Linearization Ch08

7/30/2019 Linearization Ch08

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Chapter 8

Linearization

8.1 General

We have derived twelve nonlinear, coupled, first order, ordinary differentialequations that describe the motion of rigid aircraft in a stationary atmo-sphere over a flat Earth: the equations of motion. Analytical solutions tothe equations of motion are obviously not forthcoming, so other means ofsolving them must be sought.

One such means is through numerical integration. There are several al-gorithms available that will allow the differential equations to be propagated

forward in discrete time steps. At the beginning of each time step the entireright-hand side of each equation x = . . . is evaluated, yielding the rate ofchange of x at that instant. Then, loosely, dx/dt is replaced by x/t andthe change x is approximated over the interval t. With sufficiently fastcomputers this can be done in real-time, and is the basis for flight simula-tion. Numerical integration will generate time-histories of the aircraft motionin response to arbitrary initial conditions and control inputs. These time-histories may then be analyzed to characterize the response in familiar termssuch as its frequency and damping.

Alternatively, one may consider small inputs and variations in initial con-

ditions applied at and about some reference (usually steady) flight condition.The advantage to this approach is that for suitably small regions about thesteady condition, all the dependencies may be considered linear. This will re-sult in twelve linear (though still coupled) ordinary differential equations forwhich analytical solutions are available. The process is called linearization

131


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132 CHAPTER 8. LINEARIZATION

of the equations.

Numerical integration does not require special inititial conditions or con-trol inputs; it simply approximates what happens in real life. Linearizationbrings with it the assumption that all ensuing motion will be close to thereference steady-state flight condition. (This assumption is sometimes wan-tonly disregarded and the linearized equations are treated as if they werethe equations of motion.)

Because linearization requires us to stay close to the steady flight condi-tion, it is generally reasonable to neglect altitude effects in the equations ofmotion. Therefore, none of the three navigation equations need be consid-ered, nor does the kinematic equation for (or ). We will therefore focuson the six motion variables and the remaining two Euler angles for linear

analysis.

8.2 Taylor Series

A familiar example of the linearization of a nonlinear ordinary differentialequation involves a pendulum of length , for which motion about = 0 isdescribed by

m = mg sin This second order equation may be replaced by two coupled first order

equations by introducing the angular velocity variable , whence

=

= g sin /The only nonlinearity is in the term sin , which is normally linearized by

applying the small angle approximation, sin , so

= = g/

More formally, such results as the small angle approximation may beobtained from a power series representation of a function by retaining only


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8.3. NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS 133

terms through the first power of the variables. Thus the standard series

expansion of sin about = 0 is

sin = 3/3! + 5/5! sin

In order to deal with functions of several variables it is convenient to usefor the series expansion the Taylor series representations of the functions.For a function of a single variable, f(x), about some reference value xRef wehave

f(x) = f(xRef) + df(x)dx

Ref

x + d2

f(x)dx2

Ref

x2

2!+ d

3

f(x)dx3

Ref

x3

3!+

In this expression, x = xRef + x. So for example, f(x) = sin x aboutxRef = 0 yields

sin x = sin xRef + cos xRefx sin xRefx2/2! cos xRefx3/3! + = x x3/3! + x5/5!

Or, through the first order term, sin x x.A Taylor series for a function ofn independent variables, f(x1, x2, . . . , xn)

about reference conbditions (x1, x2, . . . , xn)Ref, through the first order termsis:

f(x1, x2, . . . , xn) f(x1, x2, . . . , xn)Ref+

f

x1

Ref

x1 +f

x2

Ref

x2 +

+f

xn

Ref

xn + H.O.T.

0

(8.1)

8.3 Nonlinear Ordinary Differential Equations

To be as general as possible we will define the function to be linearized bymoving everything to the left-hand side of the equation and equating it to


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zero. For example, the body axis kinematic equation for bank angle will be

defined as

f( ,,,p,q,r) p (qsin + r cos )tan 0There are two advantages to this approach. First, the function evaluates

to zero for any reference condition, and therefore the first term in the Taylorseries will always vanish. Second, even though we have solved the equationsof motion for the explicit derivatives of the variables, some of the forces andmoments may depend on these derivatives. By defining the functions asshown all occurences of such terms will be accounted for when derivativesare taken.

