8. Статическая устойчивость

При рассмотрении статической устойчивости, возникает вопрос:, имеет ли возникающий относительно Центра Тяжести аэродинамический момент тангажа тенденцию возвращать самолет к устойчивому углу атаки, или напротив увеличивать его до момента сваливания, при условии что самолет выведен из устойчивого состояния по тангажу? На этот вопрос можно ответить, определив признак (знак) производной dM/d , либо dC_m/d .

Обращаясь к рис. 1.5-4, вращающий момент направлен по часовой стрелке (кабрирование, нос вверх) относительно центра тяжести, и определяется следующм образом:

где . Таким образом коэффициент момента тангажа будет равен:

Коэффициент подъёмной силы для всего самолёта

Мы предположим, что эффективность хвостового стабилизатора .

Коэффициент момента тангажа может быть записан в виде:

С точки зрения безразмерной величины, называемой коэффициентом объема хвостового стабилизатора:

уравнение момента тангажа

.

Условие для продольной статической стабильности:

.

либо,

Таким образом мы должны дифференцировать уравнение момента тангажа относительно и смотреть на знак производной. Это – стандартный приём, направленный на упрощение вычислений, и принимая это:

по определению:

более корректная форма записи (в установленном случае средств управления) будет выглядеть следующим образом:

1.8.1. Стабильно установленные средства управления

Сначала рассмотрели случай устойчивого положения (фиксированной стабильности) ручки управления (рукоятки, рычага), то есть угол отклонения руля высоты установлен. Мы также предположим, что триммер не отклонён, . Вспомним, что коэффициент подъёмной силы руля высоты равен:

Из этого следует (по цепному правилу), что:

используя вышеупомянутое приближение для полного наклона кривой подъёмной силы, получим:

Мы записали их как частные производные, чтобы указать, что они отображают некоторые изменения (например, наклон C_m относительно кривой ) с учётом условия . Кроме того, мы указали “ ”, чтобы подчеркнуть использование приближенного наклона кривой подъёмной силы (хотя и ранее было сделано много других допущений).

Для случая устойчивого положения (фиксированной стабильности) ручки управления (рукоятки, рычага), мы имеем:

.

1.8.1. Стабильность без усилия на ручку управления

Стабильность без усилия на ручку управления означает, что управляющая поверхность находится в свободном плавании, то есть шарнирный момент: . Мы также примем угол . Из этого следует, что:

и что:

как и в разделе 1.7.2. Используя по существу те же самые шаги с этого времени, как в случае установленного ручкой управления положения управляющей поверхности

и следовательно:

.

Опять же, мы используем частную производную, чтобы указать, что на сей раз C_H

(а не ) является константой.

1.8.3. Neutral Points

Вспомним, что для стабильности необходмо:

.

От вышеупомянутых выражений для и очевидно, что

1. Хвостовой стабилизатор имеет стабилизирующий эффект, как замечено знаками –ve [?]

2. Влияние крыла/фюзеляжа на стабилизацию учитывается только в случае h – h_ac < 0; т.е. центр тяжести лежит впереди аэродинамического фокуса соответствующих поверхностей

Для установленной средствами управления статической стабильности:

Эквивалентно

где установленная средствами управления нейтральная точка, определяемая как:

.

Отметим, что это - то же самое выражение, которое было выведено в разделе 1.7 для положения Центра Тяжести, соответствующего нулевому углу триммирования относительно кривой C_L.

Это применяют показать, что более точная формула для установленной средствами управления нейтральной точки, которая принимает во внимание факт, что часть подъёмной силы, произведенной хвостовым стабилизатором

Нейтральная точка без учёта средств управления может быть выражена, приблизительно

Немного более точная формула, аналогичная указанной выше для установленной средствами управления нейтральной точки, может также использоваться если это требуется.

