of−attack (deg) Dynamic Gain K 1 SAFO estimate LAFO estimate

Fig. 4.7. Estimated bounds on the first kernel-state parameters and parameter functions forCZof GTA.

its inverse is the weighting matrix for the next iteration. This procedure converges to a best-fit in few iterations. The final value of error-covariance matrix is used to calculate the confidence bounds on estimated values of parameters (a(α0), K1(α0)), as given in APPENDIX A.

The experimental data and model prediction using the estimated values of parameters are found to match accurately as seen in Fig.(4.6). The goodness-of-fit for the estimated values is given by parameter variance or 95% confidence bounds on estimated values of parameters, and coefficient of determination R, at each angle-of-attack α0. The confidence bounds account for measurement noise and coefficient of determination shows the variations in the measured output that are captured by the estimated model. The parameter bounds are indicated by vertical bars in Fig.(4.7). These values are acceptable for α[35o 50o_].

The coefficient of determination is relatively low at α = (30o, 55o_{), as seen in} Fig.(4.8). Unsteady aerodynamics is not sufficiently excited by the small amplitude inputs at these angles-of-attack. Hence, signal-to-noise ratio is poor, and linear relation between in-phase and out-phase derivatives is unclear, as seen in Fig.(4.5). Thus, from SAFO data, α _{∈ [30}o₅₅o_{] seems to be the region of significant unsteady normal force. However, the} LAFO data shows that the unsteady component of normal force coefficient is significant for α_{∈ [25}o60o]. Therefore, (a, K1) are estimated at α = (25o, 60o) using LAFO data in the next step.

Outside the stall angle-of-attack region, i.e. α _{∈ {[0 25}o], [60o₉₀o_{]}, C}

25 30 35 40 45 50 55 60 0.8 0.85 0.9 0.95 1 α Coefficient of Determination R

Fig. 4.8. Coefficient of determination for the estimated model parameters forCZ of GTA.

expected to be zero. This is affected by considering the magnitude of time-scale parameter to be relatively very large of about a <_{−20; while K}1is simply zero, as seen in Fig.(4.5). The linear unsteady effects in this region are captured by the damping derivative (out-of- phase derivative) term included in the aero-database.

The linear component of the model or the single-state VVM has successfully captured the frequency dependence of unsteady normal force variation due to small amplitude inputs. The identified model incorporates the dependence of in-phase and out-of-phase derivatives on the frequency of input in time-domain. Thus, it is useful for simulation of loads due to small amplitude change in angle-of-attack. For large amplitude change in angle-of-attack, variation of CZ(t) is nonlinear in nature.

In the region of unsteady aerodynamics, i.e.α _{∈ [25}o60o] deg, the time-constant of flow dynamics on the wings change continuously with angle-of-attack. This is also reflected in the estimated values of (a, K1) in Fig.(4.7). But, these parameters are estimated at 5o intervals. This resolution is insufficient to realize the actual continuous model parameter functions. Hence, the parameter functions (a(α), K1(α)) are defined by the estimated values of parameter bounds at the node-points. This is refined in the next step using LAFO data.