Nonlinear aeroelastic analysis of the folding fin with freeplay under thermal environment

Honn HE, Hong TANG, Kiping YU,*, Jinze LI, Ning YANG,Xiolei ZHANG

a Department of Astronautical Science and Mechanics, Harbin Institute of Technology, Harbin 150001, China

b Beijing Institute of Electronic System Engineering, Beijing 100854, China

KEYWORDS Aeroelasticity;Fixed-interface modal synthesis method;Folding fin;Freeplay;Thermal environment

Abstract The nonlinear aeroelastic behavior of a folding fin in supersonic flow is investigated in this paper. The finite element model of the fin is established and the deployable hinges are represented by three torsion springs with the freeplay nonlinearity. The aerodynamic grid point is assumed to be at the center of each aerodynamic box for simplicity. The aerodynamic governing equation is given by using the infinite plate spline method and the modified linear piston theory.An improved fixed-interface modal synthesis method, which can reduce the rigid connections at the interface,is developed to save the problem size and computation time.The uniform temperature field is applied to create the thermal environment. For the linear flutter analyses, the flutter speed increases first and then decreases with the rise of the hinge stiffness due to the change of the flutter coupling mechanism.For the nonlinear analyses,a larger freeplay angle results in a higher vibration divergent speed. Two different types of limit cycle oscillations and a multiperiodic motion are observed in the wide range of airspeed under the linear flutter boundary. The linear flutter speed shows a slight descend in the thermal environment, but the effect of the temperature on the vibration divergent speed is different under different hinge stiffnesses when there exists freeplay.

1. Introduction

The folding fin has been widely used to save storage space and to improve transport efficiency of missiles. Due to machining and manufacturing errors, the freeplay is unavoidable in the hinges of folding mechanisms which presents the characteristics of nonlinear stiffness. This kind of nonlinearity cannot be eliminated completely and it may result in significant effects on dynamic performance of the lifting surfaces. Firstly, the existence of freeplay plays a key role in complicated dynamic behaviors in the ground vibration test, such as jump, multiharmonics, and frequency shift, etc.1,2Secondly, the freeplay makes the nonlinear phenomena occur in some flight conditions, such as Limit Cycle Oscillation (LCO) and chaos.3-5They do not cause abrupt failure of a structure, but can lead the structure into fatigue problems and affect its manipulation stability.In practical engineering,the angle of freeplay is often reduced by improving machining and manufacturing accuracy,which is difficult and costly to be realized.Although the linear analysis could be employed,the flutter speed obtained is often too conservative when the freeplay is neglected. In other words, the structure designed based on the linear analysis is hard to meet the requirements of lightweight. Therefore, it is necessary to establish the accurate dynamic model of a folding fin and the modelling of connections considering the freeplay is essential.

In addition, the atmospheric fight at a supersonic speed causes severe aerodynamic heating effects. Generally, two main thermal effects should be considered:(A)the temperature revolution may change material properties; (B) the thermal expansion may introduce in-plane thermal stresses inside constrained structures, and alter the stiffness distribution.6Control surfaces such as folding fins are active components with weak stiffness of supersonic missiles, which are more susceptible to aerothermal effects. The stability of control surfaces directly determines the maneuverability and controllability.Therefore, it is of huge importance to understand the nonlinear aeroelastic responses under the thermal environment of a flight vehicle during the design stage.

The model of a folding wing/fin can be divided into twodimensional and three-dimensional model.7The first one is an ideal model assuming that the section remains the same in the direction of the span and the structure along the chord direction does not bend,which means the motion of the structure can be described by the following three generalized coordinates, the plunge of the rigid center, the pitch around the rigid center and the rotation of the control surface.This model can only simulate the bending and torsion vibration of the flight structures. Although it differs from the real wing/fin, it is easily used in the theoretical study of various nonlinear aeroelastic problems.8-15Bae et al.8performed linear and nonlinear aeroelastic analyses of a two-dimensional control fin with the root-locus method and the time-integration method respectively, and found that the asymmetric bilinear spring has the characteristics of both softening and hardening springs for LCO amplitude variations. Tang and Dowell9studied the aeroelastic responses of a typical wing with nonzero angle of attack excited by gust in subsonic flow, and found that the amplitude of LCO was sensitive to the initial pitch angle.Kholodar10revisited the nature of freeplay-induced oscillations, and physically explained the bifurcation airspeeds and oscillation frequencies. Liu and Ding11analytically studied the principal resonance response of a two-dimensional wing with freeplay in plunge via the average method. He et al.12addressed the flutter buzzing of a two-dimensional wing with freeplay between control surfaces in transonic flow by the describing function method, and observed the phenomenon of double limit cycles. Sazesh and Shams13studied the flutter of a two-dimensional wing with freeplay by using a stochastic method. Al-Mashhadani et al.14added the tab motion to the traditional wing-flap motion, and presented the analytical and experimental investigation of the wing section considering the freeplay at the tab.Vasconcellos et al.15assessed three representations for the nonlinear stiffness of a control surface,which are discontinuous, hyperbolic tangent and polynomial representations, and found that the first two representations could accurately predict the instability and nonlinear responses characteristics, while polynomial representation failed to predict chaotic motion observed in the experiments. In general,the two-dimensional model described in the above work can qualitatively study the effect of freeplay on aeroelasticity, but cannot provide quantitative predictions.