The reference conditions we will use will be those previously discussedin 7.4 for steady flight. Mathematically this is not necessary, but for ourpurposes it is. In any event the meaning of terms such as x should alwaysbe construed to mean (dx/dt) and not d(x)/dt or the reference conditionsof x will be lost, since

d(x)/dt = d(x xRef)/dt = dx/dtand on the other hand,

(dx/dt) = dx/dt (dx/dt)RefOf course, if (dx/dt)Ref = 0 the two are the same.

8.4 Systems of Equations

The formal procedure for dealing with several equations in several variablesis as follows. First, three vectors x, x, and u are defined. The x vectorrepresents each of the variables that appear as derivatives, the x vector rep-

resents each of those variables as a derivative, and the u vector representsall of the controls. If n state variables and m controls are being considered,then x and x are nvectors and u is an mvector. There will be n ordinarydifferential equations which are placed in the form fi(x, x, u) = 0, i = 1 . . . n.A vector-valued function f(x, x, u) is formed by considering the n functions


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8.4. SYSTEMS OF EQUATIONS 135

thus defined as a vector. We then write the Taylor series of f(x, x, u) through

the first order terms,

f(x, x, u) =f

x

Ref

x +f

x

Ref

x +f

u

Ref

u = 0

Here we have used f(x, x, u) = 0 and f(x, x, u)Ref = 0 by definition. Thederivative of the nvector of functions f(v) with respect to a pvector v isa Jacobian and is defined as

fv

f1/v1 f1/v2 f1/vpf2/v1 f2/v2 f2/vp

......

......

fn/v1 fn/v2 fn/vp

Thus, f/x and f/x are square nn matrices and f/u is an nm

matrix. The matrix f/x is generally non-singular. We then solve for thevector x,

x =

f

x1

Ref

f

xRef

x + f

uRef

u (8.2a)This equation is often written

x = Ax + Bu (8.2b)

With the obvious meaning of A and B.To see how this works, consider our pendulum problem, to which we add

an externally applied torque M to be used as a control,

= = g sin / + M

The two derivative terms are and (n = 2) and the single control is M(m = 1). We therefore define


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x =

x1

x2

, u = {u1} {M}

Two scalar functions are defined to constitute the vector-valued functionf(x, x, u):

= 0 f1 (x, x, u) x1 x2 = 0 + g sin / M = 0 f2 (x, x, u) x2 + g sin x1/ u1 = 0

f(x, x, u)

f1 (x, x, u)

f2 (x, x, u)

=

x1 x2

x2 + g sin x1/ u1

=

0

0

The derivatives are calculated and then evaluated at reference conditionsxRef = xRef = 0 and MRef = 0 (from which also x = x, x = x, andu = u):

f

x f1/x1 f1/x2

f2/x1 f2/x2 = 1 0

0 1

f

x

f1/x1 f1/x2

f2/x1 f2/x2

=

0 1

g cos x1/ 0

,

f

x

Ref

=

0 1

g/ 0

f

u

f1/u1

f2/u1

=

0

1

The final result becomes

x =

0 1

g/ 0

x +

0

1

u


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8.5. EXAMPLES 137

This is identical to the two linearized scalar equations

=

= (g/) + M

The motivation for this approach proceeds from the similarity of the setof linear, ordinary differential equations in x = Ax + Bu with its scalarcounterpart with forcing function u, x = ax + bu. Since solutions to thescalar equations are quite well known, it is reasonable to think that we canfind similar solutions to the systems of equations.

The problem with this approach from our point of view is that we have

eight equations with four controls to linearize, so f/x and f/x will be8 8 matrices and f/u will be an 8 4 matrix. However, in the formwe have derived the equations of motion, in which we have already solvedfor the explicit derivative term, the matrix f/x will be very nearly theidentity matrix (non-unity on the diagonal and non-zero off-diagonal entrieswill come from force and moment dependencies on , and possibly ).

