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A kind of high-aspect-ratio shape memory alloy (SMA) composite wing is proposed to reduce the wing’s fluttering. The nonlinear dynamic characteristics and optimal control of the SMA composite wings subjected to in-plane stochastic excitation are investigated where the great bending under the flight loads is considered. The stochastic stability of the system is analyzed, and the system’s response is obtained. The conditions of stochastic Hopf bifurcation are determined, and the probability density of the first-passage time is obtained. Finally, the optimal control strategy is proposed. Numerical simulation shows that the stability of the system varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the reliability of the system is improved through optimal control, and the first-passage time is delayed. Finally, the effects of the control strategy are proved by experiments. The results of this paper are helpful for engineering applications of SMA.

High-aspect-ratio wings are applied in uninhabited air vehicles (UAV) widely. They have a great bending and torsion deformation under the flight loads, which induce the wing’s fluttering. In this paper, a kind of shape memory alloy (SMA) composite wing is proposed to reduce the bending deformation. Shape memory alloy is a kind of smart material. It has many special properties, such as shape memory effect and damping characteristics. Moreover, SMA can be controlled by heating and cooling to achieve the structure deformation, which is helpful for reducing the wing’s fluttering and ensuring the UAV’s safety.

Several scholars have studied the application of shape memory materials in the wings. Yang et al. analyzed the characteristics of smart composite wing with SMA actuators [

A high-aspect-ratio SMA composite wing can be regarded as a cantilever SMA composite beam. There are also many researchers studying the dynamic characteristics of SMA beam. Lau discussed firstly the vibration characteristics of SMA beams with different boundary conditions [

This paper aims to offer a kind of analysis method to the nonlinear dynamical characteristic of a cantilever SMA composite beam subjected to stochastic excitation. Nonlinear differential items are introduced to explain the hysteretic phenomenon of strain-stress curve of SMA, and the hysteretic nonlinear dynamic model of a cantilever SMA composite beam subjected to stochastic excitation is developed. The stochastic stability of the system is analyzed, and the system’s response is obtained. The conditions of stochastic Hopf bifurcation are determined, and the probability density of the first-passage time is obtained. Finally, the optimal control strategy to improve the system’s reliability is proposed.

For obtaining the dynamic characteristics of a Ti-Ni SMA composite wing subjected to stochastic excitation, it is necessary to establish an accurate constitutive model of Ti-Ni SMA. The strain-stress curves of Ti-Ni SMA are presented in Figure

Strain-stress curves of Ti-Ni SMA.

Van der Pol equation is a nonlinear equation. It can be shown as follows:

The item

However, the strain-stress curves of SMA are not pure parabolic lines. In this paper, a new differential item, which is developed from Van der Pol hysteretic model, is introduced to describe the strain-stress curves of SMA as follows:

Equation (

Firstly, we obtained the stress-strain data of Ti-Ni SMA materials by experiments.

Secondly, we choose the constructive model

Please note that the coefficients

Thirdly, we input the constructive model and the stress-strain data into the partial least-square regression software SIMCA-P; then the software will calculate the variable importance (VIP) and the coefficient values of each item.

When the coefficient values of each item were given, the software can predict the stress-strain data according to the constructive model.

Finally, the software can compare the stress-strain data from prediction and that of real stress-strain curves.

The analysis results of the principal component based on the experimental data are shown in Figure

Variable importance of each item.

Coefficient values of each item.

The result of forecast test to (

Results of the forecast test for the fitting effect of (

The research object is the Chinese “Wing Loong” UAV, which is shown in Figure

Chinese “Wing Loong” UAV.

Mechanical model of a high-aspect-ratio SMA composite wing.

The boundary conditions of the composite wing can be written as follows:

Thus, the vibration mode

Considering the complex characteristics of composite materials, we introduce Hamilton’s principle to dynamic modeling of the system. The Hamilton function can be presented as

According to the Hamilton principle,

Thus, the nonlinear dynamic model of the SMA composite wing can be shown as follows:

According to (

When the wing’s length is more than its width, which means

Equation (

Let

The nondamping autonomous system from (

Its Hamilton function is

According to the quasi-nonintegrable Hamiltonian system theory, the Hamiltonian function

The averaged FPK equation of (

Equation (

Now, we can calculate the transition sets of the system response. Let

Since

Bifurcation sets:

Thus, the bifurcation set of the system is

Hysteretic sets:

Thus, the hysteretic sets of the system are

Double limited sets:

Thus, the double limited sets of the system are

The transition sets of the system’s response are shown in Figure

Stationary probability density (SPD) of the system response in different parameters.