The three-dimensional model takes into account the elastic deformation of the structure and can have nonlinearity in multiple locations, which is more in line with actual structures.However, the computation scheme is more complex when it comes to the structure with more Degrees of Freedom(DOFs).Some scholars have semi-analytically established the differential equations of motion of the folding structures by different methods, such as the linear plate theory with component modal analysis,16the Euler-Bernoulli beam theory,17and the Ritz method with penalty function.18These methods operate well in simple and regular flight structures,but have difficulties in handling complex structures. Therefore, most scholars incorporate finite element method to conduct nonlinear simulation of the actual engineering structures with the freeplay.Wang et al.19studied the nonlinear responses of an aircraft wing and found that the amplitudes of the generalized displacement considering freeplay effect decreased compared with that not considering freeplay. The matrix scale of the threedimensional structure is huge, which makes the model reduction a necessity. However, it is more complex to reduce the nonlinear structures than linear structures. The fictitious mass method proposed by Karpel and Raveh20is a choice to simplify the nonlinear model.It imposes a virtual mass with great inertia on the DOF where the stiffness changes greatly, and obtains the same series of modes that can represent the physical coordinates of different nonlinear subdomains,thus transforming the nonlinear problem into the modal space for solution.Bae et al.21,22utilized this method to analyze the nonlinear aeroelasticity of a three-dimensional airfoil with freeplay between control surfaces. Limit cycles and chaos were observed when the flight speed was lower than the linear flutter speed. Then, they researched the nonlinear aeroelastic response of a deployable missile control fin and observed three different types of LCOs in the wide range of airspeed over linear flutter boundary.23Subsequently, they further performed numerical and experimental studies on the above control fin with the consideration of structural nonlinearity as well as the dynamics of an actuator.24Similarly, Lee and Chen25applied this method to folding wings with freeplay of morphing aircrafts.

The above fictitious mass method functions well in folding structures with lumped nonlinearity. However, Yang et al.26pointed out that the parameters of fictitious mass can only be selected within a recommended range, and the nonlinear stiffness cannot be directly expressed in the nonlinear governing equations. Alternatively, they established the aeroelastic equation of a folding fin by using the dynamic substructure method, and gave solutions in the time domain and frequency domain respectively.26,27The method first established the substructure model of inboard and outboard fins, and then connected them with nonlinear springs. The model reduction was accomplished by selecting the incomplete set of principle modes in the first coordinate transformation. Recently, Zhao and Hu28proposed a parameterized aeroelastic model of a folding wing by the substructure synthesis method. In this model, the aeroelastic equations for various folding angles can be automatically formed by simply changing the values of the parameterized variables. Hu et al.29studied the folding wing during the morphing process by combining the substructure method with the flexible multi-body dynamics approach.In order to accurately model the connections of two substructures, Wu et al.30,31further described the identification of the nonlinear stiffness at the hinge via the Hilbert transformation and the conditioned reverse path method, respectively.

Besides the nonlinearity,the aerothermal heating also plays a significant role in affecting aircraft’s aerodynamics. McNamara and Friedmann32provided a comprehensive survey of the current state of hypersonic aeroelastic and aerothermoelastic research, and an insight into important challenges and future direction.McNamara et al.33developed an aerothermoelastic methodology incorporating the heat transfer between the fluid and the structure to study the low-aspect-ratio wing.They found that the flutter boundary of the heated wing witnessed significant reductions.In order to reduce the extreme computation costs resulted from taking aerodynamic heating effects into consideration in the fluid-structural coupling analysis, Chen and Zhao34provided a method of thermal modal reconstruction to directly generate the mode shapes and frequencies.With this method,they also obtained the conclusion that the aerodynamic heating can result in earlier transition to flutter.Although many studies have been performed on nonlinear aeroelasticity and aerothermoelasticity, there is still a lack of research in nonlinear flutter analyses of the folding fin considering both the structural nonlinearity and thermal effects.

