Identification of rotor wake inflow finite-state models for flight dynamics simulations

Abstract

The aim of this work is the development of dynamic, finite-state modelling of wake inflow generated by kinematic perturbations of rotors in steady flight conditions. Extracted from responses of high-fidelity aerodynamic solvers, it is suited for flight dynamics applications. A three-step identification procedure is proposed: (1) evaluation by a high-fidelity solver of the wake inflow due to harmonic perturbations of rotor kinematics, (2) determination of the corresponding inflow coefficient transfer functions, and (3) rational approximation of the transfer functions. Wake inflow models related to rotor loads (like the well-known Pitt–Peters model) are obtained, as well, as by-products of that proposed. Considering aerodynamic simulations provided by a solver based on a boundary element method for potential flows, the numerical investigation presents the validation of the proposed finite-state wake inflow modelling, along with the examination of identified models related to rotor loads, for a rotor in steady flight conditions subject to arbitrary perturbations.

Appendices

Appendix 1: Nonlinear, separable-variable least-square approach for RMA

In this appendix, the numerical approach applied for the rational matrix approximation of transfer matrices is outlined. Considering, for instance, the rational matrix form in Eq. (4), the constant-coefficient matrices are obtained as solution of the following nonlinear, least-square problem

$$\begin{aligned} \min _{\mathbf{A}_1,\mathbf{A}_0,\mathbf{A},\mathbf{B},\mathbf{C}}\left( \sum _{n=1}^{N_f}\left\| \mathbf{Q}(i\omega _n) \right\| _F^2\right) \end{aligned}$$

(11)

where $\bigl \Vert \cdot \bigr \Vert _F$ represents Frobenius norm of a matrix, $\omega _n$ is the discrete set of $N_f$ frequencies at which the transfer matrix, $\mathbf{H}(s)$, is known (samples), whereas $\mathbf{Q}$ is the error matrix defined as (see Eq. (4))

$$\begin{aligned} \mathbf{Q}(s) = s\mathbf{A}_1+\mathbf{A}_0+\mathbf{C}\left( s\mathbf{I}-\mathbf{A}\right) ^{-1}\mathbf{B}- \mathbf{H}\left( s\right). \end{aligned}$$

(12)

To solve this minimization problem several methods have been proposed in the literature (see [11] for a brief review of them). The minimization problem solution proposed here is an extension of the approach of [26], that uses the separable variables approach.

Considering Eq. (12) it can be observed that, for a given frequency $\omega _i$, matrix $\mathbf{Q}$ depends linearly on the polynomial contribution matrices (namely, $\mathbf{A}_k$ for $k=0,1$) and that, for given $\mathbf{C}$ and $\mathbf{A}$ matrices, it is linearly dependent also on matrix $\mathbf{B}$. This feature of the approximation rational form applied is exploited to separate the minimization variables into two sets, a linear one and a nonlinear one.

First, the linear set of unknown variables is defined as the solution of the following algebraic problem derived from Eq. (12)

$$\begin{aligned} \begin{bmatrix} i\omega _1\mathbf{I}&\mathbf{I}&\mathbf{C}\left( i\omega _1\mathbf{I}-\mathbf{A}\right) ^{-1}\\ i\omega _2\mathbf{I}&\mathbf{I}&\mathbf{C}\left( i\omega _2\mathbf{I}-\mathbf{A}\right) ^{-1}\\ \vdots&\vdots&\vdots \\ i\omega _{N_f}\mathbf{I}&\mathbf{I}&\mathbf{C}\left( i\omega _{N_f}\mathbf{I}-\mathbf{A}\right) ^{-1} \end{bmatrix} \begin{bmatrix} \mathbf{A}_1\\ \mathbf{A}_0\\ \mathbf{B}\end{bmatrix} = \begin{bmatrix} \mathbf{Q}\left( i\omega _1\right) \\ \mathbf{Q}\left( i\omega _2\right) \\ \vdots \\ \mathbf{Q}\left( i\omega _{N_f}\right) \end{bmatrix} + \begin{bmatrix} \mathbf{H}\left( i\omega _1\right) \\ \mathbf{H}\left( i\omega _2\right) \\ \vdots \\ \mathbf{H}\left( i\omega _{N_f}\right) \end{bmatrix} \end{aligned}$$

