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\markboth{\hfil The bowed narrow plate model \hfil EJDE--2000/27}
{EJDE--2000/27\hfil David L. Russell \& Luther W. White \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~27, pp.~1--19. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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The bowed narrow plate model
\thanks{ {\em Mathematics Subject Classifications:} 74K10, 74K30.
\hfil\break\indent
{\em Key words and phrases:} Mindlin-Timoshenk plates, Narrow Plates,
coupled structures.
\hfil\break\indent
\copyright 2000 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted December 9, 1999. Published April 12, 2000.} }
\date{}
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\author{ David L. Russell \& Luther W. White }
\maketitle
\begin{abstract}
The derivation of a narrow plate model that accommodates shearing,
torsional, and bowing effects is presented. The resulting system
has mathematical and computational advantages since it is in the
form of a system of differential equations depending on only one
spatial variable. A validation of the model against frequency
data observed in laboratory experiments is presented. The models
may be easily combined to form more complicated structures that
are hinged along all or portions of their junction boundaries or
are coupled differentiably as through the insertion of dowels
between the narrow plates. Computational examples are presented to
illustrate the types of deformations possible by coupling these
models.
\end{abstract}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Defintion}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
In this paper, we consider a class of models for
elastic structures, so-called narrow plate models [6,7], of an
intermediate nature between beams and plates. Of particular interest
are models that include linear shearing effects. Additionally, linear torsion
and quadratic bending terms reflecting narrowness of the body
are included. In \cite{r1,r2} we
introduced models of this type, whose derivation was
based on the energy form for the
Mindlin-Timoshenko plate equations, by imposing restrictions to account for the
geometry. The resulting dynamic model involves a system of linear
symmetric hyperbolic equations in two independent variables, time and
the longitudinal beam coordinate $x$, which are a special case of the
Mindlin-Timoshenko plate equations \cite{m2}. Alternatively, they may be
viewed as an extension of the familiar
Timoshenko beam system, as described in \cite{g1} for example, to include torsional
vibrations. The Mindlin-Timoshenko system is itself a generalization of
the Kirchhoff model, modified to include shearing and rotatory inertia
\cite{t1}.
In previous work, validation and parameter identification have been
carried out on the affine model in which the structure is
sufficiently narrow that only linear torsional deformations across the width
occur. In this work we present
a validation of the so-called bowed model in which the width
of the narrow plate is sufficiently large that parabolic deformations
may occur across the width. The inclusion of such terms
is significant in that it enables us to model more complicated structures
that are coupled in a variety of ways. Moreover, the mathematical justification
of coupling conditions and point loads are simplified since the resulting
models involve only one spatial dimension.
This approach enables us to model more complicated structures with
relatively simple systems. Finally, we give a model
expressing the deformation of a structure associated with a point force.
In Section 2, we present the bowed narrow plate model along with a
brief discussion of the well-posedness of solutions. In Section 3, we
give a numerical formulation of the problem and a validation of the
model using natural frequency data collected in the laboratory.
In Section 4, we
present several examples of models of structures composed of narrow plates
coupled in various ways continuously and
differentiably along sides and also at various points along the plates.
\section{Derivation of the bowed narrow plate equations}
We begin by giving the stress-strain relations obtained
under the following assumption, cf \cite{h1}.
\begin{description}
\item{(E)} The gradient of the deformation is small so that products of derivatives of the
deformation are neglected.
\end{description}
Assumption (E) results in the following linear stress-strain relations,
using the standard notation in \cite{h1}.
$$\displaylines{
\sigma_{11}={E\over{(1+\mu)(1-2\mu)}}\lbrack(1-\mu)\epsilon_{11}+\mu\epsilon_{22}
+\mu\epsilon_{33}\rbrack \cr
\hfill\sigma_{22}={E\over{(1+\mu)(1-2\mu)}}\lbrack\mu\epsilon_{11}+(1-\mu)\epsilon_{22}
+\mu\epsilon_{33}\rbrack \hfill\llap{(2.1)(i)} \cr
\sigma_{33}={E\over{(1+\mu)(1-2\mu)}}\lbrack \mu\epsilon_{11}+\mu\epsilon_{22}
+(1-\mu)\epsilon_{33}\rbrack \cr
\sigma_{12}=G\epsilon_{12},\quad \sigma_{13}=G\epsilon_{13},\quad
\sigma_{23}=G \epsilon_{23}
}$$
where $E$ is the Young's modulus, $\mu$ is the Poisson's ratio, and
$G=2E/(1+\mu)$ is the shear modulus.
