CoA Report A e r o No. 178
THE COLLEGE OF A E R O N A U T I C S
C R A N F I E L D
THE STABILITY OF THE SHORT-PERIOD MOTION
OF AN AIRFRAME HAVING NON-LINEAR AERODYNAMIC
CHARACTERISTICS IN PITCH AND SUBJECT TO A
STEP-FUNCTION ELEVATOR DEFLECTION
by
Report A e r o 178 J a n u a r y 1965.
THE COLLEGE OF AERONAUTICS CRANFIELD
The stability of the s h o r t - p e r i o d motion of an a i r f r a m e having non-linear a e r o d y n a m i c
c h a r a c t e r i s t i c s in pitch and subject to a step-function elevator deflection
by
p . A. T. C h r i s t o p h e r
SUMMARY The stability of the differential equation ~ + B (x) ^ + C(x) = Q, Q = 0, t < 0 ,
with B(x) and C(x) a s a n t i - s y m m e t r i c power s e r i e s , is shown to be d e t e r m i n e d by the nature of the s i n g u l a r i t y at the s t e a d y s t a t e value given by C(x) = Q, with c e r t a i n additional r e s t r i c -tions on the initial and final value of x. F u r t h e r , the c h a r a c t e r of the t r a n s i e n t settling down motion is d i r e c t l y r e l a t e d to the nature of this singularity and, together with the stability, i s a c c u r a t e l y predicted by c r i t e r i a d e r i v e d .
It i s shown that the previous stability c r i t e r i a can be applied to the problem of an a i r f r a m e subject to a stepfunction elevator deflection, provided that the a e r o d y n a m i c d e r i v -ative ZTJ i s negligible. When zjj is not s m a l l , special t r e a t m e n t of the stability problem i s r e q u i r e d and it i s shown that the c r i t i c a l value of the elevator step-function which will cause instability can fairly r e a d i l y be obtained from quantities taken from the elevator t r i m c u r v e s .
CONTENTS P a g e No. S u m m a r y 1.0 Introduction 1 Notation 3 2 . 0 P o i n c a r e ' s T h e o r y of Singular P o i n t s in the P h a s e - P l a n e 5
3.0 Stability C r i t e r i a for a P a r t i c u l a r Second-Order Equation 6
4 . 0 A N u m e r i c a l E x a m p l e 11 5. 0 Stability of the S h o r t - P e r i o d Motion of an A i r f r a m e Subject to
a Step-Function E l e v a t o r Disturbance 13 6 . 0 Stability of S h o r t - P e r i o d Motion When z is not s m a l l 18
7.0 Conclusions 22 R e f e r e n c e s 23 F i g u r e s
1.0 Introduction
The differential equation
^ + B ' ( x ) ^ + C(x) = Q, Q = 0. t<0 (1.1) d e s c r i b e s the motion of a c l a s s of s e c o n d - o r d e r s y s t e m s with displacement dependent stiffness
and damping and subject to a step function d i s t u r b a n c e of magnitude Q. It is c h a r a c t e r i s t i c of many physical p r o b l e m s , for example in the stability t h e o r y of synchronous e l e c t r i c a l m o t o r s (Refs. 1 and 2) and in the a e r o d y n a m i c r e s p o n s e of a i r f r a m e s having non-linear n o r m a l force and pitching moment c u r v e s (Ref. 4). The step function i s , of c o u r s e , a standard function for t e s t i n g the t r a n s i e n t r e s p o n s e of s y s t e m s and gives r i s e to many other examples whose gover-ning equation is ( 1 . 1 ) .
The equation
d * x , d x - , , „V
^ + b - + cx = Q (1.2) is a d e g e n e r a t e linear form of ( 1 . 1 ) , w h e r e b and c a r e constants, and it is of i n t e r e s t to c o m
-p a r e the linear with the n o n - l i n e a r -p r o b l e m . In the c a s e of ( 1 . 2) the solution i s made u-p of the sum of the c o m p l e m e n t a r y function, which is the solution of the homogeneous equation given by Q aO, and any p a r t i c u l a r i n t e g r a l , the l a t t e r n e c e s s a r i l y involving Q. Provided that Q i s of finite magnitude it follows d i r e c t l y from the principle of linear superposition that if the homogeneous equation, Q ^ O , can be shown to have stable solutions then (1.2) has a completely stable solution. In the l i n e a r p r o b l e m , t h e r e f o r e , stability a n a l y s i s can be r e s t r i c t e d to the homogeneous equation.
Except in those c a s e s w h e r e it is possible to s e p a r a t e the v a r i a b l e s or the equation i s exact, no explicit g e n e r a l solutions to n o n - l i n e a r differential equations a r e known. It i s of c o u r s e c e r t a i n that the principle of linear superposition is invalid for such equations and a s a r e s u l t the stability c r i t e r i a for the homogeneous, Q a O , and non-homogeneous, Ql'O, c a s e s a r e different. This m e a n s that the stability and r e s p o n s e p r o b l e m s cannot be considered s e p a r a t e l y , a s they a r e in a linear s y s t e m , but involve an a n a l y s i s of the stability of the r e s p o n s e and t h e r e -fore will depend on the n a t u r e and magnitude of the forcing t e r m Q.
When the damping t e r m is absent from (1.1) it i s usually possible to obtain a first i n t e g r a l in the form
iv» + f(Q,x) = E , ' (1.3) dx
w h e r e v = — and E i s a constant. This equation is then e x p r e s s i v e of the e n e r g y balance in the s y s t e m , which i s c o n s e r v a t i v e when the damping is a b s e n t . The t e r m |v* c o r r e s p o n d s to the kinetic e n e r g y , f(Q,x) the potential energy and E the total energy which is of c o u r s e constant. A second i n t e g r a l i s then possible by q u a d r a t u r e s . Depending on the form of C(x) the second i n t e g r a l may be analytic in t e r m s of known functions (very often elliptic integrals) or it m.ay be n e c e s s a r y to r e s o r t to n u m e r i c a l or graphical methods to evaluate the i n t e g r a l .
When damping is p r e s e n t the s y s t e m is dissipative or non-conservative and a first i n t e g r a l c o r r e s p o n d i n g to the e n e r g y balance equation ( 1 . 3) is no longer obtainable or a p p r o p r i a t e . If the damping i s not too l a r g e the solution may be obtained by an analj'tic i t e r a t i o n p r o -c e d u r e the s t a r t i n g point of whi-ch is the solution to the degenerate problem of z e r o damping. The s u c c e s s of t h i s method will depend on the r a t e at which the p r o c e s s c o n v e r g e s , rapid
con-dx v e r g e n c e being consistent with s m a l l values of the damping t e r m B* (x) —
2
-The stability of the r e s p o n s e , i . e . whether or not the s y s t e m s e t t l e s down to a steady value, or at least an oscillation of finite amplitude, is of i m p o r t a n c e in many applications and p a r t i c u l a r l y so when dealing with s y s t e m s whose damping or stiffness change sign. In the case of second or lower o r d e r s y s t e m s with a s t e p function input, it is not n e c e s s a r y to have an e x -plicit e x p r e s s i o n for the r e s p o n s e in o r d e r to d e t e r m i n e the stability of the motion since this can be t r e a t e d much m o r e conveniently and elegantly by P o i n c a r e ' s t h e o r y of singular points in the phase plane. (See R e f s . 2, 3, and 4). The advantage of t h i s method can only be fully a p p r e c i a t e d when it i s r e c a l l e d that the a l t e r n a t i v e is to consider the stability within the f r a m e -work of the i n t e r a t i o n p r o c e d u r e the conditions for convergence of which a r e generally not fully known.* T h e r e i s in fact a second a l t e r n a t i v e method known a s Lyapunov's second or d i r e c t nnethod (see Ref. 5) but t h i s will not be considered h e r e .
The object of the paper is to obtain stability c r i t e r i a for a specific equation of form (1.1) and c o m p a r e t h e s e with n u m e r i c a l solutions from a digital c o m p u t e r . Having established the usefulness of the c r i t e r i a they a r e then to be applied to the problem of the stability of the s h o r t - p e r i o d motion of an a i r f r a m e having n o n - l i n e a r a e r o d y n a m i c c h a r a c t e r i s t i c s in pitch and subject to a step-function deflection of the e l e v a t o r .
* T h i s a r i s e s b e c a u s e it i s usually i m p o s s i b l e to state the form of the general t e r m in the
NOTATION a, b , c, d a,, a^, a , b , , b 3 , . . . C j . m q t e . g . "w
constants in P o i n c a r e ' s t h e o r y of singular points
constants in the e x p r e s s i o n for c r i t i c a l initial velocity W ^ p , equation (6.9) constants in the a n t i - s y m m e t r i c functionB(x)
constants in the a n t i - s y m m e t r i c function C(x) a i r f r a m e m a s s
angular velocity about axis of pitch t i m e
^^/dt
p e r t u r b a t i o n in velocity along axis of yaw
any dependent v a r i a b l e ; often d i s p l a c e m e n t . See equation ( 1 . 1 ) . distance of c e n t r e of gravity aft of r e f e r e n c e line
m \ 3 w / w = 0 'w n i r mv(, = Z 3 . Z s . m , , m , , A,. = A3. = B B , . = B3, = B(x) = C(x) = D
t>
F MP(x,v) 1
Q(x.v) J
Q U U„ mVaqy q = m \^9T) J 77 = B U W J WB Uq
q = 0 1 ^ 3 M \ B V 9 V w 0constants in the force and moment r e l a t i o n s of equation (5.1)
(Uo+Zq)m^-mqZ^ (Uo+Zq)m3-mqZ3
moment of i n e r t i a about the a x i s of pitch (Uo+Zq)m^+mq+z^
3Z3
b , x + b j X ' +
the o p e r a t o r "^/^j^
the d i s c r i m i n a n t [ B ' ( X ) J *- 4C'(x)
the elevator forcing function r(UQ+Zg)m,j-^ m „ l H moment about axis of pitch
functions in P o i n c a r e ' s theory of singular points. See equation (2.4)
magnitude of forcing step-function, equation (1.1) velocity tangential to flight path
- 4 NOTATION (continued)
W velocity along a x i s of yaw Z force along axis of yaw
Wy « a i r f r a m e g e o m e t r i c i n c i d e n c e » /UQ • when « s m a l l . H total e l e v a t o r angle of l-p + 1 r\ p e r t u r b a t i o n in elevator angle 6 angle of pitch 3 p e r t u r b a t i o n in angle of pitch X root of c h a r a c t e r i s t i c equation (2. 7), ( 3 . 6) or ( 5 . 1 4 ) .