It is far simpler to linearize each equation as a scalar function of severalvariables and deal with and dependencies as special cases. On the otherhand the vector-matrix form will be very useful later, so after treating eachequation as a scalar problem, we will assemble them into the form of x =Ax + Bu.

8.5 Examples

8.5.1 General

In order to proceed we need to specify two things:

1. The state and control variables to be used, and

2. The reference flight condition at which the equations are to be evalu-

ated.

For purposes of discussion we will select the body axis velocities u, v, w,p, q, and r plus the two body axis Euler angles and for our states, andthe four generic controls T, , m, and n.


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Since we have picked a body axis system it is important to state which

one. The analysis is simplified somewhat by using stability axes. In that case,xS is the projection of the velocity vector in the reference flight condition ontothe plane of symmetry. Thus in the reference condition the angle betweenthe velocity vector and the xaxis is zero, we have

Ref = wRef = 0 (Stability axes)

This is true for any reference condition if stability axes are chosen. Forour reference flight condition we take steady, straight, symmetric flight. Asa consequence we have for reference conditions

uRef = vRef = wRef = pRef = qRef = rRef = Ref = Ref = 0

vRef = Ref = pRef = qRef = rRef = Ref = 0

Because vRef = wRef = 0, and because VRef =

u2Ref + v2Ref + w

2Ref, we

also have

uRef = VRef

We assume the speed and altitude at which the analysis is to be performedhave been specified, as has the climb (or descent) angle Ref. The use ofstability axes in steady, straight, symmetric flight also means that

Ref = Ref

In the force and moment dependencies we need the definitions

q =1

2V2

p =pb

2Vr =

rb

2V

q =qc

2V =

c

2V


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8.5. EXAMPLES 139

Because our linear velocity variables are u, v, and w, we also need, for

our force and moment dependencies,

V =

u2 + v2 + w2

= tan1 (w/u)

= sin1

v/

u2 + v2 + w2

In taking the partial derivatives we will apply the chain rule frequently,needing to evaluate in the end some partial derivatives repeatedly. Most arezero when evaluated in steady, straight, symmetric flight (see appendix B.1).

The derivatives are as follows:

Ref

V p q r q

u 1 0 0 0 0 0 0 RefVRef

v 0 1VRef 0 0 0 0 0 0

w 0 0 1VRef 0 0 0 0 0

p 0 0 0 b2VRef

0 0 0 0

q 0 0 0 0 c2VRef

0 0 0

r 0 0 0 0 0b

2VRef 0 0 0 0 0 0 0 0 c

2VRef0

We will encounter derivatives relating to thrust and velocity in the forceequations. The basic relationship is

T = qSCT

V , T

Recall that the functional dependency ofCT on V was introduced to allow

for the fact that thrust itself may sometimes be assumed to be invariantwith airspeed (e.g., rockets and jets). An analogous result obtains for certainengines assumed to have constant thrust-horsepower, modeled as a constantproduct T V. The derivations for CTV for both cases are at appendix B.1,and result in:


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CTV = 2CTRef (Constant Thrust)CTV = 3CTRef (Constant Thrust-Horsepower)

We will meet a few other nondimensional groupings in the process oflinearizing the equations of motion:

t tc/ (2VRef)

(Time)

D() c

2VRef

d()

dt (Differentiation)

m mSc/2

(Mass)

Iyy IyyS(c/2)3

(Moments of Inertia)

Ixx IxxS(b/2)3

Izz IzzS(b/2)3

Ixz IxzS(b/2)3

A b/c (Aspect ratio)

We will adopt the following convention to represent partial derivatives offorces and moments with respect to their independent variables, evaluatedat reference flight conditions. IfX is any force or moment that is a functionof y, then:

Xy XyRef

This notation is by no means standard; see below, 8.6 Customs andConventions.


