Modelling the Dynamics of Flight Control Surfaces Under Actuation Compliances and Losses

Transcription

1 Modelling the Dynamics of Flight Control Surfaces Under Actuation Compliances and Losses Ashok Joshi Department of Aerospace Engineering Indian Institute of Technology, Bombay Powai, Mumbai, 4 76, India Joshi, ashokj@aero.iitb.ac.in Abstract: The study investigates the effects of servo system compliances and Coulomb & viscous friction on the dynamics of an aerodynamic control surface, during its deployment through an electro-hydraulic actuation system. Starting from the pilot command, a realistic nonlinear model of the electro-hydraulic servo system based actuation system is evolved, including the applicable nonlinearities. A realistic mathematical model for the control surface motion, under the action of the actuator forces and the aerodynamic and inertia forces is postulated, using subsonic incompressible aerodynamics. The mathematical model also includes the command shaping system. The above mathematical model is first verified for its consistency, by carrying out subsystem level simulation. Next, the complete system is simulated for pilot step input and typical control surface response is obtained for the command shaper. Particular attention is paid to the control surface force time histories during deployment that have the potential to excite the elastodynamics of the control surface, as these can modify the aerodynamic loads acting on the control surface. Keywords: Actuator Models, Control Surface Dynamics, High Actuation Rate 1. Introduction It is well known that aerodynamic control surfaces are deployed in aerospace vehicles for generating control forces and moments, by giving a suitable command. In general, these commands activate an actuation system, which carries out the task of control surface deployment through a series of actions. In case of electro-hydraulic actuation system [1], hydraulic power, in conjunction with a servo valve, is used to generate the requisite forces and the motion. The desired motion is achieved through a closed loop feedback control system that senses the actual deflection and corrects it until the desired position is reached. In recent times, there has been a trend towards designing higher agility aerospace vehicles resulting in larger bandwidths, as well as higher actuation rates, of actuation systems. Such systems require an analysis that takes into account real system effects e.g. dry friction, apart from compliances of the complete actuation chain and pilot command shapers. An important by-product of such high performance actuation is the excitation of the elastodynamics of the control surface-actuator combination and the present study addresses the issue of control surface dynamics, in relation to actuator parameters that give rise to an improved performance. In literature, the problem of modelling the dynamics of hydraulic servo systems has been addressed adequately [2,3], including the various nonlinearities that are present in such systems. However, when dealing with flight control system analysis and design, the actuator system dynamics is usually approximated as a linear first (or at the most second) order system [4]. It is found that while such actuator models are sufficient for moderate actuation rate systems, there are no studies that have brought out the adequacy of such models for higher control rate actuation, which can result from using higher system pressures. The present study attempts to arrive at actuator models that need to be used for high performance control systems. The present study formulates the general flight control actuation problem, using electro-hydraulic actuation system, which includes important real system effects. A parametric study is carried out to understand the influence of specific parameters, including nonlinearities of actuation system, on the control surface excitation, as system actuation rate is increased. Results are obtained for a set of non-dimensional parameters so that generic trends can be arrived at.

