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On the application of the energy method to continuous follower load systems
COMPUTER METHODS NORTH-HOLLAND
IN APPLIED
MECHANICS
AND ENGINEERING
53 (1985) 259-276
ON THE APPLICAmON OF THE ENERGY METHOD TO CONTINUOUS FOLLOWER LOAD SYSTEMS* H.H.E. LEIPHOLZ Deparlmentsof Civil and Mechanical Engineeting, Solid Mechanics Division, Universityof Waterloo> Waterl~, Ontario, Canada N2L 3Gl
Received 25 June 1984 Revised manuscript received 7 March 1985
In this paper it is shown that, by using an appropriate energy functional which includes the work done by the nonconservative forces, an energy approach to the stability problem of continuousnonconservative systems is possible. Flutter and divergence criteria are derived. As an example, rods subjected to follower forces have been used for an application of the theory. Numerical calculations confirm that the above-mentioned criteria are practical and quite powerful.
1. ~t~uction
The treatment of problems involving mechanical systems has frequently been based on the application of classical energy methods using the Hamiltonian as a first integral of the system. The applicability of such energy methods seemed to be unlimited. However, the confidence in these methods was somewhat put into question when H. Ziegler revealed in an epoch-making paper [l] that the “classical energy method” was not applicable to flutter systems which turn up in the context of problems concerned with follower loads. This is so because flutter systems are nonconservative, and for them the Hamiltonian is therefore not a first integral any more. Later, H. Leipholz showed in a series of papers, [2-4], that an energy approach to nonconservative systems, including flutter systems, remained feasible if one did not only work with the Hamiltonian, as in the classical ease, but would add to the energy expression also the work done by the nonconse~ative forces. The above-mentioned papers were written with a finite number of degrees of freedom in mind. Using the set of Hamilton’s first-order differential equations, with a term representing the nonconservative forces included, an energy function was derived which yielded diagrams in the phase space corresponding to the various energy levels. Then, the ‘sensitivity’ of these diagrams to variations of structural parameters, like loading intensity, frequency, even time, was studied. Catastrophic situations leading to instability were predicted by deriving critical (zero or infinite) values of certain derivatives taken with respect to those parameters. The purpose of the present paper is to generalize the findings of the previous papers by developing a methodology applicable to continuous systems, i.e., systems with infinitely many degrees of freedom. * Support of this research through NSERC Grant No. A7297 is gratefully acknowledged. 004%7825/85/$3.30 @ 1985, Elsevier Science Publishers B.V. (North-Holland)
H.H.E.
260 2.
Leipholz, The energy method for continuous follower load systems
Fundamental relationships The total energy of the nonconservative
E= H-
system is given by the expression
W,,
(1)
where H is the Hamiltonian of the system and W, is the work done by the nonconservative forces. The parameters affecting E are assumed to be g, the load parameter; A = w2, the frequency parameter; and y, the mode shape. Hence, the variation of E reads SH
zsr--arv,sg sy+$sg+
sE=G
ag
-SW,.
(2)
In (2), it has been taken into account that W, depends explicitly only on g. Moreover, SH/Sy is a functional derivative so that integration by parts (integral theorem of Gauss) yields easily
where
(4 and Js) * d0 being either a line, surface, or volume integral depending on the dimension of the continuous system under investigation. In (3), D(y) is a differential operator related to the differentiai equation modelling the system, and N represents the nonconservative forces. The meaning of (3) and (4) will be explained in Appendix A. Referring back to (2), it has to be pointed out that for a rod problem, ysy=$syfl+
aw,
aw,
SW,= -sy+-Sy'.
f$Sy'+$Sy,
aY
(5)
aY’
Also, it should be noted that for rod loaded with follower forces,
(6)
D(Y) = 0 by definition. With (3), (5), and (6) in (2), one arrives at
aw,
aw,
avv,
aY
ay1
tig----sy--
6y’ .
(7)
Since the variation of y is due to a change in g and A, one has
sy=~sg+~sA,Sy’ =
J%A. zag+ aA
(8)
H.H.E. L.eipholz, The energy method for continuous follower load systems
261
Using (8) in (7) yields
(9) Quantity E is the total energy of the system. Therefore,
(10)
6E=O. Hence, it follows from (9) by virtue of (10) that aH aw,ay &$jfJ+x------
t3y aA
Q -=6A
,;dfi+z-
aH
aw,ay'
ayf aA
aw, aw,ay aw,af _--____ ag ay ag dY’ Jg
A critical situation is indicated by Sg/GA = 0. Consequently, condition dJ-j+x_-----=
ay
ah
ay'
ah
a
(11)
one has as the flutter criterion the
(1%
Even for follower load systems, flutter is not the only mode of instability, depending on the boundary conditions. For example, so-called completeIy supported systems, [5, pp. 354-3691, exhibit instability also by divergence, which is however the only mode by which conservative systems become unstable. Let it therefore ‘be shown how also respective divergence criteria can be derived by means of the previously presented theory, Let firstly the case of completely supported systems be investigated. To fix the idea of complete support, the reader may have rods in mind that are supported at both ends. Due to such support, wr.J=o.
Moreover,
03)
divergence occurs for A = 0, so that i3A=O.
Again due to the boundary
I
n
holds.
NdJ2=0
(14) conditions,
also (1%
262
H.H.E. Leipholz, T&eenergy methodfor continuousfollower load systems
By virtue of (13) and (14), equation (2) becomes
Using (4) in (16) yields (17) where Do is the operator into which the operator D changes for A = 0. Let now the mean value theorem of integration be applied to the integral on the right-hand side of (17). Then, one has instead of (17) [D,(y)+N]dfiSj%$8g,
(18)
where Sg is an appropriate mean value of 8~. But, one must keep in mind that (18) is only valid as long as the functional D,(y)+ N is sign-definite. However, this is fortunately the case for 8 E [O, &iv] as has been shown for some rod problems explicitly, [6,7]. Using (1) as well as (15) in (18) and rearranging yields
A critical situation, i.e. divergence, criterion
I
D,(y) do = 0,
occurs for Sg/Sg = 0. Hence, (19) yields as the divergence
DO(Y) = 0.
n
(20)
This criterion can be brought into the form
I
(21)
DO(y)y d&? = 0.
n
Setting D,(y) = DE(Y) - gWy> it follows from (21) that
9
(22)
H.H.E. Leipholz, The energy method for continuous follower load systems
I
DE(Y)Y
n
gdiv
263
c-1~2 7
=
I
DL(Y)Y do
n
where gdivis the divergence load, D,(y) is the operator modelling the elastic forces, and where -gD,(y) models the external forces. Hence, (23) assumes the role of Rayleigh’s quotient which one is used to encounter in the case of conservative systems. It may briefly be mentioned that by virtue of (13), the flutter criterion (12) can be simplified for completely supported systems to read
I
IV$,,+g=O.
