Stability theorems established for static and dynamic losses in discrete, linear, elastic system

Herrmann, G.

Prasad, S. N.

English

NASA Technical Reports Server

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SOME TKEOREW ON ST\BHLHTY OF DISCRETE CIRCULATORY SYSTEMS 
.k Shyam N. Prasad 
A s  s i s t ant Pro f e s s o r 
Department of Civil Engineering 
University of Mississippi 
University, Mississippi 
and 
George Herrmann 
Profess or 
Department of Civil Engineering 
Northwestern University 
Evanston 11 1 inois 
This work was supported by the National Aeronautics and Space Adminis- 
tration under Grants NsG 605 and NGR 14-007-067. 
.A. 
Formerly, Research Engineer, Department of Civil Engineering, North- 
western University, Evanston, Illinois. 
https://ntrs.nasa.gov/search.jsp?R=19680005030 2020-03-12T12:08:58+00:00Z
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Abstract  
Severa l  theorems p e r t a i n i n g  t o  the s t a b i l i t y  of d i s c r e t e ,  l i n e a r ,  c i r -  
c u l a t o r y  ( i . e . ,  nonconservat ive)  systems a r e  e s t a b l i s h e d .  One theorem 
s t a t e s  t h e  cond i t ion  under which s t a t i c  l o s s  of s t a b i l i t y  cannot occur .  The 
o the r  theorems are a s soc ia t ed  wi th  the d e s t a b i l i z i n g  e f f e c t  of v e l o c i t y -  
dependent f o r c e s .  The usefu lness  of the new theorems i n  s t a b i l i t y  a n a l y s i s  
i s  i n d i c a t e d .  
i 
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1 
Introduction 
I ’  
Hn the recent past, increased attention has been paid by numerous in- 
vestigators to the problem of stability of equilibrium of circulatory (ioem, 
nonconservative) elastic systems, as evidenced in a survey article [l]. * 
The role played by velocity-dependent forces in such problems has been 
recognized to be especially intriguing. Ziegler [ 2 ]  was first to indicate 
that linear viscous damping may have a destabilizing effect in such sys- 
tems, i.e., the critical value of the parameter associated with the exter- 
nally applied circulatory loading when even slight damping is present may 
be smaller than the corresponding value obtained in the absence of any 
damping. This discovery supplied the impetus for further studies of this 
effect t3-131,  but it appears that certain of its features can be brought 
into a still broader framework. 
In the present study we establish several stability theorems pertaining 
/ 
to a discrete, linear, elastic system with N degrees of freedom for static 
loss of stability (divergence) and for dynamic loss of stability (flutter) 
in the presence of sufficiently small velocity-dependent forces of any 
physical origin. The stability theorems, then, are shown to lead to several 
- 
... - ------- 
conclusions of some generality. In particular, the existence of the de- 
stabilizing effect of all sufficiently small velocity-dependent forces is 
brought into sharper focus, thus extending the results of Nemat-Nasser and 
Herrmann 181 e 
The new theorems contain considerable specific information concerning 
the behavior of systems under study and are thus of more immediate value 
* 
Numbers in brackets designate References at end of paper. 
' than the general Routh-Hurwitz criterion. 
The System 
In the following, a holonomic, autonomous, linear, dynamic system with 
N degrees of freedom is described by generalized coordinates q and general- 
ized velocities 4 = dqj/dt, j = 1,2, ... N. The system is subjected to a set 
= Qj(F), j = 1,2,..9N, which are defined as func- of generalized forces, 
tions of a real, finite parameter F. This parameter (0 < F < m) is associ- 
ated with the magnitude of the externally applied loadings; Q = 0 for 
F = 0. Let 
j 
Qj 
j 
= 0, j = 1,2,.*.N - qj - qj 
be the equilibrium state of the system. The kinetic energy T and the poten- 
tial (strain) energy V, which are assumed to be positive definite, are given 
by the following bilinear expressions: 
N M 
The equations of motion are then given by 
where the summation convention on sll repeated indices is implied and will 
be employed in the sequel. 