Отметим, что положение Нейтральных Точек зависит от:

1. Наклона кривой подъёмной силы

2. Градиента скоса потока

3. Геометрии самолета (крыла и области стабилизатора, плеча момента хвостового стабилизатора)

Параметры a₁, a₂ всегда положительны, в то время как b₁, b₂ обычно отрицательны. Таким образом, имеем:

стабильность не установленная о.у. < стабильность установленная органами управления (о.у)

1. Static Margin

The distance of the CG ahead of the NP can be regarded as a stability (safety) margin. The quantity is the non-dimensional form of this distance; it is called the Static Margin, denoted by .

The static margin is related to the slope of the C_m vs C_L curve. This can be shown as follows.

Now since, in the controls-fixed case,

it follows that

In other words, the downward (negative) slope of C­_m vs C­_L equals the static margin

As long as the CG is ahead of the NP, i.e.

the aircraft will be longitudinally statically stable.

2. Vehicle Aerodynamic Centre

Since, by definition, the NP is the point about which total pitching moment independent of C_L i.e.

the NP must also be the vehicle's overall aerodynamic centre. The aerodynamic effects can be represented by the lift force L acting at the NP and a constant couple whose value is M; see Fig. 1.8 -2. If the incidence increases, then so too does L. This generates a stabilizing anti-clockwise moment in (a) and a destabilizing clockwise moment in (b).

(a) CG ahead of NP => Stable

(b) CG aft of NP => unstable

Fig. 1.8‑2: Neutral Point (NP) as vehicle AC; for stability, CG must be forward of NP (Direction of flight: right to left)

2. Manoeuvre Stability

In addition to trimming the aircraft longitudinally, the elevator is also used by the pilot to change the flight-path angle, e.g. to pull ‘g’. We consider an idealized pull-up manoeuvre in which the flight path is a circular arc. During such a maneouvre, the aircraft rotates in pitch at a rate q that we assume is constant. This nose-up rotation rate generates downward motion of the horizontal stabilizer relative to the CG and this in turn increases the local AoA, thus increasing the lift and producing a pitch damping effect. Other effects are also present but it is usual to ignore them in the basic analysis. As we shall see, the effect of this damping is to alter (increase) the stability margin of the aircraft. The CG position corresponding to dC_MdC_L=0 in the manoeuvre is known as the Manoeuvre Point (h_m); it slightly aft of the Neutral Point defined in Section Error: Reference source not found.

2. Introduction to Dynamic Response and Stability

   1. Response to Gusts

Lift curve slope dC_L/d

Change in incidence () due to gust

Normal velocity component w=0 for t<0

Lift:

Increase in lift:

Equation of motion: ,

Define time constant:

=>

General solution:

Initial Condition: w(t)=0 at t=0;

Particular solution satisfying IC:

Fig. 2.1‑3: Gust response

Vertical Acceleration

Therefore peak acceleration occurs at t=0

Fig. 2.1‑4: Vertical acceleration response to gust

Gain in height

Fig. 2.1‑5: Height response

2. Short Period Simple Model

It is instructive to investigate the pitching motion caused by a disturbance from trim due to a control surface deflection. We shall assume that the velocity remains constant, and that the only effect of the control input is to cause rotation. In practice, while the primary effect of the elevator deflection will indeed be to create an unbalanced pitching moment that will rapidly generate an angular acceleration, the resulting rotation of the aircraft will lead to changes in the angle of attack, which will cause the lift and drag forces to change, which in turn will give rise to linear accelerations. These will, however, be lagged with respect to the rotational motion.

We shall assume that the angle of attack () and the pitch angle () are the same. We shall neglect downwash lag effects: in other words, the slight time-delay between downwash effects from the main wing reaching the horizontal stabilizer.

Let the aircraft rotate from its trim attitude by a small amount . Let the pitch rate be . Allowing for quasi-steady downwash, the increase in tail-plane incidence due to the attitude change will be . There will be an additional component in tail-plane incidence due to pitch rate, because the tailplane AC will be moving downwards relative to the CG with a velocity equal to where is the distance from the CG to the stabilizer AC. This downward velocity due to rotation will produce an increase in the local angle of attack at the tailplane of approximately rad, so the resultant pitching moment coefficient will be

(Note that we have used the approximate lift-curve slope a to relate C_L to ). Equating moment to rate-of-change of angular momentum, the equation of rotational motion will thus be

which can be written

Provided , this can be expressed in the standard form for a 2^nd order linear differential equation:

.