From Figure

when

when

when

in sum, stochastic Hopf bifurcation appears in the variation of the parameters.

According to the results of the above section, the stochastic Hopf bifurcation appears in the process of varying the system parameters. Bifurcation changes the system’s motion and decreases its reliability. In this section, stochastic optimal control is introduced to improve the system’s reliability.

The dynamic model of controlled system can be shown as follows:

The reliability function

The optimal control force

The stress

Substituting (

Thus, the background Kolmogorov equation (BK equation) can be shown as follows:

The initial condition is

The boundary conditions are

The probability density

The numerical simulation results of the controlled system reliability function are shown in Figure

Reliability function of the system.

Probability density of first-passage time.

From Figures

The reliability function

The system’s reliability is obviously improved when the control force increases, which means that the control strategy is effective.

The probability density of first-passage time increases with time. First passage means the system leaves the safe area, which causes the system’s instability.

First-passage time can be delayed through optimal control, which also means that the optimal control can enhance the system’s reliability.

The experimental object is a 1:26 Chinese “Wing Loong” UAV model. The model is shown in Figure

Firstly, we measure the vibration of the wingtip by vibration sensors and obtained the displacement

Secondly, we calculate

Thirdly, we calculate the optimal control force

When the optimal control force

When the control stress

Finally, we obtained the intensity of control current

Model’s parameters.

Wing’s mass (g) | 12.8 | SMA plate’s mass (g) | 2.68 |

Wing’s length (cm) | 19.4 | SMA plate’s length (cm) | 12 |

Wing’s width (cm) | 1.24 | SMA plate’s width (cm) | 0.7 |

Wing’s thickness (mm) | 1.97 | SMA plate’s thickness (mm) | 0.4 |

1:14 Chinese “Wing Loong” UAV model.

The experimental results of the SMA composite wing subjected to stochastic airflow are shown in Figures

Wingtip’s dynamic response of an uncontrolled SMA composite wing when the mean wind velocity

Frequency domain of the wingtip’s dynamic response of an uncontrolled SMA composite wing when the mean wind velocity

Wingtip’s dynamic response of a controlled SMA composite wing when the mean wind velocity

Frequency domain of the wingtip’s dynamic response of a controlled SMA composite wing when the mean wind velocity

Voltage of the control current when the mean wind velocity

Wingtip’s dynamic response of an uncontrolled SMA composite wing when the mean wind velocity

Frequency domain of the wingtip’s dynamic response of an uncontrolled SMA composite wing when the mean wind velocity

Wingtip’s dynamic response of a controlled SMA composite wing when the mean wind velocity

Frequency domain of the wingtip’s dynamic response of a controlled SMA composite wing when the mean wind velocity

Voltage of the control current when the mean wind velocity

From the experimental results, we can see that

the system’s motion is the mixture of periodic motion and stochastic motion;

the system’s vibration increases with the mean wind velocity; when the wind velocity increases from 6 m/s to 8 m/s, the vibration amplitude of wingtip increases from 3.32 mm to 4.21 mm, which rises by 26%;

the vibration amplitude of an uncontrolled wingtip is 4.21 mm when the mean wind velocity

In this paper, a kind of high-aspect-ratio shape memory alloy (SMA) composite wing is proposed to reduce the wing’s fluttering. The nonlinear dynamic characteristics and optimal control of the SMA composite wings subjected to in-plane stochastic excitation are investigated where the great bending under the flight loads is considered. The stochastic stability of the system is analyzed, and the system’s response is obtained. The conditions of stochastic Hopf bifurcation are determined, and the probability density of the first-passage time is obtained. Finally, the optimal control strategy is proposed. Numerical simulation shows that the stability of the system varies with bifurcation parameters, and stochastic Hopf bifurcation occurs in the process; the reliability of the system is improved through optimal control, and the first-passage time is delayed. Finally, the effects of the control strategy are proved by experiments. The results of this paper are helpful for engineering applications of SMA.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors gratefully acknowledge the support of the Natural Science Foundation of China (NSFC) through Grants nos. 11272229, 11302144, and 11402168, the Ph.D. Programs Foundation of the Ministry of Education of China through Grant no. 20120032120006, and the Tianjin Research Program of Application Foundation and Advanced Technology through Grants nos. 13JCYBJC17900 and 14JCQNJC05300.