linear, homogeneous functions of the generalized coordinates and are given 
The generalized forces Q are assumed to be 
j 
by 
Qj = kjkqk, j,k = 1,2,0..N 
where [k ] is a nonsymmetric matrix which vanishes identically for F = 0. 
jk 
If small velocity-dependent forces of order e are present, the equa- 
tions of motion (1) will become 
i 
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I 
, 
M 5 + eG 4 + cjkqk = Qj, j,k = 1,2 ,... N 
jk k jk k 
A Theorem on Static Loss of Stability (Divergence) 
It is known that system (1) may lose stability by either flutter (os- 
cillations with increasing amplitudes), or divergence (buckling: an adjacent 
equilibrium configuration is attained) [12]. In the present study we are 
primarily concerned with the effects of small velocity-dependent forces on 
the loss of stability by flutter, and may want to seek conditions which 
prevent divergence. In fact, for a very special class of the matrix [k 
is possible to show that buckling never occurs. For several problems in the 
] it jk 
field of aeroelasticity, especially flutter of elastic panels, the matrix 
[kjk] assumes a skew-symmetric form arising from the aerodynamic forces. 
For this particular system we may state the following theorem: 
Theorem I. If the matrix [k 
lose stability by divergence. 
] is skew-symmetric, system (1) does not 
jk 
- Proof, The loss of stability by divergence is characterized by the 
vanishing of the following determinant [12]: 
Clearly, then, if A ‘  does not change sign the only possible 
static position is the initial configuration, q t 0. The 
potential energy V was assumed to be positive-definite quad- 
ratic form in q I is always a positive quan- 
tity. We may also write 
1 1 
j 
so that detlc 1 jk 
N 
1 ’ = ?  ! ‘jkqjqk ‘7 1 ‘jkqjqk 2 - - f kjkqjqk 
j ,k=l j ,k=l j ,k=l 
because k = 0, j = k and k = -k j # k. For V to be 
jk jk kj ’ 
T 
4 
I 9 positive-definite, one of the requirements is 
Thus the determinant A '  cannot vanish. This concludes the 
proof 
The result just established could be useful in the theory of determin- 
ants and may be generalized to take the following form: 
"If a are the real elements of a determinant such that there exists 
jk 
N 
rn 
a positive-definite quadratic form ajkxjxk (ajk need not be symmet- 
j ,k=l 
ric), then det 1 a 
symmetric determinant *I '  
+ b I > 0, where b are the real elements of any skew- 
jk jk jk 
Theorems on Dynamic Loss of Stability (Flutter) 
iw t Let us assume solutions of (1) and (2 )  in the form q = , k 
i = (-I)', and obtain the following: 
2 
u) Mjk% - (cjk kjk>Ak = 0 
U, 2 MjkAk - iSWGjk\ - (Cjk kjk)% = 0 
Equations (3)  and ( 4 )  lead to the following frequency equations: 
where 
2 a = w M  - c  + k  jk jk jk jk 
without any restriction on the structure of k e ij 
= 0, equation (5) yields the natural frequencies ki j For F = 0, i.e., 
5 * 
* of f r e e  v i b r a t i o n  of t h e  system which, i n  t h e  fo l lowing ,  are assumed t o  be 
d i s t i n c t .  
t i o n  (5) y i e l d s  a t  least  one double nonzero roo t .  
indeed r e a l i z a b l e  under t h e  condi t ion  of k 
not  undertake t h e  t a s k  of e s t a b l i s h i n g  necessary cond i t ions ,  but  m e r e l y  
As P inc reases  from zero t o  a c e r t a i n  f i n i t e  va lue ,  say Fe, equa- 
Such a p o s s i b i l i t y  i s  
being nonsynnnetric. We w i l l  
j k  
assume t h a t  coalescence of two f requencies  occurs  such t h a t  beyond t h i s  
va lue  of P = F 
Therefore ,  t h e  system w i l l  o s c i l l a t e  w i th  an exponent ia l ly  inc reas ing  
ampli tude,  i . e . ,  f l u t t e r .  