The damping ratio , the undamped natural frequency and the elevator sensitivity are as follow:

This type of dynamical system can readily be simulated in the Matlab/Simulink environment: see Fig. 2.2 -6. Input to the integrator (“1/s”) block (numbered 3) is the angular acceleration . This is calculated by multiplying the resultant moment (M) by . The output from block (3) is the angular velocity , which is fed into the second integrator block (4), whose output is . Block (7) (‘eta’) represents the elevator input. It is a useful exercise to reconcile this block diagram with the equations derived above.

A matlab m-file that initializes and runs the simulation model is listed in Table 2.2 -1. The program works by setting the various constants, then computing the lift coefficient corresponding to the chosen flight condition, then using the in-built “trim” function to calculate the trim elevator deflection. The m-file then runs the block diagram model staticstab3.mdl in Fig. 2.2 -6, making a 1 deg change to the elevator at time t = 2 seconds. The responses for three different CG positions are shown in Fig. 2.2 -7, Fig. 2.2 -8 and Fig. 2.2 -9. Each of the first two sets of responses shows a typical short period mode or short period pitching oscillation: a reasonably well-damped oscillation with period of about one to two seconds; the third set of responses shows the aperiodic unstable response associated with the unstable (aft CG, negative static margin) configuration.

Fig. 2.2‑6: Computer Simulation of Pitching Dynamics

% staticstab3run.m

% file to initialize static longitudinal stability simulink model staticstab.mdl

a=4.70; % overall lift curve slope [per rad]

a1=4.60; % tailplane lift curve slope [per rad]

a2=3.1; % elevator lift curve slope [per rad]

h=0.1; % dimensionless cg position [times chord behind leading edge]

%h=0.2;

%h=0.32;

hac=0.25; % dimensionless a.c. position

c=2.041; % chord (aerodynamic mean) [m]

CMac=-0.036;% zero lift pitching moment coefficient

S=25; % wing area [m^2]

ST=0.5; % Horizontal stabilizer area

lT=11; % Tail plane lever arm

VT=ST*lT/(S*c); % tailplane volume coefficient = ST*lT/S*c

debyda=0.46; % rate of change of downwash with incidence [nondimensional]

rho=1.225; % air density [kg/m^3]

Iyy=4000; % pitch axis moment of inertia [kg m^2]

theta0=0; % initial pitch attitude [rad]

V=80; % airspeed [m/s]

m=1171; % mass [kg]

g=9.81; % acceleration due to gravity [m/s^2]

w=m*g/S; % wing loading [N/m^2]

hn=hac+VT*(a1/a)*(1-debyda); % Controls-fixed neutral point

Hn=hn-h; % Static Margin

CL=2*w/(rho*V^2); % coefficient of lift

disp(['Specified trim at CL = ',num2str(CL)])

theta0=0;eta0=0; % temporary values pending trim...

[x0,u0,y0]=trim('staticstab3',[],[],CL,[],[],1);

disp(['Trimmed at CL = ',num2str(y0)])

theta0=x0(1);

eta0=u0;

disp(['cg position h = ',num2str(h)])

disp(['Trim pitch attitude = ',num2str(theta0*180/pi),' deg'])

disp(['Trim elevator angle = ',num2str(eta0*180/pi),' deg'])

rad2deg=180/pi;

sim('staticstab3');

figure(1)

subplot(211)

plot(tout,theta*rad2deg,'-',tout,q*rad2deg,'--');

legend('theta','q')

title(['Pitch Response: Static Margin Hn= ',num2str(Hn)]),ylabel('\theta [deg], q [deg/s]'),xlabel('Time [sec]'),grid

subplot(212)

plot(tout,eta*rad2deg);

title('Elevator deflection'),ylabel('\eta [deg]'),xlabel('Time [sec]'),grid

figure(2)

subplot(211)

plot(tout,CM),ylabel('CM'),xlabel('Time [sec]'),grid

subplot(212)

plot(tout,LiftCoef),ylabel('CL'),xlabel('Time [sec]'),grid

Table 2.2‑1: Matlab m-file associated with pitching dynamics simulation model

Fig. 2.2‑7: Short Period Pitching Oscillation: Forward CG (h=0.1)