We suppose that the body occupies an open domain $\Omega$ in ${\mathbb R}^3$
given by
$$\Omega =\lbrace (x,y,z):
0 \le x \le L,\ -k(x) \le y \le k(x),\ {\rm and}\ -h(x,y)\le z \le h(x,y)
\rbrace.$$
To facilitate our analysis, we assume that
\begin{description}
\item{(i)} The functions $h$ and $k$ are bounded nonnegative piecewise continuous
with finitely many jump discontinuities.
\item{(ii)} For each $x\in [0,L]$ the mapping $y\mapsto h(x,y)$
of $(-k(x),k(x))$ into ${\mathbb R}$ is an even function.
\end{description}
\noindent The underlying assumptions for the linear plate approximation are
\begin{description}
\item{(P1)} Normal stresses in the z-direction are absorbed into the body force
\par\vskip12pt
\item{(P2)} No stretching or shearing of the neutral surface occurs.
\end{description}
\noindent
Under assumption (P1), $\sigma_{33}$ is set to zero and the resulting relation
is used to eliminate $\epsilon_{33}$ from the expressions for $\sigma_{11}$
and $\sigma_{22}.$
In this case from (2.1)(i), one obtains
$$\displaylines{
\hfill \sigma_{11}={E\over{1-\mu^2}}\lbrack\epsilon_{11}+\mu\epsilon_{22}
\rbrack \,,\hfill\llap{(2.1)(ii)} \cr
\sigma_{22}={E\over{1-\mu^2}}\lbrack\mu\epsilon_{11}+\epsilon_{22}\rbrack.
}$$
The assumption (P2) implies there are no geometric nonlinearities
resulting from large deformations that would result, for example,
in von Karman-type plate models, \cite{t1}.
The displacements in the $x$, $y$, and $z$ directions are given by the
functions
$$U = U(x,y,z,t), \quad V = V(x,y,z,t),\quad W = W(x,y,z,t)\,.$$
The strain-displacement relations under the assumption (E) are given by
$$ \displaylines{
\epsilon_{11}={{\partial U}\over {\partial x}}, \quad
\epsilon_{12}={1\over 2}({{\partial U}\over{\partial y}}
+{{\partial V}\over{\partial x}}), \cr
\hfill \epsilon_{13}={1\over 2}({{\partial W}\over {\partial x}}+ {{\partial U}
\over{\partial z}}), \quad
\epsilon_{22}={{\partial V}\over{\partial y}}, \hfill\llap{(2.2)}\cr
\epsilon_{23}={1\over 2}({{\partial W}\over{\partial y}}
+{{\partial V}\over{\partial z}})\,.
}$$
Assumptions (E), (P1), and (P2) imply the resulting model being linear.
The Mindlin-Timoshenko model assumes that the displacements
$U$, $V$, and $W$ can be expressed as
$$\displaylines{
U(x,y,z,t)=z u(x,y,t), \quad V(x,y,z,t)=z v(x,y,t), \quad
W(x,y,z,t) = w(x,y,t)\,.
}$$
In this work the displacements are further specialized by assuming
$$ \displaylines{
U(x,y,z,t)=z[\phi_0(x,t)+y\phi_1(x,t)], \cr
\hfill V(x,y,z,t)=z[\psi_0(x,t)+y\psi_1(x,t)],\hfill\llap{(2.3)} \cr
W(x,y,z,t) = w_0(x,t) + y w_1(x,t) + y^2 w_2(x,t)\,.