? ordinate in x or w r e f e r r e d to singularity away from the origin.
Suffixes
T r e f e r s to t r i m m e d condition S.S. r e f e r s to s t e a d y - s t a t e condition
M r e f e r s to m a x i m a or minima on pitching moment c u r v e . P r e f e r s to unstable singularity at point P on t r i m c u r v e .
A dot over a v a r i a b l e indicates differentiation with r e s p e c t to t i m e , whilst a p r i m e indicates differentiation with r e s p e c t to x or w.
2. O P o i n c a r e ' s T h e o r y of Singular Points in the P h a s e - P l a n e
The stability of the solution to (1.1) can be analyzed by the p h a s e - p l a n e method which is given in detail in R e f s . 2, 3 and 4 .
Writing v = -;^ , then -r-j = v — and (1.1) b e c o m e s
dt
v ^ + B ' (x)v + C(x) = Q
dx (2.1)
The g r a p h s of the solutions of t h i s equation in the xvplane, known a s the phase plane, a r e r e f e r -r e d to a s " i n t e g -r a l c u -r v e s " and th-rough each o -r d i n a -r y point in the plane t h e -r e p a s s e s only one such c u r v e . Alternatively equation (2.1) m a y be written a s an equivalent pair of equations.
^ = Q - C(x) - B ' (x). V ) dt . d_x dt ) ' ( 2 . 2 ) dv dx
which define a field vector having components — and — ; this vector i s always tangential to the i n t e g r a l curve and i n d i c a t e s the d i r e c t i o n in which t i s i n c r e a s i n g .
The s t a t i o n a r y positions of equilibrium of (2.1) or (2. 3) c o r r e s p o n d with the s i n g u l a r i t i e s of the equivalent equation
dv ^ Q - C(x) - B' (x).v ^2 3j dx V '
which defines the slope of the field v e c t o r , and a n a l y s i s of the c h a r a c t e r of t h e s e s i n g u l a r i t i e s gives considerable insight into the n a t u r e of the motion near these points. More generally, consider the s i n g u l a r i t i e s of the equation
dv dx
P(x.v)
Q(x,v) ' ( 2 . 4 )
which a r e defined by P(x,v) = Q(x,v) = 0. Since the origin can always be changed to c o r r e s p o n d with the singular point, then a n a l y s i s can be r e s t r i c t e d to singularities at the origin.
When (2.4) has a singularity at the origin then it is a s s u m e d (Poincaré) that it may be w r i t t e n in the s e r i e s form
dv _ ax + bv + p(x,v)
dx ex + dv + q(x, v) (2.5)
w h e r e p(x,v) and q(x,v) a r e the r e m a i n i n g t e r m s of s e r i e s whose lowest t e r m s a r e of second d e g r e e at l e a s t . F u r t h e r , if the constants obey the inequality ,
A = | ; ^ | = a d - b c ^ 0.
then the i n t e g r a l c u r v e s behave, in the neighbourhood of the singularity, a s if p(x, v) and q(x, v) w e r e a b s e n t .
The s i n g u l a r i t i e s of the reduced equation ax + bv
dv
dx ex + dv ( 2 . 6 )
a r e of four distinct types known a s nodes, c e n t r e s , s p i r a l points and saddles r e s p e c t i v e l y and each has a c h a r a c t e r i s t i c g e o m e t r y , s o m e t i m e s r e f e r r e d to as its "topological configuration".
4
-NOTATION (continued)
W velocity along axis of yaw Z force along axis of yaw
W)
05 airframe geometric incidence" /UQ • whence small. H total elevator angle of »Jrp + n
n perturbation in elevator angle
6 angle of pitch
3 perturbation in angle of pitch
X root of characteristic equation (2.7), (3.6) or (5.14).
C ordinate in x or w referred to singularity away from the origin.
Suffixes
T refers to trimmed condition S.S. refers to steady-state condition
M refers to maxima or minima on pitching moment curve. P refers to unstable singularity at point P on trim curve.
A dot over a variable indicates differentiation with respect to time, whilst a prime indicates differentiation with respect to x or w.
2 . O P o i n c a r e ' s T h e o r y of S i n g u l a r P o i n t s i n t h e P h a s e - P l a n e T h e s t a b i l i t y of t h e s o l u t i o n t o ( 1 . 1 ) c a n b e a n a l y z e d by t h e p h a s e - p l a n e m e t h o d w h i c h i s g i v e n i n d e t a i l in R e f s . 2 , 3 a n d 4 . ,Tr -^^ dx ., ( ? x dv J /i i \ • W r i t i n g V = -Tj- , t h e n -r—5 = v — a n d ( 1 . 1 ) b e c o m e s v ^ + B ' (x)v + C(x) = Q . d x ( 2 . 1 ) T h e g r a p h s of t h e s o l u t i o n s of t h i s e q u a t i o n i n t h e x v p l a n e , k n o w n a s t h e p h a s e p l a n e , a r e r e f e r -r e d t o a s " i n t e g -r a l c u -r v e s " a n d t h -r o u g h e a c h o -r d i n a -r y point i n t h e p l a n e t h e -r e p a s s e s o n l y one s u c h c u r v e . A l t e r n a t i v e l y e q u a t i o n ( 2 . 1 ) m a y b e w r i t t e n a s a n e q u i v a l e n t p a i r of e q u a t i o n s . ^ = Q - C(x) - E ' ( x ) . V ) dt . dx dt = V ) • ) ( 2 . 2 ) dv dx
which define a field vector having components — and — ; this vector is always tangential to the i n t e g r a l curve and indicates the d i r e c t i o n in which t i s i n c r e a s i n g .
The s t a t i o n a r y positions of equilibrium of (2.1) or (2.3) correspond with the s i n g u l a r i t i e s of the equivalent equation
dv dx
Q - C(x) - B' (x).v
( 2 . 3 )
which defines the slope of the field v e c t o r , and a n a l y s i s of the c h a r a c t e r of t h e s e s i n g u l a r i t i e s gives c o n s i d e r a b l e insight into the nature of the motion near these points. More g e n e r a l l y , consider the s i n g u l a r i t i e s of the equation
dv dx
P ( x , v )
Q(x,v) ' ( 2 . 4 )
which a r e defined by P(x,v) = Q(x,v) = 0. Since the origin can always be changed to c o r r e s p o n d with the singular point, then analysis can be r e s t r i c t e d to s i n g u l a r i t i e s at the origin.
When (2.4) has a singularity at the origin then it is a s s u m e d (Poincaré) that it may be w r i t t e n in the s e r i e s form
dv _ ax + bv + p(x,v)
dx ex + dv + q(x, v) ( 2 . 5 )
w h e r e p(x,v) and q(x,v) a r e the r e m a i n i n g t e r m s of s e r i e s whose lowest t e r m s a r e of second d e g r e e at l e a s t . F u r t h e r , if the constants obey the inequality
A = | ; ^ | = a d - b c ^ 0.
then the i n t e g r a l c u r v e s behave, in the neighbourhood of the singularity, a s if p(x,v) and q(x,v) w e r e a b s e n t .
The s i n g u l a r i t i e s of the reduced equation ax + bv
dv
dx ex + dv ( 2 . 6 )
a r e of four distinct types known a s nodes, c e n t r e s , s p i r a l points and saddles r e s p e c t i v e l y and each has a c h a r a c t e r i s t i c g e o m e t r y , s o m e t i m e s r e f e r r e d to a s its "topological configuration".
