# Category Dynamics of Flight

## Propeller and Slipstream Effects

PROPELLER NORMAL FORCE

The following method of estimating the propeller normal force is due to Ribner (1944). The normal force is expressed in terms of the derivative dCN/dap (see Sec. 3.4), which is given by

dCNp/dap = fCY;li{)

The factor / is the same for all propellers, and is given in Fig. B.7,1 as a function of Tc = TlpV2d2. The value of Су>ф0 varies with the propeller and its operating condition. The values for a particular propeller family are given in Fig. B.7,2. Extrapolation to

 SBF IBF sb h

Figure B.6,2 Effect of a fuselage on Cma. (From Royal Aeronautical Society Data Sheet Aircraft 08.01.07.) other propellers can be made by means of Fig. B.7,3, on the basis of the “side-force – factor,” SFF. This is a geometrical propeller parameter, given approximately by

SFF = 525(b/D)03 + (b/D)06] + 270(b/D)09

where (b/D) is the ratio of blade width to propeller diameter, and the subscript is the relative radius at which this ratio is measured.

Also given in Ribner (1944) are some curves which are useful for estimating the upwash or downwash at the propeller plane. These are reproduced in Fig. B.7,4.

Tc

Figure B.7,1 Variation of / with Tc. (From NACA Wartime Rept. L-25, 1944, by H. S. Ribner.)

## Tab Effectiveness, b3

The data and method that follows is taken from the USAF Datcom. It provides esti­mates of b3 [see (2.5,1)] for two-dimensional subsonic attached flow over airfoils with a control surface and tab. Corrections for part-span tabs can be made by multi­plying the result for two dimensions by the ratio of the control surface area spanned by the tab to the total control surface area (both areas being measured aft of the hinge line).

 Cftc Figure B.4,2 Effect of section lift coefficient on flap section hinge moments.

Эе

В.5 Downwash,

act

The method and data that follow are taken from the USAF Datcom. The average low- speed downwash gradient at the horizontal tail is given by

Эе

— = 4.44 [KAKXKH (cos Лс/4),/2]119 (B.5,1)

 cjc Figure B.4,3 Rate of change of angle of attack due to a change in tab deflection.

where KA, Kk, and KH are wing-aspect-ratio, wing-taper-ratio, and horizontal-tail-lo­cation factors obtained from Figs. B.5,1, B.5,2, and B.5,3, respectively. Ac/4 is the sweepback angle of the wing J chord line.

At higher subsonic speeds the effect of compressibility is approximated by

/Эе =/Э£ (ClJm

)M /low speed speed

where

( —— ] is obtained using (B.5,1)

у OQ? /]ow Speeti

(COlow speed ar|d (ClJm are the wing lift-curve slopes at the appropriate Mach

numbers, obtained by using the straight-tapered-wing method of Sec. B. l

B.6 Effect of Bodies on Neutral Point and Cmo

NOTATION

c local wing chord at center line of fuselage or nacelle

c mean aerodynamic chord

w maximum width of fuselage or nacelle

5 gross wing area

Дhn shift of neutral point due to fuselage or nacelle as a fraction of c,

positive aft

SB area of planform of body

SIIF area of planform of body, forward of 0.25c

c’ root chord of wing without fillets

(Cmo)B increment to Cmo due to a body at zero lift

6 reflex angle of fillet, i. e., angle between wing root chord and lower surface of fillet for upswept fillets, or the upper surface for down – swept fillets, positive as indicated in Fig. B.8,2

A fillet lift-increment ratio, i. e., CJCla, considering the fillet to be a

flap of chord lf

NOTES

Figure B.6,1: Дh„

The data for estimating Дhn presented in this graph were derived from wind-tun­nel tests. The forward shift in neutral point is mainly dependent on the length and width of the body forward of the wing. The values of Д/г„ given by the curves are ac­curate to within ±0.01c, and are about 5% higher for low-wing, and the same amount lower for high-wing configurations. The data are inapplicable if the wing is clear of the body. Separate values should be computed for fuselages and nacelles, and the re­sults added to obtain the total neutral-point shift.

