DYNAMIC STABILITY OF WEAK EQUATIONS OF RECTANGULAR PLATES Andrzej Tylikowski 1 ...

14 Pages·2008·0.25 MB·English

JOURNAL OF THEORETICAL
AND APPLIED MECHANICS
46,3,pp.679-692, Warsaw2008
DYNAMIC STABILITY OF WEAK EQUATIONS OF
RECTANGULAR PLATES
Andrzej Tylikowski
Warsaw Universityof Technology, Warsaw, Poland
e-mail: [email protected]
The stability analysis method is developedfor distributed dynamic pro-
blemswithrelaxedssumptionsimposedonsolutions.Theproblemismo-
tivated by structural vibrations with external time-dependent parame-
tric excitations which are controlledusing surfacemountedor embedded
actuators and sensors. The strong form of equations involves irregulari-
ties which lead to computational difficulties for estimation and control
problems.Inordertoavoidirregulartermsresultingfromdifferentiation
of force and moment terms, dynamical equations are written in a weak
form.The weakform ofdynamicalequations oflinear mechanicalstruc-
tures is obtained using Hamilton’s principle. The study of stability of a
stochasticweaksystemisbasedonexaminingpropertiesoftheLiapunov
functional along a weak solution. Solving the problem is not dependent
on assumed boundary conditions.
Key words:weakformulation,dynamicstability,differentboundarycon-
ditions, Liapunov method
1. Introduction
The strong form of plate equations involves irregularities which lead to com-
putational difficulties for estimation and control problems. In order to avoid
irregular terms resulting from differentiation of force and moment terms, dy-
namical equations are written in a weak form. The weak form of systems
is useful for development of identification methods and general computational
methods(Banks et al., 1993). We considerdynamical systemswith parametric
excitations, e.g. plates with in-plane time dependent forces. The plate motion
is described by partial differential equations that include time dependent co-
efficients implying parametric vibrations. The response of such systems can
680 A. Tylikowski
lead to a new increasing mode of oscillations and the structure dynamically
buckles. The classical Liapunov technique for stability analysis of continuous
elements is based on choosing or generating of a functional which is positive
definite in the class of functions satisfying structure boundaryconditions. The
time-derivative of the Liapunov functional has to be negative in some defined
sense. Almost sure stability of the beam equilibrium state in the strong for-
mulation was examined by Kozin (1972). The technique of stability analysis
was extended to plates and shells with in-plane or membrane time-dependent
forces (Tylikowski, 1978). Uniform stochastic stability analysis of laminated
beams and plates described by partial differential equations with time and
space dependent variables was presented in Tylikowski and Hetnarski (1996).
The weak form of a distributed controller in an active system consisting of
electroded piezoelectric sensors/actuators with suitable polarization profiles
(Tylikowski, 2005) is useful for the feedback theoretical developments.
For the purpose of active vibration and noise control, piezoelectric devi-
ces have shown great potential as elements of passive absorbers and active
control systems as they are light-weight, inexpensive, small, and can be bon-
ded to main structures. They can not be modeled as point force excitations,
and partial differential equations should be used to describe the response of
the structures driven by them. Active damping in composite structures with
collocated sensors/actuators were studied by Tylikowski (2005), Tylikowski
and Hetnarski (1996). Electrodes on sensors/actuators are spatially shaped
to reduce the spillover between circumferential modes. In order to avoid irre-
gular terms resulting from the modelling, the action of piezoelements as the
Dirac delta function concentrated on their edges and dynamical equations are
written in a weak form.
