SIMPLE UNIVERSAL NONLINEAR LONGITUDINAL FLIGHT SIMULATION WITH AVOIDING OF STATIC AND DYNAMIC STABILITY DERIVATIVES

This paper shows simple method to simulate nonlinear longitudinal flight dynamic of an aircraft in the most direct way. The method is based on physical principles of an analytical-empiric flight dynamics. Non-linearity, as stall or thrust dependent on velocity, can be included. Static and dynamic stability derivatives are not required but can be computed as an output. The model is applicable for various aircraft conceptions. Higher level model, for example, based on flight tests data, can be modified by low-level analytical methods, e.g. for modification of horizontal tail area. No special simulation software is necessary. The model is compared to linear model and flight test experiment. This model, together with valuable analytical-empiric data, might be applied for fast flight dynamic computations. Model is easily accessible and understandable even with basic knowledge of flight dynamic and computer programming. The main application of the method is conceptual design when high precision is not expected, even if VLM, CFD or flight test data can improve the precision.


INTRODUCTION
Flight dynamic has been solved more than 100 years [1]. Classic analytical-empiric theory [2] is very useful tool for an aircraft design. Disadvantage is the time-consuming design process. In the time of high-performance computers, the simulation methods of flight dynamic [3] offer faster solution. However, combination of classic analytical methods, in its derived form, with simulation can significantly slow down the process. Non-dimensional, dimensional stability and control derivatives and derivatives in normal form are necessary to estimate [4], [5], [6], [7]. This means 3 equations for each of at least 15 stability and control derivatives for longitudinal simulation. This complexity means time-consuming work with high danger of making mistakes and problems with modification of the model. Classic linear computation has a lot of limitations and simplifications. It is usually valid only for classic conception and in close surrounding of trimmed flight. Non-linearities, as thrust and stall, are not easy to include. Drag and z component are usually neglected. Rotation of the forces from the local velocity vector to flight path a. s. is neglected. The only advantage is analytical solution.

FROM CLASSIC LINEAR THEORY TO LINEAR SIMULATION
Classic analytical-empiric linear aircraft computation requires estimation of stability and control derivatives. For example, horizontal tail influence on the aircraft flight dynamics can be expressed by following equations (do not read it, see only the complexity).
After derivation of these equations by ,̅, ̅, , we obtain horizontal tail part of 8 stability and control derivatives. Each of the derivative is usually composed of wing, fuselage, tail and thrust part. Even if derivatives are very useful for comparison of different aircrafts and elimination of some parameters, the simplicity and clarity of the model can be more important. In case of using basic physics combined with simulation, we can write the same equations for horizontal tail in very simple way. There is no missing information in comparison to previous equations. Body correction constant is added only.
For complete simple longitudinal model of classic piston engine aircraft with simplifications mentioned above we need to add only several following equations: Trimming α for t=0: All the input parameters can be determined from aerodynamic analysis of the wing, body, horizontal tail and elevator and from mass and geometry analysis of an aircraft. Even if the model is very simple, experiment comparison at the end of this study shows, that it is close to measurement. In comparison to up to hundreds of the equations in the classic theory, which express the same model, it is high time save, even if simple solver and trimming algorithm is necessary to use.

NON-LINEAR MODEL
By using basic physics, we got only several equations. We can, therefore, afford level higher model in comparison to linear model. Rotation of the forces to the flight-path axis system is possible, z component as well as drag force can be included and non-linear coefficients can be used also. Every part of an aircraft can be defined by the position and by the forces which act on it. The aerodynamic forces are computed from airflow model and aerodynamic coefficients, estimated from VLM or CFD [8], [9]. Each of the model input characteristics can be linear, non-linear function or

Axis systems and notation
Axis system is defined by an origin point and x-axis angle. Upper index of a symbols describes used axis system. Lower index describes a part of an aircraft. .
Axis system origin symbol is omitted in forces symbols as the forces are independent of its origin. Axis system angle in moment symbols is omitted as the moment is independent of its angular displacement. x-axis of local air-path a. s. has a direction of air velocity vector. x-axis of local body system has a direction of incidence angle at given position, for example propeller revolution axis. If the symbol in upper index is missing, it means Θ orientation and origin. Angles and moments are controlled by right hand rule and dimensions and forces sign by axis system orientation. Plus (+) in upper index means the positive value of the vector in the picture, minus (-) means negative value of the vector in the picture. This mnemotechnic device helps to avoid mistake in derivation. Velocity in local air path axis system has opposite orientation, because we suppose, that the aircraft stands, and the air is moving. Lift and drag force have opposite orientation also, minus must be used in transformation.