Fig. 2.2‑8: Short Period Pitching Oscillation: Mid CG (h=0.2)

Fig. 2.2‑9: Unstable Pitching Response: Aft CG (h=0.32)

3. Flight Dynamics Long Period Simple Model

The phugoid is a relatively low-frequency oscillation in which kinetic and potential energies are exchanged. The mode is generally characterized by low damping and approximately constant incidence: i.e. C_L and C_D both constant. In the following simplified analysis, it is assumed that the aircraft is initially trimmed straight-and-level at a horizontal speed V₀. Lift and weight balance, as do thrust and drag.

We now assume that the velocity vector is perturbed from V=(V₀, 0) to V=(V₀+u, w) where u and w are small (relative to V₀) perturbations in the x and z directions in the inertial (i.e. non-rotating and non-accelerating) axis system oxz. Using a local tangent approximation to the quadratic dependency of lift and drag on speed, the lift and drag will increase by small increments

where is the increase in speed. Now the increase in speed is approximately u, as can be seen from the velocity triangle in Fig. 2.3 -10; the hypotenuse and the adjacent are approximately equal for small u and w. This can also be proved a bit more rigorously, as follows.

To a first-order approximation,

where the last step invokes the Binomial Theorem, which states that for . Therefore, the nett speed of the aircraft after being perturbed is, to a first-order approximation,

and hence the change in speed

,

so the increases in lift and drag are

.

The situation is now as depicted in Fig. 2.3 -10. The velocity vector has rotated through a small angle

.

So too have the lift, drag and thrust forces. The change in speed has caused the lift and drag to become

and respectively.

Fig. 2.3‑10: Perturbed Lift and Drag forces in Phugoid Mode

The horizontal and vertical forces acting on the aircraft are

Using the fact that is a small angle we can write

and substituting in the values for L and D we find that the horizontal force

and the vertical force

Invoking Newton’s 2^nd Law,

.

(Note the sign in the last equation; we have taken the velocity component as positive upwards.) These coupled ODE’s can be solved as follows: differentiate the first and substitute from the second to obtain

or equivalently

This can also be written as

.

It can also be shown that

.

Comparing with the standard 2^nd-order polynomial the undamped natural frequency

and the damping ratio

.

Note that, according to this simple model, the damping ratio is inversely proportional to the lift-to-drag ratio; an efficient wing (high C_L/C_D) means a lightly-damped phugoid mode.

4. Axis Transformations

5. Transformations between earth and body axes

To describe the orientation of a rigid aircraft in 3-D, three rotation variables are required. It is customary to employ the yaw, pitch, roll Euler angles defined in the Fig. 2.5 -11 et seq for this purpose. The aircraft (x,y,z) axes are initially alligned with the earth-based inertial axes (x_E,y_E,z_E) in which z_E is vertical and x_E points along a reference heading: for example, North.

Fig. 2.5‑11: Yaw Angle: Rotation  about z_E

Fig. 2.5‑12: Pitch Angle: Rotation  about y1

Fig. 2.5‑13: Roll Angle: Rotation  about x2

The basis vectors in the three frames of reference are related as follows.

So for example, a vector whose components are (a,b,c) in frame 1 has components

in the inertial frame (x_E,y_E,z_E).

Frames 1 and 2

and

and

i.e.

so vector whose components are (d,e,f) in frame 2 has components

in frame 1.

Frames 2 and 3

and

and

INCOMPLETE…..

