Exponential bounds and stability conditions for linear elastic column subjected to time varying axial loa

Infante, E. F.

Plaut, R. H.

English

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t 
ON THE STABILITY OF A COLUMN SUBJECTED 
TO A TIMF,-DEPENDET\TT AXIAL LOAD* 
t t t  E. F. Infante and R. H. Plaut 
Center for  Dynamical Systems 
B r 0 ~ n  University 
Providence, Rhode Island 
* 
This research was supported i n  pa r t  by the  National Aeronautics and ~ 
Space Administration, under Grant No, NGR 40-002-015, i n  part by the 
United Stated Army Research Off ice, under G r a t  No. DA-31-124-ARO-D-270, 
in pmt by the A i r  Force Office of Sc ien t i f ic  Research, under 
t NO. AF-AFOSR 693-67. 
'Associate Professor of Applied Mathematics 
t 'Assis tant  Professor of Applied Mathematics (Research) 
https://ntrs.nasa.gov/search.jsp?R=19690001641 2020-03-12T06:08:18+00:00Z
ON THE STABILITY OF A COLUMX SUBJECTED 
TO A TIME-DEPENDENT AXL4.L LOAD 
1. Introduction 
The dynamic s t a b i l i t y  of a l i nea r  e l a s t i c  column subjected t o  
a time-varying a x i a l  load i s  investigated. The form of s t a b i l i t y  con- 
sidered i s  almost sure asymptotic s t a b i l i t y  i n  t he  case t h a t  the  load 
i s  assumed t o  be s tochast ic j  if the load i s  assumed t o  be determin- 
i s t i c  t he  s t a b i l i t y  i s  asymptotic s t a b i l i t y  i n  the  sense of Liapunov. 
Studies similar t o  the  one presented here have been previously 
1 2 3 published by Caughey and Gray , Ariaratnam , and Lepore and Shah . 
these works s t a b i l i t y  conditions were obtained by the  modal approach; 
Caughey and Gray obtained conditions fo r  almost sure asymptotic s t a -  
b i l i t y ,  Lepore and Shah fo r  asyuqtotic s t a b i l i t y  i n  the  mean. 
In 
In  t h i s  note we give conditions f o r  the almost sure  asymptotic 
s t a b i l i t y  of t he  column, since it seems tha t  t h i s  i s  the  more natural. 
mathematical concept f o r  the  physical problem under consideration, 
Simultaneously with cor,ditions for  asymptotic s t a b i l i t y  we obtain 
measures of exponential decay of t he  solutions which are  of i n t e re s t  
i n  themselves. We 
but then show t h a t  
mathematic ally,  i s  
di t ions fo r  almost 
f i rs t  obtain these r e su l t s  by the modal approach, 
t h i s  procedure, 
unnecessary and 
sure asymptotie 
which is  somewhat questionable 
t h a t  ihe  same estimates and con- 
s t a b i l i t y  can be obtained i n  a simple 
and d i r ec t  winner fram the  or ig ina l  p a r t i a l  d i f f e ren t i a l  equa-tion. The 
r e su l t s  obtained are  then compared t o  previously published resu l t s .  
The approach of t h i s  paper represents an application and 
2 
4 
generalizations of the r e su l t s  of Infante 
t ions. 
on l i nea r  d i f f e r e n t i a l  equa- 
2. Basic Equations 
Consider the  displacement w(x,t) of a l inear  pinned colirm 
subjected to a time-varying ax ia l  load p ( t )  = p -t- f(t) whose 
"average" over t he  time in te rva l  [O,T] i s  
ing coefficient.  
T 
. Let f3 be the  damp- 
pT 
I n  nondimensional f c l r m  the  equation fo r  t h i s  
displacement over t he  time in te rva l  of  i n t e re s t  i s  given by 
and by the boundary conditions 
= 0, t I 0. a2w (0, t) = w ( 1 , t )  = a$ (1, t)  w ( 0 , t )  = h;? ax2 
The nondimensional time-varying load p ( t )  i s  assumed to be s t r i c t l y  
s ta t ionary and ergodic i n  the  case t h a t  it i s  assumed. to be a stochastic 
process. The average of the load- p ( t )  over the desired time in te rva l  
[0, T] i s  defined i n  such a manner t h a t  
T i e { f ( t ) ]  = - S f ( t ) d t  = o 
T o  T 
2 and it i s  assumed t h a t  e (f (t)] exis t s  and i s  posit ive.  Defining T 
3 
2 2 
E[f ( t )}  = lim eT{f(t)}, E[f (t)} = lim eTlf (t)> and p = 
T + w  T - + m  
lim pT, we obtain the standard expectations over t he  i n f i n i t e  time 
T + w  