6 -C r i t e r i a f o r d i s t i n g u i s h i n g t h e t y p e of s i n g u l a r i t y a r e o b t a i n e d f r o m t h e c h a r a c t e r i s t i c e q u a t i o n \' - X ( b + c) - (ad - be) = 0 ( 2 . 7 ) w h i c h h a s t h e r o o t g T h r e e i m p o r t a n t c a s e s a r e d i s t i n g u i s h e d , a s f o l l o w s . lOtg ..
i| (b + c) inb+c)" + 4(ad - be)! ^j
R o o t s r e a l a n d u n e q u a l . T h e s e a r i s e w h e n t h e d i s c r i m i n a n t (b + c) * + 4 ( a d - b e ) i s p o s i t i v e . T h e s u b - c a s e s a r e : if A>0 t h e n ' ^ a /x, i s n e g a t i v e a n d c o r r e s p o n d s t o a s a d d l e p o i n t , a n d if A< 0 t h e n ^ « / x , i s p o s i t i v e a n d c o r r e s p o n d s t o a n o d e . In t h e l a t t e r c a s e , if X, a n d Xj a r e b o t h n e g a t i v e t h e n o d e i s s t a b l e , w h e r e a s if X, a n d X^ a r e b o t h p o s i t i v e t h e n o d e i s u n s t a b l e . R o o t s c o m p l e x c o n j u g a t e . C o m p l e x r o o t s o c c u r w h e n t h e d i s c r i m i n a n t i s n e g a t i v e . If t h e r e a l p a r t of t h e r o o t i s n e g a t i v e t h e i n t e g r a l c u r v e s a r e s t a b l e s p i r a l s , w h e r e a s if t h e r e a l p a r t i s p o s i t i v e t h e c u r v e s a r e u n s t a b l e s p i r a l s . W h e n t h e r o o t s a r e p u r e l y i m a g i n a r y t h e s i n -g u l a r i t y is a c e n t r e ; h o w e v e r , u n d e r t h e s e c o n d i t i o n s t h e s i n -g u l a r i t y of ( 2 . 6 ) i s not n e c e s s a r i l y t h a t of ( 2 . 5 ) a n d t h e h i g h e r o r d e r t e r m s i n p ( x , v ) a n d q ( x , v ) h a v e t o b e c o n s i d e r e d . R o o t s r e a l a n d e q u a l . T h e s i n g u l a r i t y i s a n o d e a n d i t s s t a b i l i t y i s g o v e r n e d t h e s a m e w a y a s if t h e r o o t s w e r e u n e q u a l . 3 . 0 S t a b i l i t y C r i t e r i a f o r a P a r t i c u l a r S e c o n d - O r d e r E q u a t i o n . T h e s i n g u l a r i t i e s of ( 2 . 3) a r e d e f i n e d b y Q - C(x) = 0, v = 0 a n d t h e r e f o r e t h e n u m b e r of s i n g u l a r p o i n t s w i l l d e p e n d on t h e f o r m of C ( x ) . It w a s s h o w n i n R e f . 4 t h a t t h e a n t i - s y m m e t r i c f o r m f o r c(x) i s p a r t i c u l a r l y v a l u a b l e , i . e . C ( x ) = C,X + CjXS + C X5 + a n d c a n b e u s e d t o r e p r e s e n t a w i d e c l a s s of p r a c t i c a l non l i n e a r i t i e s . F o r t h e r e m a i n d e r of t h i s p a p e r a t t e n t i o n w i l l b e r e s t r i c t e d t o e q u a t i o n s i n w h i c h B ( x ) a n d C(x) a r e c a p a b l e of r e p r e s e n t -a t i o n b y p o w e r s e r i e s i n odd p o w e r s of x . T h e n u m b e r of s i n g u l -a r p o i n t s of ( 2 . 3 ) i s t h e n e q u -a l t o t h e n u m b e r of u n e q u a l r o o t s of t h e e q u a t i o n c,x -I- CjX^ + c^xs + = Q , ( 3 . 1 ) a n d t h e r e f o r e d e p e n d e n t on t h e n u m b e r of t e r m s , n, u s e d t o r e p r e s e n t C ( x ) . F o r s i m p l i c i t y o n l y t h e f i r s t t w o t e r m s a r e u s e d in t h e r e m a i n d e r of t h e a n a l y s i s , a l t h o u g h t h i s c a n r e a d i l y b e e x t e n d e d t o a n y o t h e r r e a s o n a b l e n u m b e r of t e r m s . F o r a g i v e n v a l u e of Q t h e s i n g u l a r p o i n t s of ( 3 . 1 ) a r e in f a c t t h e e q u i l i b r i u m o r s t e a d y -s t a t e p o -s i t i o n -s of t h e -s y -s t e m a n d c a n r e a d i l y b e e v a l u a t e d . S e v e r a l c a -s e -s e x i -s t d e p e n d i n g of t h e s i g n s of c, a n d Cj. T h o s e of e n g i n e e r i n g i n t e r e s t a r e : (a) c , > 0 , C3>0, c o r r e s p o n d i n g t o a " h a r d " s y s t e m i n w h i c h t h e s t i f f n e s s i s i n i t i a l l y p o s i t i v e , (b) c, >0, c , < 0 , c o r r e s p o n d i n g t o a " s o f t " s y s t e m i n w h i c h t h e s t i f f n e s s i s i n i t i a l l y p o s i t i v e a n d (c) c, <0, C3>0, c o r r e s p o n d i n g t o a " h a r d " s y s t e m in w h i c h t h e s t i f f n e s s i s i n i t i a l l y n e g a t i v e . In a d d i t i o n if B i s t a k e n i n t h e f o r m B ( x ) = b , x + bjX3 t h e n B ' ( x ) = ^ = b , -I- 3b,x2. ( 3 . 2 ) d x '
- 7
in which the c a s e s of p r a c t i c a l i n t e r e s t a r e bi>0, b3>0; bi>0, b3<0; and bi<0, b3>0.
Typical c u r v e s of the steady state values of x v e r s u s Q for the c a s e s considered a r e shown in F i g . 1. It can be seen that for a given positive value of Q t h e r e exists one singular point for c a s e (a) and t h r e e each for c a s e s (b) and (c) when Q i s r e l a t i v e l y s m a l l , reducing to one each when Q i s g r e a t e r than the maximum or l e s s than the minimum in the c u r v e . R e s t r i c t i n g attention to positive values of Q only, the singular point in (a) has a value, Xg g >0; in (b) two of the values a r e positive and one negative, whilst in (c) one value i s positive and two are negative. Only in the d e g e n e r a t e c a s e Q^^O i s the singular point at the origin. In addition, on F i g . 1, curves of B'(x) a r e shown.
Since the singular points of (2. 3) a r e in g e n e r a l away from the origin then it m a y be w r i t t e n dv Q - B'(xg g -h ?)v - C(Xg g + ?)
( 3 . 3 )
w h e r e x^ g , 0 a r e the c o - o r d i n a t e s of the singular point and J i s the displacement c o - o r d i n a t e r e f e r r e d to the singular point. Substituting for B'(x) and C(x) then gives
dv Q - C i ( x s . s . + g) - C 3 ( x s . s . + g ) ' - T b , + 3b3(xB.s. + gy] v
d | " V
Expanding, r e m e m b e r i n g that for the p r e s e n t case Q = C(x S . S . ) = c,x S . S . C j X ^
the equation for the slope of the field vector b e c o m e s
dv _ -c,g - C3(3x^s.s.g ^- 3x „ ^ g^-H g^) - [ b , + 3b,(x„ „ + Q^v dg V which in the first approximation r e d u c e s to
dv = -(c,-H 3 c , x | . s . )g - (b, -t- 3b3X»e.s.)v dg V C o m p a r i n g with the standard f o r m , equation ( 2 . 6 ) ,
a* - ( c , -I- ScjX's, s_), b«i -(b, + SbjX^g^ g ), c E 0 and d a l . The r o o t s of the c h a r a c t e r i s t i c equation a r e
T^i.ï = i [ - ( b , + SbjX^g.g.) ± [ ( b , + Sh.x's.a.)' " 4(c, + Sc^xag. g.)] ' J
( 3 . 4 ) ( 3 . 5 a ) ( 3 . 5 b )
= i [ B'(x)±D =
w h e r e j ) = [B'(xy]^ - 4C'(x), is the d i s c r i m i n a n t . ( 3 . 6 )The stability of the motion near the s t e a d y - s t a t e value of the s y s t e m , a s e x p r e s s e d by (3.6) and d i s c u s s e d in Section 2, can conveniently be s u m m a r i z e d on a d i a g r a m of the type shown in F i g . 2. Starting with Q = 0, the variation of Xg g and thereby of B''(x) and C'(x) m a y be ob-tained and the a p p r o p r i a t e c u r v e s s u p e r i m p o s e d on F i g . 2. This then p e r m i t s a r e a d y a s s e s s m e n t of the n a t u r e of the s i n g u l a r i t y at Xg. g, , 0. Typical c u r v e s for the v a r i o u s c a s e s a r e shown in F i g s . 3 and 4 . Consider each of these in t u r n :
b,>0, ci>0, C3>0
The value of b, h a s been taken r e l a t i v e l y s m a l l and the singularity is a stable s p i r a l ; a l t e r n a t i v e l y with higher damping the s i n g u l a r i t y at the origin could be a stable node, point E . With b3>0,
i n c r e a s e or d e c r e a s e of Q now gives r i s e to an i n c r e a s e of B'(x)and C* (x) along curve AB , until at B the c h a r a c t e r of the singularity changes from a stable s p i r a l to a stable node; the value of xg. s . a t
which t h i s change t a k e s place c o r r e s p o n d s to the d i s a p p e a r a n c e of the d i s c r i m i n a n t and i s given by J ) = (b, -h SbjX^s.s.)* - 4(c,-f 303X^8. s . ) = 0
or
9h,»(3^s.s.)a-h (6b,b3 - 12c,)x2s,g. -1- (b, * - 4c,) = 0 which h a s the solution
x s . s . = [ ^ 2 [ 2 c 3 - b , b 3 ± 2 ( c 3 2 + c,b3* -b,b3C3) ^] l i (3.7) in which only r e a l values a r e a p p r o p r i a t e . In the degenerate c a s e when bj = 0, the value of Xg, g.
c o r r e s p o n d i n g to ( 3 . 7) beconnes
''S.S. { i k < b , ^ - 4 c , ) ^ (3.8) which can only have r e a l values when b,2>4ci_ i . e . the initial point A on F i g . 3(a) m u s t be above
the curve £ ) = 0. Such a point i s m a r k e d E , and i n c r e a s e or d e c r e a s e of Q t a k e s place along the curve E F , F c o r r e s p o n d i n g to the value Xg g_ given by (3.8) at which the singularity changes from a stable node to a stable s p i r a l .
