Information Technology Journal

Year: 2014 | Volume: 13 | Issue: 5 | Page No.: 885-894
DOI: 10.3923/itj.2014.885.894

Modelling and Analysis of Tiltrotor Aircraft for Flight Control Design

Wang Qi and Wu Wenhai

Abstract: As known that flight control design and real-time simulation is very important in the aircraft system, in order to realize linear/nonlinear flight control design and real-time simulation, this study presents the full nonlinear flight dynamic model of tiltrotor aircraft. Primary dynamic equations of the model are developed considering nacelles tilting dynamics. The force and the moment in primary equations are decomposed and calculated by multi-body aerodynamic models, including the aerodynamic effect of rotor wake on the wing/elevator/rudder. Rotor dynamic model is developed based on the blade element theory and the gyroscopic moment caused by nacelles tilting is introduced into the blade flapping dynamics. By the linearization of full nonlinear equations of motion, a family of linear state-space models in the whole flight envelop is obtained. For the velocity control of the aircraft in hover or low speed, the nacelle angle is derived as a new control input. Because the number of unknown parameters is bigger than that of equations, two different algorithms are applied to trim the mathematical model. The results by these trimming methods are in conformity with each other and then the reference transition curve is determined. The results of trimming and linearization derivatives are identical with GTRS (Ground Test Reactor System), so the model is valid.

Keywords: transition curve, trimming calculation, motion modeling, flight dynamics and Tiltrotor aircraft

INTRODUCTION

Tiltrotor aircraft is a special vehicle combining vertical takeoff/landing capability as a helicopter and high speed cruise like a turbo-airplane which is achieved by rotating the rotor/nacelle system attached at each wingtip. Because of this advantage, tiltrotor technology has been studied and tested by many countries (Foster, 2003; Alli et al., 2003; Choi et al., 2010).

As the maturing development of tiltrotor technology, some relevant dynamic problems have been understood. Song did some researches and analysis on vibration and elastic dynamic characteristic for single nacelle/wing of tiltrotor aircraft (Song et al., 2006). Gervais studied the BVI noise phenomenon during takeoff and landing and suggested a method to suppress the noise (Gervais and Schmitz, 2002, 2003). Other research topics on tiltrotor dynamics in the past are such as, aerodynamics (McVeigh et al., 2004), aero-elastic stability (Nixon et al., 2003; Acree and Johnson, 2008), aeroacoustic (Johnson, 2001) and so on. However, researches on tiltrotor flight dynamics are not sufficient enough. Although some models were developed in the above topics, they were constraint in particular problems and could not be directly used in controller design and flight simulation.

Tiltrotor flight dynamics have been studied but still open. Carlson developed three dimension motion equations of tiltrotor but simplified too much (Carlson and Zhao, 2003, 2004). Kleinhesselink presented a thorough six degree of freedom model while the inertial force and moment of blade were handled in detail so much and other aerodynamic parts were simplified, especially not considering the nacelle tilting dynamics and interference among these parts (Kleinhesselink, 2007). Miller gave a tiltrotor aircraft model easy for flight performance evaluation, flight simulation and stability analysis (Miller and Narkiewicz, 2006). In order to get the linearization state-space model of tiltrotor, Klein gave a simplified method for the state-space model in helicopter mode and airplane mode but not including the transition mode (Klein, 1996).

The model of tiltrotor aircraft that developed for easy flight controller design is a trade off between precision and real-time simulation requirement. For model precision, many dynamic characteristics must be considered such as nacelle tilting dynamics, the change of body structure parameters during modes transition, the interference between rotor and other aerodynamic surfaces. The rotor model should be simplified for separation of control parameters and computing load reduction. For the unmodeled dynamics and the real flight dynamics that cannot be described by precise model, it could handle it by adaptive and robust controller design. In this study, it developed a tiltrotor flight dynamic model for linear/nonlinear controller design and simulation.

BASIC EQUATIONS OF TILTROTOR AIRCRAFT MOTION

Reference frames: Tiltrotor aircraft has characteristics of helicopter and airplane, so two kinds of reference frames in describing helicopter and airplane motion are needed.

The tiltrotor flight dynamic model for linear/nonlinear controller design and simulation are necessary.

Reference frames of tiltrotor aircraft are as shown in Fig. 1.

Basic assumptions: Some assumptions are given as follows:

Basic equations: All of equations are developed in the body-fixed reference frame. According to newton’s second law, equations of motion can be written as:

Where the state x = [u, v, w, p, q, r,φ, θ, ψ, x, y, z]^T, the input u = [θ₀, θ_0d, θ_1s, θ_1sd, δ_a, δ_e, δ_r]^T.