6. Angles of Attack and Sideslip

The angle of attack (AoA) is defined as the angle  between the vehicle longitudinal axis ox and the projection onto the oxz plane of the aircraft’s velocity vector relative to the wind; see Fig. 2.6 -14. Thus

The angle of sideslip is usually defined as the angle  between the vehicle longitudinal axis ox and the projection onto the oxy plane of the aircraft’s velocity vector relative to the wind. Thus

It is useful to derive expressions for the small perturbations in and that result from small changes in u, v and w. Differentiating the equation for tan ,

=>

=>

Hence for small but non-infinitessimal changes,

Differentiating the equation for tan ,

=>

=>

Fig. 2.6‑14: Angles of Attack and Sideslip

3. Dynamics of a Rigid Aeroplane moving in 3-D

The equations of motion of a rigid aircraft moving in 3-D are conveniently described in terms of a body-fixed coordinate system located at and moving with the CG and rotating with the airframe. In terms of this coordinate system, the velocity vector of the CG is

Its acceleration vector is

Since the basis vectors themselves rotate with the aircraft with an angular velocity vector

they are not constant and hence have, themselves, to be differentiated when working out the acceleration. The derivative of a rotating unit vector must be perpendicular to the vector itself. It can be shown that if (i, j, k) are unit vectors rotating with an angular velocity  then

, and

From this, it follows that the acceleration

In component form,

According to Newton’s 2^nd Law, the equation governing the translational motion of the aircraft is thus

where the terms on the LHS represent the aerodynamic and the propulsive forces and the weight, respectively.

The rotational dynamics of a rigid body can be described in terms of the angular momentum (H) and the resultant moment vector (M) about the mass centre. The moment vector about the CG is made up of components due to aerodynamic and propulsive effects; we shall denote M’s components by (L, M, N). From Newton’s Laws it can be shown that

The angular momentum vector H can be shown to be

where is the inertia matrix (tensor). For a general rigid body, the inertia matrix has the form

.

Its symmetric form means that it is completely defined in terms of six distinct elements. The terms on the leading diagonal (I_xx, I_yy and I_zz) are called the moments of inertia. The off-diagonal terms (I_xy, I_xz and I_yz) are the products of inertia. These six inertia terms, defined in terms of elemental mass elements (dm) and their location relative to the body axis coordinates (x,y,z) as follows

Moments of inertia: , ,

Product terms: , ,

describe the mass distribution of the rigid aircraft. In each case the integration is carried out over the entire body. For most aircraft, oxz is a plane of symmetry. This means that an element of mass located at (x, y, z) will be balanced by an equal mass at (x, -y, z), hence

This gives rise to the following simplification that is often used in Flight Dynamics:

The angular momentum is thus

and its rate of change is

Note that we have again used the ‘ ’ operator, because the vector H that we are differentiating is expressed in terms of a rotating coordinate system. Note, too, that since the coordinate system rotates with the airframe, the inertia terms are all constant.

The rate of change of angular momentum can thus be written

Thus, after some simplification, we obtain the Euler Equations for the symmetric aircraft

It is sometimes useful to decouple the roll-yaw Euler equations in the following sense. First, write them in the form

then solve, viz

The equations of motion are nonlinear differential equations. If the forces and moments are known functions of , , Mach numer, etc, then the equations can be solved numerically. Computer tools such as Matlab/Simulink make this relatively straight-forward. They also allow for the systems of nonlinear equations to be numerically trimmed and linearized.

4. Approximate Equations of Motion

   1. 3-DOF Longitudinal Dynamics

It is instructuive to derive the simplified equations of longitudinal motion of an aircraft in balanced flight. The angles and are the angle of attack and the pitch attitude, respectively; see Fig. 4.1 -15. We shall assume that the velocity vector V = (u, 0, w) (i.e. lies in the oxz plane), that  = p = r = = 0, and that the aircraft flies through still air. To simplify the analysis, we shall further assume that the propulsive force F acts along the x-axis and we shall neglect its contribution to the pitching moment and its dependency on all other variables except throttle setting; in short, we shall treat thrust F as constant.