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8.5. EXAMPLES 141

8.5.2 A Kinematic Equation

To complete the linearization of the function previously defined,

f( ,,,p,q,r) p (qsin + r cos )tan 0

The linearization proceeds using equation 8.1 as

f = fRef +f

Ref

+f

Ref

+ + fr

Ref

r

Since fRef = 0 by definition, and since all the s except are measuredfrom zero reference conditions, we have the linearized equation,

f = (0) + (1) + [(qcos + r sin )tan ]Ref+ [(qsin r cos )sec2 ]Ref(1)p (sin tan )Ref q (cos tan )Ref r

Evaluating the reference conditions yields

= p + r tan Ref (8.3)

In terms of the nondimensional roll and yaw rates,

=2VRef

b

pb

2VRef+

rb

2VReftan Ref

=

2VRefb

(p + r tan Ref)

This may be written as

d/dt2VRef/b

= p + r tan Ref

The term in parentheses represents one way to nondimensionalize theoperator d/dt. In fact, Etkin in the second edition of Dynamics of Flight Stability and Control did exactly that, dividing time by b/2VRef in the


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lateral-directional and by c/2VRef in the longitudinal equations. Here we have

adopted the Etkins earlier definition (inDynamics of Atmospheric Flight

) ofnondimensionalizing time by the divisor c/2VRef. This introduces the aspectratio A into the equation, since

d/dt

2VRef/b

=

d/dt

2VRef/c

b

c

= AD()

The completely nondimensional form of the bank angle equation is there-fore

D() =

1

A (p + r tan Ref) (8.4)

8.5.3 A Moment Equation

For this example we consider the body-axis rolling equation (multipliedthrough by ID),

ID p = Izz [L + Ixzpq (Izz Iyy) qr] + Ixz [N Ixzqr (Iyy Ixx)pq]Part of the linearization is easy: the explicit p, q, and r dependencies pose

no problem. The rolling and yawing moments are problematic. We need toget all the dependencies down to either states, controls, or constants. Sincethe aerodynamic data are normally available in coefficient form, we also needto relate to those. The rolling and yawing moments are:

L = qSbC (, p, r, , n)

N = qSbCn (, p, r, , n)

The only states that do not appear are and , and the two controls onwhich we have dependencies are and n. We therefore define the functionfor linearization as

fp ( p, u, v, w, p, q, r, , n) ID p IzzL IxzN+ Ixz (Ixx + Iyy Izz)pq+

Izz (Iyy + Izz) I2xz

qr = 0


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8.5. EXAMPLES 143

Before beginning the derivatives, we note that dependencies on u and w

appear only through the moments L and N (either through q, , p, or r). Wehave already determined derivatives of q, , p, and r with respect to u andw, and in particular only q/u is non-zero in the reference flight condition.However, that term will be multiplied by CRef, which is zero (LRef = 0). Weconclude therefore that when evaluated in steady, straight, symmetric flight,

fp/u = fp/w = 0

The other partial derivatives go as follows:

fpp

Ref

= ID = IxxIzz I2xzfpv

Ref

= IzzLv IxzNvfpp

Ref

= IzzLp IxzNp (Using qRef = 0)

fpq

Ref

= 0 (Using pRef = rRef = 0)

fprRef

= IzzLr IxzNr (Using qRef = 0)fp

Ref

= IzzL IxzNfpn

Ref

= IzzLn IxzNn

We may now write out the linearization of the rolling moment equationin dimensional form as follows:

p = 1ID

[(IzzLv + IxzNv) v + (IzzLp + IxzNp)p

+ (IzzLr + IxzNr) r + (IzzL + IxzN)

+ (IzzLn + IxzNn) n]

(8.5)


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If the stability axes coincide with the principal axes this simplifies to

p =1

Ixp(Lvv + Lpp + Lrr + L + Lnn) (Principal Axes)

The partial derivatives of the moments are in dimensional form. Wemay at this point take the linearized equation as-is, and evaluate each of thefactors on the right-hand side at some given altitude and speed. Alternativelywe may re-write the equation using the nondimensional derivatives. Therequired derivatives are straightforward; for example, we have:

Lv = Lv

Ref

=[qSbC (, p, r, , n)]

v

Ref

= Sb

C

q

v+ q

C

v+

Cp

p

v+

Cr

r

v

Ref

=

qRefSb

VRef

C

In Lv, CRef = 0, p/v|Ref = r/v

|Ref = 0, and /v

|Ref = 1/VRef.

