Structural Nonlinear Flutter Characteristics Analysis for an Actuator-fin System with Dynamic Stiffness

Chinese Journal of Aeronautics 24 (2011) 590-599

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Structural Nonlinear Flutter Characteristics Analysis for an Actuator-fin System with Dynamic Stiffness YANG Ning, WU Zhigang, YANG Chao* School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China Received 26 November 2010; revised 1 March 2011; accepted 18 April 2011

Abstract The flutter characteristics of an actuator-fin system are investigated with structural nonlinearity and dynamic stiffness of the electric motor. The component mode substitution method is used to establish the nonlinear governing equations in time domain and frequency domain based on the fundamental dynamic equations of the electric motor and decelerator. The existing describing function method and a proposed iterative method are used to obtain the flutter characteristics containing preload freeplay nonlinearity when the control command is zero. A comparison between the results of frequency domain and those of time domain is studied. Simulations are carried out when the control command is not zero and further analysis is conducted when the freeplay angle is changed. The results show that structural nonlinearity and dynamic stiffness have a significant influence on the flutter characteristics. Limit cycle oscillations (LCOs) are observed within linear flutter boundary. The response of the actuator-fin system is related to the initial disturbance. In the nonlinear condition, the amplitude of the control command has an influence on the flutter characteristics. Keywords: aeroelasticity; flutter; actuators; dynamic stiffness; structural nonlinearity; component mode substitution method; describing functions

1. Introduction1 When subject to high velocity flight condition, flexible structures may experience flutter, a dynamic aeroelastic phenomenon, which causes the failure of flight vehicles. In many cases, flutter is mainly affected by structural and aerodynamic properties of control fins, and the dynamic characteristics of actuators which are important parts of control systems. In order to counterbalance external disturbance and satisfy the aeroelastic stability, actuators must have suitable stiffness properties. That stiffness is called *Corresponding author. Tel.: +86-10-82313376. E-mail address: [email protected] Foundation items: National Natural Science Foundation of China (90716006, 10902006); Research Fund for the Doctoral Program of Higher Education of China (20091102110015) 1000-9361/$ - see front matter © 2011 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(11)60069-1

dynamic stiffness, since it is given as a function of the Laplace variable s or frequency [1]. Several investigators have performed flutter analysis of control fins with dynamic stiffness being considered. Yehezkely and Karpel [2] analyzed a missile having pneumatic fin actuators, and they showed that the missile maneuver commands had a strong influence on the flutter speed. Paek and Lee [3] studied the flutter characteristics of a fin considered dynamic stiffness and investigated the effect of the sweep angle. They presented an iterative method for flutter analysis with the influence of dynamic stiffness in frequency domain. Yang, et al. [4] carried out wind-tunnel flutter experiments, and predicted the flutter boundary. In most current flutter calculations of engineering practice, dynamic properties of actuators are neglected which may make a difference with actual conditions. The analysis mentioned mainly concerns the dynamic characteristics of actuators. But most actual mechanisms may include structural nonlinearities, such

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as freeplay and backlash which make the results calculated under linear assumption differ from real phenomena. The structural nonlinearity should be considered in the calculations of some structures. Much research on aeroelasticity with structural nonlinearity has been conducted. Lee, et al. [5] studied a flexible control wing with root pitch freeplay, and they found that the limit cycle oscillation (LCO) and chaotic motions were observed in different gap/amplitude ratios. Furthermore, they carried out experiments and calculations of actual fins [6-7]. Ref. [8] analyzed the influence of the nonlinearity in the folding hinge on the flutter boundary of the CF-18 aircraft. Alighanbari [9] performed an aeroelastic response study of a three degree-of-freedom (DOF) airfoil-aileron, and they predicted the bifurcation points. Refs. [10]-[11] dealt with studies of a folding wing configuration with concentrated structural nonlinearity, and results showed the transient response and LCO motion. Those studies can provide references for related analyses. As previous introduction shows, the calculation of the actuator-fin system should consider both the dynamics of the actuator and the structural nonlinearity. Shin, et al. [12] performed a flutter analysis with consideration of structural nonlinearity as well as the dynamics of an actuator and they found that the flutter boundary increased as compared with the case without considering dynamic stiffness. But in Shin’s research, the maneuver commands are neglected. In actual flight condition, flight loads are applied to the control fins, and the preload freeplay nonlinearity agrees with practical conditions. In the present study, the flutter properties of an actuator-fin system with structural nonlinearity as well as dynamic stiffness of an electric actuator are investigated. The component mode substitution method [10] is used to establish governing equations with structural nonlinearity in frequency domain and time domain. The existing describing function method and a proposed iterative method are used for the flutter analysis. The results are compared with those of simulations in time domain. The influence of the maneuver commands on the flutter characteristics is investigated. 2. Computation Scheme 2.1. Governing equations in frequency domain As general application, in the present study the free-body diagram of an actuator-fin system is shown in Fig. 1. The system consists of an electric motor, two gears, load links and a control fin, where ϕ is the plunging angle of the control fin and α the pitching angle of the fin. The model of the single motor is shown in Fig. 2. The dynamic equation of the single motor is established based on the voltage equation of coils (Eq. (1)), the moment equation (Eq. (2)) and the dynamic equation of rotors (Eq. (3)).