Transformation
C. G. is the center of rotation of an aircraft. Therefore, we need to transform all the dimensions, forces and moments from the local systems to body a. s. Transformation of the moment and forces is twostep process. Rotation of forces Figure 2 and movement of moment to C. G. Figure 3. In case, we need to transform against the arrow direction, opposite sign must be used in front of the angle or dimension value in transformation matrixes.
Note: If we transform from local air path system, we must use "minus" in front of Drag and Lift in force and moment transformation.
The schemes in the Figure 2 and Figure 3 was created as a mnemotechnic device, because there is high danger of making a mistake.

Airflow model
Airflow model for longitudinal flight dynamics can be described by undisturbed dynamic pressure velocity ∞ , AoA ∞ , and its changes as dynamic pressure change , and airflow angle change . These changes can be caused by wing -horizontal tail interaction, propeller, wind gust etc. Dynamic pressure change affects a local velocity . Airflow angle change affects a local angle of attack .
In this study we use time delayed downwash angle and angular velocity influence on local velocity and AoA. These characteristics are not usually used, but it helps to avoid using other than local dynamic stability derivatives. and are derived in following chapter.

Dynamic stability derivatives avoiding: Downwash delay and angular velocity influence.
Following part is showed on horizontal tail, but it can be general to all the aerodynamic parts on the aircraft, which can be replaced by the characteristics in one point.

Aerodynamic forces in local air path axis system
Part of an aircraft, which produces aerodynamic forces, as a wing, a fuselage, a horizontal tail, are usually described by aerodynamic coefficients. These coefficients can be a function of its angle of attack AoA, flap deflection , Reynolds number , angular velocity , Mach number , and so on.
, , , , , = ( , , , , , … ) The model in this study allows to define the aerodynamic coefficients by linear function, general function or by a table of values. The reference point is usually close to aerodynamic centre of the part. Angle of attack is referenced to the characteristic line of the part. It can be zero lift line or mean aerodynamic chord.
The forces in local air-path axis system Figure 6 can be described by these formulas.
In case of constant speed propeller, power available , is approximately constant, thrust can be expressed as = , / . Perpendicular force can be also important = − ⋅ ൫ ൯ .
Transformation to body fixed a. s. is necessary Figure 7. .

Equations of motion
The forces, transformed to body fixed axis system, must be summed.

= ∑ = ∑ = ∑
For having simple equations of motion, we can transform summed forces to flight path a. s.: Equation of motion are derived from [10]. In every step we compute new velocity and flight path angle. Velocity perpendicular to flight path velocity is assumed to be zero.
Even if first order numerical solver can be sufficient, fourth order Runge-Kutta solver with error estimation was chosen [10]. The simulation is two steps. Approximate error is determined by first simulation with rough time step. According to the error in rough simulation, the time step is refined.

Trimming by bisection method with interval expansion
Bisection method for trimming was not sufficient. Aircraft model parameters are changing during trimming process, so the solution can move out of the bisection interval. Bisection with interval expansion was developed therefore. Example of use is below.
We look for an angle of attack and we control the power of the engine and elevator deflection, so that forces and moment are at equilibrium.
Control parameters , can be zero.
Choosing minimum and maximum interval.

5.
Computing required change in power and elevator deflection from the moment and X force.

Figure 8: Comparison of experiment with linear and nonlinear simulation
Input characteristics of the models are the same and are derived from analytical methods [4]. Any of the models was not able to catch fast change of the velocity at the 10 s time. Model is probably too simple to describe the phenomena. It can be caused by fuel swing from end to end or local stall during the maneuver on examined aircraft. Both simulations are close to the experiment. Linear simulation estimated better the velocity change. Non-linear estimated better the , Θ angle. The precision of the models is on the same level.

CONCLUSION
The paper describes alternative non-linear simulation of an aircraft and compare it to the linear and an experiment. The precision of the method is comparable to the linear simulation on examined case. Non-linear simulation allows easier modifications of the model. Non-linear coefficients can be included. Simplification of the model is not necessary compared to classic theory. The physics of the model is more obvious, and number of required equations is low. Disadvantage is iterative trimming of an aircraft. Following problems were solved. Dynamic stability derivatives were succeeded by angle of attack modification with angular velocity and by delayed downwash model implementation. Bisection with interval expansion was used for trimming process. Time step refinement was used in the simulation.
Non-linear simulation of the whole aircraft is well known nowadays [11], but it is not usually used at the very beginning phase of conceptional design. Flight stability and control is not solved in the first part of conceptual design very often because it is too complex process. It was shown in introduction chapter, that linear simulation can be very simple process, can be finished in the very first part of conceptional design. Other chapters show, that non-linearities can be included quite easily.