Figure B.6,2: (Cmo)B

The curves given in this figure apply to stream-line bodies of circular or near cir­cular cross section with midwing configurations. For high – or low-wing configura­tions a positive or negative Д(Стц)в = 0.004 is added, respectively, to the value de­rived from the curves. The curves apply only for angles of attack up to about 15° for stream-line bodies where the pitching moment of the body varies linearly with angle of attack.

In the wind-tunnel tests from which the data were derived, the wings had straight trailing edges at the wing-body junction. Fillets have a large effect on Cmo, however, especially if в is large. The following equation may be used to estimate the fillet ef­fect if 0.12 < lf/c < 0.5 and 0.03 < Sf/b < 0.075:

Cmo due to fillets = [0.046 + 0m(dCJdCL)we – 0.2(c + lf)/c](w + Sf)/b

The value obtained from the curves, the fillet effect, and the effect due to wing posi­tion are added to determine (Cmn)B.

## Control Hinge Moments

NOTATION

t trailing-edge angle defined by the tangents со the upper and

lower surfaces at the trailing edge

{b)(yr theoretical rate of change of hinge-moment coefficient with

angle of attack for incompressible inviscid two-dimensional flow

(b,)0 actual rate of change of hinge-moment coefficient with angle

of attack for incompressible two-dimensional flow

 Rectangular wings infinite circular cylinder mid-wing configuration

U>)

Figure B.1,3 Body effect on lift-curve slope expressed as a ratio of lift of wing-body combination to lift of wing alone. (From “Lift and Lift Distribution of Wings in Combination with Slender Bodies of Revolution,” by H. J. Luckert, Can. Aero../., December 1955.)

(Ь2)от theoretical rate of change of hinge-moment coefficient with

control deflection for incompressible inviscid two-dimen­sional flow

(b2)о actual rate of change of hinge-moment coefficient with con­

trol deflection for incompressible two-dimensional flow

 Figure B.2,2 Flap-chord factor.

 (^l)obal’ (^2)0bal rates of change of control hinge-moment coefficients with in­cidence and control-surface deflection, respectively, in two­dimensional flow for control surfaces with sealed gap and nose balance F, aJ8 induced angle of attack correction to (&,)0 and (b2)Q, respec­tively, where F, is the value of (a,/S) [Q/C, J when cf = c F2, Д(b2) stream-line curvature correction to (bx)0 and (b2)0, respec­tively, where F2 is the value of Д(b2) when cf = c Fз factor to F2 and Д(b2) allowing for nose balance Balance ratio of control-surface area forward of hinge line to control – surface area behind hinge line

NOTES

Figures B.3,1 and B.3,2

The curves of Fig. B.3,1 were derived for a standard series of airfoils with plain controls for which tan (|)r = t/c (referred to by an asterisk). To correct for airfoils

with tan (!)t different from t/c, values of (b,)*)74 (C, a)*heory and C*a are calculated for the given t/c ratio; then (b,)0 is calculated from

Oh) 0 = (bSo + 2[(CJ theory C;*J(tan (t) – t/c). (B.3,1)

Values of (C/a)*heory and C*a may be obtained as in Appendix B. l.

The curves apply for values of angle of attack and control deflection for which there is no flow separation over the airfoil; for these conditions (h,)0 can be estimated to within ±0.05. The data refer to sealed gaps but may be used if the gap is not greater than 0.002c.

The above discussion also applies to the data given in Fig. B.3,2 for (b2)0. The subscript 1 in Eq. B.3,1 becomes a subscript 2, a becomes 8, and values of (CZs)t*heory and C*ls may be obtained from Appendix B.2.

 Cf/c Figure B.3,1 Rate of change of hinge-moment coefficient with angle of attack for a plain control in incompressible two-dimensional flow. (From Royal Aeronautical Society Data Sheet Controls, 04.01.01.)