(13)

where equation coefficients explicitly depend on the set of nonlinear variables, the unknowns are real numbers (entries of matrices $\mathbf{A}_1,\mathbf{A}_0$ and $\mathbf{B}$), whereas right hand side contributions have complex values.

Then, the set of nonlinear variables is defined as solution of a separate reformulated minimization problem, thus dealing with a drastically reduced size of solution domain. Indeed, rewriting Eq. (13) in a more compact notation as

$$\begin{aligned} \mathbf{M}\,\mathbf{X}_{lin}=\hat{\mathbf{Q}}+\hat{\mathbf{H}}\end{aligned}$$

(14)

where $\mathbf M$ is the coefficient matrix, $\mathbf{X}_{lin}$ collects the matrices of the unknown linear variables, while $\hat{\mathbf{H}}$ and $\hat{\mathbf{Q}}$ denote, respectively, transfer function and residual matrices evaluated at the sampling frequencies, the optimal value of $\mathbf{X}_{lin}$ (least-square solution of Eq. (13)) is formally given by

$$\begin{aligned} \mathbf{X}_{lin}=\mathbf{M}^*\hat{\mathbf{H}} \end{aligned}$$

(15)

with $\mathbf{M}^*=(\mathbf{M}^T\mathbf{M})^{-1}\mathbf{M}^T$ denoting the Moore-Penrose pseudoinverse of matrix $\mathbf{M}$, and the minimization problem concerning the nonlinear variables is formulated as follows

$$\begin{aligned} \min _{\mathbf{C},\mathbf{A}}\left( \left\| \hat{\mathbf{Q}}\right\| _F^2\right) \end{aligned}$$

(16)

with $\hat{\mathbf{Q}}$ not depending on $\mathbf{X}_{lin}$, as demonstrated by the combination of Eqs. (14) and (15) that yields

$$\begin{aligned} \hat{\mathbf{Q}}=\left( \mathbf{M}\mathbf{M}^*-\mathbf{I}\right) \,\hat{\mathbf{H}}.\end{aligned}$$

(17)

Equation (16) is solved by a local minimization method based on the Broyden−Fletcher−Goldfarb−Shanno (BFGS) algorithm [27]. The gradient of the objective function required by the BFGS algorithm (namely, the partial derivatives of $f_{obj}=\bigl \Vert \hat{\mathbf{Q}}\bigr \Vert _{F}^{2}$ with respect to the entries of $\mathbf{A}$ and $\mathbf{C}$) is evaluated through the method developed in [28]. Once the solution of Eq. (16) is determined, Eq. (15) directly yields the set of linear variables.

In principle, the minimization problem should be subject to a set of constraints imposing real part of poles (namely, the eigenvalues of matrix $\mathbf A$) to be negative (in order to let the finite-state form represent a stable system behaviour). However, it is observed that such constraints are automatically satisfied: this may be considered as an indication of the robustness of the presented approach, which allows application of the convenient unconstrained version of the BFGS algorithm.

Appendix 2: High-fidelity rotor aerodynamic solver

In this work, the aerodynamic responses used to determine state-space inflow models following the proposed procedure are obtained through a high-fidelity solver based on an unsteady, three-dimensional, free-wake, potential-flow formulation.