Our previous work35has established the structural model of the folding fin and detailly discussed the effects of the freeplay on the ground vibration responses. The present work aims to investigate the aeroelastic responses of it in the supersonic flow,and research the impact of freeplay on the nonlinear flutter under the thermal environment.Firstly,an improved fixedinterface modal synthesis method is developed to reduce the aeroelastic governing equations established by combining the modified linear piston theory with the infinite plate spline method. Secondly, the ground vibration test is conducted to verify the accuracy of the dynamic model. Finally, the influences caused by freeplay angle, hinge stiffness, and temperature on the linear and nonlinear aeroelastic characteristics of the folding fin are studied.

2. Computation scheme

2.1. Structural model

Fig. 1 Configuration of folding fin.

Fig. 1 shows the configuration of a folding fin which consists of an outboard fin, an inboard fin, a pin column, and a limit stop. The outboard and inboard fins are connected by the pin column.The bending moment is transferred by the contact faces of the outboard fin and the limit stop attached to the inboard fin. The freeplay can be altered via moving the limit stop. The material of the main body is aluminum, while the limit stop is made of steel.

The finite element model shown in Fig. 2 is established via the commercial software MSC.Patran/Nastran.36The pin column connecting the inboard and outboard fins is not modeled separately and the circular holes of the inserting pin column are fully filled with the same material as the main body. The limit stop is simplified as some mass points. In this way, the finite element model consists of only two parts, the inboard and outboard fins. The moment transfer of these two parts is simulated by three torsion springs in the Rxdirection, and the moment can be expressed as a function of the relative rotation angle.All the nodes at the surface of the circular holes are connected to the central node (hinge point) by Multi-Point Constraint(MPC). The RxDOFs of the adjacent hinge points are connected by torsion springs, while other five DOFs are rigidly connected. According to this modeling method, the two substructures are linear and the nonlinearity only exists in the connection between two parts. If the spring is linear,the whole model is also linear and modal characteristics can be obtained by the traditional mode superposition method.Similarly, the nonlinear spring leads to the nonlinear model and the traditional mode superposition method is no longer applicable. Instead, nonlinear methods in the time domain or frequency domain should be adopted to solve the problem.In this study, the spring is assumed to have a freeplay nonlinearity, which means there is no moment within the freeplay angle and a linear increase when the relative rotation angle reaches the freeplay, as shown in Fig. 3. The function of the freeplay nonlinearity can be expressed as

where klis the linear stiffness; γ is the relative shift; 2c is the freeplay angle.

2.2. Aerodynamic model

where x is the coordinate along the direction of air flow,t is the time,w is the downwash displacement;U is the airspeed;Ma is the Mach number;ρ∞is the air density;and the dynamic pressure is qd=ρ∞U2/2.

2.3. The interaction between the aerodynamic and structural model

2.3.1. Displacement interpolation

Fig. 2 Finite element model of folding fin.

Fig. 3 Freeplay nonlinearity.

In the aeroelastic analysis, the structural model and aerodynamic model need to be established separately. The finite element method is generally used in the structural modeling of aircrafts. Similarly, the lifting surface also needs to be discretized into many small panels in the aerodynamic analysis.Each panel is defined as an‘‘a(chǎn)erodynamic box”.The fin needs to be treated as a whole to calculate aerodynamic forces in order to take the interaction between two parts into consideration.The lifting surface of the fin is divided into 25 quadrilateral boxes in the spanwise direction and 20 quadrilateral boxes in the direction of chord,without considering the aerodynamic effect of the fin shaft. Structural and aerodynamic grids are determined based on different task requirements, so the two sets of grids are generally different. Aerodynamic forces act on aerodynamic grid points, but the differential equations of motion are established on structural nodes, so it is necessary to determine the interaction between the structural and aerodynamic model, which can be accomplished by the surface spline interpolation method. The lifting surface and the structural nodes used for interpolation are shown in Fig. 4.