Figure B.3,3

The effect of nose balance on (£,)„ and (b2)0 can be estimated from the curves given on this figure. The data were obtained from wind-tunnel tests on airfoils with control-chord/airfoil-chord ratio of 0.3. Relatively small changes in nose and trailing – edge shape, and airflow over the control surface, may have a large effect on hinge moments for balanced control surfaces, so that estimates of nose-balance effect will be fairly inaccurate. If the control-surface gap is unsealed, the hinge-moment coeffi­cients of plain and nosebalanced controls will generally become more positive.

Figure B.3,4

Two-dimensional hinge-moment coefficients for control surfaces with nose bal­ance can be corrected for finite aspect ratio of the main surface using the factors given in the curves and the following equations:

bx = (Mo(l – F,) + F2F3Cla ^2 ~ (^2)0 — (сг,/<5)(£>і)0 + Mb2)F3C, li

For plain control surfaces the above equations are used with F3 = 1. (&,)0 and (b2)0 can be obtained from Fig. B.3,1 and B.3,2, respectively, for plain controls. For nose – balanced controls, the two-dimensional coefficients (&,)0 and (b2)0 must include the effect of nose-balance. Values of C/a can be obtained from Sec. B. l, and those for Cu from Fig. B.2,1.

Lifting-surface theory was applied to unswept wings with elliptic spanwise lift distribution to derive the factors. Full-span control surfaces were assumed together with constant ratios of cf/c and constant values of (&,)0 and (b2)0 across the span. The factors apply to wings with taper ratios of 2 to 3 if c/c, (/;. )„ and (b2)0 do not vary by more than ± 10% from their average values.

Figure B.3,4 Finite-aspect-ratio corrections for two-dimensional plain and nose-balanced control hinge-moment coefficients (Cla per radian). (From Royal Aeronautical Society Data Sheet Controls 04.01.05.) “

## Data for Estimating. Aerodynamic Derivatives

This appendix contains a limited amount of data on stability and control derivatives. It is not intended to be used for design. That requires much more detail than could possibly be provided here. It is intended to display some representative orders of magnitude and trends, and to provide numerical data that teachers and students can use for exercises. All the data pertain to subsonic flight of rigid airplanes. Much of the information comes from either the USAF Datcom (USAF, 1978) or from the data sheets of the Royal Aeronautical Society of Great Britain (now out of print), which is also the source for some of the Datcom data. We have taken some liberties in extract­ing and presenting this information, but have not changed any essential content. For information about derivatives at transonic and supersonic speeds and for geometries different from those covered in the following, the reader is referred to the USAF Dat­com. When estimating derivatives, reference should also be made to Tables 5.1 and

5.2.

A. l Lift-Curve Slope, C,

A. 2 Control Effectiveness, CLs

B. 3 Control Hinge Moments B.4 Tab Effectiveness, b3 де

B.5 Downwash, —— da

B.6 Effect of Bodies on Neutral Point and Cm >

B. 7 Propeller and Slipstream Effects

B.8 Wing Pitching Derivative, Cmq

B.9 Wing Sideslip Derivatives Clp, Cnp

B.10 Wing Rolling Derivatives Clp, C„p

В.11 Wing Yawing Derivatives Cir, C„r

B. l2 Changes in Inertias and Stability Derivatives with
Change of Body Axes

NOTES

• The source of the data for airfoils and wings is USAF datcom. It applies to rigid straight-tapered wings at subsonic speeds and small angle of attack.

• The section lift-curve slope is given by

1.05

Cla = — K(Claheory (B.1,1)

where К is given in Fig. B. l,la and (C;Jtheory in Fig. B.1,1Z>. У90 and У99 are the air­foil thicknesses, in percent of chord, at 90% and 99% of the chord back from the leading edge, as illustrated, and the trailing edge angle is defined in terms of these thicknesses by

(B.1,2)

• The lift-curve slope C, a of the wing alone is given in Fig. B.1,2. The inset equation is seen to approach the theoretically correct limits of тгА/2 as A —> 0 and 27ras {A—» °°, к—> 1, Л—» 0, (3—> 1}.