When b3<0 the damping d e c r e a s e s a s Q i n c r e a s e s or d e c r e a s e s until at C, F i g . 3(a), the curve m e e t s the damping boundary B'(x) = 0. At this point the singularity at xg. g. , 0 b e c o m e s an unstable s p i r a l , t h e r e b y d e m o n s t r a t i n g that t r a n s i e n t s having values of Q which cause B* (x) to become negative a r e unstable since the motion in the neighbourhood of the s t e a d y - s t a t e value, predicted by stiffness considerations alone, F i g . 1, i s unstable. The c r i t i c a l value of x g . g . for which the s y s t e m goes unstable i s given by
B ' ( x ) = b, -^ 3b33t'g 0 = 0 or I 1 b , \ 2 * s . s . ^ 3 b^ j , > 0 , Q O , C 3 < 0 (3.9)
Again A c o r r e s p o n d s to the origin Q = 0. With i n c r e a s e or d e c r e a s e of Q the stiffness C'(x) d e c r e a s e s and provided hj i s not too l a r g e and negative, the n a t u r e of the change in the
s i n g u l a r i t i e s at xg. g. 0 a r e typified by curve ABC or A E F . The first change in the c h a r a c t e r of the s i n g u l a r i t i e s o c c u r s at B or E where the change to nodal point t a k e s p l a c e . The s t e a d y -state value for3j = 0 a r e again given by (3.7) or ( 3 . 8 ) .
With further i n c r e a s e or d e c r e a s e of Q a point is r e a c h e d , C or F , where the stiffness changes sign i . e . C ' ( x ) = c, -H 3c3Xa a . = 0 o r 3 i ' , (3.10) C3
t r a n s i e n t s having values of Q which would make xg. s . , o n the b a s i s of the c u r v e s of F i g . 1(b), lie outside the region between the m a x i m a and m i n i m a , a r e unstable.
With b3<<0 it i s possible for the s y s t e m to r e a c h the damping boundary, G, before the stiffness boundary. The value of Xg g for which t h i s o c c u r s i s given by (3.9)
b , > 0 , c , < 0 , C3>0
The singularity at the origin i s a saddle point and with bj, not too l a r g e and negative the s i n g u l a r i t i e s change along typical c u r v e s ACEF or AGHJ. At the stiffness boundary, C'(x) = 0, c o r r e s p o n d i n g to the m i n i m a and m a x i m a on the Q, Xg. g. c u r v e s of F i g . 1(c), the s i n g u l a r i t i e s become stable n o d e s . In a p r a c t i c a l s y s t e m t h i s m e a n s that the s y s t e m will never settle down to s t e a d y - s t a t e v a l u e s such a s C or H, F i g . 1(c), but will depart to one or other of the a l t e r n a t i v e stable s i n g u l a r i t i e s A or B , or E or F . The direction in which the s y s t e m m o v e s will depend on the initial a c c e l e r a t i o n .
X = Q. t = 0
i . e . the stable singularity at which the s y s t e m s e t t l e s will have a displacement of the s a m e sign a s Q. In F i g . 1(c), B and E will be the a p p r o p r i a t e settling points, r a t h e r than A or F , when subject to t r a n s i e n t s of magnitudes n u m e r i c a l l y l e s s than those corresponding to the m a x i m a or m i n i m a . This m e a n s that at Q = 0 the s y s t e m will r e s t at values given by
Q = c,x s.s.-i- C j x ' s . s . = 0, X g . s . ^ 0, or
X s . s . = C 3
the values of stiffness c o r r e s p o n d i n g to (3.11) a r e C''(x) = c, + 3 c .
C3 - 2 c .
(3.11)
(3.12) Taking E , F i g . 3, a s a typical point given by ( 3 . 1 1 ) , then application of step^function d i s t u r b a n c e s Q of the s a m e sign a s the displacement will cause the singularity at xg. g. , 0 to move along E F ; a l t e r n a t i v e l y if the sign of the d i s t u r b a n c e , Q, i s of opposite sign to the displacement then the singularity moves along EC becoming unstable at the stiffness boundary. In this l a t t e r case the instability at the stiffness boundary i s not indicative of unbounded d i s p l a c e m e n t , since further i n c r e a s e in the n u m e r i c a l value of Q will cause the s y s t e m to jump to the other stable singularity, t h i s being of opposite sign in d i s p l a c e m e n t .
b,<0, b3>0, c,>0, c,>0
All the c a s e s w h e r e b,<0, b3>0 a r e c h a r a c t e r i z e d by the possible existence of 'limit
c y c l e s " , see Ref. 6. In F i g . 4(a), A i s an unstable s p i r a l point about which oscillations of i n c r e a s i n g amplitude will develop. With i n c r e a s i n g displacement from the origin the damping, B'(x), i n c r e a s e s until it r e a c h e s z e r o . At this condition t h e r e i s established a stable oscillation known a s a limit cycle whose amplitude c o r r e s p o n d s to the displacement OL in Fig. 1(f) i . e . the limit cycle a m p -litude i s given by B'<'x) = 0, or
'LC ^ b 3 (3.13)
With vanishingly s m a l l values of Q the amplitude of the limit cycle i s that of ( 3 . 1 3 ) . I n c r e a s i n g values of Q, c o r r e s p o n d i n g to moving along curve A to B in F i g . 4(a), c a u s e s a shift in the point about which the limit cycle oscillation o c c u r s and a reduction in the amplitude.
10
-The new origin of the limit cycle motion will be at the value of Xg
i . e . at solutions of s.
given by stiffness c o n s i d e r a t i o n s
Q '1-^S. s . + C T X ' s . s .
and the amplitude will be b .
' L C ,
^-^S^
• • S . S . (3.14)When Q i s of sufficient magnitude to cause B'(Xg g ) to be z e r o , c o r r e s p o n d i n g to B , then the amplitude of the limit cycle b e c o m e s z e r o and the t r a n s i e n t r e s p o n s e b e c o m e s v e r y s i m i l a r to that obtained if the singular point at Xg g ,0 w e r e a stable node,
F o r l a r g e r values of Q the singular point at Xg g , 0 is a stable s p i r a l , or eventually a node. T h e s e a r e s i m i l a r to the c a s e s of F i g . 3(a), but the n a t u r e of the t r a n s i e n t motion will differ a s a r e s u l t of the negative damping experienced in the e a r l i e r portion of the motion. b, <0, b3>0, c, >0, C3<0
In this c a s e two p o s s i b i l i t i e s e x i s t , F i g . 4(b), With a r e l a t i v e l y s m a l l amount of n e g ative damping initially, i n c r e a s e or d e c r e a s e of Q i s a s s o c i a t e d with a limit cycle whose a m p -litude d e c r e a s e s in a s i m i l a r way tb the previous c a s e . The nature of the singularity from B through C to E i s then s i m i l a r to the second c a s e . F i g . 3(b), bj>0. Alternatively the variation m a y follow the curve FG in which the s i n g u l a r i t i e s a r e always u n s t a b l e .
b,<0, b3>0, c,<0, C3>0
The initial point F , F i g . 4(c), i s a saddle point. Small values of Q cause the s y s t e m to d i v e r g e , however, with i n c r e a s e of displacement the stiffness changes sign and the motion changes, during the t r a n s i e n t , to a limit cycle whose origin i s the value of Xg_ g_ at the stiffness boundary and amplitude c o r r e s p o n d s to the difference of xg, g. , a s given by stiffness considerations alone, at the points G and H. i . e . the origin of the limit cycle i s given by
C(x) = c 1 - S . s . + ^Tl- S . S = 0 o r
- . < ^ ) * and its amplitude is
^LC. =
( - * ^ ) * - ( • ^ ;
(3.15)
(3.16) With i n c r e a s e of Q the amplitude of the limit cycle d e c r e a s e s in a s i m i l a r way to that d i s c u s s e d in the fourth c a s e . F i g . 4(a).
A l t e r n a t i v e l y , following the curve ABCE, the damping boundary may be r e a c h e d p r i o r to the stiffness boundary. In this c a s e no limit cycle develops and the singular point variation i s s i m i l a r to that of the t h i r d c a s e , F i g . 3(c), bj>0.
T h e s e then a r e the six c a s e s of engineering i n t e r e s t and d e s c r i b e the nature of the singular points at the s t e a d y - s t a t e values given by C(x) = 0. They a r e not in t h e m s e l v e s suf-ficient to d e t e r m i n e the stability of the t r a n s i e n t motion.
11
-Returning to equation ( 3 . 5 a ) , it can be seen that the slope of the field vector in the phase plane depends on Xg^ g_ , g and v. T h i s m e a n s that if the system is initially at r e s t at a singularity X v = 0 in the phase p l a n e , then the stability can be decided by an instantaneous change in the orientation of the singular points, c o r r e s p o n d i n g to the changed values of Q. On this b a s i s the initial position of r e s t b e c o m e s an o r d i n a r y point in the phase plane and the system will either move to a stable s i n g u l a r i t y , not n e c e s s a r i l y the n e a r e s t one, diverge indefinitely or limit cycle. The t r a n s i e n t m.otion of the s y s t e m i s d e s c r i b e d by the integral curve through this o r d i n a r y initial point and the stability or o t h e r w i s e . This movement can only be completely determined by ref-e r ref-e n c ref-e to thref-e form of thref-e i n t ref-e g r a l c u r v ref-e s .