2 2. Formulation Figure 1 shows the schematic description of the flight control system, based on electro-hydraulic actuation mechanism. U Figure 1: Schematic of the Actuation System It is seen that there are three sub-systems namely; (1) the pilot station, (2) hydraulic servo-system and (3) control surface system that interact with each other, in order to perform the overall control task. In following sub-sections, the formulation details of these sub-systems are described Pilot Station The control action is initiated at the pilot station (human, auto or remote) by giving a command, which represents the desired position of a specific control surface. The command passes through a servo amplifier, which sets the overall system gain and passes through a command shaper, whose output is a demand on the movement of a servo valve. This is shown schematically below in figure 2. e E A 1/(τs 1) δ c - δ e Engine Pump S E R V O Figure 2: Schematic of Pilot Command The command shaper, shown as a standard first order lag, provides a specific time history of x v, which is consistent with overall system constraints and which minimizes the problem of cavitation in the flow, due to a sudden opening of a high-pressure valve. The above system can be represented mathematically as, x v / e = A / (τs 1) (1) δ c x v δ e 2.2. Hydraulic Servo Valve Servo valve is the device that converts commanded valve opening (x v ) into a force (F) and motion (y) at the load end. In literature [3,4], 4-way zero overlap spool valve is considered as the best option for high precision, whose action, briefly, is as follows. The servo valve opening connects the high- pressure supply to one side of valve and connects low pressure (or exhaust) to the other end. This causes flow to enter one side of Jack, after a pressure drop, and to exit from the other end. This creates a pressure difference across the piston area, resulting in a force in the direction of the flow. The basic flow equations for the valve, assuming valve openings to be ideal orifices and fluid to be compressible, are as given below [3]. K q x v = q v K c P m (2) q v = 2 A p (dy/dt) {V t /4 β} {dp m /dt) (3) Here, K q is the flow rate gain (m 2 /s), K c is the pressure gain (m 3 /sec-bar), P m is the differential pressure (bars; 1 bar = 1 5 N/m 2 ), A p is the piston area (m 2 ), is the displacement of the jack (m), V t is the total cylinder volume (m 3 ) and β is the bulk modulus of the fluid used (bars). It should be noted that K q & K c are nonlinear functions of the system pressure P s and for a standard fluid (density 87 Kgs./m 3 ) flow of turbulent type, are given as, K q = 6.7 π d (P s ) 1/2 (4) K c = 3.35 π x v d / (P s ) 1/2 (5) Here, d is the servo valve spool diameter (m). The above relations are for SI units as mentioned above wherein P s is hydraulic system pressure (bars), q v is volume flow rate (m 3 /sec) and x v is the commanded valve displacement (m). It can be seen that additionally, K c depends on x v and hence represents a time varying effect, which needs to be studied. The schematic of servo valve model is given in figure 3. x v K q P m 1/K c G(s) - q v (V t /4β) s 2A p s Figure 3: Schematic of Servo Valve Operation

3 Transfer function, G(s), converts differential pressure P m into the motion of the Jack-control surface assembly, and is a function of actuator & control surface related forces, as described in next section Load Model The load on the servo actuator consists of; (1) external aerodynamic reactions from control surface, (2) elastic forces due to actuator & joint compliances, (3) inertia forces from the jack and the control surface, (4) viscous forces and (5) dry friction forces. In the present study two-dimensional subsonic incompressible aerodynamic model is used to represent the aerodynamic forces that act on the control surface. The applicable expressions for these forces are as given below. F aerodynamic = (C mδ Q S c / x h ) δ e (6) F inertia = (M j x h I c / x h ) d 2 δ e /dt 2 (7) F friction = sign(d δ e / dt) F c max (8) F viscous = B x h dδ e / dt (9) Here, C mδ is the control surface hinge moment derivative, Q is the flight dynamic pressure (bars), S is the reference surface area (m 2 ), c is the reference chord (m), x h is offset of actuator connection point on the control surface from its hinge line (m), M j is the effective mass of the actuator jack (Kgs.), I c is control surface moment of inertia about hinge line (Kg.-m 2 ), F c max is the dry friction force (N), B is the viscous force coefficient (N-sec/m) and δ e is control surface deflection (rad). The equilibrium of above reaction forces, with the force generated by the actuator, is shown in figure 4 schematically. To Servo Valve Jack α = (M j x h I c / x h ) (1) η = B x h (11) φ = (C mδ Q S c / x h ) (12) It may be mentioned here that application of actuator force on the control surface is assumed to have negligible influence on the main lifting surface. 3. Numerical Simulations The present section provides the influence of a few of the parameters, of the generic model of an electrohydraulic actuator system, on the load time histories of the control surface forces. Initially, the simulations are performed at the sub-system level, in order to verify and validate the mathematical model, after which, the complete system simulation is carried out Pilot Sub-system Simulation In the present analysis, pilot is assumed capable of providing a step input to the actuation system, which is modulated by the command shaper. Figure 5 given below shows the effect of command shaper time constant on the time history of the servo valve input. Normalized Servo Valve Command, x v /A Command Shaper Time Constant Varying from.2 sec. to 5. sec. τ=5. s τ=.2 s P m A 1/(αs 2 p ηs φ) - δ e x h Figure 5: Effect of Command Shaper Time Constant on Normalized Servo Valve Position Coulomb s Friction To Pilot Station Figure 4: Actuator Force Equilibrium Here, the feedback block represents the non-linear effect of dry friction, whose magnitude remains constant, but direction changes depending upon the direction of motion. Terms α, η & φ are given as, It can be seen from the above figure that while for values of τ around.2 seconds, the servo valve opens fully within 1 second, for values of τ around 5 seconds, the servo valve does not open fully, even after 1 seconds. Therefore, a shaper system with a time constant between.2-5. seconds can meet speed requirements of most practical hydraulic systems Servo Valve Sub-system Simulation It can be seen from figure 3 that servo valve input translates into a force at the actuator Jack.