t h e  system (1). 
equat ion  (5) y i e l d s  a p a i r  of complex conjugate  r o o t s .  e 
We s h a l l  r e f e r  t o  Fe a s  t h e  c r i t i c a l  load f o r  
Equat ion ( 6 )  may be expanded i n  powers of e as follows: 
2 
I f ,  i n  t h e  above expansion, w e  neg lec t  terms con ta in ing  O(e ) and h ighe r ,  w e  
are l e f t  w i t h  
AI = A(ajk) - ism - G = O  aa j k  
j k  
which i s  an  approximate form of the  frequency equat ion  of system (2 ) .  
Before w e  proceed f u r t h e r  i t  i s  e s s e n t i a l  t o  estimate t h e  r o o t s  of ( 7 )  
i n  terms of t h e  r o o t s  of (5). 
nomial i n  M2 of degree N, and - G 
Le t  u s  s u b s t i t u t e  t h e  fol lowing f o r  c l a r i t y  of l a t e r  expos i t ion :  
The determinant A(a. ) gives r ise t o  a poly- 
i s  a polynomial i n  w of degree N-1 . 
Jk  
2 
ba j k  
j k  
4 
and o b t a i n  from (7)  
I .  
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6 
There are s e v e r a l  ways of e s t ima t ing  the r o o t s  of (8), but  t h e  method ex- 
p l a ined  below has been found p a r t i c u l a r l y  u s e f u l ;  it i s  based on t h e  fol lowing 
theorem : 
Theorem 11. Let  CY! be t h e  roo t s  of (8), and f cy 
PN(W ) = 0.  Then, &' = f cy. + if3 where f3 i s  obtained 
j = 1 , 2 , #  . N ,  those  of 
J j '  
2 
j J j '  j 
from t h e  fol lowing expression:  
I 
k# j 
Proof .  From t h e  theory of equat ions w e  may w r i t e  (8) as fol lows:  
I f  w e  s u b s t i t u t e  w = cy i n  (10) and no te  t h a t  cy' = f cy. + i p  , w e  
j j J j 
ob ta in  
k# j 
We a l s o  know t h a t  8 .  a r e  of  t he  o rde r  of magnitude O(e) .  There- 
f o r e ,  t o  be c o n s i s t e n t ,  t h e  above express ion  may be approximated 
J 
as 
k# j 
Had we s u b s t i t u t e d  for w = -cy. i n  t h e  above, w e  would have 
J 
der ived  an i d e n t i c a l  r e s u l t  and, t h e r e f o r e  
7 
kf j 
It is also of interest to study higher order effects in e ,  especially the 
quadratic terms. The above technique may be used to derive explicit expres- 
sions of any order 
appendix . 
For distinct, 
the expression for 
1 CY > a  > ... CY N N-1 
of magnitude and we establish several such relations in the 
2 real roots of P (u, ) = 0, we note an important property of 
. If we arrange CY in descending order of magnitude, 
N 
9 j 
> 0, we find that the denominator of the right-hand side of 
(9) alternates sign,starting with a positive quantity when j takes on the 
values of N, N-1, ... (p,, being the discriminant of the kinetic energy, is a 
%-1 (w ) = 0 will have N-1 roots f a'', and if we assume positive quantity.). 
that these roots are real and distinct such that Cy" 
efficient of the highest power in s-, is a positive quantity, then ' 
tained from (9) will always be positive if the inequality cyN > CY:-, 
> . . . > 
2 
1 
> aim2 > . . . and the co- M- 1 
ob- 
> ON-1 
j 
> cyl holds. For p. > 0, we immediately conclude that system (2) is 
J 
stable. It is also obvious that if any one of the above requirements is 
violated, at least one member of the set f!, will be a negative quantity. 
