Aerothermoelastic Control of Lifting Surfaces

Synonyms

Aerothermoelasticity; Freeplay; Hypersonic speed; Nonlinear 2D wing models; Nonlinear aerothermoelastic control analysis; Piston theory aerodynamics; Thermal loading

Overview

The interest toward the development and implementation of active control technology was prompted by the new and sometimes contradictory requirements imposed on the design of the new generation of the flight vehicle that mandated increasing structural flexibilities, high maneuverability, and at the same time, the ability to operate safely in severe environmental conditions. Designing reentry space vehicles and high-speed aircraft requires special attention to the nonlinear thermoelastic and aerodynamic instability of their structural components. The aerodynamic heating effects are usually estimated from the adiabatic wall temperature due to high-speed airstreams. The thermal effects are important since temperature environment critically influences the static and dynamic behaviors of flight structures in supersonic/hypersonic regimes and is likely to cause instability, catastrophic failure, and oscillations, resulting in structural failure due to fatigue.

Active aerothermoelastic control strategies provide solutions to a large number of problems involving aerospace flight vehicle structures. To prevent damaging phenomena produced by thermal effects on both flutter boundary and post-flutter behavior, linear/nonlinear active control methods should be implemented. A serious loss of torsional stiffness may induce the dynamic instability; consequently, in the design process, the loss of torsional stiffness that may be incurred by lifting surfaces subject to axial stresses induced by aerodynamic heating should be considered.

Active control can be used to expand the flutter boundary and convert unstable limit cycle oscillations (LCO) into the stable LCO and/or to shift the transition between these two states toward higher flight Mach numbers. The advances of active control technology have rendered the applications of active flutter suppression and active vibrations control systems feasible in the last two decades [1, 2]. A great deal of research activity devoted to the aeroelastic active control and flutter suppression of flight vehicles has been accomplished. The state-of-the-art advances in these areas are presented in [3, 4]. The reader is also referred to a sequence of articles [5, 6] where a number of recent contributions related to the active control of aircraft wing are discussed at length. In the next sections, the nonlinear aerothermoelastic governing equations for the control of lifting surfaces are presented [7, 8] along with the solution methodology adopted and the analysis of selected example cases.

Nonlinear Aerothermoelastic Control Equations

The structural model considered is of a double-wedge two degrees-of-freedom (2-DOF) plunging/pitching lifting surface. The model is free to rotate in the $xOz$ plane and free to translate in the vertical direction as shown in Fig. 1. While a linear model can be obtained considering linear flexural and torsional stiffnesses, herein the nonlinear restoring force and moment from bending and torsional springs accounting for freeplay in both degrees-of-freedom have been considered. The nonlinear aeroelastic governing equations can be written as:

Fig. 1

Two degrees-of-freedom double-wedge airfoil geometry

$$ m\ddot{h} + {S_{\alpha }}\ddot{\alpha } + {c_h}\dot{h} + F(h) = - L(t) $$

(1)

$$ {S_{\alpha }}\ddot{h} + {I_{\alpha }}\ddot{\alpha } + {c_{\alpha }}\dot{\alpha } + G(\alpha ) = {M_{EA }}(t) $$

(2)

Herein $m$ is the airfoil mass per unit wing span, $h$ is the plunging displacement at the elastic axis (EA), positive in the downward direction, ${S_{\alpha }}$ is the static unbalance moment about the elastic axis per unit wing span, $\alpha$ is the pitch angle, positive rotation nose up, ${c_h}$ and ${c_{\alpha }}$ are the linear viscous damping coefficients in plunging and pitching, respectively, $L$ is the unsteady lift per unit wing span, $t$ is the physical time variable, ${I_{\alpha }}$ is the cross-section mass moment of inertia about its elastic axis per unit span, ${M_{EA }}$ is the unsteady aerodynamic moment about the elastic axis per unit wing span, and $\mathop{()}\limits^{\cdot }, \;\mathop{()}\limits^{{\cdot \cdot }}$ are first and second time derivatives. The active nonlinear control can be represented in terms of the moment ${M_C}$ in (1) as [7, 8]:

