The characteristics of systems which are nearly in a state of neutral static stability

^TECHNISCHE HOGESCH VDIEGTUIGBOUWKUNDE REPORT No. 34

12 Juli1950

o

'^'^^T DELFT

Kluyverweg 1 - 2629 HS DELR

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE CHARACTERISTICS OF SYSTEMS WHICH

ARE NEARLY IN A STATE OF NEUTRAL

STATIC STABILITY

0

by

Professor W. J. Duncan, D.Sc, F.R.S. of the Department of Aerodynamics

TECHNISCHE HOGESCHOOL VLIEGTUIGBOirVv KUNDE REPORT No.34 Jamuary, 19 50 T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

The C h a r a c t e r i s t i c s of Systems which a r e N e a r l y i n a S t a t e of N e u t r a l S t a t i c S t a t i l i t y

b y

-P r o f e s s o r W.J. Dimcan, D . S c , -P . R . S ,

STOBIARY

It is s.hown that the rate of subsidence or divergence )y of a system which is near a state of neutral static stability can easly be calculated from a knowledge of the mode of displacement in the neutral state and this mode is found "by solving a set of linear algebraic equations. The first order correction to the mode can also be found and jn important cases this can be made the basis for calculating a second approximation to >> j if necessary a further correction to the mode can now be found and from this a still more acciorate root can bo calculated. The method can bo extended to com:inuous systems having infinitely many degrees of freedom,

— o O o —

CONTENTS

• * • t * t • • •

1. Introduction

2. Exactly Neutral Systems

3. Nearly Neutral Systems

4. A Simple Numerical Example _{' • • • '}

Page

5. Details of the Procedure in Dealing

ViTith a Nearly Neutral System . • 10

6. Treatment of Continuous Systems *.• 10

Systems which are nearly in a state of neutral static stability are of considerable interest'and technical importance, Exactly neutral systems are specially simple in as much as the neutral mode of displacement can be found merely by solving a set of simultaneous linear algebraic equations. It is shovm in this paper that the small rate of divergence or subsidence ?^. of a nearly neutral system can be very easily calculated to the first order of

small quantities from a knov/ledge of the neutral mode; the first order corrections to this mode can readily be found also. ^i^en the matrices of the inertias, darapings and stiffnesses arc

syrametric the node correct to the first ordex- can be made the basis of a calculation of A correct to the second order. For a continuous system ViTith infinitely many freedoms the first order approximation to A can be derived from the values of the potential energy and rate of dissipation of energy in an imaginary divergent notion proportional to the neutral node and having ?^ equal to imity,

2. Exactly Neutral Systems

The matrices of the inertias, damping coefficients and stiffnesses are, as usual, denoted by A, B and C respectively, These arc square no.trices of order n where n is the number of degrees of freedom of the system. The static displacements q corresponding to static loads Q are given by

Cq = Q (2,1) or q = C"^ Q. (2,2) If the determinant of C becones very small the elements of C

become very large and the displacements q corresponding to given loads Q also become very large. V/hen I C \ vanishes the system is in a state of neutral static stability and then finite displacements can occur in tlie absence of applied loads. Since the constant torn in the determinantal equation nf the system is eqioal to |c|, the equation has the root 7\ = 0 v/hen ! C I vanishes. Thus the

/velocities ...

V/e can suppose t h i s notion s t a r t e d by an extrenelj'- small

-4-velocities ^ and accelerations ^ are zero in the mode of

displacement corresponding to the root /\ = O, There.are, however, (n-1) other nodes v/hich, in general, correspond to non-vanishing roots A and in these nodes the velocities and accelerations are not, in general, zero. Cases of nultiple zero roots can arise but are rare, Y/'hen a displacement OCCIK-S in the neutral node the forces are always balanced. Hence the total work done in a neutral displacement is zero. As an exar.iple, when the angle of incidence of an a.ircraft varies slowly v/hen the centre of mass is at the longitudinal neutral point the resultant aerodynamic force is constantly eq\aal to the v/cight of the aircraft,

3« Nearly Neutral Systems

Suppose next that the system is near but not in the state of neutral static stability. There are then sone simple nethods for finding the characteristics of the nearly neutral mode,

(a) The matrices A, B and G are all syxietric

In this instance v/e can apply an extension of Rayleigh' s Principle. Suppose that in a datuiu state of the system

there is a characteristic root A and a corresponding nodal colunn q. Then ?\ must satisfy exactly the

quadratic equation

;\^(q'Aq) + A (q'Bq) + q'C q = 0 (3, 1) where the coefficients are quadratic forms in the elements

of q. This equation is obtained from the dynamical equation in natrix form

;\ " Aq +/\Bq + Cq = 0

by prenultiplication by q', the transposed of q, i.e. the TOW' matrix having the sane elements as the colunn q^ the equation is true v/hether A, B and C are symnetric or not. Now let A, B and C be syi-inetric and let the

elements of q depart from their true values by small quantities of the first order. Then according to the principle the value of the root 7\ obtained from the qioadratic equation (3,1) will only be in error by a small quantity of the second order" , Accordingly, let A, B and C refer to the near neutral state and