4 In practice, this force is resisted by aerodynamic, inertia and friction forces present in the system, resulting in control surface response δ e (t) of the loaded actuator. However, it is possible to derive a lot of understanding about actuator behaviour even under the condition no external load. In this sub-section, simulations are carried out for studying the influence of system pressure, P s and hydraulic fluid compressibility factor β, while considering only the jack inertia as G(s). Thus, a simplification of the block diagram, given in figure 3, is carried out and a transfer function between y and x v is obtained. It is found that above transfer function is of third order and type 1 so that steady state error does not become zero for both step and ramp inputs. Instead, it is possible to investigate the transfer function between dy/dt and x v, which is as follows. {sy/x v }=(K q /A p ) / [(V t M j /(4β A p 2 )s 2 {(M j K c )/A p 2 }s 1] (13) The natural frequency and damping factor from above transfer function can be derived as, ω n 2 = ( 4 A p 2 β) / (M j V t ) (14) ξ = (K c / A p ) (β M j / 2V t ) (15) It is seen from above two equations that system natural frequency is independent of the system pressure and depends only on the piston area, cylinder volume fluid bulk modulus and mass of the Jack. However, damping factor additionally depends on the system pressure as well as on valve opening (through gain K c ) and represents a variable coefficient of servo valve dynamics. Further, it can be shown that steady state unit step response of the above transfer function is, (P m ) ss = 6.7 π d M j P s / (2 A p 2 ) (18) It is seen from the above relation that steady state differential pressure increases both with increase in system pressure as well as the mass of the Jack. This indicates that a higher load, after the control surface is attached to the Jack, demands a larger pressure drop and, consequently, a higher flow rate from servo valve. A parametric study is carried out for understanding the transient response of the valve system, using the transfer functions described by equations (13) and (17), by defining a group of parameters as follows. P s = P s / β V t = V t / (26.8 π d 2 β) (19) M j = 6.7 M j π d 2 / (2 A p 2 β) Typical results are obtained for unit step response in case of actuation rate parameter and unit ramp response in case of differential pressure parameter, using typical data for a generic hydraulic system given in Appendix. With regard to the treatment of time varying damping term, the following strategy is adopted. In the case of actuation rate simulation, valve opening is given as a constant step of 1 mm so that a constant damping term is calculated using this value of x v. This renders the sub-system transfer function as time-invariant (see Appendix for details) and results of simulation are compared with the results of a time varying simulation of the same problem, to establish the adequacy of the model, using SIMULINK. In the case of simulation for P m, only the time varying simulation for a ramp input is carried out, as the valve displacement x v does not remain constant during simulation. 6 Bode Diagrams (dy/dt) ss = 6.7 π d (P s / β) / A p (16) It is found that actuation rate increases with increase in system pressure or decrease in bulk modulus. Similarly, when the transfer function between the differential pressure and servo valve movement is considered, it is found that there is a zero at the origin, which ensures zero steady state for a step input in x v. In this case, it is more appropriate to either consider ramp input for x v, which can be studied through the following transfer function. Phase (deg); Magnitude (db) {P m /sx v }=K q / {(V t /4β)s 2 (K c )s2a p 2 /M j } (17) The steady state output for differential pressure, for a unit step in d x v /dt, can be shown to be, Frequency (rad/sec) Figure 6: Effect of system pressure P s on the frequency response of actuation rate parameter

5 Actuation Rate, dy/dt, in mm/sec System Pressure varying from 1 to 5 bars Figure 7: Unit Step Response of Actuator Rate as a Function of System Pressure Differential Pressure, P m, in bars System Pressure varying from 1 to 5 bars Figure 8: Unit Ramp Response of Differential Pressure as a Function of System Pressure It can be seen from figures 6-8 that servo valve actuation rate and differential pressure increase with increase in the system pressure. However, it is also seen that in both these cases, the system damping reduces with increase in system pressure, resulting in larger peak overshoot as well as longer settling time. However, it should be noted that these results are without taking into account the viscous damping of the fluid, which would reduce both peak overshoot and settling time even at higher system pressures Control Surface Motion Simulation It can be seen from figure 4 that control surface deflection dynamics depends on the both force and motion equilibrium inasmuch as that P m results in (or δe(t)) which in turn, effects the value of P m itself, as brought out in figure 3. Therefore, load simulation cannot be done in isolation, but needs to be coupled to the servo valve subsystem. This is achieved by combining figures 3 and 4 into a single block diagram, as shown in figure 9 below. x v P m Figure 9: Combined Servo Valve-Control Surface Dynamic Model It can be seen from above figure that, similar to subsection 3.2., it is possible to obtain the effect of system pressure on the control surface deflection rate ((dy/dt)/x h ), However, in addition to Jack inertia, only the aerodynamic reaction is included as this constitutes significant reaction force from the control surface. Figure 1 below presents these results. Actuation Rate, dy/dt, in mm/sec. K q 1/K c A p /(αs 2 φ) q v (V t /4β) s 2A p s Aero. Load Constant varying from 1 to 1 Increasing Aero. Load Figure 1: Effect of Aerodynamic Load on the Actuation Rate Response It is seen from above the plot that there is a drastic reduction in the actuation rate, in the presence of aerodynamic load and that the frequency of oscillations of the servo system is now controlled by the aerodynamic stiffness term. The above plot is obtained for the highest system pressure of 6 bars and this shows that a much higher system pressure is essential to achieve reasonable actuation rates, while operating at high control forces. Aerodynamic inertia would reduce the frequency as well as actuation rates.