$$ {M_C} = {f_1}\alpha (t) + {f_2}{\alpha^3}(t) $$

(3)

where ${f_1}, \;{f_2}$ are the linear and nonlinear control gains, respectively. A third-order expansion form of the PTA is used to study the behavior of the nonlinear aerothermoelastic system in supersonic/hypersonic aeroelastic analyses. The system of governing equations of a supersonic/hypersonic double-wedge controlled airfoil featuring plunging/pitching coupled motion can be cast as [7]:

$$\begin{array}{ll}\xi ''(\tau ) &+ {\chi_{\alpha }}\alpha ''(\tau ) + 2{\zeta_h}(\bar{\omega }/{U^{*}})\xi '(\tau ) \cr& + {{(\bar{\omega }/{U^{*}})}^2}{{\bar{F}}_a}(\xi )\xi (\tau ) + {{(\bar{\omega }/{U^{*}})}^2}{{\bar{F}}_b}({\xi_s}) \cr& + {{(\bar{\omega }/{U^{*}})}^2}{{\bar{F}}_c}(\xi )[{\xi^3}(\tau ) + 3{( - 1)^n}{\xi_s}{\xi^2}(\tau ) \cr& + 3\xi_s^2\xi (\tau ) + {( - 1)^n}\xi_s^3] = \bar{L}(\tau )\end{array}$$

(4a)

$$\begin{array}{ll}({\chi_{\alpha }}/r_{\alpha}^2)\xi ''(\tau ) &+ \alpha ''(\tau ) + (2{\zeta_{\alpha }}/{U^{*}})\alpha '(\tau ) \cr& + (1/{U^{*}}^2){{\bar{G}}_a}(\alpha )\alpha (\tau ) + (1/{U^{*}}^2){{\bar{G}}_b}({\alpha_s}) \cr& + (1/{U^{*}}^2){{\bar{G}}_c}(\alpha )[{\alpha^3}(\tau ) + 3{( - 1)^n}{\alpha_s}{\alpha^2}(\tau ) \cr& + 3\alpha_s^2\alpha (\tau ) + {( - 1)^n}\alpha_s^3] = {{\bar{M}}_{EA }}(\tau ) \cr& - (1/{U^{*}}^2)({\varphi_1}\alpha (\tau ) + {\varphi_2}{\alpha^3}(\tau ))\end{array}$$

(4b)

where $\xi$ is the dimensionless plunging displacement at the elastic axis location, $\tau$ is the dimensionless time, ${x_{\alpha }}$ is the dimensionless distance between the mass center of the airfoil section and the elastic axis, ${\zeta_h}, \;{\zeta_{\alpha }}$ are the damping ratios in plunging and pitching, respectively, $\bar{\omega }$ is the dimensionless frequency ratio, $u, \;{u^{*}}$ velocity and its dimensionless counterpart (reduced velocity), respectively, ${\xi_s}$ is the dimensionless plunging freeplay magnitude, ${r_{\alpha }}$ is the dimensionless radius of gyration about elastic axis, ${\alpha_s}$ is the pitching freeplay magnitude, ${{\hat{\eta}}_h}, \;{{\hat{\eta}}_{\alpha }}$ are the normalized nonlinear stiffness coefficients in plunging and pitching, respectively, $\mu$ is the reduced mass ratio, ${\rho_{\infty }}$ is the air stream density, ${t_h}$ is the airfoil half thickness, $()', \;()''$ are the first and second time derivatives with respect to $\tau$ and

$$\begin{array}{ll}{{\bar{F}}_a}(\xi) = \left\{ \begin{array}{ll}1 \cr 0 \cr 1 \cr \end{array} \right.\begin{array}{ll}, \cr, \cr, \cr\end{array} \quad {{\bar{F}}_b}(\xi ) = \left\{ \begin{array}{ll}{ - {\xi_s}} \cr 0 \cr {\xi_s } \cr \end{array} \right.\begin{array}{ll}, \cr, \cr, \cr \end{array} \cr{{\bar{F}}_c}(\xi ) = \left\{ \begin{array}{ll} {{{\hat{\eta}}_h}} \cr 0 \cr {{{\hat{\eta}}_h}} \cr \end{array} \right.\begin{array}{ll}, \cr, \cr, \cr \end{array} \quad \quad \begin{array}{ll} \,\,\,\,\,{\xi (\tau )> {\xi_s}} \cr { - {\xi_s} \leq \xi (\tau )} \cr {\xi (\tau ) < - {\xi_s}} \cr \end{array} \leq {\xi_s}\begin{array}{ll} {,n = 1} \cr {} \cr {,n = 2} \cr \end{array}\end{array}$$

