Fractional-Order Control of Buffet-Induced Twin-Tail Vibrations with Non-Smooth Quadratic Damping: Harmonic Balance and Bifurcation Analysis

This paper investigates the nonlinear vibration suppression of buffet-induced oscillations in an F-15 twin-tail structure using a fractional-order proportional-derivative (FOPD) controller. A nonlinear coupled twin-tail model incorporating non-smooth quadratic aerodynamic damping, cubic stiffness nonlinearity, and linear structural coupling is considered. To predict the steady-state responses, a two-harmonic harmonic balance (HB) formulation is developed. The non-smooth quadratic damping is represented through a truncated Fourier expansion, while the Caputo fractional derivative is treated analytically using asymptotic approximations, leading to a closed-form set of nonlinear algebraic equations. The stability of the obtained periodic solutions is subsequently examined using Floquet theory. The proposed analytical formulation is employed to investigate both the primary and superharmonic resonance behaviors through frequency-response, force-response, and codimension-two bifurcation analyses. Comparisons with direct numerical integration demonstrate that the proposed HB solution provides excellent agreement with the numerical results and offers higher accuracy than the previously reported multiple-scales solution. The uncontrolled system is shown to exhibit pronounced nonlinear phenomena, including saddle-node bifurcations, jump phenomena, Hopf bifurcations, and zero-Hopf points under high excitation levels. The proposed FOPD controller effectively suppresses the vibration amplitudes over a wide range of operating conditions while suppressing the saddle-node bifurcations and eliminating the Hopf bifurcations responsible for dynamic instabilities. Furthermore, the influence of the fractional derivative order is examined, revealing that the optimal controller performance depends on the excitation frequency rather than always occurring at the integer-order limit. The codimension-two bifurcation analysis further confirms that the proposed controller enlarges the stable operating region by removing the Hopf boundary and substantially shrinking the remaining instability region. These results demonstrate that fractional-order control provides a powerful and flexible strategy for mitigating complex nonlinear aeroelastic vibrations in coupled twin-tail structures.