2. Weak formulation of plate dynamic equations
Consider an elastic rectangular plate of the length a, width b, thickness h,
mass density ρ and bending stiffness D subjected to the in-plane time-
dependent forces F (t) and F (t) acting in the X and Y direction, respecti-
X Y
vely.Therefore,theonsetofparametricvibrationsispossible.Inordertoavoid
developing modes of plate motion, viscous damping with the proportionality
coefficient α is introduced. The strong form of plate dynamical equation in
the transverse displacement w is given in the following form
ρhw +αw +D∆2w+F (T)w +F (T)w = 0 (2.1)
,TT ,T X ,XX Y ,YY
Dynamic stability of weak equations... 681
where (X,Y) 0,a 0,b . Introducing dimensionless variables
∈ { }×{ }
X Y T
x = y = t =
b b k
t
equation (2.1) becomes
w +2βw +∆2w+[f +f (t)]w +[f +f (t)]w = 0 (2.2)
,tt ,t ox x ,xx oy y ,yy
where (x,y) 0,r 0,1 , r = a/b is the plate aspect ratio
∈ { }×{ }
1 αk
k2 = ρhb4 β = t
t D 2ρh
1 1
f +f (t) = b2F (k t) f +f (t) = b2F (k t)
ox x X t oy y Y t
D D
and the solution should be fourth time partially differentiated with respect to
x and y. If the plate is simply supported at the end, the transverse displa-
cement and bending moment equal zero. For clamped edges, the transverse
displacement and the slope equal zero.
Similarly,wecananlyseothercombinationsofsimpleboundaryconditions,
i.e. a simply supported – clamped plate. We write the action integral of the
plate without damping in the form
t2 r 1
1
A[w] = [w2 (w +w )2 2(1 ν)(w2 w w )+
2 ,t− ,xx ,yy − − ,xy − ,xx ,yy
tZ1 Z0 Z0
(2.3)
+f w2 +f w2]dxdydt
ox ,x oy ,y
where ν is Poisson’s ratio, and applying Hamilton’s principle
d
A[w+εΦ] =0 (2.4)
dε ε=0
(cid:12)
(cid:12)
where εisarealnumber.Addingthevisco(cid:12)usdampingandthetime-dependent
components of the in-plane forces as external works, the dynamical equation
can be written in the weak form as follows for all Φ
r 1
(w +2βw )Φ+(w +νw )Φ +(w +νw )Φ +
,tt ,t ,xx ,yy ,xx ,yy ,xx ,yy
Z0 Z0 n
(2.5)
+2(1 ν)w Φ [f +f (t)]w Φ [f +f (t)]w Φ dxdy = 0
,xy ,xy ox x ,x ,x oy y ,y ,y
− − −
o
682 A. Tylikowski
where Φ is a sufficiently smooth test function satisfying essential boundary
conditions. There is no demand on the existence of higher derivatives than the
second order. As detailed in Banks et al. (1993), usual integration by parts
the terms containing derivatives of the test function Φ with respect to the
in-plane variables x and y and the assumption of sufficient smoothness of the
plate displacement leads to strong formulation (2.2).
3. Almost sure stability definition
Dynamical equations (2.2) and (2.5) contain terms explicitely dependent on
time. The time dependency of the axial forces f (t) and f (t) parametrically
x y
excites the plate and an increasing form of vibrations can occur. In determi-
nistic parametric vibrations, the stability properties are determined from the
Mathieu equation together with the corresponding Ince-Strutt diagram. If the
excitation is narrow-bandedor has onelatent periodicity, a series of wedges on
the amplitude-frequency plane can be expected, analogously to the determini-
sticparametricresonance.Thetaskismuchmorecomplexwhenthestochastic
excitation is wide-band and continuous systems with an infinite numberof na-
tural fequencies are analysed. Due to the fact that the norms and metrics in
infinite-dimensional spaces are not equivalent, the stability holds for just the
norm or the metric used in the analysis. If the parametric excitation becomes
random, the stability criteria depend on the statistical characteristics of the
excitation and the systems parameters. The present paper examines dynamic
stability dueto an action of in-planeforces in theform of stochastic physically
realizabletime-dependentprocesseswithgiven statistical properties.Equation
(2.5) with zero initial conditions posseses the trivial solution
w = w = 0 (3.1)
,t
The trivial solution to Eq. (2.2) or (2.5) is almost sure stable if with probabi-
lity 1 if the measure of distance between the perturbed solution with nonzero
initialconditionsandthetrivialonetendstozeroastimetendstoinfinity(Ko-
zin, 1972). Usually, the measure of distance is defined by a positive-definite
functional. The trivial solution is called almost sure asymptotically stable if
P lim w(,t) = 0 = 1 (3.2)
t→∞k · k
(cid:8) (cid:9)
where w(,t) is ameasureof the disturbedsolution w withnontrivial initial
k · k
conditions from the equilibrium state, and P is the probability measure. The
Dynamic stability of weak equations... 683
almost sure stability is equivalent to the clasical Liapunov stability in linear
systems.