Fig. 1

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Schematic of actuator-fin system.

Fig. 2

Model of single motor.

di = ua − Ce θ1 − ki i − ka θ1 dt Tm = ikm  J θ = T − b θ − M

iR + L

m 1

m

m 1

1

(1) (2) (3)

where i is the electric current, R the resistance of the motor, L inductance of the motor, ua the control voltage which is 0 V in most steady flight conditions, Ce the coefficient of induction electromotive force, θ1 the rotational displacement of the rotor axis, ki the feedback coefficient of the current loop, ka the feedback coefficient of the position, Tm the torque induced by the electric motor, km the coefficient of the moment, Jm the inertia of the rotor, bm the damped coefficient of the motor, and M1 the transmission torque from motor to first gear. Under most steady flight conditions, the position of the control fin is stationary. Assuming that the control voltage is 0 V and solving Eqs. (1)-(3), the transfer function of the electric motor can be represented as M 1 / θ1 = −{J m Ls 3 + [( R + ki ) J m + bm L]s 2 + [Ce km + bm ( R + ki )]s + km ka } [ Ls + ( R + ki )] (4)

Eq. (4) can be represented as a convenient form F(s). Applying iω to the transfer function F(s), the dynamic stiffness F(ω) in frequency domain is obtained. Fig. 3 shows the free-body diagram of transmission systems which consists of a decelerator and load links. The dynamic equation set can be written as ⎧ M 1 − J11θ&&1 = F1r1 , n1θ 2 = θ1 ⎪ M2 = Mt ⎪ M t = −ke (θ 2 − α ), ⎨ ⎪n F r + M = J θ&& , M 1 = F (ω ) 2 12 2 ⎪ 1 11 θ1 ⎩

(5)

where J11 is the inertia of the first gear, F1 the internal

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force between two gears, r1 the radius of the first gear, n1 the transmission ratio of the decelerator, θ 2 the rotational displacement of the second gear, Mt the torque on the root of the rudder, M2 the transmission torque from the second gear to load links, ke the connected stiffness between the second gear and load links, and J12 the inertia of the second gear.

Fig. 3 Diagram of transmission systems.

Considering the second gear to be a dynamic system, the equation of motion about parameter θ2 can be obtained: ( J12 + n12 J11 )θ&&2 − n12 F (ω )θ 2 = M 2

(6)

In order to join the actuator and the fin, the component mode substitution method is used. Based on the theory of the method [13], the elastic modes Φe with the root being fixed, the unit radian bending mode Φϕ and the unit radian pitching mode Φα are required. The vibration equation of the fin under mode coordinates is shown in Eq. (7). ⎡ mee ⎢ ⎢ mϕ e ⎢ mα e ⎣