6 3.4. Actuator Compliance Simulation In the presence of compliance of the actuation chain, the actual displacement of control surface is different from the displacement of the actuator Jack. In a simple, but adequate, manner, this compliance can be linked to the overall force equilibrium between the pressure force P m and the aerodynamic, inertia and damping forces generated by the control surface (equations 6-9). In such a case, the control surface motion y can be related to the Jack motion y, as y (t) = A p P m / K e (2) Here, K e is the equivalent spring stiffness of the complete actuator chain (N/m) Complete System Simulation The complete control system simulation is done by closing the loop at the pilot station. This simulation is carried out for one case of highest system pressure and lowest aerodynamic load, for two different values of the command shaper time constant. The resulting dynamics is shown in figure 11 below. Actuator Position Response,, in mm τ=.2 s τ=5. s Command Shaper Time Constant varying from.2 to 5 seconds Figure 11: Complete Control System Dynamics It can be seen from above figure that command shaper time constant has a significant influence on the dynamics of the actuator position, though it does not influence the final position achieved. It is found that as command shaper dynamics is the slowest dynamics in the complete control chain, it predominates the system response. 4. Conclusions The present study has investigated the problem of actuation rate dynamics for a hydraulic actuation system. The study formulates the complete servo system dynamics, which includes command shaper, servo valve and a representative aerospace vehicle control surface. The mathematical models are obtained considering the real system effects such as fluid compressibility, viscous and friction damping and servo valve non-linearities. The sub-system level simulations are carried out which clearly bring out the effect of system pressure on the dynamic performance of the actuation mechanism. These results also bring out the fact that requirement of high performance control system can be met by increasing the system pressure. However, it is also found that pressure related damping reduces as the pressure is increased, resulting in greater chatter in servo system. 5. References [1] Guillon M, Hydraulic Servo Systems Analysis and Design, Butterworth & Co (Publishers) Ltd, [2] Viersma, T. J., Analysis, Synthesis and Design of Hydraulic Servosystems and Pipelines, Elsevier Scientific Publishing Company, 198. [3] Stringer, J.D., Hydraulic Systems Analysis, The Macmillan Press Ltd., 2 nd Edition, [4] Raymond, E.T. and Chenweth, C.C., Aircraft Flight Control Actuation System Design, SAE Publications Group, [5] Chwa, D., Choi, J.Y. and Seo, J.H., Compensation of Actuator Dynamics in Nonlinear Missile Control, IEEE Trans. Control Systems Technology, Vol. 12, No. 4, , July 24. [6] Hess, R.A. and Snell, S.A., Flight Control System Design with Rate Saturating Actuators, J. of Guidance, Control and Dynamics (AIAA), Vol. 2, No. 1, 9-96, January-February [7] Joshi, A. and Jayan, P.G., Modelling and Simulation of Aircraft Hydraulic System, AIAA Paper No. AIAA , Proc. of Modelling and Simulation Technologies Conference, Monterey, CA, USA, 6-8 August 22. Appendix Representative values of the parameters, used in simulations, are as given below [3]. Servo Valve Spool Diameter, d 5 mm Fluid Bulk Modulus, β 6 bars Total Cylinder Volume, V t.15 m 3 Piston Area, A p.3 m 2 Jack Mass, M j 1 Kg P s [1,2,3,4,5,6] bars Q.882 S 1 m 2 x h.1 m The dimensions of output variables are; mm for x v and bars for P m.