Advanced Flight Dynamics with Elements of Flight Control

Visit the Taylor & Francis website at http://www.taylorandfrancis.com and the CRC Press website at http://www.crcpress.com. 45 1.9.3 Steady States of Longitudinal Flight 47 1.9.4 Flight at Constant Velocity in the Longitudinal Plane 47 1.9.5 Rolling Maneuvers at Constant Velocity 48 1.10 EQUATIONS OF MOTION IN VOID OF 48F.

MODELING THE AERODYNAMIC FORCES AND

CONTROL AND GUIDANCE FRAMEWORK FOR

SIMULATION OF 10-THRUSTER DACS FLIGHT

The physical, analytical and computational study of the 6 DOF equations of motion for an aircraft constitutes the subject of flight dynamics. Should the 6 DOF equations for the Helios be augmented by the wing bending dynamics equations.

FIGURE 1.1 Flight dynamics as the linchpin of an aerospace vehicle design/

DEFINITION OF AXIS SYSTEMS

Similarly, compared to a wind-fixed axis system (also carried on the aircraft as explained below), the axes fixed to the body reveal the orientation and relative angular velocity of the relative wind incident on the aircraft. The inertial velocity V is the speed of the aircraft (more accurately, the speed of the aircraft CG) relative to the Earth-fixed axis (that is, as observed from the ground).

FIGURE 1.4 Body-fixed axis system for an airplane and an Earth-fixed axis system. (a) Airplane-fixed axis system

DEFINITION OF VARIABLES

The relative orientation of the axes fixed by the body and the wind is defined in terms of two angles - the aerodynamic angles, β and α. Thus, the rates of change of aerodynamic angles are related to the difference between the angular velocities of the axes fixed by the body and by the wind.

FIGURE 1.7 Sketch of an aircraft in flight with X B Y B Z B axes fixed to its center of gravity (CG) O B , velocity vector V at CG, and angular velocity vector ω about the CG, axes X E Y E Z E fixed to the Earth, position vector R from origin (O E ) o

TRANSLATIONAL EQUATIONS OF MOTION

The inertial velocity of the airplane is the velocity V of its CG relative to the axes fixed on the Earth. But note that the wind-axis angular velocity components in equation 1.45 are in terms of its body-axis components.

FIGURE 1.14 Aircraft representation for the purpose of deriving equations of motion.

REPRESENTATION OF FORCES ACTING ON THE AIRPLANE

As already worked out in Equation 1.5, the components of gravity are in body solid axes. Instead, it is appropriate to define the “drag force” as the component in the direction of the component of the relative velocity vector in the XBZB plane.

FIGURE 1.15 Sense of the forces acting on the aircraft—thrust, weight, and aerodynamics.

ROTATIONAL EQUATIONS OF MOTION

In the case of the torque equations, the body-fixed axes are invariably used to write both the components of the angular momentum as well as the derivative d/dt. Inserting the form of the angular momentum vector in Equation 1.68 into the rotational dynamics equation.

REPRESENTATION OF MOMENTS ACTING ON THE AIRPLANE

Normally, the engines are placed on the airplane in such a way that the propulsive torque is as small as possible. At this point we can collect the airplane dynamics (translation and rotation) equations in Table 1.4.

FIGURE 1.17 Moment acting on the aircraft due to offset thrust line.

SELECTION OF EQUATIONS FOR SPECIFIC PROBLEMS Sometimes a particular selection of variables or a reduced set of equations

For time simulation of any aircraft maneuver in flight, the following set of equations in Table 1.5 is often used. Also note that the velocity derivative on the right side of Equation 1.59 in Table 1.9 is set to zero because of the constraint of constant velocity.

TABLE 1.4 Summary of Airplane Translational and Rotational Dynamics Equations

EQUATIONS OF MOTION IN THE PRESENCE OF WIND Figure 1.19 shows an aircraft in flight in the presence of an arbitrary wind

Note that the equation in the first line of Equation 1.59 is now an algebraic relation, not a dynamic one, because of the constant velocity constraint. In the absence of wind, the wind speed components in Equation 1.82 can be set to zero, and the original definitions in Equation 1.14 are restored. The instruments on board the aircraft are likely to measure the quantities in Equation 1.82 or, equivalently, the body-axis component of the relative wind speed on the left-hand side of Equation 1.81.