Translational equations: The difference between tiltrotor aircraft and airplane is the nacelle so that the motion dynamics of the nacelle must be considered. Suppose in any flight conditions, the position vector of the nacelle is:

where, R_N/B is the position vector of the nacelle with respect to body center, R_B is the position vector of body with respect to earth. From Eq. 2, the absolute velocity of the nacelle is as Eq. 3:

then the absolute acceleration of the nacelle is:

where the last right terms are relative acceleration, centripetal acceleration and Coriolis acceleration.

Then, motion equations can be obtained:

where VB = [u v w]^T, ω_B = [p q r]^T, m is aircraft mass, F_B is the force vector acting on the body in the body-fix axis as:

where, F^B_R, F^B_F, F^B_W, F^B_H and F^B_V are the force of rotors, fuselage, wing, horizontal tail and vertical tail respectively.

The position motion of aircraft in earth-fixed reference frame is:

where, XE = [x y z]^T, T^E_B is the transformation matrix from body-fixed frame to earth-fix frame (Leishman, 2006).

Rotational equations: Based on momentum theorem of particle system, total momentum is:

where, m₁ (m_N) is each particle (nacelle) mass, R_i (R_N is the position vector of each particle (nacelle) with respect to earth, V₁ (V_N) is the absolute velocity of each particle (nacelle).

Substitute V_N = V_N/B+ V_B, R_N = R_N/B+R_B and into Eq. 8 and . It can get Eq. 9 :

where, I_B is changing with the nacelle tilting, its element can be determined by:

I_i represents I_XX, I_yy, I_zz and I_zx respectively, K_i is the weight coefficient. I_N/B is the rotational inertia of the nacelle with respect to body-fixed frame.

Then Rotational Dynamics is obtained:

where, M_B = [l m N]^T is the moment acting on the mass center producing by several parts written as:

MULTI-BODY MODELS

Fuselage model: The force and the moment generated by the rigid fuselage are the function of angle of attack and sideslip angle, where the data is from wind-tunnel test:

where, A represents the aerodynamic force, D, Y, L and the aerodynamic moment l, m, n. Q_F are the fuselage dynamic pressure. Transform the force and the moment to body-fixed frame, the force is obtained and the moment acting on the body center:

where, T^B_A is the transformation matrix from wind axis to body-fixed frame, see in Xiao (2003). M^B_F0 is the basic part of moment and L_F is the arm matrix of the force.

Wing model: The wing area is separated into two different parts because of tilting rotor, one is the area influenced by the rotor inflow which changes as the nacelle tilting and the other is the free stream area. The first area could be simplified by Carlson and Zhao (2003):

where, is the critical velocity, a and b are selected to match the experimental data. Then, the free stream area is S_{∞, W} = S_W-S_{i, W}.

The angle of attack in these area is also different. The air velocity of the first area is:

Where:

And:

are the velocity influenced by the rotor inflow in body-fixed frame. According Eq. 18 and the computing formula of angle of attack and sideslip angle, the value of α_W,i and β_W,i in the first area can be obtained.

Then the aerodynamic force of wing is:

The aerodynamic moment of wing is:

Then, in body-fixed frame, the force and the moment generated by the wing are as Eq. 21:

Rotor dynamic model
Blade element incident velocities: In this study, it focuses on the right rotor using the blade element theory. The situation of the left rotor is just the same as the right rotor if not mentioned. Blade incident velocities are U_R, U_T and U_P which are located on O_bx_b, O_by_b and O_bx_b respectively. The blade incident velocity located on x are Leishman (2006):

where, Ω is the rotor rotating angular speed, λ is the inflow ratio, ψ_H is blade azimuth angle, β_H is blade flapping angle. The absolute rotating angular speed considering body angular velocity is:

Suppose hub velocity is the sum of the aircraft translational velocity and angular velocity, transform to shaft frame:

where, , r_s is the position vector of hub center with respect to body center (the position vector j of the left rotor is opposed to the right rotor). The transformation matrix from body-fixed frame to shaft frame is:

Then the rotor advance ratio is:

The total inflow ratio is:

where λ_c represents the inflow ratio by forward velocity, λ_i is the rotor induced inflow ratio:

For the reason of reducing the computing load, uniform inflow model is adopted (Leishman, 2006):

where C_T is the rotor thrust coefficient. After getting the value of induced inflow ratio, then the rotor induced velocity is determined by vi = λ_i·Ω'·R.