}$$
These assumptions are motivated by our desire to
develop a model of an elastic body that includes
shearing as well as torsion and bending across the width.
Substitution of equations (2.3) into equations (2.1)(ii) and (2.2)
allows us to express the strain and the stress in terms of the
displacement functions
$\phi_0$, $\phi_1$, $\psi_0$, $\psi_1$, $w_0$, $w_1$, and $w_2$
to obtain
$$\displaylines{
\epsilon_{11} = z [\phi_{0x} + y \phi_{1x} ], \quad
\epsilon_{12} = {z\over 2} [(\phi_1 + \psi_{0x}) + y \psi_{1x} ], \cr
\epsilon_{13} = {1\over 2}[(\phi_{0} + w_{0x}) +
y (\phi_{1x} + w_{1x}) + y^2 w_{2x}] ,\cr
\epsilon_{22} = z \psi_1\,,\quad
\epsilon_{23} = {1\over 2}[(\psi_0 + w_1) + y (\psi_1+2 w_2)],
}$$
and
$$\displaylines{
\sigma_{11}={{zE}\over{1-\mu^2}}[(\phi_{0x} + \mu\psi_1) + y\phi_{1x}], \cr
\sigma_{12}={{zG}\over 2}[(\phi_1+\psi_{0x})+y\psi_{1x}], \cr
\sigma_{13}={G\over 2}[(\phi_0+w_{0x})+y(\phi_1+w_{1x})+y^2 w_{2x}], \cr
\sigma_{22}={{zE}\over{1-\mu^2}}[(\mu\phi_{0x} + \psi_1) + y\mu\phi_{1x}], \cr
\sigma_{23}={G\over 2}[(\psi_0 + w_1) + y (\psi_1+2 w_2)],
}$$
We next formulate the potential energy due to strain as the quadratic
functional \setcounter{equation}{3}
\begin{eqnarray}
{\cal V}(t) &=& {1\over 2}\int^L_0\int^{k(x)}_{-k(x)}
\int^{h(x,y)}_{-h(x,y)} \big\{{{z^2E}\over{1-\mu^2}}
[(\phi_{0x}+\mu\psi_1)+y\phi_{1x}][\phi_{0x}+y\phi_{1x}] \nonumber\\
&&+{{z^2E}\over{1-\mu^2}}[(\mu\phi_{0x} + \psi_1) + y\mu\phi_{1x}]\psi_1+
z^2{G\over 4}[(\phi_1+\psi_{0x})+y\psi_{1x}]^2 \nonumber\\ %(2.4)
&&+{G\over 4}[(\phi_0+w_{0x})+y(\phi_1+w_{1x})+y^2 w_{2x}]^2 \\
&&+{G\over 4}[(\psi_0+w_1)+y(\psi_1+2 w_2)]^2\big\}\, dz\,dy\,dx\,.
\nonumber
\end{eqnarray}
Define the following functions
$$\displaylines{
K_0(x)={{2E}\over {3(1-\mu^2)}}\int^{k(x)}_{-k(x)}h^3(x,y)dy\,,\cr
\hfill K_2(x)={{2E}\over {3(1-\mu^2)}}\int^{k(x)}_{-k(x)}y^2h^3(x,y)dy\,,
\hfill\llap{(2.5)}\cr
\sigma_0(x)={G\over 2}\int^{k(x)}_{-k(x)}h(x,y)dy\,,\quad
\sigma_2(x)={G\over 2}\int^{k(x)}_{-k(x)}y^2h(x,y)dy \,,\cr
\tau_0(x)={G\over 6}\int^{k(x)}_{-k(x)}h^3(x,y)dy\,,\quad
\tau_2(x)={G\over 6}\int^{k(x)}_{-k(x)}y^2 h^3(x,y)dy\,,\cr
\sigma_4(x)={G\over 2}\int^{k(x)}_{-k(x)}y^4 h(x,y)dy\,.