interval .  O u r  choice of meaiure f o r  the function f ( t )  i s  obviously 
motivated by energy considerations. 
For the modal analysis, l e t  
w ( x , t )  = C an( t )  s i n  n m  
n= l  
which, when subst i tuted in to  (l), yields  
(4) 
n = 1,2, ..., 0 < t S T 
2 where = - It i s  immediately noted t h a t  p = x- represents the 
Euler buckling load fo r  t h i s  colwnn, 
d t  
To determine s t a b i l i t y  conditions f o r  (1) or, i n  t he  modal 
approach, f o r  (>), we follow the modification of the  Liapunov approach 
described by Infante . 
V 
equal i ty  of the type 
the  determination of s t a b i l i t y  conditions and of exponent 
4 Hence, we seek a posi t ive de f in i t e  functional 
f o r  (1) (function f o r  ( 5 ) )  such as to obtain a d i f f e r e n t i a l  in- 
t/V S X ( t ) .  Such an inequal i ty  kmediately allows 
3. Exponential Bounds and S t a b i l i t y  Conditions fo r  the Modal Approach 
Le t  us f i r s t  consider equations (3) and determine exponential 
4 
bounds for, the,r  behavior. For t h i s  purpose, l e t  these equations be 
rewrit ten i n  canonical form as 
n n where obviously x = a x = & Consider t h e  quadratic, posi t ive 1 n ' 2  n 
de f in i t e  function defined on each of these modes'by 
2 and denote by ? t h i s  expression with E{f (t)} subst i tuted for  
2 
eT{f (t)}. 
along the  solutions of (6): 
Consider now the  t o t a l  ttme d-erivative of t h i s  function 
where 
n n  
T 1, 2 
2 
n n n  Estimating tI-9 quotient (2f3xl +2xlx2)/v"(;. x ) by a proced-ure such 
as the use of Lagrange multipliers,  w e  obtain 
5 
f r o m  which (8) citn be wri t ten i n  t h e  f o r m ,  
Integrating this  expression between 0 and T and using the  Schwartz 
inequality yieids  
This expression i s  a measure of the exponential decay of the osc i l la -  
t i o n  of  the  modes; however, note t h a t  for  every d i f fe ren t  time T 
the  expression ? i s  a d i f fe ren t  one. T 
A s l i g h t l y  d i f fe ren t  expression can be obtained by l e t t i n g  
T + c o  i n  (10) and integrat ing between 0 and t. Using again the 
Schwarz inequality yields  
n n n n where (I) and a axe the  same as ch and aT w i t h  p replaced by 
p and 'e by E. Upon noting t h a t  lim et{f(t)] = 0 and 
T T 
t 3 w  T 
2 2 lim et(f ( t ) )  = E{f  (t)] we obtain fo r  suf f ic ien t ly  large t 
t + w  
6 
where E i s  m y  posi t ive number, as small as desired. Now, f o r  asymptotic 
s t a b i l i t y  it i s  suf f ic ien t  t h a t  the  term multiplying i n  the  exponential 
be negative. 
t 
A simple computation on the quantity i n  the brackets yields  
t h a t  t h i s  condition i s  
E { f 2 ( t ) }  6 4P2(1 - -&,) - E 
n7r 
where .E i s  any posi t ive number. The most r e s t r i c t i v e  condition i s  the 
one f o r  n = 1. Hence, f o r  the  asymptotic skab i l i t y  of a l l  the  modes we 
obtain as a suf f ic ien t  condition t h a t  
E{f 2 (t)} 5 4p2(l - 2) - E 
2 which also shows t h a t  we must have p < 7r . It i s  of i n t e r e s t  to note 
t h a t  the exponential estimate (12) i s  a ra ther  sharp one, since i f  
f ( t )  0 ( the system i s  autonomous) then we obtain the  estimates 
which are  the exact exponential bounds f o r  the autonomous system. 
I f  we consider f ( t )  as a random, s t r i c t l y  s ta t ionary ergodic 
process then the estimate (13) gives us a condition for almost sure 
asymptotic s t a b i l i t y  . 
7 
4. Direct Approach 
In t h i s  section we derive the  s t a b i l i t y  conditions and ex- 
ponential.bounds d i r ec t ly  from t he  or ig ina l  equation of motion (1). 
Consider f i r s t  the functional 
where v = &/a. For functions w sat isfying the boundary conditions 
(2 ) ,  one can use the  calculus of var ia t ions to show t h a t  
so t h a t  
Assume t h a t  
Then VT i s  posi t ive def ini te .  
The t o t a l  time derivative of V along the  solutions of  (1) i s  T 
easily computed to be 
8 
a2w 
2 
t T ( W , V , t )  = -2pvT(W,v) + 2 ~[ 28 2 vw+2p w +@f(t)  (g) - f ( t ) ~ ~ ] d x  (21) 