It is convenient to use the c u r v e s of F i g . 10, where for the p r e s e n t purpose x and v a r e identified with W and W r e s p e c t i v e l y , x^ is the s t a r t i n g point, Xg the position of the n e a r e s t stable singularity and Xp the n e a r e s t saddle point. Taking the c a s e s in turn
b,>0, c,>0, C3>0
T h e r e i s only one singular point for a given value of Q, fig. 10(a), and since this i s a stable L-piral or node all i n t e g r a l c u r v e s move into it.
b,>0, c,>0, c,<0
A stable singularity can only exist if Q l i e s between the points of m a x i m a and m i n i m a on the Q, x c u r v e . For Q between t h e s e l i m i t s t h e r e a r e t h r e e s i n g u l a r i t i e s consisting of a stable s p i r a l or node lying between two saddle p o i n t s . F i g . 10(b). In o r d e r that the s y s t e m will settle at the stable singularity the initial point x^ must lie between the saddle points, i . e . the initial and final positions must not be s e p a r a t e d by an i n t e g r a l curve which p a s s e s through a s a d d l e . b,>0, c,<0. c,>0
When Q lies between the minima and m a x i m a of the Q, x curve t h r e e s i n g u l a r i t i e s e x i s t , t h e s e consisting of a saddle point lying between two stable s p i r a l s or nodes. F i g . 10(c). If A
l i e s between a stable singularity and a saddle it will always settle at that stable singularity. When x^ i s n u m e r i c a l l y g r e a t e r than the x c o - o r d i n a t e of the stable singularity the s y s t e m will con-tinue to settle at this singularity until an initial displacement is r e a c h e d at which A i s s e p a r a t e d from E by an i n t e g r a l curve passing through the saddle. The i n t e g r a l curve through A now m o v e s , not into E , but to the stable singularity on the other side of the saddle.
The complete c r i t e r i a for stability of the r e s p o n s e of a step function a r e t h e r e f o r e (1) The singular point at the steady state value given by C(x) = Q must be s t a b l e .
(2) On the phase plane d i a g r a m a s s o c i a t e d with the final steady state v a l u e s , the final and initial positions of the s y s t e m must not be s e p a r a t e d by an integral curve p a s s i n g through a saddle point.
4 . 0 A N u m e r i c a l E x a m p l e .
As a check on the validity and a c c u r a c y of the stability c r i t e r i a obtained in Section 3 . 0 , a limited number of solutions of th.> equation
X + hx + c , x -H CjXï = Q , Q ~ 0 , t < 0
have been obtained on the F e r r a n t i " P e g a s u s " digital computer at The College of A e r o n a u t i c s . In these examples the damping was taken constant and positive, t h e r e b y excluding limit cycling from the solutions. The values of the coefficients used w e r e a s follows:
12 -(a) (1) b = 0 . 6 , c , = 1 . 0 , C3= 1 . 0 , Q = O t o 2 . 1 ( l i ) b = 0 . 6 , c = 0 . 0 5 , 0 3 = 0 . 0 4 , Q = 0 t o 0 . 6 (b) b = 0 . 6 , C i = 1 . 0 , C 3 = - 1 . 0 , Q = 0 t o 0 . 5 (c) b = 0 . 6 , c , = 1 . 0 , c , = 1 . 0 , Q = 0 t o l . 0 . T h e t r a n s i e n t r e s p o n s e s o b t a i n e d a r e s h o w n p l o t t e d i n F i g s . 5 , 6 , 7 a n d 8 i n w h i c h t h e d i s p l a c e m e n t c o - o r d i n a t e h a s b e e n n o r m a l i z e d b y d i v i s i o n b y t h e a p p r o p r i a t e s t e a d y s t a t e v a l u e . T a k i n g t h e c a s e s in turn: ( a ) (i) H e r e t h e i n i t i a l s t i f f n e s s i s p o s i t i v e a n d £ ) < O f o r a l l v a l u e s of Q . A s e x p e c t e d t h e s e t t l i n g down m o t i o n i s o s c i l l a t o r y for a l l v a l u e s of Q . T h e r e s p o n s e c u r v e s t e n d t o t h e l i n e a r r e s u l t a s Q-»0, i n c r e a s e of Q p r o d u c e s a n i n c r e a s e of f r e q u e n c y of t h e o s c i l l a t i o n a b o u t t h e s t e a d y - s t a t e . It i s a l s o e v i d e n t t h a t t h e a m o u n t of " o v e r s h o o t " a n d " u n d e r s h o o t " i s d e p e n d e n t on Q . T h i s f e a t u r e of t h e c u r v e s i s o u t s i d e t h e s c o p e of t h e p r e s e n t s t a b i l i t y i n v e s t i g a t i o n a n d t o a n a l y s e i t , i n t e r m s of b , c , , C3 a n d Q , w o u l d r e q u i r e a n a n a l y t i c s o l u t i o n t o t h e e q u a t i o n . (a) ( i i ) A g a i n t h e i n i t i a l s t i f f n e s s i s p o s i t i v e , b u t X ) > 0 . At l a r g e v a l u e s of Q t h e d i s c r i m i n a n t 35 b e c o m e s n e g a t i v e , c o r r e s p o n d i n g t o t h e t r a n s i e n t s e t t l i n g down m o t i o n b e c o m i n g o s c i l l a t o r y . W i t h i n c r e a s e of Q t h e m o t i o n b e c o m e s m o r e d a m p e d u n t i l a t jD=0 b o u n d a r y t h e t r a n s i e n t b e c o m e s n o n - o s c i l l a t o r y i n c h a r a c t e r . T h i s o c c u r s w h e n Xg g h a s t h e v a l u e g i v e n b y ( 3 . 8 ) , i . e . Xg g = ( i ) 2 = 0 . 5 8 0 a n d Q = 0. 05 X 0. 580 -t- 0. 04 X 0. 5 8 0 ' = 0. 0 3 7 . (b) W i t h Q s m a l l , jL)<0, a n d t h e s i n g u l a r i t y a t x g g i s a s t a b l e s p i r a l . I n c r e a s e of Q c o r r e s p o n d s t o i n c r e a s e o f ^ , u n t i l D = 0 w h e n ' • S . S . = r — ^ ( 0 . 6 2 - 4 X l ) r = 0 . 5 5 1 (_12 X 1 J a n d Q = 0 . 5 5 1 - 0 . 5 5 1 ' = 0 . 3 8 4 . F u r t h e r i n c r e a s e of Q c a u s e s a r e d u c t i o n i n s t i f f n e s s , t h e z e r o s t i f f n e s s b o u n d a r y b e i n g r e a c h e d w h e n X s . s . = < - i - J i ) ' = 0 - 5 8 0 a n d Q = 0 . 5 8 0 - 0 . 5 8 0 ' = 0 . 3 8 5
13
-F r o m t h i s it can be seen that in t e r m s of Q theXJ =0 and C'(x) = 0 boundaries a r e v e r y close t o g e t h e r . The computer r e s u l t s , F i g . 7, indicate that instability o c c u r s at the lower value Q = 0.36 0 . 3 7 . This difference i s probably due to instability in the RungeKutta i t e r a t i o n p r o
-cedure used to p e r f o r m the machine i n t e g r a t i o n . (c)
In this c a s e the singularity at the origin i s a saddle and any positive initial a c c e l e r a t i o n will cause the s y s t e m to move to the stable s p i r a l point at a positive value of Xg g The m a g -nitude of Q d e t e r m i n e s Xg g_ and has a m a r k e d influence on the " r i s e t i m e " and frequency of the settling down motion, F i g . 8. The r e s u l t s of F i g . 8 do not have v e r y much engineering s i g -nificance since if such a s y s t e m w e r e employed any t r a n s i e n t would n o r m a l l y commence from a stable condition of e q u i l i b r i u m . If for i n s t a n c e the initial condition corresponded to a stable s p i r a l point at a positive value of x, then positive step-functions of Q would produce r e s p o n s e c u r v e s s i m i l a r to F i g . 5. Small negative step-functions of Q would produce curve s i m i l a r to F i g . 7.
L a r g e negative step-functions of Q would cause the s y s t e m to jump to negative values of Xg_ g and would p r e s u m a b l y have a t r a n s i e n t s s i m i l a r to that sketched in F i g . 9.
Summarizingi the computer solutions show that the c h a r a c t e r of the t r a n s i e n t settling down motion is d i r e c t l y r e l a t e d to the n a t u r e of the singularity at the s t e a d y - s t a t e condition and i s a c c u r a t e l y p r e d i c t e d by the c r i t e r i a of Section 3 . 0 .
5. 0 Stability of the S h o r t - P e r i o d Motion of an A i r f r a m e Subject to a Step-Function E l e v a t o r D i s t u r b a n c e .