}$$
We make the positivity assumption
\begin{description}
\item{(P)} There is a positive number $\nu_0$ such that the functions
$$K_0(x),\quad K_2(x),\quad \sigma_0(x),\quad \sigma_2(x),\quad
\tau_0(x),\quad \tau_2(x),\quad \sigma_4(x)$$
are bounded below by a positive number $\nu_0$.
\end{description}
\noindent The potential energy is now expressed by
\setcounter{equation}{5}
\begin{eqnarray}
{\cal V}(t)&=&{1\over 2}\int^L_0\big\{K_0[(\phi_{0x}+\mu\psi_1)^2+(1-
\mu^2)\psi^2_1]+K_2\phi^2_{1x}\nonumber\\
&&+\sigma_0[(\psi_0+w_1)^2+(\phi_0+w_{0x})^2] \\
&&+\sigma_2[(\psi_1+2w_2)^2+(\phi_1+w_{1x})^2+2(\phi_0+w_{0x})w_{2x}]
\nonumber\\ %(2.6)
&&+\tau_0(\phi_1+\psi_{0x})^2+\tau_2\psi^2_{1x}+\sigma_4 w^2_{2x}\big\}
\, dx\,. \nonumber
\end{eqnarray}
\begin{remark} \label{Rem2.1} It will be shown that under the positivity
assumption (P), the potential energy functional given in (2.6) is positive
definite. \end{remark}
We next consider the kinetic energy of the system. Assuming a
constant density function $\rho$, the kinetic energy is expressed as an
integral by
\begin{eqnarray*}
{\cal T}(t)&=& {1\over 2}\int^L_0\int^{k(x)}_{-k(x)}
\int^{h(x,y)}_{-h(x,y)}\rho\lbrace U^2_t(x,y,z,t)+V^2_t(x,y,z,t)\\
&&+W^2_t(x,y,z,t)\rbrace\, dz\,dy\,dx\,.
\end{eqnarray*}
Define the functions
$$\displaylines{
I_{\rho,0}(x)={{2\rho}\over 3}\int^{k(x)}_{-k(x)} h^3(x,y)dy\,,\quad
I_{\rho,2}(x)={{2\rho}\over 3}\int^{k(x)}_{-k(x)} y^2h^3(x,y)dy\,,\cr
\hfill \rho_{0}(x)=2\rho\int^{k(x)}_{-k(x)} h(x,y)dy\,,\quad
\rho_{2}(x)=2\rho\int^{k(x)}_{-k(x)} y^2 h(x,y)dy\,,
\hfill\llap{(2.7)}\cr
\rho_{4}(x)=2\rho\int^{k(x)}_{-k(x)} y^4 h(x,y)dy\,.
}$$
From these assignments and from (2.3), we find, after performing integrations
with respect to $z$ and $y$, that \setcounter{equation}{7}
\begin{eqnarray}
{\cal T}(t)&=&{1\over 2}\int^L_0{1\over 6}\lbrace
I_{\rho,0}\phi^2_{0t}+I_{\rho,2} \phi^2_{1t}+
I_{\rho,0}\psi^2_{0t}+I_{\rho,2}\psi^2_{1t} \nonumber\\
&&+\rho_0 w^2_{0t}+\rho_2 w^2_{1t}+2\rho_2 w_{0t}w_{2t}+\rho_4 w^2_{2t}
\rbrace\, dx %(2.8)
\end{eqnarray}
Let us suppose that a body force, $F,$ is exerted normal to the $x$-$y$ plane.
The work due to this force is
$${\cal W}(t)= \int_0^L \int_{-k(x)}^{k(x)} \int_{-h(x,y)}^{h(x,y)}
F(x,y,z,t) W(x,y,z,t)\, dz\, dy\, dx\,.$$
Define the functions
$$\displaylines{
F_0(x,t)=\int^{k(x)}_{-k(x)} \int^{h(x,y)}_{-h(x,y)} F(x,y,z,t)\,
dz\, dy\,, \cr
\hfill F_1(x,t)=\int^{k(x)}_{-k(x)} \int^{h(x,y)}_{-h(x,y)} y F(x,y,z,t)
\, dz\, dy\,,\hfill\llap{(2.9)} \cr
F_2(x,t)=\int^{k(x)}_{-k(x)} \int^{h(x,y)}_{-h(x,y)} y^2 F(x,y,z,t)\,
dz\, dy\,.
