Frequency domain approach for probabilistic flutter analysis using stochastic finite elements

Abstract

In this work, a stochastic finite element method based on first order perturbation approach is developed for the probabilistic flutter analysis of aircraft wing in frequency domain. Here, both bending and torsional stiffness parameters of the wing are treated as Gaussian random fields and represented by a truncated Karhunen–Loeve expansion. The aerodynamic load on the wing is modeled using Theodorsen’s unsteady aerodynamics based strip theory. In this approach, Theodorsen’s function, which is a complex function of reduced frequency, is also treated as a random field. The applicability of the present method is demonstrated by studying the probabilistic flutter of cantilever wing with stiffness uncertainties. The present method is also validated by comparing results with Monte Carlo simulation (MCS). From the analysis, it is observed that torsional stiffness uncertainty has significant effect on the damping ratio and frequency of the flutter mode as compared to bending stiffness uncertainty. The probability density functions of damping ratio and frequency using perturbation technique and MCS are also discussed at various free stream velocities due to stiffness uncertainties. Furthermore, the flutter probability of the cantilever wing is studied by defining implicit limit state function in conditional sense on flow velocity for the flutter mode. Both perturbation and MCS are considered to study the flutter probability of the wing. From the cumulative distribution functions of flutter velocity, it is noticed that the presence of uncertainty in torsional rigidity lowers the predicted flutter velocity in comparison to uncertainty in bending rigidity.

Appendix

The weak forms of the governing Eqs. (5) and (6) are obtained by multiplying weight functions $v_1$ and $v_2$ respectively and integrating over an $i^{th}$ element length as [32]:

$$\begin{aligned} \int _{y_i}^{y_{i+1}}v_1\left( m\ddot{h}+mx_{\alpha }b\ddot{\alpha }+\frac{\partial ^2}{\partial y^2}\left( EI\frac{\partial ^2h}{\partial y^2}\right) +L\right) dy= \,& {} 0 \end{aligned}$$

(43)

$$\begin{aligned} \int _{y_i}^{y_{i+1}}v_2\left( I_p\ddot{\alpha }+mx_{\alpha }b\ddot{h}-\frac{\partial }{\partial y}\left( GJ\frac{\partial \alpha }{\partial y}\right) -M\right) dy= \,& {} 0 \end{aligned}$$

(44)

Substituting the expressions for lift (L) and moment (M), and performing integration by parts, the above equations can be expressed as:

$$\begin{aligned}&\int _{y_i}^{y_{i+1}}v_1m\ddot{h}dy+\int _{y_i}^{y_{i+1}}v_1\pi \rho _{\infty }b^2\ddot{h}dy+\int _{y_i}^{y_{i+1}}v_1mx_{\alpha }b\ddot{\alpha }dy\nonumber \\&\quad -\,\int _{y_i}^{y_{i+1}}\pi \rho _{\infty }b^3a\ddot{\alpha }dy+\int _{y_i}^{y_{i+1}}v_1UC(k)2\pi \rho _{\infty }b\dot{h}dy\nonumber \\&\quad +\,\int _{y_i}^{y_{i+1}}v_1U\pi \rho _{\infty }b^2\dot{\alpha }dy+\int _{y_i}^{y_{i+1}}v_1UC(k)\pi \rho _{\infty }b^2(1-2a)\dot{\alpha }dy\nonumber \\&\quad +\,\int _{y_i}^{y_{i+1}}v_1U^2C(k)2\pi \rho _{\infty }b{\alpha }dy+\int _{y_i}^{y_{i+1}}\frac{\partial ^2v_1}{\partial y^2}\left( EI\frac{\partial ^2h}{\partial y^2}\right) dy\nonumber \\&\quad +\,\left( -v_1S^b-\frac{\partial {v_1}}{\partial {y}}M^b\right) \arrowvert _{y_i}^{y_{i+1}}=0 \end{aligned}$$

(45)

$$\begin{aligned}&\int _{y_i}^{y_{i+1}}v_2I_p\ddot{\alpha }dy+\int _{y_i}^{y_{i+1}}v_2\pi \rho _{\infty }b^4\left( \frac{1}{8}+a^2\right) \ddot{\alpha }dy\nonumber \\&\quad +\,\int _{y_i}^{y_{i+1}}v_2mx_{\alpha }b\ddot{h}dy-\int _{y_i}^{y_{i+1}}v_2\pi \rho _{\infty }b^3a\ddot{h}dy\nonumber \\&\quad -\,\int _{y_i}^{y_{i+1}}v_2U\pi \rho _{\infty }b^3\left( -\frac{1}{2}+a\right) \dot{\alpha }dy\nonumber \\&\quad -\,\int _{y_i}^{y_{i+1}}v_2UC(k)\pi \rho _{\infty }b^3\left( \frac{1}{2}+a\right) \left( 1-2a\right) \dot{\alpha }dy\nonumber \\&\quad -\,\int _{y_i}^{y_{i+1}}v_2UC(k)2\pi \rho _{\infty }b^2\left( \frac{1}{2}+a\right) \dot{h}dy\nonumber \\&\quad -\,\int _{y_i}^{y_{i+1}}v_2U^2C(k)2\pi \rho _{\infty }b^2\left( \frac{1}{2}+a\right) {\alpha }dy\nonumber \\&\quad +\,\int _{y_i}^{y_{i+1}}\frac{\partial {v_2}}{\partial {y}}\left( GJ\frac{\partial {\alpha }}{\partial {y}}\right) dy-v_2\tau \arrowvert _{y_i}^{y_{i+1}}=0 \end{aligned}$$