(24)
n
Let now the case of conservative systems be discussed. For these, relations (13) and (14) remain certainly valid. But, condition (15) can be replaced by the stronger condition D,(y)+N=fi,=o.
(25)
Following the same argumentation (25) in mind, one has SE =
IR
as for completely
supported
rods and keeping in addition
da + $6g
&,(y)by
(26)
instead of (17) and SE =
IR
do 67 + $6g
f&(y)
(27)
instead of (18), where (27) holds as long as the functional Do(y) is sign-definite. Using (10) in (27) and rearranging yields
Q
I
%(Y
n
G=
>do
--aHlag
(19*)
’
As before, divergence occurs for Sg/6jj = 0 yielding
I
f&(y)
dJ2 = 0 ,
&(y)
R
Hence, one can substitute (20*) by
= 0 .
(2W
H.H.E.
264
I
Leipholz,
f&,(y)ydfl =
The energy method for continuous follower load systems
G3*1
0.
n
Now, one has
6,(y) = MY)
-
gh(y)
(28)
instead of (22), where D, is a ~elf~~joi~toperator,
I
n
gdiv
WY)Y
da .
=
I
R
Hence,
(29)
DL(Y)Y da
By virtue of (25), the flutter criterion for a conservative (24) that ~YH/ilh= 0. However,
system would require according to
py2dLM0,
(30)
where y is the always positive unit mass density. Therefore, exhibit flutter.
conservative
systems do not
3. Interpretation of the flutter and divergence criteria for rod problems For elastic rods of length 1, flexural stiffness Q[, linear mass density ,u, and distributed follower forces q, the differential equation for the lateral deflections W(X,t) reads /!.lG + awn” + q(Z - x)w” = 0 .
(31)
w(x, t) = e’“‘y(x) ,
(32)
However,
where y(x) represents vibration modes. Therefore, -/.&y
+ ay”” + g(l -
with (32), equation (31) changes into
x)y” = D(y) = 0 .
(33)
The boundary conditions are y(0) = y”(O) = y(Z) = y”(l) = 0
for the pinned-pinned
rod,
y(0) = y’(0) = y(l) = y’(l) = 0
for the clamped-clamped
y(0) = y’(0) = y(Z) = y”(l) = 0
for the clamped-pinned
y(0) = y’(O) = y”(l) = y”(t) = 0
for the clamped-free
rod, rod,
rod.
(34) (35) (36) (37)
H.H.E. Leiphoiz, The energy method for continuous follower load systems
265
Let first the clamped-tree rod be discussed. For this system, E =
4
and integration
I0
’
[-/uo2y
+ ay"" + g(Z - x)y"]y dx
by part yields
E = +
I’ [-,wo2y2
+ a(~“)’ - g(Z - x)y’)“]
dx + 4 1’ gy’y dx .
0
(38)
0
fn (38),
6
I
o’[-/.uJy*
+ a(y")"- g(Z - x)y’)‘]
dx = H,
(39)
1 $
_f0
gy’ydx=-$
I,’ (-gy’)y dx = - $ I,’ Ny dx = - W, ,
w-9
and
(41)
Iv = -gy’
represents the nonconservative follower forces. Hence, with (39) and (40), relationship yields, as predicted, equation (l), i.e., E=H-
Furthermore,
F
(38)
W,.
according to (3) and (39), sy =
1’(cxy” Sy”-
g(Z - x)y’ Sy’ - ,w2y Sy) dx ,
(42)
0
an expression which by integration y
Sy =
by parts can be transformed
’[-,uw2y + tyy”” + g(Z -
x)y” - gy’] Sy dx .
By virtue of (33) and (41), equation (43) can be written as E&y=jl[D(y)+N]Sydr, 0 SY which confirms the correctness of (4). By means of (40) one has for expression (5) explicitly -swN=$
I0
‘gy’Sydx+;gySy’dx.
into
(43)
H.H.E. Leipholz, The energy method for continuous follower load systems
266
Moreover,
1’
awr4
Y'Y dx, 0 ae=-T5
(45)
and, because of (39), and keeping in mind that A = CM’,also dH -=
at
-- ; ‘(I-x)(y’)‘dx, I0
g=
(W
-jlipy2dx. 0
With (41), (#), (4% and (46), equation (7) reads 6E =
$
f 0
’[-gy’
6y + gy 6~’ -
(I - x)(y’)” Sg - py2 8h + y’y Zig] dx .
(47)
Using (8) in (47) yields &E=+
t
-gy -
ay
-WgY’Tp+gY
JY
ag
(I - x)(y’)” 6g - py* 6h + y’y
6g I
dx .
(48)
By virtue of (lo), equation (48) yields after rearranging py2-gy$+gy$
dx I
%+
f
6h=
(49)
gy$gy$+(Z-x)y”y]dx which can easily be identified to represent obtain (49) from (47), also the identity
j’ [-(I
- x)(y’)‘+
(11). It may be mentioned
though, that in order to
y’y] dx = I,’ (t - x)y”y dx
0
has been applied. From (49) follows the flutter criterion 1
f0
py2 dx =
gy
$$-gy’$
Let it be shown that (51) corresponds
1dx. to (12). For this purpose write (12) using (39), (40) and
H.H.E. Leipholz, The energy method for continuous follower load systems (41).
267
One then obtains aY (-gy’),-fpy2+~gy’~+~gy~] dx = 0.
This expression can be rewritten as 1
-2
1dx=O,
py’+gy$-gyg
from which (51) follows immediately. Condition (51) shall be used later on to derive the smallest flutter load of the clamped-free rod numerically. Let now the completely supported rods be considered for which the boundary conditions (34), (35), and (36) hold. By virtue of these boundary conditions, indeed, IV,=-$g
which corresponds
I0
I0
‘Y’ydx=-:g[y’(Z)-y2(0)]=0,
to (13), and
1 N dx = -g
'y'y dx
= - ;g[y2(1)-
y2(0)] = 0,
I 0
which corresponds to (15). Let now the condition for divergence be checked, which occurs for A = w2 = 0, 8h = 0. Going back to (2), one has first because of (13) and (14),
(52) With (39) and for A = o2 = 0, one finds that
6H6y = I ’[ay” 8~" - g(Z 6Y
Integration
x)y’ 6y’] dx .
(53)
0
by parts performed
on (53) yields
[a~“” + g(Z - x)y” - gy’] 6y dx.
The operator Do is supposed that therefore
to follow from operator
(54) D for A = w2 = 0. Equation
(33) shows,
268
H.H.E.
Leipholz,
The energy method for continuous follower load systems
D,(y) = a~““+ g(Z - x)y”.