The Infinite Plate Spline(IPS)method proposed by Harder and Desmarais38is an improvement of the two-dimensional interpolation method proposed by Rodden.39The IPS method is very suitable for flat structures. It requires the structural nodes and aerodynamic grid points to be in the same plane.Suppose there are m aerodynamic grid points:（xia，yia）， i=1，2，...，m, and n structural nodes:（xis，yis）， i=1，2，...，n. According to work,40the downwash displacement of aerodynamic grid points(wa)can be expressed as an interpolation function of the downwash displacement of structural nodes (ws) as follows:

Fig. 4 Lifting surface and structural nodes used for interpolation.

2.3.2. Force interpolation

Two ways will be illustrated here to calculate the aerodynamic force, and the main difference lies in the choice of the aerodynamic grid points. In Method 1, four nodes of each aerodynamic box are considered to be aerodynamic grid points,while Method 2 is much easier assuming that the aerodynamic grid points are located at the center of the aerodynamic boxes.

The isoperimetric formulation will be used in the first method to get the equivalent force at each node of aerodynamic boxes. Considering the four-node quadrilateral element in Fig. 5, the coordinate interpolations are

Fig. 5 Four-node two-dimensional element.

2.4. Model reduction

The fixed-interface modal synthesis method42is a traditional substructure method, which is suitable to obtain the differential equations of motion of linear structures. Yang et al.27extended the method to nonlinear structures, and carried out the nonlinear aeroelastic analysis of a folding wing with nonlinear stiffness at multiple DOFs in the direction of chord.The core of the method is two coordinate transformations.The incomplete principal modes are selected to construct the first coordinate transformation matrix, so as to achieve the modal reduction of the substructure.In the second coordinate transformation, substructures are connected via springs. In fact, only the DOFs of elastic connections at the interface are related to nonlinearity, but all DOFs including both the elastic DOFs (the rotation DOFs of each hinge point) and rigid DOFs (other 5 DOFs of each hinge point) are retained in this method. Sometimes, the number of rigid DOFs at the interface is more than that of retained principal modes of the substructure, which causes much computational burden.

This paper develops an improved fixed-interface modal synthesis method which can reduce rigid DOFs while retaining elastic DOFs at the interface. The method is first proposed in our previous work35and applied to the aeroelastic problems here. Different from the traditional method, the finite element modelling of the overall linear structure rather than the substructures will be carried out here.The overall linear structure means the structure excluding interfacial DOFs of elastic connection which have nonlinearities. The first coordinate transformation directly acts on the linear model to achieve the modal reduction and then elastic DOFs are connected in the second coordinate transformation by nonlinear springs.Therefore,the advantage of the proposed method compared with the traditional fixed-interface synthesis modal method lies in the smaller matrix size due to the reduction of rigid DOFs at the interface. The detailed description of the present method is shown as follows.

The differential equations of motion of the folding fin are written as

2.5. Linear flutter analysis

where s is a complex frequency parameter. The eigenvalues of Eq. (39) are calculated at a series of airspeeds. When the airspeed is low, real parts of all eigenvalues are negative and the motion is damped. When the airspeed is high, some eigenvalues have positive real parts and the motion is divergent.The airspeed at which the motion happens to be harmonic is the flutter speed.

2.6. Nonlinear flutter analysis

If there is a nonlinear term in Eq. (37), there are two methods that can be used to calculate the nonlinear aerodynamic responses. The first one is the harmonic balance method in the frequency domain, via which the nonlinear stiffness is linearized,and then the eigenvalue method is used to calculate the flutter speed (The flutter speed here is called nonlinear flutter speed). The method assumes that the response is simple harmonic,so to speak,an LCO in the aeroelastic analysis.If only the first harmonic is considered, the relative shift of the hinge can be written as follows

The second method is the numerical integration algorithm in the time domain, which is used to directly calculate the displacement, velocity or acceleration responses of Eq. (37). The airspeed at which the LCO appears is called the LCO onset speed, while the speed at which the motion becomes divergent is called the vibration divergent speed.