(46)

where $M^b$, $S^b$, and $\tau$ are bending moment, shear force, and torsional moment respectively and can be written as:

$$\begin{aligned} M^b= \,& {} EI\frac{\partial ^2h}{\partial y^2}\\ S^b= \,& {} -\frac{\partial }{\partial {y}}\left( EI\frac{\partial ^2h}{\partial y^2}\right) \\ \tau= \,& {} GJ\frac{\partial {\alpha }}{\partial {y}} \end{aligned}$$

Now by using finite element approximation functions, the heave (h) and pitch $(\alpha )$ displacements can be expressed as:

$$\begin{aligned} h(y,t)= \,& {} \lfloor N_w\rfloor \{w_e(t)\}\nonumber \\ \alpha (y,t)= \,& {} \lfloor N_\alpha \rfloor \{\alpha _e(t)\} \end{aligned}$$

(47)

where $\lfloor N_w\rfloor =\lfloor N_1, N_2, N_3, N_4\rfloor$ and $\lfloor N_\alpha \rfloor =\lfloor \bar{N}_1,\bar{N}_2\rfloor$ are Hermite and Lagrange shape functions [32] respectively and $\{w_e\}$ and $\{\alpha _e\}$ are the bending and torsional degrees of freedom respectively. The weight functions $v_1$ and $v_2$ can be also written as $v_1= \lfloor N_w\rfloor ^T$ and $v_2=\lfloor N_\alpha \rfloor ^T$ respectively. On substitution of the above approximation functions and weight functions, the elemental equations can be written in notational form as:

$$\begin{aligned}&\left[ M_{SB}\right] \{\ddot{w}_e\}+\left[ M_{AB}\right] \{\ddot{w}_e\}+\left[ M_{SC}\right] \{\ddot{\alpha }_e\}+\left[ M_{AC}\right] \{\ddot{\alpha }_e\}\nonumber \\&\qquad +\,\left[ C_{AB}\right] \{\dot{w}_e\}+\left[ C_{AC}^{12}\right] \{\dot{\alpha }_e\}+\left[ K_{AC}\right] \{\alpha _e\}+\left[ K_{SB}\right] \{w_e\}\nonumber \\&\quad =\{F_i\} \end{aligned}$$

(48)

$$\begin{aligned}&\left[ M_{ST}\right] \{\ddot{\alpha }_e\}+\left[ M_{AT}\right] \{ \ddot{\alpha }_e\}+\left[ M_{SC}\right] \{\ddot{w}_e\}+\left[ M_{AC}\right] \{\ddot{w}_e\}\nonumber \\&\qquad +\,\left[ C_{AT}\right] \{\dot{\alpha }_e\}+\left[ C_{AC}^{21}\right] \{\dot{w}_e\}+\left[ K_{AT}\right] \{\alpha _e\}+\left[ K_{ST}\right] \{\alpha _e\}\nonumber \\&\quad =\{\tau _i\} \end{aligned}$$

(49)

where $\{F_i\}=\lfloor -S_{y_{i}}^{b}, -M_{y_{i}}^{b}, S_{y_{i+1}}^{b}, M_{y_{i+1}}^{b}\rfloor ^T$ and $\{\tau _i\}=\lfloor -\tau _{y_i},\tau _{y_{i+1}}\rfloor ^T$. The terms involved in Eq. (48) are:

$$\begin{aligned} \left[ M_{SB}\right]= \,& {} m\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_w\rfloor dy}\nonumber \\ \left[ M_{AB}\right]=\, & {} \pi \rho _{\infty }b^2\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_w\rfloor dy}\nonumber \\ \left[ M_{SC}\right]=\, & {} mx_{\alpha }b\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ M_{AC}\right]=\, & {} -\pi \rho _{\infty }b^3a\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ C_{AB}\right]=\, & {} UC(k)2\pi \rho _{\infty }b\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_w\rfloor dy}\nonumber \\ \left[ C_{AC}^{12}\right]= \,& {} U\pi \rho _{\infty }b^2\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\&+\,UC(k)\pi \rho _{\infty }b^2(1-2a)\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ K_{AC}\right]= \,& {} U^2C(k)2\pi \rho _{\infty }b\int _{y_i}^{y_{i+1}}{\lfloor {N_w}\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ K_{SB}\right]=\, & {} \int _{y_i}^{y_{i+1}}{EI\lfloor {N_w^{''}}\rfloor ^T\lfloor N_w^{''}\rfloor dy} \end{aligned}$$

(50)

and the terms in Eq. (49) can be written as:

$$\begin{aligned} \left[ M_{ST}\right]= \,& {} I_p\int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ M_{AT}\right]= \,& {} \pi \rho _{\infty }b^4\left( a^2+\frac{1}{8}\right) \int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ M_{SC}\right]= \,& {} mx_{\alpha }b\int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_w\rfloor dy}\nonumber \\ \left[ M_{AC}\right]= \,& {} -\pi \rho _{\infty }b^3a\int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_w\rfloor dy}\nonumber \\ \left[ C_{AT}\right]= \,& {} -U\pi \rho _{\infty }b^3(-0.5+a)\int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\&-\,UC(k)\pi \rho _{\infty }b^3(0.5+a)(1-2a)\int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ C_{AC}^{21}\right]= \,& {} -UC(k)2\pi \rho _{\infty }b^2(0.5+a)\int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_w\rfloor dy}\nonumber \\ \left[ K_{AT}\right]= \,& {} -U^2C(k)2\pi \rho _{\infty }b^2(0.5+a)\int _{y_i}^{y_{i+1}}{\lfloor {N_\alpha }\rfloor ^T\lfloor N_\alpha \rfloor dy}\nonumber \\ \left[ K_{ST}\right]= \,& {} \int _{y_i}^{y_{i+1}}{GJ\lfloor {N_\alpha ^{'}}\rfloor ^T\lfloor N_\alpha ^{'}\rfloor dy} \end{aligned}$$

(51)