The end result of derivatives of L and N with respect to the states andcontrols is that the non-zero expressions are:

Ref

L N

vqRefSb

VRef

C

qRefSb

VRef

Cn

pqRefSb

2

2VRef

Cp

qRefSb

2

2VRef

Cnp

r

qRefSb

2

2VRef Cr

qRefSb

2

2VRef Cnr

(qRefSb) C (qRefSb) Cnn (qRefSb) Cn (qRefSb) Cnn

(8.6)

When related to the nondimensional coefficients we have (see appendixB.2):


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8.5. EXAMPLES 145

D(p) = 1AID

IzzC + IxzCn

+

IzzCp + IxzCnp

p

+

IzzCr + IxzCnr

r +

IzzC + IxzCn

+

IzzCn + IxzCnn

n (8.7)

In this expression A is the aspect ratio and ID = IxxIzz I2xz. If thestability axes coincide with the principal axes,

D(p) =1

AIxp C+ Cp p + Cr r + C + Cnn (Principal Axes)8.5.4 A Force Equation

For this example we take the body-axis Z-force equation,

w =1

m(Z+ Tsin T) + g cos cos + qu pv

Define the function

fw ( w,u,v,w,p,q,r,,, T, m, n) = w 1m

(Z+ Tsin T)

g cos cos qu + pvSince this is a longitudinal equation, the presence of dependencies on

lateral-directional independent variables v, p, r, , and n is not desired.The dependencies on v and are explicit. The dependencies on v, p, r, andn arise through the force dependencies, which are

T = qSCT V , TZ = qSCZ

The latter is technically complicated by the mixed system of wind- andbody-axis forces (see equation 6.1). It may be shown, however, that lineariz-ing the correct relationship for CZ in a symmetric flight conditions, yields


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the same result as linearizing the simplified relationship (equation 6.2) in the

same flight condition. That is, the linearization of

CZ = CD sin sec CY sin tan CL cos

leaves no terms involving lateral-directional independent variables (v, p, r,, , or n) when evaluated at the reference flight condition, and gets thesame result as setting = 0 and linearizing

CZ =

CD sin

CL cos

In general, however, one should never apply reference conditions prior totaking the derivatives.

The remaining derivatives evaluate as follows:

fww

Ref

= 1 Zw/m

fwu

Ref

= (Zu + Tu sin T) /m (Using qRef = 0)

fww

Ref

= Zw/m (Using V/w = 0 in CT)

fwq

Ref

= Zq/m VRef (Using uRef = VRef)

fw

Ref

= g sin Ref (Using cos Ref = 1)

fwT

Ref

= TT sin T/m

fwmRef

= Zm/m

The dimensional form of the linearized equation is then


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8.6. CUSTOMS AND CONVENTIONS 147

w = 1m Zw [(Zu + Tu sin T) u + Zww + (Zq + mVRef) q

mg sin Ref + TT sin TT + Zmm](8.8)

Nondimensional derivatives are evaluated at appendix B.3, and are:

Ref

Z T

w qRefSc

2V2Ref

CL 0

u qRefSVRef 2CLRef MRefCLM qRefS

VRef 2CTRef + CTVw

qRefS

VRef

CDRef + CL

0

q qRefSc

2VRef

CLq 0

m (qRefS) CLm 0T 0 (qRefS) CTT

(8.9)

The nondimensional form of the equation (see B.4) becomes

D() =1

2m + CL (2CW cos Ref + CTV sin T MRefCLM) V CDRef + CL + 2m CLq q CW sin Ref

+CTT sin TT CLmm (8.10)

8.6 Customs and Conventions

8.6.1 Omission of.

In the linearized equations it is customary to drop the symbol whetherthe reference conditions are zero or not. In general any linear differential

equation appearing in flight dynamics will have been obtained through thelinearization of more complicated equations, and it may be assumed that allthe variables are small perturbations from some reference. After the equationsare solved it is important to remember to add the reference values to the sobtained in order to get the actual values.


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8.6.2 Dimensional Derivatives.