• Figure B.1,3 gives some theoretical values of the body effect on CLa for unswept wings in mid-wing combination with an infinite circular cylinder body. For values of A < 1, the theory also applies to delta wings with pointed tips.

In Fig. B. l,3a the wing angle of attack is the same as that of the fuselage; that is, e = 0. In this case the lift of the wing-body combination increases to a maximum value, then decreases with increasing body diameter. Where there is a wing setting,

i. e., e Ф 0, and aB = 0 (Fig. В. 1,3i>), the lift of the combination decreases with in­creasing a.

B.2 Control Effectiveness, CLs

SECTION DATA

Figure B.2,la presents theoretical values of the two-dimensional control derivative C, s for simple flaps in incompressible flow. These values can be corrected by the em-

 4 [P2+ tan2 Лс/2]2 Figure B.1,2 Subsonic wing lift-curve slope.

pineal data of Fig. B.2,1 b for the strong effect of nonideal lift-curve slope of the main surface to which the control is attached.

SURFACE DATA

The derivative CLs for a finite lifting surface with a part span control flap is obtained from the section derivative by

where CLo and Cla are as defined in B. l, C, s is the corrected value from Fig. B.2,1 b and Kl and K2 are the factors given in Figs. B.2,2 and B.2,3. In these figures the para­meter (as)Cl is the rate of change of zero-lift angle with flap deflection, given by the inset graph, and Aw and A are, respectively, the aspect ratio and taper ratio of the main surface.

## Velocity and Acceleration in an Arbitrarily. Moving Frame

Since in many applications, we want to express the position, inertial velocity, and in­ertial acceleration of a particle in components parallel to the axes of moving frames, we need general theorems that allow for arbitrary motion of the origin, and arbitrary angular velocity of the frame. These theorems are presented below.

Let F^Oxyz) be any moving frame with origin at О and with angular velocity to relative to F,. Let r = r0 + r’ be the position vector of a point P of FM (see Fig. A.6). Let the velocity and acceleration of P relative to F, be v and a. Then in F,

v, = r, a/ = І/

We want expressions for the velocity and acceleration of P in terms of the compo­nents of r’ in FM. Expanding the first of (A.6,1)

V/ = r0/ + r;

= v0, + r;

where v0 is the velocity of О relative to FThe velocity components in FM are given by

= W/V/ = LM/(v0/ + ry) — + L Mli,

From the rule for transforming derivatives (A.4,22)

Ljvf/Г/ — iM + ojmvm (A.6,3)

whence

Vm = VoM + Гм+ "лАм (A.6,4)

The first term of (A.6,4) is the velocity of О relative to F„ the second is the velocity of P as measured by an observer fixed in FM, and the last is the “transport velocity,” that is the velocity relative to F, of the point of FM that is momentarily coincident with P. The total velocity of P relative to F, is the sum of these three components. Following traditional practice in flight dynamics, we denote

(A.6,5)

(When necessary, subscripts are added to the components to identify particular mov­ing frames.)

The scalar expansion of (A.6,4) is then

vx = v0r + x + qz – ry

vv = v0r + у + rx – pz (A.6,6)

vz = u0. + z + РУ – qx

These expressions then give the components, parallel to the moving coordinate axes, of the velocity of P relative to the inertial frame.

On differentiating v7 and using (A.6,4) we find the components of inertial accel­eration parallel to the Fm axes to be

ам Lw, v, vM + toMM

= v0m + FM + + "мГдг + + <aMi’M + (Ьмымг’м

= a0„ + r’M + ojMrM + 2tbMr’M + ojMwMr’M (A.6,7)

where a0vf = v0m + 6>mv0m = LjWvl); is the acceleration of О relative to F,.