(55)
By means of (55) and (41) one realizes that (54) yields ‘[D,((y)+N]6ydx, which, substituted 6E=
in (52) leads to
I0
‘[D,(y)+N]Sydx+$%g
(56)
confirming (17). For A = w2 = 0, (39) yields dH -=
JET
-$ I ‘(I-x)(y’)2dx.
(57)
0
With (54) and (57), relationship
SE = I 0
(52) becomes equal to
’[a~""+ g(Z- x)y"- gy’] Sy dx - ;l’
(I - x)(y’)‘dx 6g = 0.
Applying (10) and using the mean value theorem of integration
sg-2
sj;-
I’
[a~““+
gy’]
g(Z - x)y”-
allows one to change (58) into
dx ,.
O
(58*)
1 I0
Yet,
(I - x)(y’)’ dx
1 I 0 gy’ dx = dy(O - YKOI= 0
(59)
due to the boundary conditions. Therefore,
I’
[ay”“+
Q-2 G-
(3
0
g(Z - x)y”]
relationship
dx ,
O
(58*) can be simplified to read
’ (I - x)(y’)’ dx which corresponds by virtue of (55) and (57) to (19). Divergence occurs for Sg/Sy = 0. Hence,
(60)
H.H.E. Leipholz, The energy method for continuourrfollower toad systems
’[a,“” + g(E -
x)y"]dx = 0 ,
f0
must hold simultaneously. Conditions Conditions (61) are equivalent to
I
/
g(Z - x)y"]y dx
[a~““+
269
(61)
ffy*lf+ g(I - x)y” = 0
(61) confirm (20).
(62)
= 0,
0
which corresponds
to (21). One realizes easily, Yhat
I
I
0
I0
’ (I - x)y”y dx
i
a(y”)‘dx
0
Es
gdiv =
-
I
cxy”“y dx
(63)
I’ (l- x)(y’)’ dx 0
Since DE(y) = my”” and D, = -(E - x)y”, relationship (63) confirms (23). As far as the flutter criterion (24) for completely supported systems is concerned, for rods with (39) and (41) 1 ,
py2dx+2 f
0 gy
f9 -dx=O.
tw
dh
Whether flutter will occur in addition to divergence for completely the behaviour of the functional 1
1
9=
I
py*dX+2 gy
it reads
dY -ddx
supported
rods depends on
(65)
dh
on the left-hand side of (64). If this functional is not sign-definite and can assume the value zero, then flutter will take place, However, if the functional is sign-definite, condition (64) cannot be satisfied, and flutter is ruled out. This happens for example for the pinned-pinned rod which is a sheer divergence system. Finally, the case of conservative systems shall be investigated, For these systems, (13), (14), and (25) hold, i.e. W,=-;
‘gy’ydx=O, I0
8h = SW*“0
and D,(y) + N = “y”” -t g(l - x)y”- gy’ = ay” + g[(t - x)y’]’ =Do(y)=O. Concluding
from (25*) that
t25*)
H.H.E. Leipholz, ‘171eenergy methodfor continuousfollower load systems
270
D,(y) = try”,
D, = -[I - x)y’]’ ,
one has for the relationship
I
(29) explicitly
1
I
ay”“y dx
0
gdiv
I
LY (y “)’ dx
0
=
=
L’
-
[(Z
x)y’]‘y dx
-
(66)
9
Jo’(I - x)(y’)* dx
which is indeed Rayleigh’s quotient for the buckling load g&v+ 4. Numerical examples Let first the flutter criterion (51) be applied. For this purpose consider the clamped-free rod. The coordinate functions to be used in a Galerkin approach are given in [8, p. 961. By means of +I and d;* one can approximate Y=
(67)
al#l++2.
With this assumption
for y, one can set up the Galerkin equations
a&b + g%) f gQi12 = 0,
@J)
(A2+ g%) + wqliz, = 0,
where Al = -pd+
c~(12.41-~),
A2 = -/.uo* + ct(485.W4),
@22= -6.65/-1,
@,, = 0.431-l ,
(6% @j,, = -4.343-l
,
lrpz, = 1.181-1,
ski =
’ (t - x)#:&
dx,
J-O
?Pkz=
’&#k
dx ,
YF,, - PI2 = @, - 523, .
The numerical values in (69) have been determined Setting Ai + @ii = Di
9
using the information
in [8, pp. 96-971.
(70)
the first equation in (68) yields al = -gat121Dl.
Moreover, one has
(71)
H.H.E.
Leipholz, The energy method for continuous follower load systems
271
(72) By virtue of (67) and (71), and the orthonormality I
1
,uy2dx=p
I0
(al&
Using in (73) the relationship
I
of the 4i,
I
/-v2dx=p
0
+ $2)2 dx = p(af
+ 1) .
(73)
(71) yields
g2&, + D: D2
Again, by virtue of (67), the orthonormality the !Pki in (69), one obtains aY
gy$gytz
(74)
.
1
of the +i, and in addition due to the definition of
1
dx=g$+&,-1Y~z)=/q2
@12(%2-
D:
%I)
(75)
*
Hence, (51) reads
5
(g2@:2+
D:)
= 5
1
g2@12Wi2-
(76)
P21).
1
After rearranging
and using the last identity in (69), one arrives at
g2Q12Q21+ D: = 0 as the flutter criterion. It is obvious that this relationship g2@,,@,,+D~=0,
i=l,2.
can be generalized
to read (77)
This is the final form of the flutter criterion. Introducing the new notation Yi = @i +
@12@21
as well as (70), criterion (77) can be transformed g =
(78)
,
$I [-cDii2 (- @12@2J’2].
into
(79)
Using in (79) the values given in (69) for Ai, 3/i, @ii, Q12, and @21, one can derive the
272
H.H.E. Leipholz,
The energy method for continuous follower load systems
following sets of equations
g=
0.372/~0~1- 4.606al-“ , -0.54Jj~w’l+ 6.768&-‘,
(80)
and -0.1 14/Lw21+ 54.469al-’ ) g=
-0.232p.w21+ 110.664al-” .
(81)
Equating the right-hand sides of (SO), and (Sl), yields w2 = 121.55cu/.-‘1-j
(82)
Equating the right-hand sides of (SO), and (8l)2 yields correspondingly
as a flutter frequency. the flutter frequency
o2 = -329.829~/l.-‘l-~.
(83)
Using (82) either in (SO), or in (Sl), yields the flutter load g, = 40.61cul-“ . Using correspondingly
(84) (83) in either (SO), or (Sl), yields the other flutter load
g, = 18J.18cul-3.