(5)

Similar expression for $\bar{G}'s$ by replacing $\xi (\tau )\Leftrightarrow \alpha (\tau )$. The unsteady aerodynamic lift and moment appearing in (4a and 4b) can be expressed as:

$$\begin{array}{ll}\bar{L}(\tau ) =& - \frac{\eta }{{12{M_{\infty}}\mu }}\Bigg[ {12(\xi ' - a\alpha ' + \alpha ) - 3(\gamma + 1)\hat{\tau}\eta {M_{\infty }}(\alpha ')} \cr& + {M_{\infty}^2(\gamma + 1)\eta^2}\bigg\{ (\xi ' - a\alpha ' + \alpha )\Big[ {{{(\xi ' - a\alpha ' + \alpha )}}^2} + 3{{\hat{\tau}}^2} + (\alpha')^2 \Big]\bigg\}\Bigg]\end{array}$$

(6a)

$$ \begin{array}{ll} {{\bar{M}}_{EA }}(\tau ) =& \frac{\eta }{{12\mu {M_{\infty }}r_{\alpha}^2}}\bigg[ {12[a\xi ' - \left( {\frac{1}{3} + {a^2}} \right)\alpha ' + a\alpha ] + 3(\gamma + 1)\hat{\tau}\eta {M_{\infty }}(\xi ' - 2a\alpha ' + \alpha ) } \cr& -\,M_{\infty}^2(\gamma + 1){\eta^2}\left\{ {\frac{1}{5}{{{(\alpha ')}}^3} - a(\xi ' - a\alpha ' + \alpha )\left[ {{{{(\xi ' - a\alpha ' + \alpha )}}^2} + 3{{\hat{\tau}}^2}} \right] } \right. \cr& {+ \,\left. {\alpha '\left[ {{{{(\xi ' - a\alpha ' + \alpha )}}^2} + {{\hat{\tau}}^2} - a\alpha '(\xi ' - a\alpha ' + \alpha )} \right]} \right\}} \bigg] \end{array}$$

(6b)

The two normalized linear and nonlinear control gain parameters ${\varphi_1},{\varphi_2}$ are defined as ${\varphi_1} = {f_1}/{K_{\alpha }},{\varphi_2} = {f_2}/{K_{\alpha }}$, respectively.

Solution Methodology

To perform the nonlinear aerothermoelastic analysis in the time domain, (4a, 4b) is transformed into state-space matrix form:

$$\begin{array}{lll} \dot{{\bf y}} \left( \tau \right) =& \left[ \begin{array}{lll}\quad\quad\quad\quad\quad\quad\quad\quad{\bf 0}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{\bf I} \cr {{{{\bf M}}^{- 1 }}\left( {{\bf QL}{{{\bf 2}}_{{\bf ext}}} + {\bf QNL}{{{\bf 2}}_{{\bf ext}}} - {\bf KL} - {\bf KNL} - {{{\bf M}}_{{\bf control}}}} \right)} \quad\quad {{{{\bf M}}^{- 1 }}\left( {{\bf QL}{{{\bf 1}}_{{\bf ext}}} + {\bf QNL}{{{\bf 1}}_{{\bf ext}}} - {\bf C}} \right)} \cr\end{array} \right]{\bf y}\left( \tau \right) \cr& - \left[ \begin{array}{ll} {\bf 0} \quad {\bf 0} \cr {\bf 0} \quad {{{{\bf M}}^{- 1 }}} \cr \end{array} \right]{\bf R}\left( {\xi_s, {\alpha_s}} \right) \cr {\text where}\;{\bf y}\left( \tau \right) =& \left\{ \begin{array}{ll} {\xi \left( \tau \right)} \cr {\alpha \left( \tau \right)} \cr {\dot{\xi}\left( \tau \right)} \cr {\dot{\alpha}\left( \tau \right)} \cr \end{array} \right\},{\bf R}\left( {\xi_s, {\alpha_s}} \right) = \left\{ \begin{array}{ll} 0 \cr 0 \cr {{{{\text Q}}_{{\text f}}}\left( {1,1} \right)} \cr {{{{\text Q}}_{{\text f}}}\left( {2,1} \right)} \cr \end{array} \right\},{{{\bf M}}_{{\bf control}}} = \left[ \begin{array}{ll} 0 & 0 \cr 0 & {(1/{U^{*}}^2)({\varphi_1} + {\varphi_2}{\alpha^2}(\tau ))} \cr \end{array} \right]\end{array}$$