3. Presentation of results

3.1. Ground vibration test and free vibration analysis

The accuracy of the present method without considering the aerodynamic effects has been verified in our previous work35by comparing the frequencies calculated by the present method and software NASTRAN at different hinge stiffnesses. The ground vibration test is then implemented to update the finite element model. The modal parameters of the folding fin are obtained by the hammering method. As shown in Fig. 6, the accelerometer is attached to point 21, and the hammer is moved from point 1 to point 25 in turn. Using M+P Vibration Test System to collect and analyze the excitation and response signals, 25 Frequency Response Functions (FRFs)can be obtained.The experimental and synthetic driving point FRFs with five clear and sharp resonance peaks are shown in Fig. 7. After being identified by the Polyreference Time-Domain (PTD) method,43,44the first five frequencies are employed to update the finite element model. The unknown parameters in the finite element model are three torsion spring stiffnesses, which are determined by the model updating method based on the genetic algorithm.45Table 1 shows the comparison of the experimental and simulation results. Both the frequencies and mode shapes are in good agreement, so it can be concluded that the established finite element model can simulate the dynamic characteristics of the actual structure very well and can be used for the following aeroelastic analysis.

Fig.8 shows the variation trend of the first four frequencies and mode shapes with the hinge stiffness. It can be seen that(A) with the increase of the stiffness of the hinge, the first frequency approaches the second one which remains steady; (B)the first frequency remains stable when the stiffness increases to the vicinity of 6×108N.mm.rad-1; (C) the third and fourth modes cross each other near the stiffness of 2×108N.mm.rad-1; (D) the second and fourth modes are independent of the hinge stiffness, because the second mode shape is a torsion mode and the fourth mode is an in-plane mode,both of which do not have the relative shift at the hinge.

Fig. 6 Hammering modal test.

Fig. 7 FRF of driving point.

3.2. Aeroelastic analysis (linear case)

Linear case means c=0.000 rad. In the following numerical examples, unless otherwise stated, the air density is taken as ρ∞=1.225 kg/m3, the Mach number is set to be 4, and three torsion springs share the same stiffness. Firstly, the effect of the number of retained modes on the linear flutter speed is studied. Table 2 shows the variation of the flutter speed with the number of retained normal modes under two different torsion spring stiffnesses (Aerodynamic forces are calculated via Method 2). The flutter speeds obtained by retaining the first four and five modes are the same.In other words,five normal modes are enough to obtain the convergent flutter results.

Then,the aeroelastic modelling accuracy is verified by comparing the results of the present method and software ZAERO.46Table 3 compares the linear flutter speeds at different Mach numbers when the torsion spring stiffness is k=1010-N.mm.rad-1.The results obtained using Method 1,Method 2,and ZAERO are in good agreement with each other,thus verifying the correctness and accuracy of the present method considering aerodynamic effects. Moreover, using Method 1 or Method 2 makes the results nearly no difference.The equivalent aerodynamic forces are easily obtained in Method 2,so Method 2 will be adopted in the following analysis.

Fig. 9 shows the variation of the linear flutter speed with the hinge stiffness.The flutter speed increases initially and then decreases with the increment of the stiffness, because the flutter mechanism changes at the critical stiffness(7.5×108N.mm.rad-1).The flutter on the left side of the critical stiffness is arising from the coupling of the second and third modes, while the flutter on the right side of the critical stiffness is arising from the coupling of the first and second modes. Particularly, the variation of the lowest three frequencies with the airspeeds at a small and large torsion stiffnesses is shown in Fig.10.It can also be seen from Fig.8 that when the stiffness is lower than the critical stiffness, the second frequency is closer to the third frequency, thus the second and third modes are more easily coupled in the airflow. However,when the stiffness is higher than the critical stiffness, the second frequency is closer to the first frequency making the first and second modes more likely to be coupled.

3.3. Aeroelastic analysis (nonlinear case)

The harmonic balance method in the frequency domain is adopted first to qualitatively study the aeroelastic responsecharacteristics of the structure.The nonlinear connection stiffness alters with the change of the generalized coordinates.Fig.11 shows the relationship between the equivalent stiffness calculated by Eq. (42)and the limit cycle amplitude. The stiffness ratio, which means the equivalent stiffness is divided by the linear stiffness, is zero before the vibration amplitude reaches the freeplay angle, but when the amplitude is larger than that, the stiffness ratio gradually increases, tending to 1.

Table 1 The first four modes obtained from experiments and simulation.

Fig. 8 Variation of the first four frequencies and mode shapes with hinge stiffness.

Table 2 Linear flutter speeds under different numbers of retained modes.