⎡ kee ⎢ 0 ⎢ ⎣⎢ 0

0 kϕ 0

meϕ mϕϕ mαϕ

meα ⎤ ⎡ q&& ⎤ ⎥ mϕα ⎥ ⎢⎢ϕ&& ⎥⎥ + mαα ⎥⎦ ⎢⎣α&& ⎥⎦

0 ⎤ ⎡q ⎤ ⎡q ⎤ 1 ⎥ ⎢ ⎥ 2 ⎢ ⎥ 0 ⎥ ⎢ϕ ⎥ = ρV Q ⎢ϕ ⎥ 2 ⎢⎣α ⎥⎦ kα ⎦⎥ ⎢⎣α ⎥⎦

meϕ mϕϕ mαϕ

meα ⎤ ⎡ q&& ⎤ ⎡ kee ⎥ mϕα ⎥ ⎢⎢ϕ&& ⎥⎥ + ⎢⎢ 0 mαα ⎥⎦ ⎢⎣α&& ⎥⎦ ⎢⎣ 0

0 0⎤ ⎡ q ⎤ 0 0 ⎥⎥ ⎢⎢ϕ ⎥⎥ = 0 0 ⎥⎦ ⎢⎣α ⎥⎦

⎡q ⎤ ⎡ 0 ⎤ 1 ρV 2 Q ⎢⎢ϕ ⎥⎥ + ⎢⎢ − M ϕ ⎥⎥ (8) 2 ⎢⎣α ⎥⎦ ⎢⎣ − M t ⎥⎦ where −Mt is the moment applied to the axis of the fin, and Mt=kαα in linear condition. −Mϕ is the bending moment applied to the root chord of the fin, and Mϕ = kϕ ϕ in linear condition. Integrating Eq. (6) and Eq. (8), the vibration of the integrated actuator-fin system can be expressed as ⎡θ&&2 ⎤ ⎡θ 2 ⎤ 2 ⎡ J12 + n1 J11 0 ⎤ ⎢ q&& ⎥ ⎡ − n12 F (ω ) 0 ⎤ ⎢ q ⎥ ⎢ ⎥⎢ ⎥+⎢ ⎥⎢ ⎥ = ⎢ ⎥ 0 0 M K ⎦⎥ ⎢ ϕ ⎥ ⎣⎢ ⎦⎥ ⎢ ϕ&& ⎥ ⎣⎢ ⎢ ⎥ ⎢⎣ α&& ⎥⎦ ⎣α ⎦ ⎡θ 2 ⎤ ⎡ M 2 ⎤ ⎢ ⎥ ⎡0 0 ⎤ ⎢ q ⎥ ⎢ 0 ⎥ 1 ⎢ ⎥ + (9) ρV 2 ⎢ ⎥ ⎢−Mϕ ⎥ 2 ⎣⎢ 0 Q ⎦⎥ ⎢ ϕ ⎥ ⎢ ⎢ ⎥ −M ⎥ t ⎥ ⎣ α ⎦ ⎢⎣ ⎦ where M and K are the mass matrix and the stiffness matrix of the control fin. Eq. (9) can be represented as a simple form: % +F % && + KX % = QX MX (10) As the theory of traditional component mode substitution method, the connection condition is the equality of the coordinates of the interface. But if there is a connection area between two structures, the coordinates are not equal. Consider the transmission error δ to be a generalized coordinate, and the relationship can be represented as follows: θ2 = α + δ (11)

where δ is the error between the axes of the gear and the fin. The second coordinate transformation matrix can be written as (7)

where mee is the modal mass corresponding to the elastic modal Φe, mϕϕ the modal mass corresponding to the bending mode Φϕ , mαα the modal mass corresponding to the pitching mode Φα. meϕ , meα and mϕα are the coupled modal masses. kee is the modal stiffness corresponding to the elastic modal Φe, kϕ and kα are the physical plunging and pitching stiffness respectively. ρ is the air density of the flight altitude, V the flight speed, Q the generalized aerodynamic influence coefficient matrix which is obtained by the doublet-lattice method (DLM) [14-15], q the generalized coordinates of elastic modes. The vibration equation can be written as ⎡ mee ⎢ ⎢ mϕ e ⎢ mα e ⎣

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⎡θ 2 ⎤ ⎡1 ⎢ q ⎥ ⎢0 ⎢ ⎥=⎢ ⎢ ϕ ⎥ ⎢0 ⎢ ⎥ ⎢ ⎣ α ⎦ ⎣0

0 0 1 ⎤ ⎡δ ⎤ I 0 0 ⎥⎥ ⎢⎢ q ⎥⎥ = TP 0 1 0 ⎥ ⎢ϕ ⎥ ⎥⎢ ⎥ 0 0 1 ⎦ ⎣α ⎦

(12)

where T is the second transformation matrix of the general coordinates, P the second general coordinates. Substituting Eq. (12) into Eq. (9) and premultiplicated by TT, Eq. (9) can be changed into ⎡ − n12 F (ω ) 0 ⎢ kee 0 % && + ⎢ T T MTP ⎢ 0 0 ⎢ 2 ⎢⎣ − n1 F (ω ) 0