(7)

where ${\bf y}$ is the state vector and ${\bf M}$ is the mass matrix. ${\bf KL}\;{\text and}\;{\bf KNL}$ in (7) represents the linear and nonlinear stiffness matrices, while the aerodynamic damping and stiffness matrices ${\bf QNL}{{{\bf 1}}_{\bf ext }}$ and ${\bf QNL}{{{\bf 2}}_{\bf ext }}$ contain both uncoupling and coupling nonlinear quadratic and cubic terms, respectively. The matrices ${\bf QL1} \ {\text and}\;{\bf QL2}$ include the damping and stiffness aerodynamic linear terms, respectively, ${\bf R}$ and ${{\bf Q}}_{\text f}$ are the freeplay force/moment vectors. ${{{\bf M}}_{{\bf control}}}$ represents the active linear and nonlinear control moment matrix. A numerical simulation using the fifth to sixth Runge-Kutta Fehlberg time integration scheme with step size control is carried out for the system in (7).

Aerothermoelastic Control of Lifting Surfaces: Analysis

Before we apply any control and to emphasize the importance of aerodynamic heating on the nonlinear aerothermoelastic behavior of the examined aerothermoelastic system, the influence of the loss in effective torsional stiffness of a solid thin double-wedge wing under various parameters such as flight condition, thickness ratio, pitch freeplay, and pitching stiffness nonlinearity has been analyzed. Unless otherwise stated, the numerical simulations consider the baseline parameters which are listed in Table 1.