The convention we adopted for dimensional derivatives is consistent withEtkins usage, in which

Xy Xy

Ref

Several authors use the same notation, but include in the definition di-vision by mass or some moment of inertia. Thus one frequently sees, forexample,

Zw Z/wm

Ref

, Lp L/pIxx

Ref

,

In the force equations, and in the pitching moment equation, such nota-tion marginally simplifies the expressions. However, in the rolling and yawingequations, unless the analysis is performed in principal axes, the notation ac-tually complicates the resulting expressions.

8.6.3 Added Mass.

Any time a force or moment is dependent upon a state-rate (such as or) the result is a modification to the mass or moment of inertia factor ofthe derivative term in the linearized equation. Thus we had the expression(m Zw) w instead of simply mw in the Zforce equation. Such terms as Zware often referred to as added mass parameters (they are usually negative) forobvious reasons. In ship dynamics several such terms arise, and the intuitivedescription usually offered is that the terms represent the mass of water beingdisplaced by the ships motion.

8.7 The Linear Equations

It is important to remember that the linearization was performed at a partic-ular reference flight condition: steady, straight, and symmetric. If any otherflight condition is to be analyzed the linearization will have to be performedover again.


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8.7. THE LINEAR EQUATIONS 149

8.7.1 Linear Equations

Dimensional Longitudinal Equations

u =1

m[(Xu + Tu cos T) u + Xww mg cos Ref

+TT cos TT + Xmm] (8.11a)

w =1

m Zw [(Zu + Tu sin T) u + Zww + (Zq + mVRef) q

mg sin Ref + TT sin TT + Zmm] (8.11b)

q =1

Iyy

Mu +

Mw (Zu + Tu sin T)

m Zw

u +

Mw +

MwZwm Zw

w

+

Mq +

Mw (Zq + mVRef)

m Zw

q

mgMw sin Refm Zw

+

MwTT sin T

m Zw

T +

Mm +

MwZmm Zw

m

(8.11c)

= q (8.11d)

Dimensional Lateral-Directional Equations

v =1

m[Yvv + Ypp + (Yr mVRef) r + mg cos Ref + Ynn] (8.12a)

p = 1ID

[(IzzLv + IxzNv) v + (IzzLp + IxzNp)p

+ (IzzLr + IxzNr) r + (IzzL + IxzN)

+ (IzzLn + IxzNn) n] (8.12b)


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r = 1ID

[(IxzLv + IxxNv) v + (IxzLp + IxxNp)p

+ (IxzLr + IxxNr) r + (IxzL + IxxN)

+ (IxzLn + IxxNn) n] (8.12c)

= p + r tan Ref (8.12d)

Nondimensional Longitudinal Equations

D(V) = 12m

(2CW sin Ref MRefCDM + CTV cos T) V

+

CLRef CD

CW cos Ref+CTTT CDmm

(8.13a)

D() =1

2m + CL

(2CW cos Ref + CTV sin T MRefCLM) V

CDRef + CL

+

2m CLq

q CW sin Ref

+sin T

CTT

T

CLm

m (8.13b)

D(q) =1

Iyy

MRefCmM

+Cm (2CW cos Ref + CTV sin T MRefCLM)

2m + CL

V

+

Cm

Cm

CDRef + CL

2m + CL

+

Cmq +

Cm

2m CLq

2m + CL

q

CWCm sin Ref2m + CL

+

Cmm CmCLm2m + CL

m

(8.13c)

D() = q (8.13d)


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Nondimensional Lateral-Directional Equations

D() =1

2m

CY+ CYp p + (CYr 2m/A) r

+CW cos Ref + CYnn

(8.14a)

D(p) =1

AID

IzzC + IxzCn

+

IzzCp + IxzCnp

p

+

IzzCr + IxzCnr

r +

IzzC + IxzCn

+

IzzCn + IxzCnn

n

(8.14b)

D(r) =1

AID

IxzC + IxxCn

+

IxzCp + IxxCnp

p

+

IxzCr + IxxCnr

r +

IxzC + IxxCn

+

IxzCn + IxxCnn

n

(8.14c)