0 −n12 F (ω) ⎤ ⎥ 0 0 ⎥P = ⎥ 0 0 ⎥ 0 −n12 F (ω ) ⎥⎦

⎡ M2 ⎤ ⎢ ⎥ 0 1 % ⎥ ρV 2T T QTP +⎢ ⎢ −Mϕ ⎥ 2 ⎢ ⎥ ⎣M 2 − M t ⎦

(13)

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The item M2−Mt is zero, because M2 and −Mt are mutual acting force. And the item M2 can be represented as –f(δ ). In linear condition, f(δ )=kδ . At nonlinear case, the stiffness ke is a function of generalized coordinate δ . In the present study, the moment in the bending direction is linear. When the nonlinear moment can be denoted as the multiplication of the nonlinear stiffness and the generalized coordinate, the nonlinear vibration equation can be represented as ⎡ − n12 F (ω) + ke (δ ) 0 ⎢ 0 kee % && + ⎢ T T MTP ⎢ 0 0 ⎢ 2 ⎢⎣ −n1 F (ω) 0 1 % ρV 2T T QTP 2

0 0 kϕ 0

− n12 F (ω) ⎤ ⎥ 0 ⎥ P= ⎥ 0 ⎥ −n12 F (ω) ⎥⎦

(14)

where ke(δ ) is the nonlinear stiffness between the second gear and load links. In the present study, a preload freeplay structural nonlinearity is assumed in the load links shown in Figs. 4-5, in which Et is the freeplay angle, Ep the preload angle, k the stiffness of the linear part, α0 the static offset angle, and Z the vibration amplitude.

Fig. 4

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M2 and the maneuver command ua to be the input and the rotational displacement of the second gear θ2 to be the output, the state-space equation of the electric motor and the decelerator is written as ⎧ ⎡θ&2 ⎤ ⎪d ⎢ ⎥ ⎪ ⎢θ 2 ⎥ = ⎪ dx ⎢ i ⎥ ⎪ ⎣ ⎦ ⎪ ⎡ ⎤ n12 bm n1km ⎪ ⎢− 0 ⎥ 2 2 ⎪ ⎢ n1 ( J m + J11 ) + J12 n1 ( J m + J11 ) + J12 ⎥ ⎪ ⎢ ⎥⋅ 1 0 0 ⎪ ⎢ ⎥ ⎪ ⎢ Ce n1 n1ka R + ki ⎥ − − − ⎪ ⎢ ⎥ L L L ⎪ ⎣ ⎦ ⎨ 1 ⎪ ⎡ ⎤ ⎪ ⎡θ& ⎤ ⎢ n 2 ( J + J ) + J 0 ⎥ 11 12 ⎪ ⎢ 2⎥ ⎢ 1 m ⎥ ⎡M ⎤ ⎪ ⎢θ 2 ⎥ + ⎢ 0 0⎥⎢ 2⎥ ⎪ ⎢ ⎥ ⎢ ⎥ u 1⎥⎣ a ⎦ ⎪ ⎣i⎦ ⎢ 0 ⎪ ⎢ L ⎥⎦ ⎣ ⎪ ⎪ ⎡θ&2 ⎤ ⎪ ⎡M 2 ⎤ ⎢ ⎥ ⎪θ 2 = [0 1 0] ⎢θ 2 ⎥ + [0 0] ⎢ ⎥ ⎣ ua ⎦ ⎪ ⎢i⎥ ⎣ ⎦ ⎩ (16)

In order to establish the state-space equation of the fin, minimum state rational function approximations of unsteady aerodynamic force coefficient matrix are used [16]. The form of this method in Laplace domain is written as

Position of nonlinearity

A( s ) = Q0 + Q1s + Q2 s 2

(17)

where Qi (i=0, 1, 2) is calculated from a least square fit. The state-space equation of the rudder is established as a similar process. The input and the output of the formulas are the transmission torque Mt and the pitching angle α respectively. Eq. (7) can be transformed into a state-space equation in time domain as

Fig. 5

Schematic of freeplay nonlinearity.