(85)
These results are in good agreement with those given in [S, pp. 99-1001 which have been derived in a different way. Such agreement of results confirms the applicability of the flutter criterion (51). Equations in (80) and (81) represent straight lines in the eigenvalue plane, i.e. in the g, 02-plane. Therefore, a simple geometrical interpretation of the calculation of flutter loads is possible, providing the designing engineer with a convenient tool: the points in the eigenvalue plane having flutter loads and flutter frequencies as coordinates are obtained as the points of intersection of the above mentioned lines. An outline of the geometric procedure can be found in [9]. Secondly, the flutter criterion (64) for completely supported rods shall be discussed. Considering the specific boundary conditions for rods, one realizes easily that
I
I
I
gy$dx=
-
gygdx.
0
For this reason, criterion (64) can be written as
I
1
0
py2
dx =
I
I
1
0
gygdx-
gy$dx,
H.H.E. Leipholz, The energy method for continuous follower load systems
which is identical with (51). Therefore, the result will be again (79), i.e. [- ~ii + (-
g = t
273
all the calculations carried out before remain valid, and
~D1*~*1)l’*] .
(79)
I
However, the last relationship in (69), i.e. P21 - !Py,,= @*I- @I*, has now to be replaced by @21- @I*= -2?P,,. This fact is again true due to the boundary conditions of completely supported rods. From [S, pp. 96-971, one obtains for the clamped-pinned rod A, = -/NJ* + c~(237.721-~),
A2 = -/.Lw* + c~(2496.491-~), (86)
d&1 =
-4.291-l)
-18.411-1,
cD2*=
Q12 =
0.921-l ,
a&=
-5.081-l )
so, y1 = dr,,@,, + @i:, =
13.731-l )
y2 = @&‘21+
@z2 =
334.261-* .
Using these values in (71) yields
(87) and (88) Equating (87), and (88), yields o* = -22.974~~/.-~1-~,
(89)
and equating (87), with (88)2 yields w* = -1255.59i~/_-‘l-~. Using these flutter frequencies yields the flutter loads g, = 122.495~&~
wo
either in (87),, (88), respectively,
and
or in (87),, (88)2 respectively,
gfl = 231.49~~~~.
The first one has been reported in [8, p. 100, Fig. 201. It corresponds well to the above one calculated here. If one would try to repeat such calculations for the pinned-pinned rod, one would find with the numerical values reported in [8], i.e. CD12 = -1.781-l )
@21=
-7.121-l)
274
H.H.E. Leipholz, The energy method for continuous follower load systems
that (79) would yield complex values for g. The conclusions must be that no flutter loads exist for the pinned-pinned rod. This result has been established in different ways in [5,8]. The pinned-pinned rod is indeed an example for so-called pseudo-divergence systems, (see [8, pp. G-74]) and conservative systems of the second kind (see [5, pp. 111, 217-223; 8, pp. 70-741). These are systems which fail by divergence only, although nonconservative follower forces are present. Finally, let the divergence criterion (63) for completely supported rods be checked. In terms of the previously introduced notations it reads
Setting i = 1 and using the numerical values given in [8, pp. S-973, one has a237.721-4 gdiv
=
4.291-l
=
This result is an approximate 991.
55.4;. one, which is in good agreement with the value reported in [8, p.
Appendix A
The basic expression to be considered
is
(A.11 where X is the density of the Hamiltonian. For a rod:
Therefore, 6X = -pw2y sy + cyynSy" - g(1 - x)y' Sy’
(A-3)
and g
6y =
f
’ [-,uw*y 6y + ay” 6~” - g(f - x)y’ Sy’j dx .
For all possible boundary conditions, SH 6y
Sy = I 0
’[-/.m*y
+ a~‘“‘+
(A.4) yields through integration g(l-
x)y”-
gy’] Sy dx.
(A4
by parts
6W
H.H.E. Lkpholz,
275
The energy method for continuous follower load systems
However, D(y) = -ploy
+ cyy”“+ g(1- x)y”
and
N = -gy’ .
Hence, indeed, ySy=
‘[D(y)+N]6ydx. I0
as stated in (3). For a rectangular, fully supported plate:
64.6)
j-fl.dQ=jF.dx,dx,, where F is the surface of the plate. Moreover, parallel to the x,-direction, x = -
b02Y2
(YXIXl + JL,,)‘+
+ M
for a plate with compressive
2(1 - 4(Y’x,*,
follower forces
(A-7)
-
where y is the deflection perpendicular to the middle surface of the plate, R the flexural rigidity, g the load parameter, a the sidelength of the plate in the q-direction, /_Lthe density per unit surface, and Y Poisson’s parameter. Obviously,
IE
Sy
dx, dx, =
6X dxI dx2
I=8y
=
I
F [(-PU’Y
+ RA*y) 6~ +
da - xl)yxlby, Sy,,] dxl dx2.
G4.8)
But
I da - ~~~~~~ dx, dx2+ I [da + I Ma 6~1dC, SY,,
F
c
F
XI)Y~
- u,l
SY
dx,dx,
xl)~xl
(A-9)
where C indicates that the respective expression is to be taken along the contour of the plate. Due to the boundary conditions of the plate along C,
I [da - xl)yx,W dC = C
0.
Hence, with (A.9) and (A.lO) in (A.8), and because of (A.l) one obtains
(A. 10)
H.H.E. Lzipholz, The energy rnet~o~for contirtuous follower Ioad systems
276
SH 6y
6~ =
I (---Pw~Y+ RA2y+ da - x2hx1- gy,) Sy dxl h. F
However, D(y) = -,uo2y f RA2y + g(a - xl)yxlx, and
N = -gy,, .
Hence, again,
SH6y = I 6Y
[D(y) + IV] Sy dxl dx, , F
confirming (3).
References [l] H. Ziegler, Die Stabilitltskriterien der Elastomechanik, Ingr.-Arch. 20 (1952) 49-56. [2] H. Leipholz, On the application of the energy method to the stability problem of nonconservative mechanical systems, Acta Mech. 28 (1977) 113-138. [3] H. LeiphoIz, An energy approach to stability probfems of nonconse~ative systems, in: Development in Theoretical and Applied Mechanics, Proceedings Ninth Sectam, Nashville, TN (1978) 427-437. [4] H. Leipholz, Stability analysis of nonconservative systems via energy considerations, Mech. Today 5 (1980) 193-214. [!I] H. Leipholz, Stability of Elastic Systems (Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980). [6] H. Leipholz, On the spillover effect in the control of continuous elastic systems, Mech. Res. Comm. 11 (1984) 217-226. [7] H. LeiphoIz, On the control of structural systems with distributed parameters, in: Proceedings 3rd International Conference on Systems Engineering, Wright State University, Dayton, OH (1984) 160-166. [8] H. Leipholz, Direct Variational Methods and Eigenvalue Problems in Engineering (Noordhoff, Leyden, 1977). [9] H. Leipholz, On the calculation of flutter loads by means of a variational technique, Paper No. 105, Solid Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada, February 1972.