The total inertial acceleration of P is seen to be composed of the following parts:

 aoM; *’м’- A* f ^ 2a>MrM: m’-

the acceleration of the origin of the moving frame the acceleration of P as measured by an observer fixed in the mov­ing frame

the “tangential” acceleration owing to rotational acceleration of the frame FM

the Coriolis acceleration the centripetal acceleration

Three of the five terms vanish when the frame FM has no rotation, and only r’M re­mains if it is inertial. Note that the Coriolis acceleration is perpendicular to coM and
t’M, and the centripetal acceleration is directed along the perpendicular from P to to. The scalar expansion of (A.6,7) gives the required inertial acceleration components of P as

ax = a0x + x + 2qz ~ 2ry – x(q2 + r ) + y( pq – r) + z(pr + q)

ay = a0y + >’ + 2rx – 2pz + x(pq + r) – y(p2 + t2) + z(qr – p)

az = a0z + z + 2py – 2qx + x(pr – q) + y(qr + p) – zip2 + q2)

## Computation of Eigenvalues and Eigenvectors

Some software packages provide for the calculation of eigenvalues and eigenvectors of matrices directly. The software used for many of the computations in this book is the Student Version of Program CC,3 which does not do this. However, these impor­tant system properties can readily be obtained from it, as shown in the following.

Program CC is oriented to the calculation of transfer functions and presents them in various forms; one is the pole-zero form. Any transfer function of the system, for example, that from elevator angle to pitch rate, when displayed in this form, will show the eigenvalues in the denominator. That is how we obtained the eigenvalues presented in Chap. 6.

’Available from Systems Technology, Inc., 13766 South Hawthorne Blvd., Hawthorne, CA, 90250­7083 U. S.

For the eigenvectors, we turn to the expansion theorem (A.2,10). Consider a case where the input to the system is 8C = 8(f), Dirac’s delta function. The response of the ith component of the state vector to this input in the mode corresponding to eigen­value A is

The ratio of this component to x, for the same input 8(f) is

x,(t) = A,(A) *,(0 A, (A)

This ratio gives the ith component of the eigenvector for the mode associated with A. Any component can be chosen for reference instead of xl5 as illustrated in Figs. 6.3 and 6.15.

## TRANSFORMATION OF THE DERIVATIVE OF A VECTOR

Consider a vector v that is being observed simultaneously from two frames Fa and Fh that have relative rotation—say Fh rotates with angular velocity ы relative to Fa, which we may regard as fixed. From (A.4,3)

The derivatives of ya and h are of course

(A.4,13)

where vai = (d/dt)(vai), and so forth. It is important to note that v„ and yb are not simply two sets of components of the same vector, but are actually two different vec­tors.

Now because Fb rotates relative to Fa, the direction cosines are changing with time, and the derivative of (A.4,3) is

ifh = L.„v„ + L h/yn

or alternatively

V, = L ahyb + L abyh

the second terms representing the effect of the rotation.

Since L must be independent of v, the matrix Lab can readily be identified by considering the case when vb is constant (see Fig. A.5.). For then, from the funda­mental definitions of derivative and cross product, the derivative of v as seen from Fa is readily shown to be

dy

— = Ы X V dt

The corresponding result from (A.4,14) is

Va = tabv„ (A.4,17)

It follows from equating (A.4,16) and (A.4,17) that

L аЬУЬ = "aVa

or

Kb^b = &аКьУь (A.4,18)

for all yb. Whence — <UaLafe

^ab^ba

Finally if the above argument is repeated with Fb considered fixed, and Fa having an­gular velocity – to, we clearly arrive at the reciprocal result

Lba = – d>bLba (A.4,19)

From (A.4,18) and (A.4,19), recalling that to is skew-symmetric so that to’ = —to, the reader can readily derive the result

From (A.4,14), (A.4,18), and (A.4,19) we have the alternative relations

Vft = Lbaxa – 6>hxh xa = L ahvh + ыаха

with two additional permutations made possible by (A.4,20). A particular form we shall finally want for application is that which uses the components of xa transformed into Fh, viz.

