Flutter analysis of a viscoelastic tapered wing under bending–torsion loading

Abstract

The dynamic stability of a tapered viscoelastic wing subjected to unsteady aerodynamic forces is investigated. The wing is considered as a cantilever tapered Euler–Bernoulli beam. The beam is made of a linear viscoelastic material where Kelvin–Voigt model is assumed to represent the viscoelastic behavior of the material. The governing equations of motion are derived through the extended Hamilton’s principle. The resulting partial differential equations are solved via Galerkin’s method along with the classical flutter investigation approach. The developed model is validated against the well-known Goland wing and HALE wing and good agreement is obtained. Different solution methods, namely; the k method, the p-k method, and the flutter determinant method are compared for the case of elastic wing. However, when the viscoelastic damping is introduced, the k and p-k methods become less effective. The flutter determinant method is modified and employed to carry out non-dimensional parametric study on the Goland wing. The study includes the effects of parameters such as the taper ratio, the density ratio, the viscoelastic damping of wing structure and many other parameters on the flutter speed and flutter frequency. The study reveals that a tapered wing would be more dynamically stable than a uniform wing. It is also observed that the viscoelastic damping provides wider stability region for the wing. The investigation shows that the density ratio, bending-to-torsion frequency ratio, and the radius of gyration have significant effects on the dynamic stability of the wing. Based on the obtained results, a wing with an elastic center and inertial center that are located closer to the mid-chord would be more dynamically stable.

Appendix A

Coefficients of the equations of motion.

$$a_{11} = \mathop \smallint \limits_{0}^{1} \left( {\left[ {1 - c_{t} \xi } \right]f^{\prime\prime}} \right)^{''} f\,d\xi$$

$$a_{12} = \frac{{\eta_{E} }}{\sigma }\mathop \smallint \limits_{0}^{1} \left( {\left[ {1 - c_{t} \xi } \right]f^{\prime\prime}} \right)^{''} f\,d\xi$$

$$a_{13} = \frac{1}{{\sigma^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]f^{2} \,d\xi$$

$$a_{14} = \frac{1}{{2\mu \sigma^{2} }}\frac{{dC_{L} }}{d\theta }\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]f^{2} \,d\xi$$

$$a_{15} = \frac{\pi }{{4\mu \sigma^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{2} f^{2} \,d\xi$$

$$a_{21} = \frac{{y_{a} }}{{r_{a}^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{2} f\phi\,d\xi$$

$$a_{22} = \frac{1}{{2\mu r_{a}^{2} }}\frac{{dC_{L} }}{d\theta }\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{2} \left( {y_{0} - \frac{1}{4}} \right)f\,\phi d\xi$$

$$a_{23} = \frac{\pi }{{4\mu r_{a}^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{3} \left( {y_{0} - \frac{1}{2}} \right)\phi f\,d\xi$$

$$b_{11} = \frac{{y_{a} }}{{\sigma^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{2} f\phi\,d\xi$$

$$b_{12} = \frac{1}{{2\mu \sigma^{2} }}\frac{{dC_{L} }}{d\theta }\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]f\phi\,d\xi$$

$$b_{13} = \frac{1}{{2\mu \sigma^{2} }}\frac{{dC_{L} }}{d\theta }\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{2} \left( {\frac{3}{4} - y_{0} } \right)f\phi\,d\xi$$

$$b_{14} = \frac{\pi }{{4\mu \sigma^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{2} \phi f\,d\xi$$

$$b_{15} = \frac{\pi }{{4\mu \sigma^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{3} \left( {y_{0} - \frac{1}{2}} \right)\phi f\,d\xi$$

$$b_{21} = \mathop \smallint \limits_{0}^{1} \left( {\left[ {1 - c_{t} \xi } \right]\phi^{\prime}} \right)^{'} \phi\,d\xi$$

$$b_{22} = \eta_{G} \mathop \smallint \limits_{0}^{1} \left( {\left[ {1 - c_{t} \xi } \right]\phi^{'} } \right)^{'} \phi\,d\xi$$

$$b_{23} = \mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]\phi^{2}\, d\xi$$

$$b_{24} = \frac{1}{{2\mu r_{a}^{2} }}\frac{{dC_{L} }}{d\theta }\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{2} \left( {y_{0} - \frac{1}{4}} \right)\phi^{2}\, d\xi$$

$$b_{25} = \frac{1}{{2\mu r_{a}^{2} }}\frac{{dC_{L} }}{d\theta }\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{3} \left( {y_{0} - \frac{1}{4}} \right)\left( {\frac{3}{4} - y_{0} } \right)\phi^{2}\,d\xi$$

$$b_{26} = \frac{\pi }{{4\mu r_{a}^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{3} \left( {\frac{3}{4} - y_{0} } \right)\phi^{2}\,d\xi$$

$$b_{27} = \frac{\pi }{{4\mu r_{a}^{2} }}\mathop \smallint \limits_{0}^{1} \left[ {1 - c_{t} \xi } \right]^{4} \left( {\frac{9}{32} + y_{0}^{2} - y_{0} } \right)\phi^{2}\,d\xi$$