D() = 1A (p + r tan Ref) (8.14d)

8.7.2 Matrix Forms of the Linear Equations

Before we place these equations in the form x = Ax + Bu, we notethat equations involving derivatives of longitudinal states are functions onlyof longitudinal states and controls, and that equations involving derivativesof lateral/directional states are functions only of lateral/directional statesand controls. That is, for the dimensional equations, we have differentialequations for u, w, q, and that are functions of the states u, w, q, and

the controls T and m. Similarly, the differential equations for v, p, r, and are functions of the states v, p, r, , and the controls and n. The sameholds for the nondimensional equations except the states are V, , q, and for the longitudinal equations, and , p, r, and for the lateral/directionalequations.


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Thus the longitudinal equations are decoupled from the lateral/directional

equations, and vice-versa. Instead of dealing with eight equations all atonce, we may break them up into two sets of four equations each. Thereforedefine the longitudinal state and control vectors xLong and uLong, and thelateral/directional state and control vectors xLD and uLD:

xLong

u

w

q

, uLong

T

m

xLD

v

p

r

, uLD

n

Likewise we have the nondimensional states xLong and xLD:

xLong

V

q

xLD

p

r

Control deflections are normally defined as non-dimensional quantities(throttle from zero to one, and radians for flapping surfaces, for example),so there is no need for separate definitions of uLong or uLD.

Further assumptions. For our subsequent purposes it is sufficient to fur-ther simplify the linear equations of motion with four assumptions:

1. The aircraft is in steady, straight, symmetric, level flight (SSSLF), orRef = 0.

2. The engine thrust-line is aligned with xB, so T = 0.

3. The body-fixed coordinate system is aligned with the principal axes,so Ixz = 0.

4. There are no aerodynamic dependencies on (or w).

While these assumptions are not necessary, they do permit us to focus onthe more significant effects typically seen in the study of aircraft dynamicsand control.


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Dimensional Longitudinal Equations (SSSLF, T = 0)

xLong = ALongxLong + BLonguLong (8.15a)

xLong

u

w

q

, uLong

T

m

(8.15b)

ALong =

Xu+Tum

Xwm 0 g

Zum

Zwm

Zq+mVRefm 0

MuIyy

MwIyy

MqIyy

0

0 0 1 0

(8.15c)

BLong =

TTm

Xmm

0Zmm

0 MmIyy0 0

(8.15d)

Nondimensional Longitudinal Equations (SSSLF, T = 0)

xLong = ALongxLong + BLonguLong (8.16a)

xLong

V

q

(8.16b)


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ALong =

MRefCDM+CTV

2mCLRefCD

2m 0 CW

2m2CWMRefCLM

2m

CDRefCL

2m

2mCLq2m

0MRefCmM

Iyy

CmIyy

Cmq

Iyy0

0 0 1 0

(8.16c)

BLong =

CTT2m

CDm2m

0CLm

2m

0CmmIyy

0 0

(8.16d)

Dimensional Lateral-Directional Equations (SSSLF,Ixz = 0)

xLD = ALDxLD + BLDuLD (8.17a)

xLD

v

p

r

, uLD

n (8.17b)

ALD =

Yvm

Ypm

YrmVRefm

gLvIxx

LpIxx

LrIxx

0NvIzz

NpIzz

NrIzz

0

0 1 0 0

(8.17c)

BLD =

0

Ynm

LIxx

LnIxx

NIzz

NnIzz

0 0

(8.17d)


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Nondimensional Lateral-Directional Equations (SSSLF, Ixz = 0)

xLD = ALDxLD + BLDuLD (8.18a)

xLD

p

r

(8.18b)

ALD =

Cy2m

Cyp2m

Cyr2m/A

2mCW2m

C

AIxx

Cp

AIxx

CrAIxx

0Cn

AIzz

Cnp

AIzz

CnrAIzz

0

0 1A

0 0

(8.18c)

BLD =

0

CYn2m

CIxx

CnIxx

CnIzz

CnnIzz

0 0

(8.18d)


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Linearization Ch08

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Transcript of Linearization Ch08