FIGURE 1.19 Aircraft in the presence of external wind.

DEFINITION OF AERODYNAMIC COEFFICIENTS

What distinguishes one from the other is the nature and magnitude of the aerodynamic forces and moments generated during flight. Therefore, accurate modeling and estimation of aircraft aerodynamic properties is critical, without which the results of a 6 DOF simulation may also be meaningless. At first glance, there appears to have been little or no change in the state of the art regarding the exterior aerodynamics of the aircraft.

MODELING OF AERODYNAMIC COEFFICIENTS

Likewise, the angular velocity of the body's fixed axes relative to the ground also has no influence on the aerodynamic forces and moments. The static term is a function of the relative flow Mach number Ma, the aerodynamic angles α, β, and the control surface deviations δ. The dynamic term is a function of the three components of the relative angular velocity between the body and windage axes, appropriately non-dimensionalized.

FIGURE 2.1 3-View drawings of different airplanes (clockwise from top left):

STATIC AERODYNAMIC COEFFICIENT TERMS .1 Longitudinal Coefficients with Angle of Attack

Note that for larger α in Figure 2.2, the lift coefficient continues to increase, albeit with a smaller value of the slope. Thus, the variation of Cm with α is comparable (almost linear at low angles of attack) to that of the lift coefficient CL. The variation of the side force coefficient CY with the Mach number is in principle similar to that of CL.

FIGURE 2.2 Longitudinal aerodynamic coefficients as function of α for a typical military airplane.

DYNAMIC AERODYNAMIC COEFFICIENT TERMS

As sketched in Figure 2.8, (qb− qw) is the relative angular velocity (relative pitch) of the body axis XB with respect to the velocity vector V. These corresponding velocity derivatives are labeled Cnr1 and Clr1, and they are both proportional to VTVR. There is another source of moment and yaw in response to a relative yaw rate (rb - rw) and that is due to the wing.

FIGURE 2.8 Sketch showing induced flow and hence induced angle of attack at the horizontal tail due to a relative pitch rate between body- and wind-fixed axes.

FLOW CURVATURE COEFFICIENT TERMS

Also, parts of the wing on the inside of the winding experience reduced relative flow velocity, smaller by rwy, where y is the distance along the span, as shown in Figure 2.9. Similarly, wing sections on the outside of the turn experience increased relative flow velocity. The value of the derivative Clp2 is largely dependent on the slope of the lift curve of the wing section (airfoil) and the wing geometry.

Figure 2.10 shows an airplane with velocity vector V pointing into the plane of the paper having a roll rate p w about the wind X W axis (aligned to the velocity vector)

DOWNWASH LAG TERMS

A change in the angle of attack at the tail due to the effect of washout lag is approximated as. The increase in tail lift due to the change in the angle of attack of the tail due to the washout lag effect is modeled as. Through these expressions, the effect of the landing lag appears as a damping in the direction of the normal acceleration given by the derivative.

SAMPLE SIMULATION CASES

This simulation has shown that trim state 1 is indeed a stationary state of the aircraft (for the data set under consideration). Plot the following aircraft data and analyze the trend of the variation of the static longitudinal coefficients with angle of attack. Plot the following aircraft data and analyze the trend of variation of the static lateral coefficients with angle of attack (in degrees).

FIGURE 2.12 Result of numerical integration of example ODE showing time history.

TYPES OF STEADY STATES

The other steady state at the top of the hoop (θ=π) is unstable for all values of γ. In fact, there are an infinite number of distinct cyclic solutions in the case of the simple pendulum. In the space of the state variables (state space) it appears to be either a point (equilibrium) or a closed loop (periodic state) or a toroidal surface (quasi-periodic state) or a strange attractor (chaotic state).

STABILITY OF STEADY STATES

Homework: For the dynamics of the simple pendulum, we have seen that the equilibrium state (θ=0,θ =0) is stable in the Lyapunov sense. As long as all the eigenvalues of the Jacobian matrix lie in the left half-plane. Similar to the case of the equilibrium point, which we have already seen, for the fixed point x*.