IN APPLIED
MECHANICS
AND ENGINEERING
53 (1985) 259-276
ON THE APPLICAmON OF THE ENERGY METHOD TO CONTINUOUS FOLLOWER LOAD SYSTEMS* H.H.E. LEIPHOLZ Deparlmentsof Civil and Mechanical Engineeting, Solid Mechanics Division, Universityof Waterloo> Waterl~, Ontario, Canada N2L 3Gl
Received 25 June 1984 Revised manuscript received 7 March 1985
In this paper it is shown that, by using an appropriate energy functional which includes the work done by the nonconservative forces, an energy approach to the stability problem of continuousnonconservative systems is possible. Flutter and divergence criteria are derived. As an example, rods subjected to follower forces have been used for an application of the theory. Numerical calculations confirm that the above-mentioned criteria are practical and quite powerful.
1. ~t~uction
The treatment of problems involving mechanical systems has frequently been based on the application of classical energy methods using the Hamiltonian as a first integral of the system. The applicability of such energy methods seemed to be unlimited. However, the confidence in these methods was somewhat put into question when H. Ziegler revealed in an epoch-making paper [l] that the “classical energy method” was not applicable to flutter systems which turn up in the context of problems concerned with follower loads. This is so because flutter systems are nonconservative, and for them the Hamiltonian is therefore not a first integral any more. Later, H. Leipholz showed in a series of papers, [2-4], that an energy approach to nonconservative systems, including flutter systems, remained feasible if one did not only work with the Hamiltonian, as in the classical ease, but would add to the energy expression also the work done by the nonconse~ative forces. The above-mentioned papers were written with a finite number of degrees of freedom in mind. Using the set of Hamilton’s first-order differential equations, with a term representing the nonconservative forces included, an energy function was derived which yielded diagrams in the phase space corresponding to the various energy levels. Then, the ‘sensitivity’ of these diagrams to variations of structural parameters, like loading intensity, frequency, even time, was studied. Catastrophic situations leading to instability were predicted by deriving critical (zero or infinite) values of certain derivatives taken with respect to those parameters. The purpose of the present paper is to generalize the findings of the previous papers by developing a methodology applicable to continuous systems, i.e., systems with infinitely many degrees of freedom. * Support of this research through NSERC Grant No. A7297 is gratefully acknowledged. 004%7825/85/$3.30 @ 1985, Elsevier Science Publishers B.V. (North-Holland)
H.H.E.
260 2.
Leipholz, The energy method for continuous follower load systems
Fundamental relationships The total energy of the nonconservative
E= H-
system is given by the expression
W,,
(1)
where H is the Hamiltonian of the system and W, is the work done by the nonconservative forces. The parameters affecting E are assumed to be g, the load parameter; A = w2, the frequency parameter; and y, the mode shape. Hence, the variation of E reads SH
zsr--arv,sg sy+$sg+
sE=G
ag
-SW,.
(2)
In (2), it has been taken into account that W, depends explicitly only on g. Moreover, SH/Sy is a functional derivative so that integration by parts (integral theorem of Gauss) yields easily
where
(4 and Js) * d0 being either a line, surface, or volume integral depending on the dimension of the continuous system under investigation. In (3), D(y) is a differential operator related to the differentiai equation modelling the system, and N represents the nonconservative forces. The meaning of (3) and (4) will be explained in Appendix A. Referring back to (2), it has to be pointed out that for a rod problem, ysy=$syfl+
aw,
aw,
SW,= -sy+-Sy'.
f$Sy'+$Sy,
aY
(5)
aY’
Also, it should be noted that for rod loaded with follower forces,
(6)
D(Y) = 0 by definition. With (3), (5), and (6) in (2), one arrives at
aw,
aw,
avv,
aY
ay1
tig----sy--
6y’ .
(7)
Since the variation of y is due to a change in g and A, one has
sy=~sg+~sA,Sy’ =
J%A. zag+ aA
(8)
H.H.E. L.eipholz, The energy method for continuous follower load systems
261
Using (8) in (7) yields
(9) Quantity E is the total energy of the system. Therefore,
(10)
6E=O. Hence, it follows from (9) by virtue of (10) that aH aw,ay &$jfJ+x------
t3y aA
Q -=6A
,;dfi+z-
aH
aw,ay'
ayf aA
aw, aw,ay aw,af _--____ ag ay ag dY’ Jg
A critical situation is indicated by Sg/GA = 0. Consequently, condition dJ-j+x_-----=
ay
ah
ay'
ah
a
(11)
one has as the flutter criterion the
(1%
Even for follower load systems, flutter is not the only mode of instability, depending on the boundary conditions. For example, so-called completeIy supported systems, [5, pp. 354-3691, exhibit instability also by divergence, which is however the only mode by which conservative systems become unstable. Let it therefore ‘be shown how also respective divergence criteria can be derived by means of the previously presented theory, Let firstly the case of completely supported systems be investigated. To fix the idea of complete support, the reader may have rods in mind that are supported at both ends. Due to such support, wr.J=o.
Moreover,
03)
divergence occurs for A = 0, so that i3A=O.
Again due to the boundary
I
n
holds.
NdJ2=0
(14) conditions,
also (1%
262
H.H.E. Leipholz, T&eenergy methodfor continuousfollower load systems
By virtue of (13) and (14), equation (2) becomes
Using (4) in (16) yields (17) where Do is the operator into which the operator D changes for A = 0. Let now the mean value theorem of integration be applied to the integral on the right-hand side of (17). Then, one has instead of (17) [D,(y)+N]dfiSj%$8g,
(18)
where Sg is an appropriate mean value of 8~. But, one must keep in mind that (18) is only valid as long as the functional D,(y)+ N is sign-definite. However, this is fortunately the case for 8 E [O, &iv] as has been shown for some rod problems explicitly, [6,7]. Using (1) as well as (15) in (18) and rearranging yields
A critical situation, i.e. divergence, criterion
I
D,(y) do = 0,
occurs for Sg/Sg = 0. Hence, (19) yields as the divergence
DO(Y) = 0.
n
(20)
This criterion can be brought into the form
I
(21)
DO(y)y d&? = 0.
n
Setting D,(y) = DE(Y) - gWy> it follows from (21) that
9
(22)
H.H.E. Leipholz, The energy method for continuous follower load systems
I
DE(Y)Y
n
gdiv
263
c-1~2 7
=
I
DL(Y)Y do
n
where gdivis the divergence load, D,(y) is the operator modelling the elastic forces, and where -gD,(y) models the external forces. Hence, (23) assumes the role of Rayleigh’s quotient which one is used to encounter in the case of conservative systems. It may briefly be mentioned that by virtue of (13), the flutter criterion (12) can be simplified for completely supported systems to read
I
IV$,,+g=O.
