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A modified incremental harmonic balance method is presented to analyze the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity. The current method, which combines the traditional incremental harmonic balance method and a fast Fourier transform, can be used to obtain the higher-order approximate solution for the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity using significantly fewer linearized algebraic equations than the traditional method, and the dominant frequency components of the response can be obtained by a fast Fourier transform of the numerical solution. Thus, periodic solutions can be obtained, and the calculation process can be simplified. Furthermore, the nonsmooth nonlinearity was expanded into a Fourier series. The procedures of the modified incremental harmonic balance method were demonstrated using systems with hysteresis and free play nonlinearities. The modified incremental harmonic balance method was validated by comparing with the numerical solutions. The effect of the number of harmonics on the solution precision as well as the effect of the free-play and stiffness ratio on the response amplitude is discussed.

Nonlinear phenomena can be frequently encountered in the aerospace industry. It is important to predict the nonlinear aeroelastic characteristic of airfoils [

The harmonic balance (HB) method was first used to analyze the nonlinear aeroelastic responses of structures by Woolston et al. [

The incremental harmonic balance (IHB) method [

Most of the studies mentioned above involved systems with smooth nonlinearities, such as quadratic, cubic, or even higher-order nonlinearities, which could be solved using the HB or IHB methods. It is well known that it is easier to handle smooth nonlinearities than nonsmooth ones. Furthermore, as the number of degrees of freedom (DOFs) of the system changes, the difficulty of the analysis and calculations dramatically increases, which is inconvenient for obtaining solutions. Moreover, nonsmooth nonlinearities exist widely in engineering fields, such as free-play and hysteresis nonlinearities, which can be expressed in the form of nonsmooth functions. Not surprisingly, systems with free-play or hysteresis nonlinearities can rarely be solved by the HB or IHB methods due to the difficulty of handling nonsmooth functions [

In this paper, a modified incremental harmonic balance method based on the fast Fourier transform and the traditional incremental harmonic balance method is presented for two-degrees-of-freedom aeroelastic airfoil motion with nonsmooth structural nonlinearities, in which the solution is based on multiple harmonics. After the alternative incremental and iterative processes, the analytic algebraic equations are obtained by applying the proposed IHB, in which the nonsmooth nonlinearity is expanded into a Fourier series. Finally, the procedures of the modified incremental harmonic balance method were demonstrated using systems with hysteresis and free-play nonlinearities. The modified incremental harmonic balance method was validated by comparing with the numerical solutions obtained by the Runge–Kutta method. The effect of the number of harmonics on the solution precision and the effect of the free-play and stiffness ratio on the response amplitude are analyzed.

A two-dimensional airfoil is shown in Figure

Typical section of a two-dimensional airfoil.

Consequently, the aeroelastic equations of the airfoil can be recast in a nondimensional form as follows [

Discrete model of vortex lattice for a two-dimensional airfoil section [

Generally, the aerodynamic force based on the unsteady vortex lattice model is referred to as the reduced-order force [

If the nonlinearity is due to backlash in loose or worn control surface hinges, the nonlinearity exhibits free-play. Furthermore, if the friction and backlash must be considered, the nonlinearity is usually in the form of hysteresis [

Illustration of nonsmooth function.

The first step in the IHB method is the Newton–Raphson procedure. Equation (

It is assumed that

Equations (

Substituting equations (

In equation (

The assumed solution given in the previous section (equation (

The basic idea of the modified incremental harmonic balance method is that a fast Fourier transform is carried out to extract the dominant frequency components in the response of the system using a numerical method, which determines the harmonic number and the dominant frequency of the approximate solution. The dominant frequency components are then applied to the incremental harmonic balance method to solve the linearized equations, which effectively determines the approximate solution of the nonlinear system and improves the solution precision.

To reduce the dimensions of matrices in equation (

According to the steps discussed above, the solutions for an aeroelastic system with a nonsmooth structural nonlinearities can be obtained. Figure

Flowchart of the proposed method.

There are two key differences between the two IHM methods:

If the solutions of the multi-DOF system contain high-order harmonics, the number of linearized algebraic equations in terms of

When the nonlinear term is nonsmooth, its Fourier series expansion in the modified IHB method is much simpler, as an effective and general method for the Fourier series expansion of nonsmooth nonlinearities was developed. The explicit expressions of the first-order Taylor series approximations of the nonsmooth nonlinear term were derived, i.e., equations (

A description of a two-dimensional airfoil and the corresponding parameters can be found elsewhere [

The incremental harmonic balance method was mainly used to determine the periodic motion of nonlinear systems. To demonstrate that the modified incremental harmonic method can be used to determine the response of the aeroelastic system with a nonsmooth nonlinearity, the plunge and pitch responses of the two-dimensional airfoil were obtained through two examples: one with a hysteresis nonlinearity, and one with a free-play nonlinearity.

For the example with a hysteresis nonlinearity, the following parameters were employed:

Numerical simulation and the frequency spectrum (hysteresis nonlinearity). (a) Numerical simulation of

As shown in Figure

When the assumed solution contained 1,

The numerical solutions were obtained by the Runge–Kutta method. Figures

Numerical and analytical solutions (assumed solution in modified IHB contained 1,

Numerical and analytical solutions (assumed solution in modified IHB contained 1,

For the free-play nonlinearity, the following parameters were employed:

Numerical simulation and the frequency spectrum (free-play nonlinearity). (a) Numerical simulation of

As shown in Figure

For the solution containing 1,

For the solution containing 1,

The numerical solutions were obtained using the Runge–Kutta method. Figures

Numerical and analytical solutions (the modified IHB approximation contained 1,

Numerical and analytical solutions (the modified IHB approximation contained 1,

To analyze the effects of

The effects of

Comparison of amplitude (

Comparison of amplitude (

Comparison of amplitude (

Comparison of amplitude (

As shown in Figures

As shown in Figures

The effect of

Comparison of amplitude (

Comparison of amplitude (

When

As shown in Figures

In this paper, a modified incremental harmonic balance method is presented for an aeroealstic system with a nonsmooth structural nonlinearity. The procedure for the modified incremental harmonic balance method was demonstrated for systems with hysteresis and free play nonlinearities. The validity of the modified incremental harmonic balance method was demonstrated by comparing with the numerical solutions. In addition, the influence of the parameters on the nonlinear aeroelasticity was also studied. The following conclusions were drawn:

A modified incremental harmonic balance method was presented to analyze the responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity. Not only could the periodic solutions be obtained, but also the calculation process was simplified.

The proposed approach can be applied in other nonsmooth cases, especially those arising in aeroelastics. The application of the incremental harmonic balance method was extended.

For a given

Since the response of the aeroelastic model in this paper is stable, the method in this paper is limited to the stable periodic solution. Going forward, more studies are required, including the unstable periodic solution. The homology method may obtain the unstable periodic solution. Therefore, it may be suitable for the response of aeroelastic system with nonsmooth structural nonlinearities by integration of the incremental harmonic balance method and the homotopy method. The application of the incremental harmonic balance method may be expanded.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.