In Ref. 4 the author has shown how to introduce non-linear n o r m a l force and pitching moment c h a r a c t e r i s t i c s into the equations of motion of an a i r f r a m e whose dontiinant mode of oscillation i s the " s h o r t - p e r i o d " motion. F o r t h i s purpose the c h a r a c t e r i s t i c s a r e taken to be of a n t i - s y m m e t r i c form and a r e e x p r e s s e d analytically a s
Z(w) = z ^ W -I- Z j W ' + ZsW' -I-and M(w) B m ^ W + m j W ' -1- nijW' -H (5.1)
Upon substituting t h e s e e x p r e s s i o n s into the equations of motion and eliminating 3 between t h e m , the equation of motion in the v e r t i c a l velocity, W, b e c o m e s
^ - [ (Uo + Zq)mvt + mq + Zw + ^ZjW' . • • •] \^ " [(U,, + Zq)m^ - mqZ^^ W
- [(Uo+ V " 3 - m q Z 3 ] W ' = [z„D-f (Uo + Zq)m„ - z „ m q ] H , (5.2) w h e r e
W = w,j, -f w, (5.3) B =ri^ + n (5.4) and suffix rr, r e f e r s to initial t r i m m e d conditions. In the p r e s e n t problem the i n c r e m e n t in v e r
-t i c a l veloci-ty, w, a r i s e s from -the applica-tion of a s-tep-func-tion d i s -t u r b a n c e of -the e l e v a -t o r , ?} . R e s t r i c t i n g (5.1) to two t e r m s , then (5.2) b e c o m e s
14
-w h e r e
w - [ B , -I- B3(w,p + v/)^]^ - [ A,(w,j, + v/) + Aj(v/rj, + w ) ' ] = [ z ^ + (UQ + Zq)m;j - ZrjmqJ (r)rp +r]), A, = (UQ -I- Zq)mw - mqZ^ A , = (UQ + Zq)m, - mqZ3 B , = (U^ + Zq)m^ + "iq + ^w B3 = 3Z3 (5.5) (5.6)
It i s now convenient to r e f e r the w c o - o r d i n a t e to the final s t e a d y - s t a t e value Wg_ c. Write
" S . S . + g , (5.7) (5.8) s. s . 5)+A3(Wg_g_ + g ) ] and (5.5) b e c o m e s w - [ B , + B3(Wg_g_ + g ) ' ] w - [A,(W = [ Z T P + (UQ -)- ZQ)m„- z^mqj (;j,p +v) The s t e a d y - s t a t e condition i s defined by
-AiWg.g. -A3w'g_g_ = [ ( U Q + Z q ) m ^ - z ^ i q ] (n^ + r,), which upon substitution into (5.9) gives
w- - [ B , + B3(Wg_ g+ g)^j w - [ A , g -f A3 [3W^g_ 6. ? + 3Wg_ g_ g % g^] J = 2„ ri
(5.9)
(5.10)
(5.11) Following P o i n c a r é , an e x p r e s s i o n i s sought for the slope of the field vector in the g,g plane in the r e g i o n of the singularityW , 0 . F o r t>0, ri , which i s a D i r a c delta function, i s z e r o , t h e r e f o r e in the first approximation (5.11) b e c o m e s
g - ( B , + B3W'g_g )g - ( A , + 3A3W'g g )g = 0 (5.12) The problem d e s c r i b e d by (5.11) differs from that of Section 3 b e c a u s e of the t e r m
Zr)'^, which gives r i s e to an initial velocity
^ t = 0 1 = 1 ^») I
W (5.13)
and can have an important influence on the s y s t e m stability. (See Ref. 7 for a discussion on the determination of initial conditions). F o r many a i r f r a m e s with s m a l l e l e v a t o r s situated well away from the c e n t r e of gravity the value of z„ i s s m a l l and the effect of initial velocity may in many
15
-c i r -c u m s t a n -c e s be negle-cted. F o r moving wing -configurations the effe-ct is dominant and r e q u i r e s special t r e a t m e n t . In the p r e s e n t d i s c u s s i o n attention will be focussed on the f o r m e r , whilst c e r t a i n a s p e c t s of the effect of initial velocity on the stability will be d i s c u s s e d in Section 6. 0
Neglecting then the initial velocity, the stability of the r e s p o n s e to a stepfunction e l e -v a t o r deflection m a y be t r e a t e d in a s i m i l a r way to the problem of Section 3 . The r o o t s of the
c h a r a c t e r i s t i c equation become
X , _ , = i [ ( B , + B 3 W \ . g _ ) ±[(B, +B3W%.g.)^-f 4(A, + 3 A 3 W % . g . ) ] ^ J (5.14) W r i t e the equation of t r i m , ( 5 . 1 0 ) , in the form
F = [ ( U o + Zq)m^ - z^mq 1 (rirj, + n) = -A,Wg_g_ - A j W ^ ^ g , ;
identify F , W, -A, , - A j , - B , and -B3 with Q, x, c^, Cj, b , a n d 3 b 3 r e s p e c t i v e l y in Section 3 and the d i s c u s s i o n of a i r f r a m e stability can then proceed on the b a s i s of F i g s . 1 and 3 .
F o r a n o r m a l a i r f r a m e (aeroplane or m i s s i l e ) B, is negative, t h e r e b y excluding the possibility of a change from negative to positive damping and the a s s o c i a t e d limit cycling. The c a s e s to be considered a r e t h e r e f o r e s i m i l a r to the first t h r e e d i s c u s s e d in 3 . 0 . Taking these in t u r n :
B.<0, A,<0, A3<0
The dominant t e r m in A i s U g m ^ . Since m.^^, i s proportional to the c e n t r e of gravity m a r g i n then A, <0 i m p l i e s that the a i r f r a m e i s statically stable at low incidence. The t e r m Ugmj is dominant in A3 and with A3<0 the a i r f r a m e i n c r e a s e s its static stability with i n c r e a s e of incid-e n c incid-e . A l t incid-e r n a t i v incid-e l y t h i s m a y bincid-e d incid-e s c r i b incid-e d by saying that thincid-e a incid-e r o d y n a m i c stiffnincid-ess i n c r incid-e a s incid-e s with incidence,thereby constituting a " h a r d " s y s t e m .
B3 can be of e i t h e r sign. Many a i r f r a m e s having wings of low a s p e c t r a t i o of a x i -s y m m e t r i c body configuration-s exhibit W, C^, c h a r a c t e r i -s t i c -s who-se -slope i n c r e a -s e -s
over the whole of the useful incidence r a n g e , c o r r e s p o n d i n g to Z3 and B3 being negative. O t h e r s have wings of higher a s p e c t - r a t i o which stall at r e l a t i v e l y s m a l l i n c i d e n c e s , an effect which can be r e p r e s e n t e d a p p r o x i m a t e l y by taking Z3>0. The approximation involved is satisfactory provided the t r a n s i e n t motion does not cause the incidence to i n c r e a s e v e r y much above the s t a l l . When oscillating through the s t a l l it is fairly c e r t a i n that aerodynamic h y s t e r e s i s will o c c u r , ( i . e . the W, Cg curve followed during the nose-up swing of the a i r f r a m e will not be r e - t r a c e d during the subsequent nose-down swing) and the form used in (5.1) to r e p r e s e n t the forces and m o m e n t s will be inadequate.
The point A, in F i g . 3(a), now c o r r e s p o n d s to the origin of the F , W curve and the problem r e s o l v e s itself into deciding the stability of t r a n s i e n t s when moving from any typical singular point, c o r r e s p o n d i n g to the final t r i m m e d condition, along curves such a s AB or AC. With B3 <0 the c h a r a c t e r of the settling down motion can change from o s c i l l a t o r y to heavily damped and the boundary between t h e s e conditions is given by
(B,-I- B3W^g_g_)% 4(A, -f- 3A,W%_g_) = 0
[W,. [ - ^ ^ - S i S 3 i 6 [ A 3 + i A 3 B , B 3 - i A , B ; ] * | ƒ (5.15) or
16
-With B3>0 the curve AC will meet the damping boundary when -B, - B 3 W ' g . g . =0, or { B j N i r ( U o + zq)m-vfr -^ mg -f zw T ,:• i«v Now from ( 5 . 1 ) Z(w) „ , ,1,3 - ^ = z ^ W + Z3W
which upon differentiation with r e s p e c t to W gives 1 dZ(w) . m dW ^w ^ "^'^'^ •
The s t a l l , which o c c u r s at s y m m e t r i c a l values about z e r o incidence, is defined by dZ(w)
dW or
= 0 ,
Wgtall = i-^S" (5.17) The values of z^^,, mq and m^ will n o r m a l l y be negative and for this c a s e Zj will have to be
positive. It follows that the s t e a d y s t a t e incidence for which the damping b e c o m e s z e r o is n u m e r -ically g r e a t e r than the stalling incidence by an amount
^[l,-|.,i| - i i g / l ] ,
W/
w h e r e the incidence has been taken to be /u„ • T h i s r e s u l t m u s t , of c o u r s e , be t r e a t e d with s o m e r e s e r v a t i o n since it is unlikely that the "damping in pitch" t e r m
(UQ + Zq)mv5, -I- mq
will be constant for an oscillation which includes the s t a l l . However, it i s equally unlikely that this t e r m will change sign and t h e r e f o r e it m a y be concluded that the previous statement r e g a r d -ing the damp-ing boundary is qualitatively t r u e , but leaves uncertainty a s to the amount the damp-ing boundary e x c e e d s the stalling incidence.
B,<0. A.<0, A,>0
Again the a i r f r a m e i s statically stable at low incidence, but now the static stability d e c r e a s e s with incidence, thereby constituting a "soft" s y s t e m . A notable example cf such an a i r f r a m e i s the c a n a r d m i s s i l e configuration which e x p e r i e n c e s considerable n o n - l i n e a r body lift and, a s a r e s u l t of the c e n t r e of p r e s s u r e of the non-linear body lift being ahead of the centre of gravity, develops a nose-up pitching m o m e n t . This configuration would n o r m a l l y have A3>0 a s s o c i a t e d with B,<0.