Table 1 Baseline parameters of 2-DOF plunging/pitching airfoil

A number of bifurcation diagrams were constructed from the amplitude of the pitch LCO as a function of the flight Mach number for a plunging/pitching airfoil with a freeplay structural nonlinearity in pitch, cubic pitch structural nonlinearities subjected to supersonic/hypersonic flow which induced also aerodynamic heating are presented in Fig. 2. Because of symmetric pitch LCO amplitude and to have a better graphical representation, some of the plots in Fig. 2 have been presented in positive or in negative side of the LCO curve as shown later. In Fig. 2a case #1 (positive side of LCO curve) is for the system with no aerodynamic heating and ${{\hat{\eta}}_h} = {{\hat{\eta}}_{\alpha }} = 0$, such that ${M_{LF }} = 17.4$. Case #2 (negative side) is for the system with no aerodynamic heating also, but ${{\hat{\eta}}_h} = 0, \;{{\hat{\eta}}_{\alpha }} = 10$ (hard structural nonlinearities) and the flutter speed is the same as case #1. Note that the simulations are restricted to cases where the pitching displacement is within $\pm {20^{\circ }}$ to remain within the limits of validity of the proposed model and approach. The aerothermoelastic system exhibits a bifurcation behavior for these two cases at ${M_{\infty }}\approx 1.7$ due to the presence of freeplay in pitch direction. For the speed range ($1.7 < {M_{\infty }} \leq 7$), different types of response behavior (periodic, quasiperiodic, or chaotic) will occur. Within the speed ranges ($7 < {M_{\infty }} \leq 17$) for case #1, ($7 < {M_{\infty }} \leq 21$) for case #2, a stable LCO is experienced; its amplitude increases with the increase of the flight Mach number. At ${M_{\infty }}\approx 16$, the case #1 exhibits a pitch LCO with amplitude of about ${9.9^{\circ }}$, while the case #2, the LCO has a pitching amplitude about ${6.6^{\circ }}$. It appears that cubic structural nonlinearities significantly decrease the LCO amplitude, while the linear flutter speed remains constant, besides case #1 has maximum amplitude of ($\approx\ {13^{\circ }}$) at ${M_{\infty }}\approx 16.2$ compared with ($\,\approx\, {17^{\circ }}$) at ${M_{\infty }}\approx 21$ for case #2. Figure 2b shows the effect of aerodynamic heating. Case #1 in Fig. 2b is the same as case #2 in Fig. 2a but in positive side. The result of case #2 reveals that the flutter speed (${M_{LF }} = 13.65$), as well as the LCO behavior, is affected by the loss of the torsional stiffness. In both cases, the pitching structural nonlinearities are considered (${{\hat{\eta}}_h} = 0, \;{{\hat{\eta}}_{\alpha }} = 10$). Under pitch active control, a considerable change in the amplitude of the LCO is significantly observed in Fig. 3. Case #1 is the same as case #2 in Fig. 2b but in positive side and does not include any active control (${\varphi_1} = {\varphi_2} = 0$). Case #2 (${\varphi_1} = 0.1,{\varphi_2} = 10{\varphi_1}$), #3 (${\varphi_1} = 0.3,{\varphi_2} = 10{\varphi_1}$), #4 (${\varphi_1} = 0.8,{\varphi_2} = 10{\varphi_1}$), and #5 (${\varphi_1} = 1,{\varphi_2} = 10{\varphi_1}$) present a shift of the bifurcation behavior to ${M_{\infty }}\approx 3.8,\;8.0,\;12.5,\;{\text and} \ 13.5$, respectively. The unstable LCO including the chaotic region in case #1 (until ${M_{\infty }}\approx 7$) has been suppressed as shown in case #3. Figure 3 also shows the phase portraits and time histories for various flight Mach numbers which represent the uncontrolled (case #1) and controlled system (such as case #4 and #5), respectively. Clearly, increasing the linear pitch gain can extend the flutter boundary and convert the unstable LCO into stable LCO and/or shift the transition between these two states toward higher flight Mach numbers with suppression of LCO.

Fig. 2

Bifurcation pitch diagrams for the double-wedge airfoil with nonlinearities

Fig. 3

Pitch LCO amplitude versus flight Mach number for a 2-DOF system with all nonlinearities. Time histories and phase portraits represent the uncontrolled and controlled system, respectively

Figure 4 shows the effect of nonlinear active control gain with zero linear gain (${\varphi_1} = 0,{\varphi_2}\ne 0$). It shows that increasing ${\varphi_2}$ alone (for case #2, ${\varphi_2} = 50$ and for case #3, ${\varphi_2} = 100$) is less effective in stabilizing the aerothermoelastic system than for the linear one. This leads to a practical application of the control mechanism on actual and future generation aerospace vehicle lifting surfaces.

Fig. 4

Effect of nonlinear active control for system encompassing all nonlinearities

Consideration About the Aerothermoelastic Control Behavior of Lifting Surfaces

The influence of aerodynamic heating on the nonlinear aerothermoelastic behavior of a solid thin double-wedge airfoil encountered all nonlinearities (structural freeplay and cubic stiffness, aerodynamic third-order piston theory) in supersonic/hypersonic flight speed regime is highlighted in the preceding sections. The nonlinear aerothermoelastic analysis of aero-surfaces is an important aspect of design. Linear and nonlinear active control can extend the flutter boundary and convert the unstable aerothermoelastic behavior into stable one and/or shift the transition between these two states toward higher flight Mach numbers with suppression of LCO. Moreover, the analysis presented can serve as a guideline for selecting appropriate control gains to maximize performance. Active control can be produced via a device behaving similarly to a linear/nonlinear spring. The issue of the design of the controller is not addressed here. Only a theoretical analysis of the nonlinear active control of aerothermoelastic phenomena of a lifting surface at supersonic/hypersonic flight speed regimes is presented. The application of various controllers using different linear/nonlinear control theories such as optimal control (LQR and others) for more robust control strategy is an active area of current research.