Blade element force: The lift force and the drag force of the blade element dx located x from the origin are Bramwell et al. (2001):

where, is the section lift-curve slope, C_D is the section drag coefficient, c is the blade chord, θ_b is the blade pitch angle. θ_b can be described as the Fourier series and neglect high order terms:

where θ₀ is the blade collective pitch, θ_1c is the lateral cyclic pitch, θ_1s is the longitudinal cyclic pitch, θ_tw is the linear blade twist rate, θ_tw0 is the blade twist at the origin.

Then the force perpendicular and parallel to the rotor disk is:

where, φ_b≈U_P/U_T.

Blade flapping: Rotating blade element will flap up with respect to the hub and reach a steady equilibrium position under the action of several forces, such as aerodynamic force, centrifugal force, inertial force and gyro force causing by nacelle tilting. Taking moments about the flapping hinge, flapping equation of motion is shown as Eq. 33:

where, M_CF is the centrifugal moment, M_I is the inertial moment, M_G is the gravitational moment, M_A is the aerodynamic moment, M_S is hub spring moment, M_gyro is the gyro moment. Each moment is:

where, I_b and M_b are the blade inertial and mass inertial moment, is the flapping spring constant, β_p is the rotor pre-cone angle.

The flapping equation of motion is impossible to be solved and always expanded as the Fourier series. Generally, the flapping angle is less than 10° such that the high harmonics can be neglected. The solution for flapping motion in the first harmonics is:

where, β₀ is the coning angle, β_1c is the longitudinal tip path plane tilt angle, β_1s is the lateral tip path plane tilt angle.

Substituting Eq. 31 and 35 into Eq. 34 and combining with Eq. 36, it can get the final equations. Transfer the result high periodic (sine and cosine) terms to the first order and cast terms with n·Ψ_H (n≥2), let the constant term, coefficients of Ψ_H and cosΨ_H be zero, then expressions of β₀, β_1c and β_1s can be obtained.

Rotor forces and moments: The rotor thrust is simply the average of the blade lift during one revolution multiplied by the number of blades. In shaft frame, the rotor thrust coefficient is:

where, σN_bc/(πR) is the rotor solidity and r = x/R.

The rotor drag force is:

where, dF_R = ρU²cC_L/2.

The rotor side force is:

The pitch moments generated by both rotors are the same value but different directions:

The rotor torque and roll moment are cancelled so not given. Note that the β_1s coefficients of both rotors are different; the coefficients of differential pitch θ_0d and differential longitudinal cyclic pitch θ_1sd are also different.

Integral above equations, the rotor force and moment in body-fixed frame are obtained:

Interference between rotor and wing/horizontal and vertical tail: The airframe will be influenced by the rotor airflow, mainly on wing/nacelle and horizontal/vertical tail. The absolute airspeed and angle of attack which are the sum of actual airspeed and the interference airspeed determine multibody aerodynamics. In nacelle frame, the interference airflow by the rotor is as Eq. 41 and if transfer it to body-fixed frame, the Eq. 42 can be obtained:

where, T^B_N is the transformation matrix from nacelle frame to body-fixed frame, v_i^N is the interference airflow vector.

Borizontal tail model: Consider rotor influence, the airspeed of horizontal tail is:

From Eq. 43, it can get the horizontal tail angle of attack α_H and sideslip angle β_H. Then the lift and drag force of horizontal tail is:

Transfer it to body-fixed frame, the force and the moment of horizontal tail are:

Vertical tail model: The expression of influence airflow is just the same as Eq. 43. The side force and the drag force of vertical tail can be described as:

Then the force and the moment to the body center are:

LINEARIZATION

The linearization of equations is based on the Taylor series of Eq. 1 initialized in some trimmed value (x_e, u_e). The differentiation in the first order of the Taylor series gives:

where, are the Jacobian matrix of f with respect to the state x and the input u about the initial condition respectively. f can be divided into two parts as:

where, f₁ (x, t) is the basic terms without control inputs while f₂ (x, u, t) is the remainder terms. The linear state space model can be written as:

The forces and the moments are constituted by many different aerodynamic parts so the derivatives are calculated by each part and then combined together. For the force in O_B, x_B, the derivative can be described as:

where the derivatives of rotor forces with respect to the state are very tedious involving the partial derivatives of μ, λ, β₀, β_1c and β_1s for each state.