17 -T h i s c a s e w i l l h a v e b o u n d a r i e s s i m i l a r t o t h o s e e x p r e s s e d b y ( 5 . 1 5 ) a n d ( 5 . 1 6 ) , a d d i t i o n w i l l h a v e a z e r o s t i f f n e s s b o u n d a r y g i v e n b y but S A j W ^ . g . 0, W s . s . / 0. o r W S . S . ' 3 • A, V
r 1 1 i
" 3 - (Ur. + z q ) m w - mqZw 2 ( U Q + Zq)m3 - m q Z , J ( 5 . 1 8 ) F r o m ( 5 . 1 ) M(w) m = m ^ W 4 m3W a n d t h e s y m m e t r i c a l l y d i s p o s e d p o i n t s of m a x i m a a n d m i n i m a on t h e p i t c h i n g m o m e n t v e r s u s i n c i d e n c e c u r v e w i l l o c c u r a t W M ( 5 . 1 9 ) Now z ^ a n d m „ w i l l n o r m a l l y b e n e g a t i v e a n d it f o l l o w s t h a t if Z3<0 t h e n t h e s t e a d y - s t a t e i n c i d e n c e W M f o r w h i c h t h e s t i f f n e s s b e c o m e s z e r o i s n u m e r i c a l l y g r e a t e r t h a n U . W h e n Z3>0, t h e z e r o s t i f f n e s s b o u n d a r y m a y b e a t s m a l l o r g r e a t e r v a l u e s of i n c i d e n c e t h a n M d e p e n d i n g on t h e r e l a t i v e m a g n i t u d e of z ^ a n d Zj. U B , < 0 , A , > 0 , A^<0 H e r e t h e a i r f r a m e i s s t a t i c a l l y u n s t a b l e a t low i n c i d e n c e . W i t h F = 0 t h e a i r f r a m e w i l l t r i m out a t v a l u e s of Wg g_ , g i v e n by F = - A , W g . g . - A 3 W g . g . 0, w , ^ 0, o r W . •A^^i A, a n d t h e c o r r e s p o n d i n g v a l u e of t h e s t i f f n e s s w i l l b e^1
-'^AA;=-'^'-( 5 . 2 0 ) ( 5 . 2 1 ) T h e l i n e C ' ( x ) = - 2 c , , in F i g . 3 ( c ) , i s now i n t e r p r e t e d a s a l i n e of c o n s t a n t s t i f f n e s s - 2 A , . I n c r e a s i n g v a l u e s of F now c a u s e t h e s i n g u l a r i t y t o m o v e a l o n g E F o r H J . If F i s n e g a t i v e t h e n t h e m o v e m e n t i s a l o n g E C o r H G , b e c o m i n g u n s t a b l e a t t h e s t i f f n e s s b o u n d a r y a s g i v e n by ( 5 . 1 8 ) . S i n c e m.^ >0 a n d m3<0, t h e v a l u e of W c o r r e s p o n d i n g t o t h e s t i f f n e s s b o u n d a r y c a n b e e i t h e r g r e a t e r o r s m a l l e r t h a n Wjyj a n d i n p a r t i c u l a r w h e n z ^ < 0 , Z3<0 t h e n W a t z e r o s t i f f n e s s w i l l b e >^j^-T h e i n s t a b i l i t y a t t h e s t i f f n e s s b o u n d a r y c o n s i s t s of a j u m p t o t h e r e m a i n i n g s t a b l e s i n g u l a r i t y , c o r r e s p o n d i n g t o a c h a n g e in s i g n a n d i n c r e a s e of m a g n i t u d e of t h e t r i m m e d i n c i d e n c e . O b v i o u s l y t h i s s o r t of b e h a v i o u r i s out of t h e q u e s t i o n for a n a e r o p l a n e , but p o s s i b l e on a m i s s i l e h a v i n g no a u t o m a t i c c o n t r o l s y s t e m . A s i n d i c a t e d i n Ref. 8, w h e n d e a l i n g w i t h a r e a r - c o n t r o l l e d m i s s i l e a u s e f u l i n c r e a s e i n a e r o d y n a m i c gain,(W/^)g_ g ^ c a n b e o b t a i n e d by m a k i n g t h e a i r f r a m e s t a t i c a l l y u n s t a b l e a t s m a l l i n c i d e n c e . T h i s w i l l b e o f f s e t b y h a v i n g a r e g i o n a r o u n d W = 0 f o r w h i c h a s t a b l e t r i m c o n d i t i o n c a n n o t b e o b t a i n e d , a t l e a s t not w i t h o u t t h e u s e of a c l o s e d - l o o p c o n t r o l s y s t e m .
18
-It was shown in Section 3 . 0 that in addition to having a stable singularity at Xg g , the conditions for complete stability imposed limitations on the initial and final values of t r i m , i . e . limited the magnitude of the step function imposed. This condition has not been included in the p r e s e n t section since it will be seen to constitute a special c a s e of the c r i t e r i a developed in Section 6 . 0 .
6. 0 Stability of S h o r t - P e r i o d Motion When ZT; i s Not Small.
As indicated in Section 5 . 0 , when z^ i s not s m a l l , the initial velocity | W^ = Q I "" I ^T? I can under c e r t a i n c i r c u m s t a n c e s have an important influence on the stability. Again the stability i s t r e a t e d by a s s u m i n g an instantaneous change in the orientation of the singular points, c o r r e s -ponding to a step-function change in F . When Zj, is not s m a l l t h e r e will be an a p p r e c i a b l e value of W|. _ Q and t h e r e b y the initial position of r e s t i s t r a n s f o r m e d to an o r d i n a r y point in the p h a s e -plane with a known displacement from the new s i n g u l a r i t i e s and an initial velocity W^ _ Q. Consider each c a s e in turn:
B, <0, A, <0, A,<0
F o r each value of F t h e r e e x i s t s only one singularity, which will be stable if the damping is p o s i t i v e . In this c a s e the s y s t e m will always settle to the singularity r e g a r d l e s s of the sign of W^ = 0 . although the settling t i m e will be l e s s for negative values of W, shown a s W, in F i g . 10(a).
B, <0, A, <0.A,>0
F o r s m a l l positive values of F t h r e e singular points e x i s t . F i g . 10(b), one stable s p i r a l at positive W, one saddle at l a r g e r positive W and another saddle at negative W. Now when z_ and hence W, or Wg a r e z e r o then a sufficient condition for stability is that the dashed line, r e p -r e s e n t i n g the initial value of W, shall lie between the saddle points, b e a -r i n g in mind that the orientation changes with F , Obviously when W, and W^ a r e s m a l l the s a m e c r i t e r i o n i s a p p r o x -i m a t e l y t r u e . W-ith -i n c r e a s -i n g values of W^ = Q SL po-int -is r e a c h e d at wh-ich the -i n t e g r a l c u r v e s no longer s p i r a l in to the stable s i n g u l a r i t y . When W, i s n u m e r i c a l l y g r e a t e r than a c r i t i c a l value, (W^)^. which lies on the i n t e g r a l curve p a s s i n g through the saddle point at negative W, then the motion d i v e r g e s indefinitely in the negative s e n s e . A s i m i l a r divergence o c c u r s if
%>(%)cThe i n t e r e s t i n g problem h e r e is to d e t e r m i n e the c r i t i c a l values of W^ _ Q or the c o r r e s -ponding value of zjj for a given step change of F or v. Alternatively if z^ i s fixed the problem i s to d e t e r m i n e the magnitude of the step change in F or r; for which W^ = o r e a c h e s a c r i t i c a l v a l u e . As will be seen from Ref. 2, pp. 6 1 - 8 0 , t h i s problem is v e r y s i m i l a r to that of determining the c r i t i c a l d i s t u r b a n c e of a damped pendulum or the "pull-out t o r q u e " of a synchronous m o t o r .
A s s u m i n g that the a i r f r a m e is t r i m m e d at a s m a l l positive incidence and z_ i s negative (this i s always t r u e for the convention adopted), then for a r e a r controlled m i s s i l e an i n c r e a s e of incidence is obtained by making 7?more negative and the a s s o c i a t e d value of Wj- = o = "^n will be positive. F o r a canard or moving wing a r r a n g e m e n t an i n c r e a s e of incidence is achieved by making 77 m o r e positive and t h e r e f o r e W^ = 0 = '^r} will be negative.
Consider the case of a canard a i r f r a m e , for which the conditions A^<0, A3>0 a r e c h a r a c -t e r i s -t i c (a fea-ture a r i s i n g from -the c e n -t r e of gravi-ty being behind -the cen-tre of p r e s s u r e of -the n o n - l i n e a r body lift) and a s s u m e that it i s t r i m m e d at a positive incidence corresponding to the point A on F i g . 10(b) and below the m a x i m u m on the W , F c u r v e . A negative step-function of V is applied to the elevator and the new t r i m value will be at the c e n t r e , E , of the stable s p i r a l shown. However, in this c a s e W^ = Q = -Zj^ is positive and will oppose the motion toward the
19
stable condition. If W^ = g * ^ A c ^^^ c r i t i c a l value, the r e s p o n s e will finally settle at the
stable s i n g u l a r i t y , w h e r e a s if W^ = Q ^^Ar^ the a i r f r a m e will become unstable.