/substitute' .,, * See p.300 of 'Elementary Matrices' by R.A. Frazer, W.J. Duncan

-5-substitute for q the neutral modal col\ann q (v\rhich is

^ o very easy to calculate). The root v/ill be a small

quantity of the first order and ^ ' will be of the same order as the terms neglected and therefore to be discarded. Hence we derive the approximation

q' C q ^0 ^o which is easy to compute.

Let C reduce to G in the exactly neutral state so that

C q = 0 . ,..(3,3)

0 0

Also let C = C + 6 0 (3,^) where the elements of 6C are small quantities of the first

order and C is symmetric. Then by (3,3) and (3,4-)

q' C q = q' 50 q . (3,5) ^o o 0 ^o

and (3,2) becomes

q' 60 q

^o

^ %

which is correct to the first order of small quantities. Prom knowledge of this approximation to X v/e can calculate the first order correction to the modal column q merely by solving a set of simultaneous linear equations (see eqiiation (3,13)).

The approximation to 7t can nov/ be improved by use of the corrected modal column and the eqioation (3>l). Since /^ is a small quantity of the first order equation (3,1) "vd-ll be correct to the second order if q'Bq is correct to the first order and q'Gq to the second order, Nov/ suppose that all the elements of 5C are proportional to p, v/hich is thus of the first order, and let

ÖC = p C^. (3,7)

Suppose also that to the second order

q = q^ + pq>i + p qg» ' •**•" ^^»^)

-6-where q. is no^v known but q„ is not. Then to the second order of small quantities

q'Cq = P (q; 0^ q^ + q; C^ q^)

9

+ P (q; C^ q^ + q; C^ q^ + q^ C^ q^ + q^ C^ q^)

(3,9)

in view of i^l),!)). But equation (3,3) gives by transposition

q; c^ = 0 (3,10)

s i n c e C i s symmetric. Hence (3>9) becomes q' Cq = p q ; C^ q^

+ P (q^ C^ O] + q^ C^ q^ + qjj C^ q^) . . . . ( 3 , 1 l ) Nov/ this does not contain q^, and we conclude that

equation (3,1) v/ill be correct to the second order of

small qiiantities v/hen v/e substitute for q the approximate modal column v/hich is correct to the first order. Hence

the small root of (3j1) will nov/ be correct to the second order,

The improved approximation to 7\ obtained from the quadratic equation (3>1) can be used to obtain an improved approximation to the corresponding mode by use of the dynamical equations. Then this recalculated mode may, if desired, be used to obtain a still better approximation to ?\ by means of (3,l). The whole process can be repeated as often as desired,

(b) The matrices A, B and 0 are general, except that C is nearly singular

The dynamical equations in matrix f orr.i yield

(>\^ A + A B + C^ + 6C)(q^ + 8q) = 0 (3,12)

v/hile {j)fj)) i s a g a i n s a t i s f i e d . I f we nov/ n e g l e c t second o r d e r teiras we o b t a i n

A B q ^ + C ^ 6 q = - 6 C q ^ , (3,13)

-7-This represents n linear scalar equations and there are apparently (n + 1) unknov/ns namely ~)\ and the n elements of 6q, Hov/ever, the modal colunn is arbitrary to a scalar

multiplier and we nay accordingly assign one of the elements of 6q, say make one element zero. We have no\~r n unknov/ns, which can be calculated,

We shall nov/ shov/ that v/hen C is symmetric this

procedure leads exactly to the value of A given by (3,2) or (3,6), Premultiply (3,13) by q', IVc obtain

A q ' B q = - q' (O 6q + 6Cq )

^o o ^o o ^o

= - q; 8Cq^

by (3,10) and this is identical with (3,6).

4, A Simple Numerical Example The inertia matrix is

"4 2 1

A =

₅

2

(4,1)

The daiuping matrix is

B

The stiffness natrix i s

0 = 0 + 6 0 o

(4,3)

v/here

2 3 1

3 5 1

1 1 1

(4,4)

and i s singiilar, v/hile

60 = p

0 0 0

0 0 0

0 0 1

(4,5)

v/here p is supposed to be small. The matrices are all symmetric so both methods described above are applicable. To facilitate

^

-8-comparisons with the approxiiTiations v/hich we shall obtain we give the expansion of the exact determinantal equation for ^ * namely

