3.4.1
Requirements for Validity of the Model Structure
The features of unsteady aerodynamics are summarized in Section (3.2), while VVEs were derived in Chapter 2. For application of VVEs to model variations in CZ(t) and Cm(t) in the stall angle-of-attack regimes, these coefficients should meet the assumptions made in derivation of VVEs.
The Volterra series is used to model the systems with memory effect, i.e the systems for which instantaneous output depends on the history of inputs. As presented in the previous sections, the variation in longitudinal loads depends on initial angle-of-attack α0 and trajectory of α(t) input. Thus, the longitudinal force or moment coefficient is determined by history of input α(t).
When the aircraft comes to rest at a particular angle-of-attack, CZ(t) and Cm(t) always converge to a steady state value, which is known from static wind tunnel tests. Thus, they meet the essential system requirement that the system must have a single stable steady state for given inputs.
The VVEs are equivalent to Volterra series for a system which is analytic in state and input-affine, in addition to the conditions stated above. From static wind tunnel tests data, it is known that the CZ(α) and Cm(α) of the delta wing aircraft are smooth enough to be approximated by a polynomial in α. There is no static hysteresis or any other bifurcation. Therefore, longitudinal coefficients are analytic. In the model structure formulation presented in the next section, the system of aerodynamic coefficients is assumed to be input-affine with ˙α as input.
Physical systems should have fading-memory in order to model it using Volterra series truncated to a finite number of kernels. This is evident if the system produces a periodic response to a periodic input [68]. In forced oscillation tests the normal force and pitching moment coefficients settle to a periodic variation in response to sinusoidal inputs. This shows that the coefficients have fading memory and depend only on a finite history of motion α(t). Thus, longitudinal coefficients exhibit fading memory effect.
Thus, CZ(t) and Cm(t) for an aircraft maneuvering in the stall regime satisfy all the conditions essential to model them in the form VVM.
3.4.2
Proposed Model Structure
A classical aero-model, commonly referred to as aero-derivative model or an aero- database, is built using data from static wind tunnel tests. It is in the form of data-tables which consist of incremental change in coefficient as a function of flow-angles and control surface deflections. The effect of unsteady aerodynamics is incorporated in the form of so called damping derivatives. At the angles-of-attack before stall, damping derivatives are included in a look-up table with angle-of-attack as the look-up parameter.
An aero-database is used for performing both flight dynamic analysis and for real- time flight simulation studies. If the aero-database has been validated from flight test data, it is expected that the model is kept intact for any further modeling or analytical studies. Therefore, the proposed model structure for CZ and Cmto account for unsteady aerodynamic effects, is formulated as an add-on over the aero-database.
CZ(t) = CZst(α(t)) + CZq(α(t))
q¯c
2V + Cdyn(α(t), ˙α(t)) (3.1) Consider the splitting of CZ(t) into the components of corresponding steady state value and incremental unsteady variations as given in Eq.(3.1). CZst(α) is the steady state
value of CZ at each angle-of-attack, and it is known from static wind tunnel tests. CZq
represents the incremental effect of steady pitching motion on CZ. Cdyn(t) represents the unsteady variation due to rate of change in flow incidence angles. In this thesis, we consider the unsteady effect of change in angle-of-attack only. This component is required to be identified from dynamic wind tunnel test data.
CZq is the pitch-rate derivative that represents the damping effect due to pure pitching
motion of the aircraft. This corresponds to vertical rotational motion of a particular radius, in which angle-of-attack remains constant ( ˙α = 0). In an aero-derivative model, it is estimated using empirical methods. It can also be estimated from flight test data by system identification techniques [71]. But this is valid only for low angles-of-attack or the regions of linear aerodynamics. It cannot be estimated directly from the small amplitude forced oscillation test data, as in this test the pitching motion of the wind tunnel model simultaneously produces pitch-rate and change in angle-of-attack, such that ˙α = q. The contribution of CZq to CZ is usually found to be of small magnitude, while Cmq can
produce significant contribution to Cm.
A recent study, by Khrabrov and Greenwell, presented two methods for computing the effect of steady pitch rate on the normal force and pitching moment coefficients of a 2D airfoil [85]. In these models the derivatives CNq and Cmq are accurately modeled as a
function of flow separation point xsand the non-dimensional pitching frequency, in stall angle-of-attack region. This model was not considered in the proposed model formulation as the flow separation point (or vortex break-down location) is not included in it. In the proposed VVM formulation, the pitch-rate derivatives are estimated simultaneously with parameters of the model of Cdyn(t) using SAFO data.