Specifically, wake inflow and blade loads are evaluated by a boundary element method for the solution of a boundary integral equation approach, suited for the analysis of potential flows around helicopter rotors in arbitrary flight conditions, included those where strong blade-vortex interactions occur [12]. It introduces the decomposition of the potential field into an incident field, $\varphi _{_{I}}$, and a scattered field, $\varphi _{_{S}}$. The scattered potential is generated by sources and doublets over the bodies surfaces and by doublets over portions of the wakes surfaces that are very close to the trailing edges from which they emanated (near wake, $S_{_{W}}^ {N}$). The incident potential is generated by doublets over the complementary wake regions that compose the far wakes, $S_{_{W}}^ {F}$. The scattered potential is discontinuous across $S_{_{W}}^ {N}$, whereas the incident potential is discontinuous across $S_{_{W}}^ {F}$. As demonstrated in [12], for $\varphi =\varphi _{_{I}}+\varphi _{_{S}}$, the scattered potential at an arbitrary observer position, $\mathbf{y}_*$, is given by

$$\begin{aligned} \varphi _{_{S}}(\mathbf{y}_*,t) = \int _{S_{_{B}}} \!\! \left[ G\left( \chi - \chi _{_{I}} \right) -\varphi _{_{S}}{\frac{\partial {G}}{\partial {n}}}\right] \!\mathrm{d}S(\mathbf{y}) - \int _{S_{_{W}}^ {N}} \!\Delta \varphi _{_{S}}\,{\frac{\partial {G}}{\partial {n}}}\, \mathrm{d}S(\mathbf{y}) \end{aligned}$$

(18)

where $G=-1/4\pi \Vert \mathbf{y}-\mathbf{y}_*\Vert$ denotes the unit-source solution of the three-dimensional Laplace equation, $\chi =\mathbf{v}_{_{B}}\cdot \mathbf{n}$ accounts for the impermeability boundary condition ($\mathbf{v}_{_{B}}$ is the body velocity due to rigid and elastic kinematics, and $\mathbf{n}$ is the surface outward unit normal), whereas, for the far wake discretized through N panels, $\chi _{_{I}}=\mathbf{u}_{_{I}}\cdot \mathbf{n}$, with the induced velocity, $\mathbf{u}_{_{I}}$, given by [12]

$$\begin{aligned} \mathbf{u}_{_{I}}(\mathbf{y}_*,t) \approx -\sum _{n=1}^N \Delta \varphi _{_{S}}(\mathbf{y}^ {TE}_{_{W_n}},t-\tau _n) \int _{C_n} \nabla _{_{\mathbf{x}}} G \times \mathrm{d}\mathbf{y}\end{aligned}$$

(19)

where $C_n$ denotes the contour line of the n-th far wake panel, $\mathbf{y}^ {TE}_{_{W_n}}$ is the trailing edge position where the wake material point currently in $\mathbf{y}_{_{W_n}}$ emanated at time $t-\tau _n$.

Equation (19) represents the velocity field given by the Biot-Savart law applied to the vortices having the shape of the far-wake panel contours and intensity $\Delta \varphi _{_{S}}(\mathbf{y}^ {TE}_{_{W_n}},t-\tau _n)$; it is applied to evaluate both the term $\chi _{_{I}}$ in Eq. (18) and, once extended to the whole wake, the velocity field for the free-wake analysis, and wake inflow over the blades. The vortices at the panels contours are assumed to have a finite-thickness core (Rankine-vortex type vortices), in order to assure stable and regular solutions even in body-vortex impact conditions [12].

Equation (18) is solved numerically by boundary elements, i.e., by dividing $S_{_{B}}$ and $S_{_{W}}^ {N}$ into quadrilateral panels, assuming $\varphi _{_{S}}$, $\chi$, $\chi _{_{I}}$ and $\Delta \varphi _{_{S}}$ to be piecewise constant (zero-th order BEM), and imposing the equation to be satisfied at the centre of each body element (collocation method).

Once the potential field is known, the Bernoulli theorem yields the pressure distribution that, suitably integrated over the blades surfaces, provides the aerodynamic loads (see [25] for details on the application of the Bernoulli theorem to the potential field decomposed into incident and scattered components).