4. Stability analysis in weak formulation
In order to examine the almost sure stability of the plate equilibrium (the
trivial solution), the Liapunov functional is chosen in the form
r 1
1
V = w2 +2βww +2β2w2+(w +w )2+
2 ,t ,t ,xx ,yy
Z0 Z0 h
(4.1)
+2(1 ν)(w2 w w ) f w2 f w2 dxdy
,xy ,xx ,yy ox ,x oy ,y
− − − −
i
The functional is positive-definite if the constant components of the in-
planeforces f and f fullfilthestaticbucklingcondition,i.e.aresufficiently
ox oy
small. Therefore, the measure of disturbedsolutions is chosen as a squareroot
of the functional V
w(,t) = √V (4.2)
k · k
As trajectories of the solution to equations (2.5) are physically realizable,
classical calculus is applied to find the time-derivative of functional (4.1). Its
time-derivative is given by
r 1
dV
= (w +βw)w +βw2 +2β2ww +
dt ,t ,tt ,t ,t
Z0 Z0 h
+w w +w w +2(1 ν)w w + (4.3)
,xx ,xxt ,yy ,yyt ,xy ,xyt
−
+ν(w w +w w ) f w w f w w dxdy
,xx ,yyt ,yy ,xxt ox ,x ,xt oy ,y ,yt
− −
i
Substituting βw and w as the test functions in Eq.(2.5), we have two
,t
identities, respectively
r 1
(w +2βw )βw+(w +νw )βw +(w +νw )βw +
,tt ,t ,xx ,yy ,xx ,yy ,xx ,yy
Z0 Z0 h
+2(1 ν)βw2 (f +f (t))βw2 (f +f (t))βw2 dxdy = 0
,xy ox x ,x oy y ,y
− − −
(4.4)
i
r 1
(w +2βw )w +(w +νw )w +(w +νw )w +
,tt ,t ,t ,xx ,yy ,xxt ,yy ,xx ,yyt
Z0 Z0 h
+2(1 ν)w w (f +f (t))w w (f +f (t))w w dxdy = 0
,xy ,xyt ox x ,x ,xt oy y ,y ,yt
− − −
i
684 A. Tylikowski
Subtracting identities (4.4), from the time-derivative of functional (4.3),
we obtain the following form
1
dV
= βw2 β[w2 +w2 +2w w +2(1 ν)(w2 w w )]+
dt − ,t− ,xx ,yy ,xx ,yy − ,xy − ,xx ,yy
Z0 n
(4.5)
+βf w2 +βf w2 +f (t)(w +βw )w +f (t)(w +βw )w dxdy
ox ,x oy ,y x ,xt ,x ,x y ,yt ,y ,y
o
After some algebra, we rewrite the time-derivative of functional as
dV
= 2βV +2U (4.6)
dt −
where U is an auxiliary functional given by
1
1
U = 2β2ww +2β3w2+
,t
2
Z0 h
(4.7)
+f (t)(w +βw )w +f (t)(w +βw )w dxdy
x ,xt ,x ,x y ,yt ,y ,y
i
Now we attempt to construct a bound
λV U (4.8)

where the stochastic function λ is to be determined. In an explicite notation,
the function λ has to satisfy the following equation for arbitrary functions w
and w satisfying suitable boundary conditions
,t
r 1
λ[w2 +2βww +2β2w2+(w +w )2+
,t ,t ,xx ,yy
Z0 Z0 n
+2(1 ν)(w2 w w ) f w2 f w2]+ (4.9)
,xy ,xx ,yy ox ,x oy ,y
− − − −
[2β3w2+2β2ww +f (t)(w +βw )w +f (t)(w +βw )w ] dx 0
,t x ,xt ,x ,x y ,yt ,y ,y
−
o
It should be noticed that the way to obtain estimation (4.8) is purely alge-
braic contrary to systems described by strong equations, where the derivation
of stability conditions is based on integrations by parts and manipulations
with higher order partial derivatives. Usually, the Liapunov stability analysis
of plates is performed for all four simply supported edges (Tylikowski, 1978).