ax 0 
Using t h e  method of Lagrange mult ipl iers ,  we can obtain the  inequal i ty  
where 
2 2 2  2 4 4  2 2  2 1/2 r =  { [% -n T f(t)]  / (n  T -n T p,+B ) ]  . 
1. n=l, 2,. . . 
Theref ore 
Integrat ing t h i s  expression between 0 and T, and using (3) and the  
Schwartz inequality, we f ind  t h a t  
To This expression gives an  exponznttal bound f o r  t h e  funct ional  V 
If we l e t  T -+ 03 and then in tegra te  (24) between 0 and t, 
we obtain f o r  su f f i c i en t ly  la rge  t 
9 
where V denotes the  functional V with p replaced by p. The 
t e r m  multiplying t i n  the  exponential w i l l  be negative i f  
T T .  
E[f2(t)}. 5 @'(I - ?$) - E 
where E i s  any pos i t ive  number. It follows t h a t  (27) i s  a suf f ic ien t  
condition f o r  almost sure asymptotic s t a b i l i t y  of t h e  column f o r  the  
case of a s t r i c t l y  s ta t ionary and ergodic load 
s t a b i l i t y  i n  the case of a deterministic load. This condition i s  the  
same as t h a t  obtained by a modal analysis i n  the previous section. 
f ( t ) ,  and f o r  asymptotic 
'We can obtain stronger exponential bounds than (25)  and (26) i f  
we consider the functional 
0 
where 
K = [ (n  4 4  T -n 2 2  T pT-B 2 2  ) -t n 4 4  .rr eTIf 2 ( t )} ]  1/2 
n 
yields  equations (11) and (1.3) 
vT 
Following the  same procedure as f o r  
with n = 1 and V replaced by W 1 T T' 
3 .  Discussion of Results 
We have derived exponential bounds and s t a b i l i t y  conditions for  
a pinned column subjected to a time-dependent ax ia l  load. In  particular,  
i f  the  l o a 1  i s  given by p + f(t) with E { f ( t ) }  = 0, then equation 
(13) gives an exponential bound f o r  each mode, with 
a bound f o r  any motion 
(27) give a su f f i c i en t  condition f o r  s t a b i l i t y .  
n = 1 providing 
of t he  colwnn, and equations (15) and w(x,t) 
This s t a b i l i t y  con- 
d i t i on  i s  shewn i n  Fig. 1. 
conditions on E ( f  (t)} 
references 1 and 3 .  
For comparison, we have also depicted 
which can be obtained using the methods of 2 
A s  can be seen from Fig. 1, the s t a b i l i t y  condition derived 
here i s  a s ign i f icant  improvement on the previously published r e su l t s  
2 2 of other authors. If p < 7~ , it i s  possible to l e t  E ( f  (t)} -+ w 
as the damping increases, whereas previous r e su l t s  have indicated a 
2 
E l f  (t)} l imi t  to the  value of 
ing. 
no matter what the magnitude of damp- 
We note again the d i r ec t  approach of Section 4, which el-iminates 
the mathematical uncertainty involved when judging the s t a b i l i t y  of a 
column by the  s t a b i l i t y  of each mode separately. This approach i s  a l -  
so of par t icu lar  advantage when dealing with boundary conditions f o r  
which the  modes a re  not as simple as s i n  nm. 
References 
[l] Caughey, T.K., and Gray, A.H., Jr., "On t h e  Almost Sure Sta-  
b i l i t y  of Linear Dynamic Systems With Stochastic Coefficients", 
Transactions of t h e  American Society of Mechanical Engineers, 
Series E: Journal of Applied Mechanics, Vol. 32, No. 2, June 
1969, PP. 363-3720 
[ 2 ]  Ariaratnam, S.T., "Dynamic S t a b i l i t y  of a Column Under Random 
Loading", Dynamic S t a b i l i t y  of Structures, Proceedings of 
In te rna t iona l  Conference, Pergamon Press, New York, 1967, 
pp. 267-284. 
[ 3 ]  Lepore, J.A., and Shah, H.C., "Dynamic S t a b i l i t y  of Axially 
Loaded Columns Subjected to Stochastic Excitations", AIAA 
Journal, Vol. 6, No. 8, August 1968, pp. 1715-1721. 
[4] Infante, E. F., "On the  S t a b l l i t y  of Some Linear Nonautonomous 
Random Systems", Transactions of the  American Society of 
Mechanical Engineers, Series E: Journal of Applied Mechanics, 
V o l .  33, No. 1, March 1968, pp. 7-12. 

Caption for Illustration 
Figure 1. Stability Conditions 


On the stability of a column subjected to a time-dependent axial load

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