L bJa = + 6>bxh (A.4,22)

TRANSFORMATION OF A MATRIX

Equation (A.4,20) is an example of the transformation of a matrix, the elements of which are dependent on the frame of reference. Generally the matrix of interest A oc­curs in an equation of the form

v = Au (A.4,23)

where the elements of the (physical) vectors u and v and of the matrix A are all de­pendent on the reference frame. We write (A.4,23) for each of the two frames Fa and Fh, that is,

v„ = A„u(,

v,, = Ahub

and transform the second to

= A,,Lfc„uu

Premultiplying by Lab we get

Va = f a/)A/)L/,„U„

By comparison with (A.4,24a) we get the general result

A a = LahAbLba

## PROPERTIES OF THE L MATRIX

Since va and vh are physically the same vector v, the magnitude of a must be the same as that of vb, that is, v2 is an invariant of the transformation. From (A.4,3) this requires

v2 = lvh = TaJhaLhay a = Tau (A.4,5)

It follows from the last equality of (A.4,5) that

KaUa = I (A.4,6)

Equation (A.4,6) is known as the orthogonality condition on L^r From (A.4,6) it fol­lows that and hence that |Lfe,| is never zero and the inverse of Lha always exists. In view of (A.4,6) we have, of course, that

I T _ І -1 _ t

*^ba *^ah

that is the inverse and the transpose are the same. Equation (A.4,6) together with (A.4,3b) yields a set of conditions on the direction cosines,

It follows from (A.4,8) that the columns of Lba are vectors that form an orthogonal set (hence the name “orthogonal matrix”) and that they are of unit length.

Since (A.4,8) is a set of six relations among the nine liJt then only three of them are independent. These three are an alternative to the three independent Euler angles for specifying the orientation of one frame relative to another.

THE L MATRIX IN TERMS OF ROTATION ANGLES

The transformations associated with single rotations about the three coordinate axes are now given. In each case Fa represents the initial frame, Fh the frame after rota­tion, and the notation for L identifies the axis and the angle of the rotation (see Fig. A.4). Thus in each case

Уь = L,(X,)va

By inspection of the angles in Fig. A.4, the following matrices are readily verified.

’10 0 L^X,) = 0 cosX, sinX, 0 — sinXj cosX,

‘cosX2 0 — sinX2 L2(X2) =010

sin X2 0 cos X2

cosX3 sinX3 0 L3(X3) = -sinX3 cosX3 0 0 0 1

*»l *ai

The transformation matrix for any sequence of rotations can be constructed readily from the above basic formulas. For the case of Euler angles, which rotate frame FE into FB as defined in Sec. 4.4, the matrix corresponds to the sequence (X3, X2, Xt) = (iff, в, Ф), giving

 L,() • L2(0) • L3(ф) (A.4,11) [The sequence of angles in (A.4,11) is opposite that of the rotations, since each trans­formation matrix premultiplies the vector arrived at in the previous step.] The result of multiplying the three matrices is cos в cos ф cos в sin ф — sin в sin ф sin в cos ф — cos ф sin ф sin ф sin в sin ф + cos ф cos ф sin ф cos в (A.4,12) cos ф sin в COS Ф + sin ф sin ф cos ф sin в sin ф —sin ф cos Ф cos ф cos в

## The Convolution Integral

The response of any linear system to any arbitrary input f(t) can be obtained from in­tegrals of the two basic response functions h(t) and A(t). h(t) is the response to the unit impulse 5(f), and A(t) is the response to the unit step 1(f). The system is assumed to be initially quiescent. If not, the transient associated with nonzero initial condi­tions must be added to the following integrals. The response to /(f) is then given by Duhamel’s integral, or the convolution integral:

(a)

t (A.3,1)

x[t) = f A{t – t)/(r) dr (/(0) = 0) ib)

■4=0

When /(0) is not zero, then there must be added to (A.3,15) a term to allow for the initial step in /(f); i. e.,

xit) = f(0)A(t) + f Ait – t)/(t) dr (A.3,2)

Jt=0

The physical significance of these integrals is brought out by considering them as the limits of the following sums

xit) = 2/i(f – t)/(t) At ia) ^

xit) = Ait)fi0) + b4(f – t)/(t) At (5) C ’

Typical terms of the summations are illustrated in Figs. A. l and A.2. The summation forms are quite convenient for computation, especially when the interval At is kept constant.