A^ A + A B + C I

3 8 3 A ^ + 9 6 A ^ + (l6p + 144)A^

+ (l4p + 86) A ^ + (21p + 11)) h'^

+ (8p + 6) /\ + p = 0. (4,6)

When p is zero this has a zero root in accordance v/ith C being singular. The nodal colui'-in q corresponding to the zero root is obtained at once from the linear equation

0 q o ^0 and is found to be = 0 •- 2 1 1 (4,7)

v/here the last element has been assigned the value xxnity. When p is given the value 0,1 v/e find from (4,6) the true value of the nur.ierically small root and the corresponding nodal column to be

A = - 0.01597

1

2,22535

1.13840

L 1.0

The first approximation to the snail root as given by (3,6)

IS

>\ = - I =.--0,01667, (4,10) Equation (3,13) yields the s.ane value of 7\ and v/hen we assign the

zero value to the final element in 6 q v/e obtain the approximate modal colunn '- 2,21667 1.13333 1.0 (4,11) /On ,.,

»9>-On s u b s t i t u t i o n of t h i s value of q in equation (3,1) we obtain the

qviadratic eqxiation

22.12778 >, + 7.20500 ;^

+ 0.10944, = 0

4

(4,12)

of v/hichnthe smaller root i s

X

= - 0,01597,.

(4,13)

This differs from the true root only by about 1 part in 5,000» The error in the modal colimn is of the order of -2 per cent and the

departure from the neutral modal column is about 10 per cent,

For comparison, the calculation has been repeated with the inertia matrix reduced to one quarter of that given in (4,1), i.e. v/ith all the inertia coefficients reduced to one quarter of their former values. The first order approximations to }\ and to the mode remain as in (4,10) and (4, II) respectively, Hov/ever, the quadratic

equation (3,l) giving the second approximation to 7\ is obtained from (4,12) by reducing the coefficient of A to one q-uarter of the value in (4,12) and the second approximation is

A = - 0 . 0 1 5 3 7 ^ . (4,14). Now the 'exact' root as obtained from the detcnninantal equation is

?\ = - 0.01537. (4,15)

so the error is only about 1 part in 15,000 or say one third of the former error. The 'exact' mode is

q =

- 2.22377 1.13745 1.0

(4,16)

so the first approxiraation to the mode is in error by roughly 0.3 per cent,

-10-5, Details of the Procedure in Dealing v/ith a Nearly Neutral System We suppose that the determinant | C | of the stiffness matrix has been evaluated and found to be small, so the system is near a state of neutral static stability. It is now convenient to consider an artificial state of the system which is exactly neutral and we can derive this merely by varying one element of 0; this element may be chosen so as to simplify the calculation. The increment of the chosen clement' is - j C j /k v/here K is its cofactor. We next derive, merely by solving a set of linear equations, the neutral mode q and nay then proceed as explained in §3. There is some advantage in choosing the element of the modal colunn v/hich is arbitrarily kept fixed so that the product of this clement and the varied element of C occurs in the calculations»

6, Treatment of Continuous Systems

The nethods given above p.re not, in general, ii:miodiately applicable to continuous systems having infinitely nany degrees of freedom. We nay, hov/ever, treat such systens with any desired degree of approximation by the 'Lagrangian' method in v/hich the continuous system is represented by another having a finite nixiber of degrees of freedom , This method, also knov/n as 'the method of semi-rigid representation', has been much used in dealing with aero-elastic problems and is of great utility. The methods explained in the present paper can be applied at once to the semi-rigid representation of the sytem,

Another method can be applied directly when the forces dependent on the displacements are derivable from a single-valued potential, for then the stiffness natrix is symnetric. We saw in §3 under iten (b) that equation (3,2) is valid to the first order of snail quantities when 0 is syrxietric whatever nay be the nature of A and B, Nov/ 0 v/ill be syrxietric in the circunstances we have postulated so (3,2) is applicable hov/ever nany degrees of freedom the system may possess. To assist the application of the formula v/e nov/ give a physical interpreto.tion of the nunerator and denominator. We know that the potential energy stored in the system v/hen the displacenent is q is

'• ^ o

V = i q; C q^. (6,1)

Next, the tii.ic r a t e of rlissipation of energy i n the system i s

D = ^' B4 (6,2)

_ ^ / and . . .

See, for example. Chap. V of 'Mechanical Admittances and their Applications to Oscillation Problems' by V/. J, Duncan,

1 1

-and i f we suppose t h a t

q = % ° (^»3)

this becomes

D(,^) = A % ^ B q ^ . (6,4)

Thus we see that

q; B q^ =D(1)

(6,5)

where D ( I ) is the rate of dissipation of energy when

q = q^ e .

iè>^)

Accordingly eqi^ation (3,2) becomes

A = - ^ i (6,7)

D(1)

The numerator aind denominator can be calculated for a continuous

system when the neutral mode of displacement is known