Dynamic stability of weak equations... 685
In order to extend the field of possible applications, let us assume the fol-
lowing combinations of boundary conditions: a) c-c-c-c, b) c-c-c-s, c) c-c-s-c,
d) c-c-s-s, e) s-c-s-s, f) s-s-s-s, shown in Fig.1, where ”s” denotes a simply
supported edge, and ”c” denotes a clamped edge. Contrary to the Levy me-
thod of determination of displacements of a rectangular plate with two simply
supported opposite edges, the proposed technique of determining the stability
domains can be applied to plates with arbitrary combinations of simply sup-
ported and clamped edges. In the first combinations of boundary conditions
for plates with all four edges simply supported, we have
∞
mπx
w(x,y,t) = w (t)sin sin(nπy) (4.10)
mn
r
m,n=1
X
If the plate with mixed simply supported-clamped edges is considered, the
plate displacement is written in the following form
∞
x
w(x,y,t) = w (t)X Y (y) (4.11)
mn m n
r
mX,n=1 (cid:16) (cid:17)
where X and Y are beam functions depending on boundary conditions for
m n
x = 0, r and y = 0, 1, respectively. For example, if all four edges of plate are
clamped, the beam functions in Eq. (4.11) have the following form
β x β x
n n
X = sin sinh (cosβ coshβ )+
m n n
r − r −
(cid:16) (cid:17)β x β x
n n
(sinβ sinhβ ) cos cosh
n n
− − r − r
(cid:16) (cid:17) (4.12)
Y = (sinγ y sinhγ y)(cosγ coshγ )+
n n n n n
− −
(sinγ sinhγ )(cosγ y coshγ y)
n n n n
− − −
Integrating, we have the following equality
r r 1 1
X2 dx = κ β2 X2 dx Y2 dy = χ γ2 Y2 dy (4.13)
m,xx m m m,x n,yy n n n,y
Z0 Z0 Z0 Z0
Using the orthogonality condition, we can also write
r r 1 1
β2 γ2
X2 dx = m X2 dx Y2 dy = n Y2 dy (4.14)
m,x κ m n,y χ n
Z0 m Z0 Z0 m Z0
686 A. Tylikowski
where the first few constants κ and χ used in equalities (4.13) and (4.14)
m n
are given in Table 1 for the most commonly used boundary conditions: a
clamped-clamped beam (c-c), and a clamped-simply supported beam (c-s).
The constants tend to 1 as n . In order to unify the notations in the
→ ∞
case of the simply supported beam, we assume β = nπ and κ = 1. Using
n n
properties of the functions X and Y , we have
m n
r 1 ∞ r 1
w (x,y,t)w (x,y,t) dx = w (t)w (t) X2 dx Y2 dy
,xt ,x mn,t mn m,x n
Z0 Z0 mX,n=1 Z0 Z0
(4.15)
Substituting Eq. (4.14) , yields
2
∞ r 1
w (t)w (t) X2 dx Y2 dy =
mn,t mn m,x n
mX,n=1 Z0 Z0
(4.16)
∞ β2 r 1
= mw (t)w (t) X2 dx Y2 dy
κ mn,t mn m n
mX,n=1 m Z0 Z0
Similarly, we have
r 1
w (x,y,t)w (x,y,t) dxdy =
,xx ,yy
Z0 Z0
(4.17)
∞ ∞ r 1
= w (t)w (t) X X dx Y Y dy
mn ij m,xx i n j,yy
mX,n=1iX,j=1 Z0 Z0
Integrating by parts the terms in the right-hand-side of Eq. (4.17) for