 V* = Cbaya (a) (A.4,3) Ua – Щ (Ь)

where (A.4,4)

I = T 1

^ab — *-*ba

When a vector is successively transformed through several frames of reference, for example, Fa, Fb, Fc. . . then

vfc = L haa

and c = Lcbb = LJLh(ya)

Since also vc = Ltav(„ then it follows that

Lea = Lc/)Lfca

The sequence of subscripts in the preceding expression should be noted, as it provides a convenient mnemonic for remembering these relations.

## METHODS FOR THE INVERSE TRANSFORMATION The Use of Tables of Transforms

Extensive tables of transforms (like Table A. l) have been published that are use­ful in carrying out the inverse process. When the transform involved can be found in the tables, the function x(t) is obtained directly.

The Method of Partial Fractions

In some cases it is convenient to expand the transform x(s) in partial fractions, so that the elements are all simple ones like those in Table A. 1. The function x(t) can then be obtained simply from the table. This procedure is illustrated with an example. Let the second-order system of Sec. 7.3 be initially quiescent, that is, x(0) = 0, and i(0) = 0, and let it be acted upon by a constant unit force applied at time t = 0. Then fit) = 1, and f(s) = l/s (see Table A. l). Then (see (7.4,1))

‘To avoid ambiguity when dealing with step functions, t = 0 should always be interpreted as t = 0+.

Let us assume that the system is aperiodic; that is, that £ > 1. Then the roots of the characteristic equation are real and equal to

A, 2 = n ± со’ (A.2,6)

where

n = -£a)n со’ = Шяа2 ~ 1)1/2

The denominator of (A.2,5) can be written in factored form so that

1

s(s – A,)(s – A2) Now let (A.2,7) be expanded in partial fractions,

Heaviside Expansion Theorem

When the transform is a ratio of two polynomials in s, the method of partial frac­tions can be generalized. Let

m

D(s)

where N(s) and D(s) are polynomials and the degree of D(s) is higher than that of N(s). Let the roots of D(s) = 0 be ar, so that

D(s) = (s – a{)(s – a2) ••• (s – an)

Then the inverse of the transform is

– ar)N(s) l

——— I ё

D(s) s=ar

The effect of the factor (5 — ar) in the numerator is to cancel out the same factor of the denominator. The substitution s = ar is then made in the reduced expression.2

In applying this theorem to (A.2,7), we have the three roots a, = 0, a2 = A,, a3 = A2, and N(s) = 1. With these roots, (A.2,9) follows immediately from (A.2,10).

The Inversion Theorem

The function x(t) can be found formally from its transform x(s’) by the application of the inversion theorem Jaeger (1949) and Carslaw and Jaeger (1947). It is given by the line integral

1 ry+ito

x{t) = —– lim es‘x(s) ds (A.2,11)

ІТГІ Jy-іш

where у is a real number greater than the real part of all values of s for which x(s) di­verges. That is, s = у is a straight line on the s plane lying parallel to the imaginary axis, and to the right of all the poles of x(.s). This theorem can be used, employing the methods of contour integrals in the complex plane, to evaluate the inverse of the transform.

Extreme Value Theorems

Equation (A.2,2) may be rewritten as

 J

fCC

e ‘‘x(t) dt

0

= lim [ e~s‘x(t) dt

T—Jfj

We now take the limit s —» 0 while T is held constant, that is, —x(0) + lim sx(s) = lim lim e~s, x(t) dt

■ s—»0 T-* 00 Jq s—*0

= lim f x(t) dt = lim [x(T) – x(0)]

T—>CO Jq T—»cc

Hence lim ix(i) = lim x(T) (A.2,12)

s—>0 T-^ 00

This result, known as the final value theorem, provides a ready means for determin­ing the asymptotic value of x{t) for large times from the value of its Laplace trans­form.

!For the case of repeated roots, see Jaeger (1949).

In a similar way, by taking the limit j —» °o at constant T, the integral vanishes for all finite x(t) and we get the initial value theorem.

lim sx(s) = x(0) (A.2,13)