Therefore, the system will oscillate with increasing amplitude (flutter). The 
j 
above is a set of necessary and sufficient conditions for the system to be 
stable, and, as we will discuss later, they are indeed fulfilled for certain 
types of the matrix [G 3.  Hence, the following theorem may be stated: 
jk 
Theorem 111. For F < Fe, a necessary and sufficient condition for the 
system (2) to be stable is that the coefficient of the 
highest power of the polynomial be positive and its 
roots separate those. of P N' 
8 
'L 
E t  may be worthwhile t o  poin t  out t h e  e s s e n t i a l  d i f f e r e n c e s  between t h e  
above s t s b f l i t y  theorein and t h e  Routh-Hurwitz c r i t e r i o n .  The Routh-Burwitz 
c r i t e r i o n  sets down t h e  necessary and s u f f i c i e n t  cond i t ions  t h a t  a l l  t he  r o o t s  
of a polynomial l i e  i n  t h e  l e f t  h a l f  of t h e  complex p lane .  These condi t ions  
have very  complex dependence on t h e  c o e f f i c i e n t s  of t h e  polynomial and thus  
are not  capable  of y i e l d i n g  any f u r t h e r  information wi thout  ca r ry ing  ou t  ex- 
t e n s i v e  c a l c u l a t i o n s .  The r eade r s  are r e f e r r e d  t o  t h e  c l a s s i c a l  t reat ise  by 
Routh [ 1 4 ]  and a recent  work by Gantmacher [15] f o r  a comprehensive d i scus -  
s i o n  of t h i s  c r i t e r i o n .  By c o n t r a s t ,  t he  s t a b i l i t y  theorem j u s t  e s t a b l i s h e d  
i s  s u f f i c i e n t l y  s p e c i f i c  t o  permit one t o  draw s e v e r a l  conclus ions  regard ing  
t h e  e f f e c t  of veloci ty-dependent  forces  as i l l u s t r a t e d  below. 
As a consequence of t he  e s t ab l i shed  s t a b i l i t y  theorem, t h e r e  ar ises  an 
i n t e r e s t i n g  s p e c i a l  case .  For F E 0,  t h e  polynomial P by v i r t u e  of t h e  
s ta tement  of t h e  problem, possesses  nonzero, d i s t i n c t ,  real  r o o t s ,  and it  i s  
e n t i r e l y  p o s s i b l e  t h a t  one o r  severa l  of t h e  requirements s t a t e d  i n  Theorem 
119 are a i o l a t e d  by the  polynomial 
system w i t h  nega t ive  damping so  t h a t  some of @ obta ined  from (9 )  are nega- 
t i ve  q u a n t i t i e s .  I n  t h i s  s p e c i a l  s i t u a t i o n  t h e  c r i t i c a l  load of  t h e  system 
( 2 )  is zero .  
M' 
h f a m i l i a r  exsmple i s  t h a t  of a %- 1 
j 
r -  
If F is a nonvanishing va lue  of t he  c r i t i c a l  load of system ( 2 1 ,  then  d 
w e  may s t a t e  t h e  fol lowing c o r o l l a r y :  
Coro l l a ry .  For a l l  s u f f i c i e n t l y  s m a l l  veloci ty-dependent  forces;, 
0 < Fd S Pe. 
The proof of t h e  above c o r o l l a r y  i s  q u i t e  elementary i f  one bea r s  i n  
%- 1 mind t h e  f a c t  t h a t ,  by t h e  reguirement c f  Theorem 111, every r o o t  of 
i s  bounded by t h e  two ad jacen t  r o o t s  of P when F = 0. N 
functiorr of I?, y i e l d s  a t  least  two roots  which approach each o t h e r ,  and, 
PN, regarded as a 
9 
when F = Fe, complete coalescence occurs.  
have a t  least  one common r o o t ,  say when F = Fdo As soon as F > Fd2 t h e r e f o r e ,  
a t  l e a s t  one of B .  w i l l  be gega t ive ,  which i n d i c a t e s  o s c i l l a t i o n  w i t h  fncreas-  
ing  amplitude.  i n  
conjugate  p a i r s  whose real  p a r t s  w i l l  be nega t ive  q u a n t i t i e s .  