(24)
n
Let now the case of conservative systems be discussed. For these, relations (13) and (14) remain certainly valid. But, condition (15) can be replaced by the stronger condition D,(y)+N=fi,=o.
(25)
Following the same argumentation (25) in mind, one has SE =
IR
as for completely
supported
rods and keeping in addition
da + $6g
&,(y)by
(26)
instead of (17) and SE =
IR
do 67 + $6g
f&(y)
(27)
instead of (18), where (27) holds as long as the functional Do(y) is sign-definite. Using (10) in (27) and rearranging yields
Q
I
%(Y
n
G=
>do
--aHlag
(19*)
’
As before, divergence occurs for Sg/6jj = 0 yielding
I
f&(y)
dJ2 = 0 ,
&(y)
R
Hence, one can substitute (20*) by
= 0 .
(2W
H.H.E.
264
I
Leipholz,
f&,(y)ydfl =
The energy method for continuous follower load systems
G3*1
0.
n
Now, one has
6,(y) = MY)
-
gh(y)
(28)
instead of (22), where D, is a ~elf~~joi~toperator,
I
n
gdiv
WY)Y
da .
=
I
R
Hence,
(29)
DL(Y)Y da
By virtue of (25), the flutter criterion for a conservative (24) that ~YH/ilh= 0. However,
system would require according to
py2dLM0,
(30)
where y is the always positive unit mass density. Therefore, exhibit flutter.
conservative
systems do not
3. Interpretation of the flutter and divergence criteria for rod problems For elastic rods of length 1, flexural stiffness Q[, linear mass density ,u, and distributed follower forces q, the differential equation for the lateral deflections W(X,t) reads /!.lG + awn” + q(Z - x)w” = 0 .
(31)
w(x, t) = e’“‘y(x) ,
(32)
However,
where y(x) represents vibration modes. Therefore, -/.&y
+ ay”” + g(l -
with (32), equation (31) changes into
x)y” = D(y) = 0 .
(33)
The boundary conditions are y(0) = y”(O) = y(Z) = y”(l) = 0
for the pinned-pinned
rod,
y(0) = y’(0) = y(l) = y’(l) = 0
for the clamped-clamped
y(0) = y’(0) = y(Z) = y”(l) = 0
for the clamped-pinned
y(0) = y’(O) = y”(l) = y”(t) = 0
for the clamped-free
rod, rod,
rod.
(34) (35) (36) (37)
H.H.E. Leiphoiz, The energy method for continuous follower load systems
265
Let first the clamped-tree rod be discussed. For this system, E =
4
and integration
I0
’
[-/uo2y
+ ay"" + g(Z - x)y"]y dx
by part yields
E = +
I’ [-,wo2y2
+ a(~“)’ - g(Z - x)y’)“]
dx + 4 1’ gy’y dx .
0
(38)
0
fn (38),
6
I
o’[-/.uJy*
+ a(y")"- g(Z - x)y’)‘]
dx = H,
(39)
1 $
_f0
gy’ydx=-$
I,’ (-gy’)y dx = - $ I,’ Ny dx = - W, ,
w-9
and
(41)
Iv = -gy’
represents the nonconservative follower forces. Hence, with (39) and (40), relationship yields, as predicted, equation (l), i.e., E=H-
Furthermore,
F
(38)
W,.
according to (3) and (39), sy =
1’(cxy” Sy”-
g(Z - x)y’ Sy’ - ,w2y Sy) dx ,
(42)
0
an expression which by integration y
Sy =
by parts can be transformed
’[-,uw2y + tyy”” + g(Z -
x)y” - gy’] Sy dx .
By virtue of (33) and (41), equation (43) can be written as E&y=jl[D(y)+N]Sydr, 0 SY which confirms the correctness of (4). By means of (40) one has for expression (5) explicitly -swN=$
I0
‘gy’Sydx+;gySy’dx.
into
(43)
H.H.E. Leipholz, The energy method for continuous follower load systems
266
Moreover,
1’
awr4
Y'Y dx, 0 ae=-T5
(45)
and, because of (39), and keeping in mind that A = CM’,also dH -=
at
-- ; ‘(I-x)(y’)‘dx, I0
g=
(W
-jlipy2dx. 0
With (41), (#), (4% and (46), equation (7) reads 6E =
$
f 0
’[-gy’
6y + gy 6~’ -
(I - x)(y’)” Sg - py2 8h + y’y Zig] dx .
(47)
Using (8) in (47) yields &E=+
t
-gy -
ay
-WgY’Tp+gY
JY
ag
(I - x)(y’)” 6g - py* 6h + y’y
6g I
dx .
(48)
By virtue of (lo), equation (48) yields after rearranging py2-gy$+gy$
dx I
%+
f
6h=
(49)
gy$gy$+(Z-x)y”y]dx which can easily be identified to represent obtain (49) from (47), also the identity
j’ [-(I
- x)(y’)‘+
(11). It may be mentioned
though, that in order to
y’y] dx = I,’ (t - x)y”y dx
0
has been applied. From (49) follows the flutter criterion 1
f0
py2 dx =
gy
$$-gy’$
Let it be shown that (51) corresponds
1dx. to (12). For this purpose write (12) using (39), (40) and
H.H.E. Leipholz, The energy method for continuous follower load systems (41).
267
One then obtains aY (-gy’),-fpy2+~gy’~+~gy~] dx = 0.
This expression can be rewritten as 1
-2
1dx=O,
py’+gy$-gyg
from which (51) follows immediately. Condition (51) shall be used later on to derive the smallest flutter load of the clamped-free rod numerically. Let now the completely supported rods be considered for which the boundary conditions (34), (35), and (36) hold. By virtue of these boundary conditions, indeed, IV,=-$g
which corresponds
I0
I0
‘Y’ydx=-:g[y’(Z)-y2(0)]=0,
to (13), and
1 N dx = -g
'y'y dx
= - ;g[y2(1)-
y2(0)] = 0,
I 0
which corresponds to (15). Let now the condition for divergence be checked, which occurs for A = w2 = 0, 8h = 0. Going back to (2), one has first because of (13) and (14),
(52) With (39) and for A = o2 = 0, one finds that
6H6y = I ’[ay” 8~" - g(Z 6Y
Integration
x)y’ 6y’] dx .
(53)
0
by parts performed
on (53) yields
[a~“” + g(Z - x)y” - gy’] 6y dx.
The operator Do is supposed that therefore
to follow from operator
(54) D for A = w2 = 0. Equation
(33) shows,
268
H.H.E.
Leipholz,
The energy method for continuous follower load systems
D,(y) = a~““+ g(Z - x)y”.