The component Cdyn(t) represents the incremental component of force due to unsteady aerodynamics. This is modeled based on VVM. Any change in angle-of-attack produces a non-zero value of Cdyn(t). Hence, consider ˙α(t) to be the input and Cdyn(t) to be the output of the system. This system has a stable equilibrium [Cdyn(t), ˙α(t)) = (0, 0)] over the domain α ∈ [−90o, 90o). The input ˙α has an upper-bound which depends on operational conditions of the given aircraft type. From Eq.(3.1), it is evident that CZ(t) converges to CZst(t) with a finite time delay when ( ˙α = 0), and its time-scale is the same
as that governing Cdyn(t). Thus, Cdyn(t) sufficiently represents the unsteady component of load for modeling using VVM.
The VVEs are given as equations (2.19-2.21), in Chapter 2. The response of vortex breakdown location to change in angle-of-attack is known to be similar to first-order linear differential system [74], and all the models except the ONERA Dynamic Stall model can be reduced to an equivalent first-order differential equation [42]. Hence, the differential operator in VVEs is taken to be of first-order only, i.e. F (d/dt) = d/dt + a. The
differential equations of only first three kernel states are used in this formulation because three states were found to be sufficient in the case studies in Chapter 4 and the equations of higher order kernel states are much more complicated. In order to avoid confusion between the equivalence of PoDE model parameters and VVEs as extensively dealt with in Chapter 2, different notations for the parameters of VVM are used in the formulations presented here onwards. Input gain parameter b1 = K1 and the time-scale parameter a1/p1 = −a = 1/τ. Other parameters have been redefined in terms of Ki as, a2 = K21, b2 = K22, a3 = K31, b3 = K32. Taking into account all these aspects, the Volterra variational model of Cdyn(t) is given by,
Cdyn(t) = x1(t) + x2(t) + x3(t)
˙x1(t) = a(t)x1(t) + K1(t)u(t), x1(0) = 0
˙x2(t) = a(t)x2(t) + K21(t)x21(t) + K22(t)x1(t)u(t), x2(0) = 0 ˙x3(t) = a(t)x3(t) + K21(t)x31(t) + K31(t)x1(t)x2(t) +
K32(t)x2(t)u(t) + K22(t)x21(t)u(t), x3(0) = 0 (3.2) For a time-invariant system, all the parameters of the model can be considered to be constants. In case of unsteady aerodynamics, the variations in loads depend on initial angle-of-attack, and its characteristic time-scale is a function of angle-of-attack. This angle-of-attack dependence is accounted for by considering the model parameters as function of angle-of-attack. As presented in the previous chapter, any constant value of the parameters can reproduce amplitude and frequency dependence. Therefore, we consider all the parameters to be functions of instantaneous angle-of-attack α. Thus, the model for Cdyn(t) becomes as in Eq.(3.3).
Cdyn(t) = x1(t) + x2(t) + x3(t)
˙x1(t) = a(α)x1(t) + K1(α) ˙α(t), x1(0) = 0
˙x2(t) = a(α)x2(t) + K21(α)x12(t) + K22(α)x1(t) ˙α(t), x2(0) = 0 ˙x3(t) = a(α)x3(t) + K21(α)x31+ K31(α)x1(t)x2(t) +
K32(α)x2(t) ˙α(t) + K22(α)x21(t) ˙α(t), x3(0) = 0 (3.3) The VVM gives differential equations for second and third kernels containing nonlinear terms in x1, x2, x3. The parameter values in these equations give a certain definite kernel shape, and this has been obtained computationally in [86]. Further, an interpretation of each of these terms as the effect on time-constant of a step response is also provided. Using these interpretations it is possible to choose or reject a certain term in the model structure beforehand. This is especially useful when the parameter of
a term is found to be insignificant and hence can be removed from the model structure. Excluding a certain nonlinear term implies that the kernel shape is constrained. However, reducing the parameters to be estimated in a three state model can improve the accuracy of the model.
It is important to note that, in this formulation there is no assumption of the type of data implicitly or explicitly, as done in other model structures proposed in literature. Therefore, this model structure is generic and can be further extended to any application, and estimated from any data using an appropriate approach.
In the next step, the parameter functions are required to be estimated from small and large amplitude forced oscillation wind tunnel test data, as discussed in the next section.