In the following aeroelastic computation, unless otherwise stated, the undetermined parameters in Eq. (1) are assumed as c=0.001 rad and k=108N.mm.rad-1. All the initial disturbances of three torsion springs are set to be 0.002 rad.It is necessary to assume the amplitude of the aeroelastic response first when using the harmonic balance method to obtain the equivalent stiffness of the torsion spring.The amplitudes of three springs are assumed to be the same in the research,although it is unrealistic.Fig.12 shows the nonlinear flutter speeds under different limit cycle amplitudes. With the increase of the limit cycle amplitude, nonlinear flutter speed increases firstly and then decreases. In the positive part of the curve slope,the larger amplitude corresponds to the larger nonlinear flutter speed, so the LCO is stable. However, in the negative part, the larger amplitude corresponds to the smaller flutter speed, which means the LCO is unstable and the structure will be divergent after a disturbance.

Table 4 gives the limit cycle amplitudes of three hinges at different airspeeds calculated by the Bathe two substep implicit composite algorithm in the time domain.47,48It can be found that the leading hinge has the largest amplitude while the middle hinge has the smallest amplitude when the LCO is stable.With the increase of the airspeed, the amplitudes of three hinges increase gradually. The amplitudes in Table 4 can be used to obtain the nonlinear flutter speed via the harmonic balance method. Fig. 13 compares the results in the frequencydomain and the time domain at each case of Table 4. The results are basically the same, which is a mutual verification of two methods.

Table 3 Comparison between present method and ZAERO.

Fig. 9 Variation of linear flutter speed with hinge stiffness.

Different airspeeds will cause different aeroelastic responses of the structure.The Bathe algorithm is used again to calculate the angular displacement responses of the trailing hinge. Five typical forms of motion can be observed in the wide range of airspeeds:(A)decay motion;(B)LCO 1,one-period limit cycle oscillation;(C)LCO 3,three-period limit cycle oscillation;(D)multiperiodic motion;(E)divergent motion.Fig.14 shows the amplitude variation of the trailing hinge with airspeeds.Decay- LCO 3 - decay - LCO 3 - multiperiodic motion - LCO 3 -LCO 1 - multiperiodic motion - LCO 1 occur successively with the increase of airspeed. The amplitude of LCO 3 varies little, but the amplitude of LCO 1 increases rapidly with the increase of airspeed. Figs. 15-17 show the angular displacement responses of the trailing hinge in both the time domain and the frequency domain at different airspeeds. The phase graph of Fig.15(a)has three closed orbits,and there are three resonance peaks in Fig. 15(b), which indicates that LCO 3 motion has been induced under the airspeed of 400 m.s-1.The phase graph of Fig. 16(a) has multiple orbits, and Fig. 16(b) has multiple resonance peaks, which indicates that the airspeed of 635.5 m.s-1has caused the multiperiodic motion of the structure. The phase graph of Fig. 17(a) has one closed orbit, and there is only one resonance peak in Fig.17(b),which indicates that the LCO 1 motion is generated by the airspeed of 800 m.s-1.

Table 5 gives the aeroelastic results of the folding fin with different freeplay angles under two torsion stiffnesses. There is an LCO when the hinge stiffness is large, while no LCO is observed when the hinge stiffness is small.The vibration divergent speed increases with the increase of the freeplay angle,but the LCO onset speed increases first and then decreases. In other words, the increase of the freeplay angle may cause the structure to enter the long-term constant-amplitude limit cycle earlier,thus endangering the fatigue stability and attitude control accuracy of the structure.

3.4. Aeroelastic analysis (thermal environment)

Fig. 10 Variation of the lowest three frequencies with airspeed.

Fig. 11 Equivalent stiffness of nonlinear spring.

Fig. 12 Variation of nonlinear flutter speed with limit cycle amplitude.

Fig. 13 Nonlinear flutter speeds obtained via time-domain and frequency-domain methods.

Fig. 14 Variation of amplitude of trailing hinge with airspeed.