The preload freeplay f(δ ) is expressed as ⎧ kδ δ ≤ Ep ⎪ f (δ ) = ⎨kEp Ep < δ < Ep + Et ⎪k (δ − E ) δ ≥ Ep + E t t ⎩

(15)

2.2. Governing equations in time domain The actuator-fin system can be separated as the actuator and the control fin which are connected by the load links. The same method is used to establish the state-space equations. Assuming the transmission torque

⎧ ⎡ ⎡0⎤ ⎤ ⎪ ⎢ −1 ⎢ ⎥ ⎥ −1 −1 M − ⎡ ⎤ −M C −M K ⎪d ⎢ ⎢0⎥ ⎥ M ⎥x+⎢ ⎪ x=⎢ ⎢⎣1 ⎥⎦ ⎥ t 0 ⎥⎦ ⎨ dx ⎢⎣ I 2×2 ⎢ ⎥ ⎪ 0 ⎢ ⎥⎦ ⎣ ⎪ ⎪α = [0 0 0 0 0 1] x ⎩

(18) where 1 2 1 ⎧ ⎪ M = M − 2 ρ b Q2 , C = − 2 ρVbQ1 ⎪ ⎨ k 0⎤ 1 2 ⎪K = ⎡ y ⎢ ⎥ − ρV Q0 , x = [q& ϕ& α& ⎪⎩ ⎣ 0 0⎦ 2

q ϕ α ]T

Finally, the state-space equations Eq. (16) and

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Eq. (18) are coupled based on the relationship which is shown as M 2 = − M t = ke (δ )δ

(19)

Combining Eq. (16) and Eqs. (18)-(19), a simulation framework is established (see Fig. 6).

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The equivalent linear equation is expressed as ⎡ −n12 F (ω ) + keq ⎢ 0 T % && ⎢ T MTP + ⎢ 0 ⎢ 2 ⎢⎣ − n1 F (ω )

0 0

0 kee 0

kϕ

0

0

−n12 F (ω ) ⎤ ⎥ 0 ⎥P = ⎥ 0 ⎥ −n12 F (ω ) ⎥⎦

1 % ρV 2T T QTP 2

(22)

The value of parameter Z decides the stiffness and flutter speed. The flutter results are obtained by p-k method in frequency domain. According to the curve Z-Vf (where Vf is the flutter speed), the qualitative results can be obtained. 2.3.2. Iterative method Fig. 6

Simulated flowchart of actuator-rudder system.

The 4th-order Runge-Kutta algorithm is used for the integration. 2.3. Methods of solution in frequency domain

Referring to the frequency response of Eq. (4), the phase and amplitude of the stiffness are changed along with the change of frequency shown in Fig. 7. An iterative method is developed to solve the vibration equation in frequency domain [17].

2.3.1. Describing function method According to the describing function method, the vibration amplitude Z of the transmission error δ should be given. There is a static offset angle α0, because the restoring force is not odd symmetric. When the vibration amplitude Z is given, the static offset angle α0 can be obtained. Then the equivalent stiffness is determined by Eq. (20). f (δ ) = keqδ = Z < Ep − α 0 ⎧ kδ ⎪ ⎪k ⎛ 1 + ϕ2 + 1 sin 2ϕ ⎞ δ Ep − α 0 ≤ Z ≤ Et + Ep − α 0 ⎪ ⎜ 2⎟ ⎨ ⎝ 2 π 2π ⎠ ⎪ ⎪k ⎛ 1 + ϕ2 − ϕ1 + sin 2ϕ2 − sin 2ϕ1 ⎞ δ Z > E + E − α t p 0 ⎟ ⎪⎩ ⎝⎜ π 2π ⎠ (20)

where keq is the equivalent linear stiffness, and E + Ep − α 0 Ep − α 0 ϕ1 = arcsin t , ϕ2 = arcsin Z Z Because the work of restoring force in a cycle is zero, the static offset angle α0 can be written as ⎧Z ⎛ π ⎞ ⎪ π ⎜ cos ϕ2 + ϕ2 sin ϕ2 − 2 sin ϕ2 ⎟ ⎠ ⎪ ⎝ ⎪ Ep ≤ Z < E p + E t ⎪ α0 = ⎨ Z ⎛ π ⎪ π ⎜ cos ϕ2 + ϕ 2 sin ϕ 2 + 2 sin ϕ1 − cos ϕ1 − ⎪ ⎝ ⎪ π ⎞ Z ≥ Ep + E t ⎪ ϕ1 sin ϕ1 − sin ϕ 2 ⎟ 2 ⎠ ⎩ (21)

Fig. 7

Frequency response of motor.