Tiltrotor forward speed control in the low-speed condition is realized by the nacelle tilting while the longitudinal cyclic pitch and the elevator control are used to trim the attitude angle. So new control variables β_M and should be introduced and then Eq. 51 becomes:

CALCULATION AND DISCUSSION

Trim results: The purpose of trim is to find the equilibrium points of the states and the inputs under desired flight condition at a given nacelle angle. In the trim equations, Eq. 5 and 12 will be used and the navigation variables are given zeros such as the heading angle ψ and position parameters (x, y, z). Suppose the desired flight condition is straight and level flight that is γ = 0°, χ = 0° and the angular rate is zero and consider the relationship between u, v, w and α_F, β_F, V where V is given from outside, then the unknown variables to be determined are α_F, β_F, φ, θ and the inputs θ₀, θ_0d, θ_1s, θ_1sd, δ_a, δ_e, δ_r. Since there are only 6 equations but Eq. 11 unknown variables, the addition equations are needed as:

From above equations in which new unknown variable φ_V is introduced, 9 equations and 12 unknown variables are gotten. Next, it uses two algorithms to trim the equations using the data from Harendra et al. (1973).

Trim results 1: Trim results 1 is calculated by using Levenberg-Marquardt algorithm to solve the equations under the condition where the number of unknown variables is more than the number of equations. The longitudinal trimmed control inputs are shown in Fig. 2 while the lateral variables approximately to zero are not given. At low nacelle angle (0~15°), the rotor control of tiltrotor is just the same as a helicopter that the pitch angle is increasing then decreasing as the velocity increasing. When the nacelle tilts and the wing becomes more efficient, the bigger pitch angle is needed as velocity increasing. Trim results in helicopter mode and airplane mode are not far from the result of GTRS model shown by cross. In this trim method, aerodynamic surface controls are independent of rotor controls, as shown by (b) and (c) of Fig. 2. At low nacelle angle (β_M<15°) and under low airspeed condition, elevator deflection is small; as nacelle tilting forward (β_M<45°), efficient of aerodynamic surfaces is low at low speed so the deflection is small and then increasing and deflecting backward as airspeed increasing and this will not happen as nacelle angle bigger than 45°. At a fixed nacelle angle, the longitudinal cyclic pitch angle is increasing as airspeed increasing and as the efficient of aerodynamic surface increasing the longitudinal cyclic pitch angle will be backward. In trim results 1, the deflection of elevator does not saturate the value of ±20° and the longitudinal cyclic pitch angle is less than ±10°.

Trim result 2: The second trim method is based on the control mixing concept in which the controls of each axis are connected. The attitude controls of tiltrotor aircraft contain rotor control and aerodynamic surfaces.

By the control mixing, the controls of rotor and surfaces are dependent on each other by a function of nacelle angle.

In the open-loop, the relationship between rotor controls and pilot control is:

By lateral stick control δ_LAT, longitudinal stick control δ_LON and pedal control δ_PED, the rotor and aerodynamic surfaces are connected:

Through Eq. 56, the number of unknown variables is just the same as the number of equations and then the normal trim method can be used as shown in Fig. 3. The longitudinal stick control is less than ±2 cm; trim results of pitch angle at β_M = 0°, β_M = 45° and β_M = 90° are not far different with GTRS.

Reference transition corridor: From the trim results 1 and pick the trimmed points as the pitch angle is zero, then the reference transition curve as shown in Fig. 4 and the correspond control inputs shown in Fig. 5. Linearization models on these points can be used for the flight control design.

Linearization validation: Using the linearization method introduced in 3, it can get a series of linear state space models of tiltrotor aircraft in the whole flight envelop.

Because of the paper length constraint, it only gives the comparison linearization result of this paper with GTRS model as Table 1. And the Stability derivatives are mentioned in the above Eq. 50-56.

CONCLUSION

In the study, it has developed a tiltrotor aircraft mathematical model considering the nacelle tilting dynamics. In this model, the multi-body parts are modeled properly for real-time simulation and model preciseness. The linearization of nonlinear equations is presented, including the new control variable.

It adopts Levenberg-Marquard method in the equations of motion when the number of unknown variables is larger than the number of equations. From the comparison of the trim result and GTRS trim result, the model is then proved be valid.

Based on the control mixing concept, it adopts use ordinal method to trim the equations and get the same result as the above method. From the trimmed results, the reference transition curve is determined by the trimmed points setting the pitch angle be zero. Linearization along this reference curve, a series of linear state space models of tiltrotor aircraft for linear controller design can be obtained.

The full nonlinear tiltrotor model for flight control is developed which can be used not only for nonlinear controller design and simulation but also for linearization of nonlinear model and linear controller design.

REFERENCES