I n  t h i s  i n t e r v a l ,  P and %_, w i l l  N 
J 
When F > Fe, equat ion (9)  w i l l  y i e l d  complex numbers f o r  B 
j 
This  concludes 
the  proof (we indeed included F = 0 a s  a c r i t i c a l  load ,  as discussed  earlier.) .  d -  
So f a r ,  w e  have no t  i nd ica t ed  the phys ica l  o r i g i n  of veloci ty-dependent  
f o r c e s .  I n  view of t h e  f a c t  t h a t  the theorems e s t a b l i s h e d  above may have 
a p p l i c a t i o n s  i n  va r ious  branches of engineer ing sc i ence ,  it i s  not  d e s i r a b l e  
t o  a s s i g n  any d e f i n i t e  form t o  [G. 3. I n  the  f i e l d  of 
Jk 
p o s s i b l e  o r i g i n s  may be those  assoc ia ted  wi th  v iscous  damping o r  gyroscopic  
e f f e c t s .  
d e f i n i t e  form, 
-
I n  the  case of  v i scous  damping t h e  matrix [G ] assumes a p o s i t i v e -  
For t h i s  ca se ,  as was shown by Routh [ 1 4 ] ,  t h e  requirements  
jk 
of t h e  s t a b i l i t y  Theorem 111 are indeed m e t  when F = 0,  s o  t h a t  t he  i n e q u a l i t y  
B .  > 0 is  s a t i s f i e d .  The d e s t a b i l i z i n g  e f f e c t  of v i scous  damping, t h e r e f o r e ,  
fo l lows  from t h e  c o r o l l a r y  s t a t e d  above. This  s p e c i a l  r e s u l t  w a s  f i r s t  
J 
ob ta ined  by Nemat-Nasser and Herrmann [ 8 ]  
10 
Appendix 
The determinant  Al, when expanded i n  powers of C ,  may be w r i t t e n  i n  t h e  
fo l lowing  form: 
n 
3 I f  w e  n e g l e c t  o rde r s  of O(C ) and h ighe r ,  t h e  fol lowing approximate express ion  
is obtained:  
(11) 
A1 = PN(w 2 ) - iecuffN-, (w 2 ) - 7 1 2 2  e w SN-2(w 2 ) = 0 
where, i n  a d d i t i o n  t o  t h e  des igna t ions  def ined  before ,  the fo l lowing  s u b s t i t u -  
t i o n  h a s  been made: 
aa a2A aa GjkGQm = "N-2W 2(N-2) + . " 0  + so = sN-2(w2) 
j k  Am 
2 L e t  CY! be t h e  r o o t s  of (11) and f CY j = 1,2,. . .M, t hose  of P (w ) = 0. 
J j '  N 
Then w e  may w r i t e  
t o  t h e  
n i t u d e  
u r e  as 
I f ,  i n  
2 
CY' = f CY. + icB.  f e e 
j J J j 
2 accuracy of c Mote t h a t  B .  and 0 .  w i l l  be of t h e  same order  of mag- 
J J 
as a . To ob ta in  express ions  f o r  fl and 0 .  w e  fol low t h e  same proced- 
j j J 
be fo re ,  and w e  o b t a i n  
t h e  above, w e  l e t  w = CY w e  have 
j '  
The r ight-hand s i d e  of t h e  above express ion  may, a f t e r  some c a l c u l a t i o n s ,  be 
wr i t ten  i n  t h e  fol lowing form t o  t h e  accuracy of c : 2 
11 
Therefore, equating the coefficients of like powers of e, we have 
Note that by substituting w = -a. and carrying out the above procedure, the 
J 
expression on the right-hand side of (13) will be the same, but with a 
negative sign in front of it. 