The iterative procedure is as follows. (1) Assume the oscillation amplitude and get the equivalent linear stiffness. (2) Give a series of frequencies and calculate the corresponding flutter speeds and frequencies. (3) Draw the curve which is about flutter frequencies along with the given frequencies. When the flutter frequency is equal to the given frequency, the result is the true frequency. (4) Draw the curve of flutter speeds changing with the computing frequencies. When the frequency is true vibration frequency, the speed is the true flutter speed. (5) Change the oscillation amplitude and repeat Step (1). The influence of the frequency is only on the dynamic characteristics of the electric motor. Therefore, the iterative process does not affect the describing function method.

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3. Results and Discussion In the present study, the model used is a low-speed proof control fin which is shown in Fig. 8. The linear flutter speed is 46.4 m/s.

Fig. 8

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tion amplitude increases. In contrast, LCOs occur when the initial disturbed angle is in the range B1-C1. When the disturbed angle is less than E1, the flutter speed is higher than D1. As a result, if the flight velocity is D1 and the disturbed angle is less than E1, the vibration converges. For the same reason, when the disturbed angle is between E1 and F1, the vibration amplitude increases. The closer the flight speed to the linear flutter boundary, the greater the vibration amplitude is. As shown in Fig. 10, when LCOs occur, the results in time domain agree with those in frequency domain. The results in time domain are shown from Figs. 11-14.

Geometry of control fin.

Based on the flowchart of Fig. 6, the simulated framework is established in MATLAB/Simulink® which is shown in Fig. 9. The calculations when neglecting and considering maneuver commands are conducted and the change of the preload angle is considered.

Fig. 9

Fig. 10

Flutter boundary calculated by discribing function.

Fig. 11

Responses of transmission error δ and pitch angle α (velocity: 44 m/s, disturbed angle: 0.57°).

Framework for flutter simulation.

The actuator-fin system parameters to be used are given in Table 1. Table 1

Values of parameters

Parameter

Value

Parameter

Value

n1

50

Ce

2.58×10−2

2.7

L/H

Jm/(kg·mm ) J11/(kg·mm2)

1.35 10

R/Ω

1.95×10−4 1.5×10−2 2.36

J12/(kg·mm2)

1 000

km/( kg·mm2·s−2·A)

20 000

bm 2

ki

3.1. Results when command is zero and the preload is fixed In this section, the maneuver command is 0°. The preload angle and the freeplay angle are assumed to be 0.2° and 0.5° respectively. The results in frequency domain and time domain are obtained (see Fig. 10). As shown in frequency domain, the flutter boundary changes obviously because of the freeplay. Within a range of flight velocities within the linear flutter boundary, LCOs are observed. When the initial disturbed angle is in the range A1-B1, LCOs do not occur because the flutter speed decreases when the oscilla-

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YANG Ning et al. / Chinese Journal of Aeronautics 24(2011) 590-599

Responses of transmission error δ and pitch angle α (velocity: 44 m/s, disturbed angle: 0.58°).

Fig. 14

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LCOs of transmission error δ and pitch angle α (velocity: 44 m/s, disturbed angle: 2.0°).

Based on the results of simulations in time domain, the prediction of flutter boundary in frequency domain is correct which can provide reference for simulations. Because of the existence of the preload angle, the average of the vibration angle is not zero, which agrees with the results in frequency domain. When the flight velocity increases, the initial disturbed angle which can make the system generate LCOs decreases. The results are shown in Table 2. Table 2 Minimum value of disturbed angle which can generate LCOs Velocity/(m·s−1)

40

41

42

43

44

45

Angle/ (°)

1.47

0.7

0.66

0.62

0.58

0.55

As shown in Table 2, the closer to the linear flutter boundary, the smaller the disturbed angle is needed to generate LCOs. 3.2. Results when command is zero and the preload is changed

Fig. 13

Responses of transmission error δ and pitch angle α (velocity: 44 m/s, disturbed angle: 2.0°).