(55)
By means of (55) and (41) one realizes that (54) yields ‘[D,((y)+N]6ydx, which, substituted 6E=
in (52) leads to
I0
‘[D,(y)+N]Sydx+$%g
(56)
confirming (17). For A = w2 = 0, (39) yields dH -=
JET
-$ I ‘(I-x)(y’)2dx.
(57)
0
With (54) and (57), relationship
SE = I 0
(52) becomes equal to
’[a~""+ g(Z- x)y"- gy’] Sy dx - ;l’
(I - x)(y’)‘dx 6g = 0.
Applying (10) and using the mean value theorem of integration
sg-2
sj;-
I’
[a~““+
gy’]
g(Z - x)y”-
allows one to change (58) into
dx ,.
O
(58*)
1 I0
Yet,
(I - x)(y’)’ dx
1 I 0 gy’ dx = dy(O - YKOI= 0
(59)
due to the boundary conditions. Therefore,
I’
[ay”“+
Q-2 G-
(3
0
g(Z - x)y”]
relationship
dx ,
O
(58*) can be simplified to read
’ (I - x)(y’)’ dx which corresponds by virtue of (55) and (57) to (19). Divergence occurs for Sg/Sy = 0. Hence,
(60)
H.H.E. Leipholz, The energy method for continuourrfollower toad systems
’[a,“” + g(E -
x)y"]dx = 0 ,
f0
must hold simultaneously. Conditions Conditions (61) are equivalent to
I
/
g(Z - x)y"]y dx
[a~““+
269
(61)
ffy*lf+ g(I - x)y” = 0
(61) confirm (20).
(62)
= 0,
0
which corresponds
to (21). One realizes easily, Yhat
I
I
0
I0
’ (I - x)y”y dx
i
a(y”)‘dx
0
Es
gdiv =
-
I
cxy”“y dx
(63)
I’ (l- x)(y’)’ dx 0
Since DE(y) = my”” and D, = -(E - x)y”, relationship (63) confirms (23). As far as the flutter criterion (24) for completely supported systems is concerned, for rods with (39) and (41) 1 ,
py2dx+2 f
0 gy
f9 -dx=O.
tw
dh
Whether flutter will occur in addition to divergence for completely the behaviour of the functional 1
1
9=
I
py*dX+2 gy
it reads
dY -ddx
supported
rods depends on
(65)
dh
on the left-hand side of (64). If this functional is not sign-definite and can assume the value zero, then flutter will take place, However, if the functional is sign-definite, condition (64) cannot be satisfied, and flutter is ruled out. This happens for example for the pinned-pinned rod which is a sheer divergence system. Finally, the case of conservative systems shall be investigated, For these systems, (13), (14), and (25) hold, i.e. W,=-;
‘gy’ydx=O, I0
8h = SW*“0
and D,(y) + N = “y”” -t g(l - x)y”- gy’ = ay” + g[(t - x)y’]’ =Do(y)=O. Concluding
from (25*) that
t25*)
H.H.E. Leipholz, ‘171eenergy methodfor continuousfollower load systems
270
D,(y) = try”,
D, = -[I - x)y’]’ ,
one has for the relationship
I
(29) explicitly
1
I
ay”“y dx
0
gdiv
I
LY (y “)’ dx
0
=
=
L’
-
[(Z
x)y’]‘y dx
-
(66)
9
Jo’(I - x)(y’)* dx
which is indeed Rayleigh’s quotient for the buckling load g&v+ 4. Numerical examples Let first the flutter criterion (51) be applied. For this purpose consider the clamped-free rod. The coordinate functions to be used in a Galerkin approach are given in [8, p. 961. By means of +I and d;* one can approximate Y=
(67)
al#l++2.
With this assumption
for y, one can set up the Galerkin equations
a&b + g%) f gQi12 = 0,
@J)
(A2+ g%) + wqliz, = 0,
where Al = -pd+
c~(12.41-~),
A2 = -/.uo* + ct(485.W4),
@22= -6.65/-1,
@,, = 0.431-l ,
(6% @j,, = -4.343-l
,
lrpz, = 1.181-1,
ski =
’ (t - x)#:&
dx,
J-O
?Pkz=
’&#k
dx ,
YF,, - PI2 = @, - 523, .
The numerical values in (69) have been determined Setting Ai + @ii = Di
9
using the information
in [8, pp. 96-971.
(70)
the first equation in (68) yields al = -gat121Dl.
Moreover, one has
(71)
H.H.E.
Leipholz, The energy method for continuous follower load systems
271
(72) By virtue of (67) and (71), and the orthonormality I
1
,uy2dx=p
I0
(al&
Using in (73) the relationship
I
of the 4i,
I
/-v2dx=p
0
+ $2)2 dx = p(af
+ 1) .
(73)
(71) yields
g2&, + D: D2
Again, by virtue of (67), the orthonormality the !Pki in (69), one obtains aY
gy$gytz
(74)
.
1
of the +i, and in addition due to the definition of
1
dx=g$+&,-1Y~z)=/q2
@12(%2-
D:
%I)
(75)
*
Hence, (51) reads
5
(g2@:2+
D:)
= 5
1
g2@12Wi2-
(76)
P21).
1
After rearranging
and using the last identity in (69), one arrives at
g2Q12Q21+ D: = 0 as the flutter criterion. It is obvious that this relationship g2@,,@,,+D~=0,
i=l,2.
can be generalized
to read (77)
This is the final form of the flutter criterion. Introducing the new notation Yi = @i +
@12@21
as well as (70), criterion (77) can be transformed g =
(78)
,
$I [-cDii2 (- @12@2J’2].
into
(79)
Using in (79) the values given in (69) for Ai, 3/i, @ii, Q12, and @21, one can derive the
272
H.H.E. Leipholz,
The energy method for continuous follower load systems
following sets of equations
g=
0.372/~0~1- 4.606al-“ , -0.54Jj~w’l+ 6.768&-‘,
(80)
and -0.1 14/Lw21+ 54.469al-’ ) g=
-0.232p.w21+ 110.664al-” .
(81)
Equating the right-hand sides of (SO), and (Sl), yields w2 = 121.55cu/.-‘1-j
(82)
Equating the right-hand sides of (SO), and (8l)2 yields correspondingly
as a flutter frequency. the flutter frequency
o2 = -329.829~/l.-‘l-~.
(83)
Using (82) either in (SO), or in (Sl), yields the flutter load g, = 40.61cul-“ . Using correspondingly
(84) (83) in either (SO), or (Sl), yields the other flutter load
g, = 18J.18cul-3.