Fig. 4.1‑15: Balanced flight: velocity vector V = (u, 0, w)

Subject to the above assumptions, the equations of motion are as follow:

Translational Motion:

Rotational Motion:

Also,

This allows us to write

The aerodynamic force components X and Z and the pitching moment M about the CG are functions of V, , (or equivalently of u, w), q and , i.e.

, ,

Let the aircraft be trimmed at a some speed V_e and flight-path angle _e where

and where the subscript ‘e’ denotes equilibrium (trim) value. This means that

Writing the velocity components, the pitch attitude and the elevator deflection in terms of perturbations (u, w, ) about their equilibrium values as follows

, , ,

it follows that

.

The force components can be expanded using Taylor’s Theorem. If the perturbations are assumed small enough that terms above 1^st order can be neglected, including products of small quantities such as , it follows that

Using the equilibrium force balance equations above and assuming  is small, it follows that

This is usually written in the form

where the following X-force dimensional stability derivatives are defined:

; ; ;

and where

is the dimensional elevator-to-X-force control derivative.

When working with linearized perturbation equations of motion it is customary to drop the ‘’ and to write (for instance) ‘u’ instead of ‘u’, so we shall write

It must, however, be understood that the linearized equations relate to small perturbations about trim.

The Z-force equation can be dealt with in a similar manner.

which implies that

and hence

This will be written as

where the Z-force stability derivatives are defined as follow:

; ; ;

and the elevator Z-force elevator control derivative is

.

Using similar approximations, the pitch rate acceleration can be written as

where

; ; ;

The above three equations for the longitudinal dynamics, plus the kinematic relationship between  and q, can be combined in the following state-space form:

This is often written as

where

= state vector

= control vector

= state transition matrix

= control coupling matrix.

Stability and Control Derivatives play an important role in the study of aircraft dynamics. Their numerical values can often be estimated from wind-tunnel tests, from full-scale flight tests, and from theory. Derivatives for a wide variety of aircraft have been put into the public domain by NASA, for use by researchers; see, for example, Teper (1969) and Heffley & Jewell (1972). Similar data for a number of helicopters are in Heffley et al (1979). Stability and control derivatives may be presented in one of several different dimensional or non-dimensional forms, and it is vital before using them properly to understand how the data are presented.

2. 3-DOF Lateral-Directional Equations

Let us assume that the aircraft is trimmed with p = r = 0, w = w_e and u = u_e both constant, pitch attitude a bank angle = . The equation governing lateral translational motion is

In the trim,

and for perturbations about trim,

The aerodynamic side force Y will be of the form

where the side force coefficient is a function principally of sideslip (), but also with dependency on roll and yaw rates (p, r) and as well as on rudder () and aileron () deflections, i.e.

Hence for small perturbations,

It is straight-forward to show that

and that

and hence that

The first term on the RHS

combines two effects, the first due to the slight increase in speed associated with the sideslip v at constant u and w, the second due to the increase in sideslip at constant speed.

The equation governing the small perturbation lateral translational motion becomes

which can be rearranged as

5. References

Anonymous: http://www.aerospaceweb.org/question/design/q0289.shtml. Last Accessed: 09 Aug 2008

Cook, M.V. (1997) Flight Dynamics Principles. Arnold, London.

Heffley, R.K and Jewell, W.F. (1972), “Aircraft Handling Qualities Data” NASA CR 2144

Heffley, R.K, Jewell, W.F., Lehman, J.M. and Van Winkle, R.A. (1979) “A compilation and analysis of helicopter handling qualities data.” Vol 1 (Data Compilation, NASA CR 3144) and Vol 2 (Data analysis, NASA CR 3145)

Kuethe, A.M. and Schetzer, J.D. (1964). (2^nd Edition) Foundations of Aerodynamics. John Wiley and Sons, New York

Russell, J.B. (1996) Performance & Stability of Aircraft. Arnold, London.

Seckel, Edward (1964). Stability and Control of Airplanes and Helicopters, Academic Press, New York

Teper, G.L. (1969), “Aircraft stability and control data”, NASA CR 96008