The problem then i s to develop an e x p r e s s i o n d e s c r i b i n g the i n t e g r a l c u r v e s through the unstable singularity P with the intention of using the e x p r e s s i o n to d e t e r m i n e W^ . If g i s the W c o - o r d i n a t e r e l a t i v e to P.then
W = Wp + g
and the i n t e g r a l c u r v e s in the region of Wp a r e d e s c r i b e d by
k' - [ B , + B3(Wp + g)*] g - [^ A,g + A3 [3W'p. g + 3Wpg + C']J = 0 (6.1)
Now g = g -rr- and (6.1) m a y be r e - w r i t t e n in the form dg
g [ ^ - [ B ^ + B3(Wp+ g)*] J= (A, + 3A,W*p) g + 3A3Wpg^+ g' (6.2) A t g = 0, W = g = 0 and g m a y be developed a s a power s e r i e s in the form
g = a,g + a,g"+ a j g ' t (6.3) which upon differentiation with r e s p e c t to g, gives
^ = a, -h 2ajg + 2a^g + 3a3g^ + (6.4) Substitution from (6.3) and (6.4) into (6.2) then gives
(a,g + a^g'-f a j g ' + ) [ ( a , - B, - B3W'p) + 2(a, - B3Wp)g
-I- (3a3 - B3)g^-^ ] = (A, 4- 3A3W'p)g + 3A3Wpg% A3g' (6.5) Since (6. 5) is to be t r u e for all values of g, then the coefficients of like powers of g m a y be
equated giving the indicial equations:
1) a ' - (B, + B j W y a , - (A, + SAjW^) = 0 or
1, = I ["(B, + B3W' )± [ (B, + B,W' f+ 4(A, + 3 A , W ' )] * 1 (6.6)
2) a,(2a2 - 2B,W ) 4- a^ia, - B, - B3W* ) = 3A,W o r
(3a34 2a,B,)W
a, = ^ 3a, - B , - B3W 1 1 p
3) a,(3a3 - B3) 4- a2(2a^ - 2B3W ) 4- a,(a, - B, - B3W^ ) = A,
(6.7)
or
A3 - 2a2(a a - B 3Wp) + a,B 3 4a, - B , - B 3 W ^ p
20
-The value of W^ will then be
WA^ = a,(wT^ - Wp) + a,(w,j,^ - W p ) * + ^ ' < * T A " V ' ^^"^^ F r o m ( 6 . 6 ) , a, has two v a l u e s . Since P i s a saddle point then (A, + 3A3W*p)> 0
and the values of a, m u s t be r e a l , positive and negative r e s p e c t i v e l y . T h e s e two values c o r r e s -pond to the initial slopes of the two i n t e g r a l c u r v e s passing through the singularity. In the p r e s e n t problem the negative value of a, is a p p r o p r i a t e .
With known a e r o d y n a m i c c h a r a c t e r i s t i c s , A, , A j , . . . e t c . , it i s now possible to d e t e r m i n e the c r i t i c a l value of W^ = Q or Zjj for a r a n g e of v a r i o u s initial and final t r i m m e d i n c i d e n c e s .
B,<0, A, >0, A3<0
When flying t r i m m e d in the lower r a n g e of positive incidence t h r e e singular points e x i s t s , F i g . 10(c). This c a s e would be typical of the r e a r controlled a i r f r a m e if it w e r e s t a t -ically unstable at low incidence.
A positive step-function of n i s applied in o r d e r to r e d u c e the t r i m m e d incidence to a point c o r r e s p o n d i n g to E . When Z77 = 0, it i s a sufficient condition for the motion to settle at E , that the s t a r t i n g point A shall not lie outside the i n t e g r a l curve p a s s i n g through the saddle point c o m p l e m e n t a r y to the stable s p i r a l at E . T h i s m e a n s that if the magnitude of the s t e p -function is too g r e a t the motion will not settle at E, but will p a s s over to the other stable s p i r a l at negative i n c i d e n c e . The limiting value can only be obtained by constructing the i n t e g r a l curve through the unstable singularity at P .
When ZTJ ^ 0, the value of W. _ „ = Zrj , which i s negative, and if this value i s n u m e r i c a l l y g r e a t e r than W A „ the a i r f r a m e will not t r i m at E but will d e p a r t to the other stable singular it J'.
F r o m t h e s e t h r e e c a s e s it can be seen that the t e r m z^. does not a l t e r the stability c r i t e r i a of Section 3 . 0 , provided t h e s e a r e i n t e r p r e t e d in a slightly m o r e g e n e r a l m a n n e r . As before the final steady state condition must be a stable singularity, and the initial and final points on the phase plane must not be s e p a r a t e d by an i n t e g r a l curve p a s s i n g through a saddle point which i s c o m p l e m e n t a r y to the stable singularity at the steady state value. W h e r e a s previously the initial value was at a point x ^ , v = 0, when z^) ^ 0 the initial value will be X. , v ^ 0. The c a s e s d e s c r i b e d in Section 3.0 and 5.0 a r e s p e c i a l c a s e s for which v = 0. A N u m e r i c a l Example
In o r d e r to d e m o n s t r a t e the calculation of W^ and show its influence on the effective stability b o u n d a r i e s , an example has been chosen of a r e a r controlled m i s s i l e conforming to the conditions B,<0, B3<0, A^>0, A3<0. The m i s s i l e is a c r u c i f o r m , a i r t o a i r type having a u s e -ful speed r a n g e of 1,500 to 3,500 f. p . s . Its operation and a e r o d y n a m i c c h a r a c t e r i s t i c s a r e given in detail in Ref. 8.
Taking a flight altitude of 60, 000 feet at a speed of 2, 000 f. p. s. and a c e n t r e of gravity position X = 0 . 5 feet, the a e r o d y n a m i c d e r i v a t i v e s become
z ^ = - 0 . 227 s e c . ' ' , Zj = - 1 . 33 x lO'^ft. "^ s e c . ,
m^= 0.00566 f t . ' ' s e c . ' \ m , = - 0 . 354 x lO'^ft. '^ s e c . , Zn = -86. 7 ft. s e c . "^, mn = -37.91 s e c . '' ,
21 -z = -0.259 ft. s e c . "* , m = -0.1134 s e c . m ^ = -0.0340 X 1 0 ' ' f t . ' ' , giving - 2 - 4 - 2 A, = 11,30 s e c . , A3 = - 0 . 7 0 8 x 10 ft. B, = - 0 . 4 0 8 s e c . ~\ B, = -3.399 x l o ' ^ f t . '^ s e c . The equation of t r i m i s -2J0OO A,oc,j, - 2 , 0 0 0 ' A ^ « ' = [ ( 2 , 0 0 0 4-z ) m^, - z„m ],n, ^ ^ , _ , - - - _^, -..,„ _ , , . „ ^ j , . , , j , o r - 2 2 . 6 0 X 10'« 4- 5.864 x 1 0 ' « ' = -75,829»» (6.10) w h e r e « and rj a r e in r a d i a n s and <*• - ^ T y ^ . The t r i m curve i s shown in F i g . 1 1 .
When the a i r f r a m e i s statically stable at low incidence ( c o r r e s p o n d i n g , in this a i r f r a m e , to x^ <0.3 feet) the values of n-p will all be negative. In the p r e s e n t c a s e , with s t a t i c instability at low incidence, a r e g i o n e x i s t s between M and N, F i g . 1 1 , w h e r e a stable t r i m m e d positive incidence i s achieved with T?rp positive. The minimum positive value of "-T, for which a stable t r i m m e d condition can exist will be given by
, . / 22.60 x 10? \ i „ , , , . ^
< V M = ( , 3 — 5 . 6 8 4 x 1 0 0 " 0-1154 r a d i a n ,
w h e r e only the positive value i s r e l e v a n t in the p r e s e n t p r o b l e m . The c o r r e s p o n d i n g value of n>p Is 4-0. 0229 r a d i a n . When «-p i s g r e a t e r than that corresponding to the point S, F i g . 1 1 , only one stable t r i m m e d condition can e x i s t , the limit being given by the m a x i m u m positive root of
- 22.60 X 10-« + 5.664 x l o ' a ^ = -75,829 x (-0.0229)
or
(« )_ = 0.232 r a d i a n . 1 S
Let E be the point c o r r e s p o n d i n g to the final t r i m condition; then if it lies between M and S t h e r e will always be a c o m p l e m e n t a r y saddle singularity at the point P . This i m p l i e s that a c r i t i c a l value of W^ = Q °^ " t = 0 =(Wt = 0^ / U w i l l exist only if E lies between M and S. A r a n g e of values of («ip)^ , between ^ « T ^ M and ( " T ^ S • ^^"^ '^'^^ ^^ selected and the a s s o c i a t e d values of (t\J) , {<x ) = W p / , a,, a^ and a , calculated. If now a r a n g e of values of the Initial t r i m m e d incidence, (*'p)j:\ . i s a s s o c i a t e d with each value of («»>_)„, then the c r i t i c a l i n t e g r a l c u r v e s , i . e . the i n t e g r a l c u r v e s through (« ) , may be evaluated and a r e shown plotted in F i g . 12 in the form W^ versusoc ^
Taking first c a s e s in which ("rp)A *• ("rp),^. . for which the minimum positive value i s 0.1154 r a d i a n and W^ = -ZTJ = 86. 7 ft. s e c . " * . It can be seen from F i g . 12 that the c r i t i c a l i n t e g r a l curve can never s e p a r a t e the initial and final positions, implying that the a i r f r a m e is stable when subjected to negative step functions of V of any magnitude within the l i m i t s of the elevator mechanical s t o p s .