In this section, the maneuver command is 0° and the freeplay angle is 0.5°. The influence of the preload angle on the flutter speed is investigated. Comparison of the condition when the preload angle is not 0° shows that the flutter speeds are smaller when neglecting preload (see Fig. 15). When the vibration

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amplitude is small and preload is neglected, the pitching stiffness is 0 kg·mm2·s−2·rad−1, which does not agree with actual flight conditions.

Fig. 17 Fig. 15

Flutter boundary when neglecting preload.

As shown in Fig. 16, because the preload angle increases, the flight speed and initial disturbed angle increase when LCOs occur. Within a range of speeds, the amplitude of LCO is smaller under a larger preload angle. Therefore, the preload angle has a suppressed influence on LCOs. When the speed increases to a certain value, the amplitudes of LCOs are equal in the conditions of different preload angles. In that case, effects of the preload angle are the same.

Fig. 16

Flutter boundaries at different preload angles.

3.3. Results when command is not zero In the above analysis, the maneuver command is assumed to be zero and the responses are calculated when the flight condition is not changed. But the control fin is used to change the flight condition according to the maneuver command. Therefore, the responses when the command is not zero should be analyzed. The response of the control fin is related to the maneuver command. At the same velocity, when faced great commands, the vibrations will be LCOs or divergence. The response boundary of different commands is shown in Fig. 17.

Relationship between maneuver commands, velocities and responses.

Due to the existence of the freeplay, different commands generate different responses. When the velocity is small, the fin deflects to the corresponding angle according to the command. When the velocity increases, the smaller command could control the fin to generate expectant response, but the greater command could make the fin generate LCO which may cause the failure. The value of the command required to generate LCO decreases along with the increase of the velocity. If the value of freeplay angle is changed, LCOs boundary is changed. As shown in Fig. 17, the speed when LCOs occur at freeplay angle of 0.2° is greater than that of at 0.5°. When the flight speed is between A2 and B2, the command required to generate LCO in the case of freeplay angle of 0.2° is greater than that of 0.5°. When the flight speed is greater than B2, the influence of the two conditions is the same. When the freeplay angle is 0.5° and the speed is 43 m/s, the maneuver commands of 0.1° and 0.5° are applied to the system. The responses are simulated, as shown in Figs. 18-19. When the command is 0.1° which is condition C2 shown in Fig. 17, the response of the fin is expected. When the command is 0.5° which is condition D2, LCOs occur and affect the control effectiveness of the

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Fig. 18

YANG Ning et al. / Chinese Journal of Aeronautics 24(2011) 590-599

Maneuver command and response of pitch angle (command 0.1°, freeplay 0.5°).

Fig. 20

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Maneuver command and response of pitch angle (command 0.5°, freeplay 0°).

tive method, the results in frequency domain have been obtained. The responses in time domain have been simulated and the influence of the maneuver command on the vibration characteristics has been analyzed. The conclusions are as follows. (1) The results in frequency domain and time domain agree with each other, and the methods are effective. (2) Affected by the structural nonlinearity, LCOs occur within the linear flutter boundary. Different disturbances lead to different responses. (3) Within a range of speeds, the preload angle has a suppressed influence on LCOs. (4) When the freeplay exists in the system, the small maneuver commands could have a corresponding deflection, but the great commands make the system generate LCOs. References [1]

[2]

[3] Fig. 19

Maneuver command and response of pitch angle (command 0.5°, freeplay 0.5°).

fin. But the results show that under the assumption of structural linearity, the actuator can control the fin precisely, as shown in Fig. 20. 4. Conclusions In this study, the governing equations with consideration of dynamic stiffness and structural nonlinearity have been established by the component mode substitution method. By use of describing function and itera-

[4]

[5] [6]

[7]

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Biographies: YANG Ning Born in 1986, he received B.E. degree from Beihang University in 2009. His main research interests are aeroelasticity and active control. E-mail: [email protected] WU Zhigang Born in 1976, he received B.E. and Ph.D. degrees from Beihang University in 1999 and 2004 respectively, and then became a teacher there. His main research interests are aeroelasticity and active control. E-mail: [email protected] YANG Chao Born in 1966, he received B.E. and Ph.D. degrees from Beihang University in 1989 and 1996 respectively, and then became a teacher there. His main research interests are aeroelasticity and active control. E-mail: [email protected]