simply supported or clamped edges, we have
r r r
r
X X dx = X X X X dx = δ X2 dx (4.18)
m,xx i m,x i m,x i,x mi m,x
0− −
Z0 (cid:12) Z0 Z0
(cid:12)
(cid:12)
where δ denotes the Kronecker delta function
mi
1 1 1
1
Y Y dy = Y Y Y Y dx = δ Y2 dy (4.19)
n j,yy n,y j n,y j,y nj n,y
0− −
Z0 (cid:12) Z0 Z0
(cid:12)
(cid:12)
Dynamic stability of weak equations... 687
Substituting Eqs. (4.18) and (4.19) into Eq. (4.17), yields
r 1 r 1
w (x,y,t)w (x,y,t) dxdy = w2 dxdy (4.20)
,xx ,yy ,xy
Z0 Z0 Z0 Z0
Table 1. Numbers κ
n
m, n 1 2 3 4 5 6 7
c-s 1.3396 1.1649 1.1086 1.0809 1.0645 0.90205 1.0537
c-c 1.8185 1.3392 1.2224 1.1648 1.1309 1.10857 1.09276
Combining Eqs. (4.9) and (4.20) and substituting expansion (4.11), we
have
∞ r 1
[λw2 +2β(λ β)w w +2β2(λ β)w2 ] X2 dx Y2 dy+
mn,t mn,t mn mn m n
− −
mX,n=1n Z0 Z0
r 1 r 1 r 1
+λw2 X2 dx Y2dy+2 X2 dx Y2 dy+ X2dx Y2 dy +
mn m,xx n m,x n,y m n,yy
(cid:16)Z0 Z0 Z0 Z0 Z0 Z0 (cid:17)
(4.21)
r 1
[(λf +βf (t))w2 +f (t)w w ] X2 dx Y2 dy+
ox x mn x mn,t mn m,x n
−
Z0 Z0
r 1
[(λf +βf (t)]w2 +f (t)w w ] X2 dx Y2 dy 0
oy y mn y mn,t mn m n,y
−
Z0 Z0 o
Using Eqs. (4.14)-(4.17), yields
∞ β2 γ2
λw2 + 2β(λ β) f (t) m f (t) n w w +
mn,t − − x κ − y χ mn,t mn
nX=1n (cid:16) m n(cid:17)
β2 γ2 β2
+ 2β2(λ β)+λ β4 +2 m n +γ4 (λf +βf (t)) m + (4.22)
− m κ χ n − ox x κ
m n m
h (cid:16) (cid:17)
r 1
γ2
(λf +βf (t)) n w2 X2 dx Y2 dy 0
− oy y χ mn m n
ni oZ0 Z0
Therefore, the variational inequality is reduced to an infinite system of
quadratic inequalities
λw2 +2A (λ,t)w w +B (λ,t)w2 0 (4.23)
mn,t mn mn,t mn mn mn
688 A. Tylikowski
where
β2 γ2
A (λ,t) =β(λ β) f (t) m f (t) n
mn x y
− − 2κ − 2χ
m n
β2 γ2
B (λ,t) = 2β2(λ β)+λg β4 +2 m n +γ4g +
mn − m κ χ n
m n
(cid:16) (cid:17)
β2 γ2
[λf +βf (t)] m [λf +βf (t)] n
ox x oy y
− κ − χ
m n
The unknown function λ is determined from the zero determinant condi-
tion for all m and n of the form
λ A (λ,t)
mn
= 0 (4.24)
A (λ,t) B (λ,t)
(cid:12) mn mn (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
After some algebraic man(cid:12)ipulations, (4.24) yield(cid:12)s
λ = max λ (4.25)
mn
m,n=1,2,...
where
β2+f (t) βm2 +f (t) γn2
x 2κm y 2χn
λ = (4.26)
mn β4 +2(cid:12)(cid:12)βm2 γn2 +γ4 f βm2 f (cid:12)(cid:12)γn2 +β2
m (cid:12)κmχn n− oxκm − oy(cid:12)χn
q
Combining inequality (4.6) and (4.8), yields an upper estimation of the
time-derivative of the functional
dV
= 2βV +2U 2(β λ)V (4.27)
dt − ¬ − −
Integratingwithrespecttotime,yieldsthefollowingupperestimationof V
t
1
V(t) V(0)exp 2 β λ(τ) dτ t (4.28)
¬ − − t
h (cid:16) Z0 (cid:17) i
Therefore
lim w(,t) = 0 (4.29)
t→∞k · k
if the exponent in (4.28) is negative
t
1
β lim λ(τ) dτ (4.30)
t→∞ t
Z0