(85)
These results are in good agreement with those given in [S, pp. 99-1001 which have been derived in a different way. Such agreement of results confirms the applicability of the flutter criterion (51). Equations in (80) and (81) represent straight lines in the eigenvalue plane, i.e. in the g, 02-plane. Therefore, a simple geometrical interpretation of the calculation of flutter loads is possible, providing the designing engineer with a convenient tool: the points in the eigenvalue plane having flutter loads and flutter frequencies as coordinates are obtained as the points of intersection of the above mentioned lines. An outline of the geometric procedure can be found in [9]. Secondly, the flutter criterion (64) for completely supported rods shall be discussed. Considering the specific boundary conditions for rods, one realizes easily that
I
I
I
gy$dx=
-
gygdx.
0
For this reason, criterion (64) can be written as
I
1
0
py2
dx =
I
I
1
0
gygdx-
gy$dx,
H.H.E. Leipholz, The energy method for continuous follower load systems
which is identical with (51). Therefore, the result will be again (79), i.e. [- ~ii + (-
g = t
273
all the calculations carried out before remain valid, and
~D1*~*1)l’*] .
(79)
I
However, the last relationship in (69), i.e. P21 - !Py,,= @*I- @I*, has now to be replaced by @21- @I*= -2?P,,. This fact is again true due to the boundary conditions of completely supported rods. From [S, pp. 96-971, one obtains for the clamped-pinned rod A, = -/NJ* + c~(237.721-~),
A2 = -/.Lw* + c~(2496.491-~), (86)
d&1 =
-4.291-l)
-18.411-1,
cD2*=
Q12 =
0.921-l ,
a&=
-5.081-l )
so, y1 = dr,,@,, + @i:, =
13.731-l )
y2 = @&‘21+
@z2 =
334.261-* .
Using these values in (71) yields
(87) and (88) Equating (87), and (88), yields o* = -22.974~~/.-~1-~,
(89)
and equating (87), with (88)2 yields w* = -1255.59i~/_-‘l-~. Using these flutter frequencies yields the flutter loads g, = 122.495~&~
wo
either in (87),, (88), respectively,
and
or in (87),, (88)2 respectively,
gfl = 231.49~~~~.
The first one has been reported in [8, p. 100, Fig. 201. It corresponds well to the above one calculated here. If one would try to repeat such calculations for the pinned-pinned rod, one would find with the numerical values reported in [8], i.e. CD12 = -1.781-l )
@21=
-7.121-l)
274
H.H.E. Leipholz, The energy method for continuous follower load systems
that (79) would yield complex values for g. The conclusions must be that no flutter loads exist for the pinned-pinned rod. This result has been established in different ways in [5,8]. The pinned-pinned rod is indeed an example for so-called pseudo-divergence systems, (see [8, pp. G-74]) and conservative systems of the second kind (see [5, pp. 111, 217-223; 8, pp. 70-741). These are systems which fail by divergence only, although nonconservative follower forces are present. Finally, let the divergence criterion (63) for completely supported rods be checked. In terms of the previously introduced notations it reads
Setting i = 1 and using the numerical values given in [8, pp. S-973, one has a237.721-4 gdiv
=
4.291-l
=
This result is an approximate 991.
55.4;. one, which is in good agreement with the value reported in [8, p.
Appendix A
The basic expression to be considered
is
(A.11 where X is the density of the Hamiltonian. For a rod:
Therefore, 6X = -pw2y sy + cyynSy" - g(1 - x)y' Sy’
(A-3)
and g
6y =
f
’ [-,uw*y 6y + ay” 6~” - g(f - x)y’ Sy’j dx .
For all possible boundary conditions, SH 6y
Sy = I 0
’[-/.m*y
+ a~‘“‘+
(A.4) yields through integration g(l-
x)y”-
gy’] Sy dx.
(A4
by parts
6W
H.H.E. Lkpholz,
275
The energy method for continuous follower load systems
However, D(y) = -ploy
+ cyy”“+ g(1- x)y”
and
N = -gy’ .
Hence, indeed, ySy=
‘[D(y)+N]6ydx. I0
as stated in (3). For a rectangular, fully supported plate:
64.6)
j-fl.dQ=jF.dx,dx,, where F is the surface of the plate. Moreover, parallel to the x,-direction, x = -
b02Y2
(YXIXl + JL,,)‘+
+ M
for a plate with compressive
2(1 - 4(Y’x,*,
follower forces
(A-7)
-
where y is the deflection perpendicular to the middle surface of the plate, R the flexural rigidity, g the load parameter, a the sidelength of the plate in the q-direction, /_Lthe density per unit surface, and Y Poisson’s parameter. Obviously,
IE
Sy
dx, dx, =
6X dxI dx2
I=8y
=
I
F [(-PU’Y
+ RA*y) 6~ +
da - xl)yxlby, Sy,,] dxl dx2.
G4.8)
But
I da - ~~~~~~ dx, dx2+ I [da + I Ma 6~1dC, SY,,
F
c
F
XI)Y~
- u,l
SY
dx,dx,
xl)~xl
(A-9)
where C indicates that the respective expression is to be taken along the contour of the plate. Due to the boundary conditions of the plate along C,
I [da - xl)yx,W dC = C
0.
Hence, with (A.9) and (A.lO) in (A.8), and because of (A.l) one obtains
(A. 10)
H.H.E. Lzipholz, The energy rnet~o~for contirtuous follower Ioad systems
276
SH 6y
6~ =
I (---Pw~Y+ RA2y+ da - x2hx1- gy,) Sy dxl h. F
However, D(y) = -,uo2y f RA2y + g(a - xl)yxlx, and
N = -gy,, .
Hence, again,
SH6y = I 6Y
[D(y) + IV] Sy dxl dx, , F
confirming (3).
References [l] H. Ziegler, Die Stabilitltskriterien der Elastomechanik, Ingr.-Arch. 20 (1952) 49-56. [2] H. Leipholz, On the application of the energy method to the stability problem of nonconservative mechanical systems, Acta Mech. 28 (1977) 113-138. [3] H. LeiphoIz, An energy approach to stability probfems of nonconse~ative systems, in: Development in Theoretical and Applied Mechanics, Proceedings Ninth Sectam, Nashville, TN (1978) 427-437. [4] H. Leipholz, Stability analysis of nonconservative systems via energy considerations, Mech. Today 5 (1980) 193-214. [!I] H. Leipholz, Stability of Elastic Systems (Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980). [6] H. Leipholz, On the spillover effect in the control of continuous elastic systems, Mech. Res. Comm. 11 (1984) 217-226. [7] H. LeiphoIz, On the control of structural systems with distributed parameters, in: Proceedings 3rd International Conference on Systems Engineering, Wright State University, Dayton, OH (1984) 160-166. [8] H. Leipholz, Direct Variational Methods and Eigenvalue Problems in Engineering (Noordhoff, Leyden, 1977). [9] H. Leipholz, On the calculation of flutter loads by means of a variational technique, Paper No. 105, Solid Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada, February 1972.