The atmospheric flight at a high speed can cause severe aerodynamic heating on the structure. Thermal environment affects the aeroelastic characteristics through modifying structural modes in two ways: altering material properties (mainly the elastic modulus) and inducing thermal stress. The following analysis compares the effects of two factors on thermal modes of the root-fixed folding fin and the effects of two factors on the first four frequencies are considered alone respectively in Table 6. Firstly, assuming the elastic modulus of aluminum to be the one of 300°C, the first four frequencies are calculated without considering the effect of thermal stress by NASTRAN using SOL 106 (Normal modes). Then, the 300°C uniform temperature field is created but the elastic modulus is assumed to be the one at the room temperature.The thermal stress calculated by SOL 101 (Linear static) is regarded as the initial stress for the next modal computation in SOL 106. From Table 6, we can know that the effect of the thermal stress on the structure is very small, and the frequency is mainly influenced by the change of the elastic modulus with the temperature. It is because the thermal stress can be released from free boundaries and only little exists at the root of the fin. Finally, the thermal modal experiment is conducted. As shown in Fig. 18, the 300°C uniform temperature field is created by double-sided quartz lamp arrays and the laser vibrometer(PolyTech)is utilized to measure the displacement signals under the base excitation. The experimentalresults corroborate the conclusion that the impact of the thermal stress on the dynamics of the fin model can be neglected in the following thermal flutter analysis.

Table 4 Limit cycle responses of different airspeeds at freeplay of 0.001 rad.

Fig. 15 Aeroelastic response at air speed of 400 m.s-1.

Fig. 16 Aeroelastic response at air speed of 635.5 m.s-1.

Fig. 17 Aeroelastic response at air speed of 800 m.s-1.

Based on the above thermal modal analysis,only the elastic modulus needs to be changed according to the predetermined temperature in the computation of the thermal flutter, so the method in the room temperature illustrated in Section 2.4 is still applicable here. Actually, after experiencing a sustained flight in a fixed state,the vehicle will eventually reach the thermodynamic equilibrium.Therefore,it is reasonable to perform the aeroelastic analysis at each equilibrium state. In this research, the folding fin is assumed to operate at a uniform steady temperature field. Table 7 shows the effect of the temperature on the nonlinear flutter when the freeplay angle is0.001 rad as well as on the linear flutter under different spring stiffnesses. It can be found that the temperature has less than 5% effect on the linear flutter speed. However, the results of the nonlinear flutter are quite different from those of the linear flutter.When the spring stiffness is small,the LCO onset speed increases and the vibration divergent speed decreases at a higher temperature. However, when the stiffness is large, no LCO occurs and there is a slightly rise of the vibration divergent speed at a high temperature. Therefore, special attention should be paid to the thermal environment in the presence of freeplay.

Table 5 Nonlinear aeroelastic responses under different freeplays at stiffness of 108N.mm.rad-1.

Table 6 Frequencies obtained considering variation of elastic modulus and thermal stress respectively (Hz).

Table 7 Effect of temperature on linear and nonlinear aeroelastic responses.

Fig. 18 Configuration of thermal modal experiment.

4. Conclusions

The linear and nonlinear aeroelastic characteristics of the folding fin are investigated by both the time-domain and frequency-domain methods. The connections are represented by three torsion springs with the freeplay nonlinearity. The governing equations considering thermal effects are obtained by combining the infinite plate spline method and the modified linear piston theory. The established finite element model is reduced by the developed model reduction method, which can reduce the inelastic DOFs at the connection compared with the traditional fixed-interface modal synthesis method.The free vibration analysis and linear flutter analysis support the validity of the present method. The main conclusions are highlighted as follows.

(1) The reduction of the hinge stiffness will change the flutter mechanism from first- and second-order modal coupling to second- and third-order modal coupling, thus making the linear flutter speed increase first and then decrease.

(2) The vibration divergent speed of the fin with freeplay is larger than the linear flutter speed,so the flutter computation by the linear analysis is conservative. If there is freeplay, LCO will occur during the flight under linear flutter speed, but LCO only exists when the hinge stiffness is small.

(3) Two different types of LCOs are observed in the wide range of airspeed under linear flutter speed. The amplitude of LCO 1 increases, while the amplitude of LCO 3 remains stable with the rise of the airspeed.

(4) The thermal environment has different impacts on the linear and nonlinear aeroelasticity.The heat will slightly decrease the linear flutter speed of the fin, basically less than 5%. However, the effect of heat on the vibration divergent speed of the nonlinear structure is concerned with the hinge stiffness.

In the future research, the wind tunnel test will be carried out to verify the numerical results presented in this paper. In order to simulate the actual aerodynamic force, the fin will be installed on a missile body, and the boundary condition of the finite element model will be updated accordingly.Apart from the accelerometer to measure the vibrational response,the high-speed camera will also be deployed to provide a visual record of the model behaviour.However,it is hard to integrate the quartz lamp arrays into the wind tunnel directly to analyze the thermal aeroelasticity,so the combination of these two systems will be further investigated from the practical view.