FIGURE 3.5 Depiction of the different possible types of steady states of any dynamical system

BIFURCATIONS OF STEADY STATES

In the case of the saddle-node bifurcation in Figure 3.13, for a negative value of μ, there are two equilibrium states—one stable and the other unstable. Imagine a dynamical system with a saddle-node bifurcation with its state on the stable branch in Figure 3.13. How does it differ from the shape of the bifurcation diagram for the jump in Figure 3.14b.

FIGURE 3.11 Phase portrait of the van der Pol oscillator.

CONTINUATION ALGORITHMS

We want the continuation algorithm to solve for steady states and periodic states of dynamical systems of the form Equation 3.1. Note that the approximation in Equation 3.28 is used here only to illustrate the use of the continuation and bifurcation procedure. This is obtained by calculating the eigenvalues of the Jacobian matrix of the system in equation 3.28.

FIGURE 3.22 (a) Predictor step and (b) corrector step in a continuation.

CONTINUATION FRAMEWORK FOR MULTIPARAMETER SYSTEMS

It is clear that the relative degree of an output function cannot exceed the order of the system n. Analyze fixed points/stability and branches of the two-dimensional system with respect to the parameter μ. The second of equation 4.5 is more usefully rearranged in terms of the flight path angle γ.

FIGURE 3.25 Parameter schedule a = a(b) for the phugoid model in Equation 3.39 with the constraint, top: γ = 0 and bottom: γ = 0.4 rad (solid lines: stable solutions; dashed lines: unstable solutions; HB: Hopf bifurcation).

LONGITUDINAL STEADY STATES (TRIMS)

Then there are two other control inputs—the pitch deflection δe and the thrust T, which is usually determined by the throttle setting η and is also a function of the aerodynamic parameters, as we'll see shortly. A typical variation of turbojet thrust as a function of Mach number for different throttle settings and at two different altitudes is shown in Figure 4.1. A typical variation of SFC of a turbojet engine as a function of Mach number for different throttle settings and at two different altitudes is shown in Figure 4.1.

LONGITUDINAL TRIM AND STABILITY ANALYSIS

Likewise, the steady state in Figure 4.2 at the diamond symbol is not level flight unlike the trim state indicated above. Referring back to the aerodynamic data in figure 2.12, it can be seen that the derived Cmq changes sign at precisely that value of the angle of attack. Note that the initial steady state marked by the triangle in Figure 4.3 is not a planar trim.

FIGURE 4.2 Longitudinal trim states for F-18 data with varying elevator deflec- deflec-tion (throttle setting held fixed at η = 0.346)

LEVEL FLIGHT TRIM AND STABILITY ANALYSIS

Figure 4.5, for example, suggests that all three are on the "front side" of the power curve (as T vs. Note that these unstable regions are gaps on the "back side" of the power curve in Figure 4.5, where the unstable phugoid mode is understandable. The minima in Figure 4.7 correspond to the flight at the (VL/D)max condition that provides the best range (optimal cruise conditions).

FIGURE 4.5 Trim thrust for F-18 data with varying elevator deflection to satisfy level flight constraint plotted as a function of (a) trim velocity and (b) trim angle of attack (full line: stable, dashed line: unstable).

CLIMBING/DESCENDING FLIGHT TRIM AND STABILITY ANALYSIS

For many aircraft, the typical value of the angle of attack in best range is 4 ± 1 degrees (as is also the case in Figure 4.7). The speed and angle of attack for minimum power flight can be read from figure 4.8 to be respectively approx. 100 m/s and 5.5 degrees. In Figure 4.10, the SBA analysis reveals that the fastest pitch trim is stable, while the steepest pitch trim occurs at a point of phugoid instability.

FIGURE 4.9 Analysis of climbing flights by SBA, throttle η = 1, and elevator deflection varied as the parameter (full line: stable, dashed line: unstable).

PULL-UP AND PUSH-DOWN MANEUVERS

Let us begin with the longitudinal dynamics equations in the form given by equation 4.5 (reproduced below). 4.5) Consider a pull-up maneuver (a push-down is very similar). From the second of equation 4.5, still considering the flight condition at the bottom of the curve, where γ= 0, we have. Note, however, that perturbations in the angle of attack will also affect the lift coefficient in equation 4.15 and thus perturb the load factor nz and thus the curvature of the flight path (equation 4.13).