1 2  
References 
1. Herrmsnn, G., "Stability of Equilibrium of Elastic Systems Subjected to 
Nonconservative Forces," Appl. Mech. Rev., 2, No. 2, 1967, pp. 103-108. 
2 .  Ziegler, H., "Die Stabilitztskriterien der Elastomechanik," Ingen.-Arch., 
- 20, 1, 1952, pp. 49-56. 
3. Bolotin, V. V., Nonconservative Problems of the Theory of Elastic Stabil- 
a, MOSCOW, 1961; English translation published by Pergamon Press, Icnc., 
New York, New York, 1963. 
4. Herrmann, 6. and Jong, 1. C . ,  "On the Destabilizing Effect of Damping in 
Noxonservative Elastic Systems," J. Appl. Mech,, 32, 3, 1965, pp. 592- 
593. 
5. Herrmann, 6. and Jong, I. C . ,  "On Nonconservative Stability Problems of 
Rlastic Systems with Slight Damping,'' J. Appl. Mech., 33, 1, 1966, pp. 
125- 133 
6. Bottema, O . ,  "On the Stability of the Equilibrium of a Linear Mechanical 
System," -' ZAW - 6, 2 ,  1955, pp. 97-104. 
7 a Leipholz, W. , %her den Einf luss der Dzmpfung bei nichtkonservatfven 
Stabflbt3t~problemen elastischer Stzbe," 1ngen.-Arch., - 33, 5 ,  1964, pp. 
308-321 
8. Nernat-Nasser, S o  and Herrmann, G., "Some General Considerations Concern- 
ing the Destabilizing Effect in Nonconservative Systems," - ZAMP, l7, 2 ,  
1966, pp. 305-313; also Northwestern University Structural Mechanics 
Laboratory Technical Report No. 65-2, June 1965. 
13 
w 
- '  9 ,  Nemat-Masser, S., Prasad, S. No and Herrmann, G., "Destabilizing Effect 
of Velocity-Dependent Forces in Nonconservative Continuous Systems," AHAA 
- J,, 4, 7, 1966, pp. 1276-1280; also Northwestern University Structural 
Mechanics Laboratory Technical Report No. 65-4, December 1965, 
-
10. Prasad, S. N. and Herrmann, G., "Complex Treatment of a Class o f  Noneon- 
servative Stability Problems," Northwestern University Structural Mechan- 
ics Laboratory Technical Report No. 67-1, January 1967. To be published, 
11. Memat-Nasser, S o ,  "On the Stability of Equilibrium of Nonconservative 
Systems with Slight Damping," J. Appl. Mech., 34, 2, 1967, pp. 344-348. 
12.  Herrmann, 6;. and Nemat-Nasser, S o ,  "Energy Considerations in the Analysis 
of Stability of  Nonconservative Structural Systems," Proc. Int. Conf. - on 
Dynamic Stability of Structures, ed. G. Werrmann, Pergamon Press, 1967, 
pp. 297-308; also Northwestern University Structural Mechanics Laboratory 
Technical Report No. 65-5, December 1965. 
13, Prasad, S. Ea,, Nemat-Nasser, S. and Herrmann, G o ,  "Bending-Torsional 
Flutter of a Swept Wing in a High-Density, Low-Speed Plow,"  A M  J., 5, 
2 ,  1967, pp. 316-321, 
14, Wouth, E. J., Advanced Dynamics of a System of Rigid Bodies, Dover Pub- 
lications, Hnc., New York, New York, 1955, pp. 221-231, 238-239, 
15. Gantmacher, F. R., Matrix Theory, Vols. I and II, Chelsea, New York, New 
York, 1959. 


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