}$$
From (2.3) the work may be expressed upon integration as
$${\cal W}(t)=\int_0^L\lbrace F_0 w_0+F_1 w_1+F_2 w_2\rbrace dx\,.
\eqno(2.10)$$
The Lagrangian is given by
$${\cal L}(t) = {\cal T}(t) -{\cal V}(t)+ {\cal W}(t).$$
Hamilton's principle indicates that the deformation experienced by the
body is obtained as an extremal of the integral
$\int^t_0 {\cal L}(s) ds$, cf \cite{m1}.
That is, the deformation assumed by the body has the property that
the variation of the Lagrangian functional is zero:
$$\delta\int^t_0 {\cal L}(s) ds=0$$
with respect to functions $\delta\phi_0$, $\delta\phi_1$, $\delta\psi_0$, $\delta\psi$,
$\delta w_0$, $\delta w_1$, and $\delta w_2$ satisfying the
essential boundary conditions and equal to zero at times $0$ and $t$. Upon
integration by parts with respect to time and the spatial variable, we
obtain the following equations of motion.
$$\displaylines{
I_{\rho,0}\phi_{0tt}-(K_0(\phi_{0x}+\mu\psi_1))_x+\sigma_0(\phi_0+w_{0x})
+\sigma_2 w_{2x}=0\cr
I_{\rho,2}\psi_{1tt}-(\tau_2\psi_{1x})_x+\mu K_0(\phi_{0x}+\psi)+\sigma_2
(\psi_1+w_2)=0\cr
\rho_0 w_{0tt}+\rho_2w_{2tt}-(\sigma_0(\phi_0+w_{0x}))_x
-(\sigma_2w_{2x})_x=F_0\cr
\hfill \rho_2w_{0tt}+\rho_4w_{2tt}-(\sigma_4 w_{2x})_x-2(\sigma_2(\phi_0
+w_{0x}))_x=F_2\hfill\llap{(2.11)}\cr
I_{\rho,2}\phi_{1tt}-(K_2\phi_{1x})_x+\sigma_2(\phi_1+w_{1x})
+\tau_0(\phi_1+\psi_{0x})=0\cr
I_{\rho,0}\psi_{0tt}-(\tau_0(\phi_1+\psi_{0x}))_x+\sigma_0(\psi_0+w_1)=0\cr
\rho_2w_{1tt}-((\sigma_2(\phi_1+w_{1x}))_x+\sigma_0(\psi_0+w_1)=F_1
}$$
where we have written the equations in an order which emphasizes
the couplings between them. \smallskip
The boundary conditions at $0$ and $L$ are associated with conditions
$$\displaylines{
(\phi_{0x}+\mu\psi_1)\delta\phi_0\vert^L_0=0\,,\quad
\psi_{1x}\delta\psi_1\vert^L_0=0\,,\cr
(\sigma_0(\phi_0+w_{0x})+\sigma_2w_{2x})\delta w_0\vert^L_0=0\,,\cr
(2 \sigma_2(\phi_0+w_{0x})+\nu_0 w_{2x})\delta w_2\vert^L_0=0\,,\cr
\phi_{1x}\delta\phi_1\vert^L_0=0\,,\quad
(\phi_1+\psi_{0x})\delta\psi_0\vert^L_0=0\,,\cr
(\phi_1+w_{1x})\delta w_1\vert^L_0=0\,.
}$$
For example, if the narrow plate is clamped at $x=0$ and free at $x=L$,
we see that
$$\phi_0(0)=\phi_1(0)=\psi_0(0)=\psi_1(0)=w_0(0)=w_1(0)=w_2(0)=0$$
and
$$\displaylines{
(\phi_{0x}+\mu\psi_1)(L)=0\,,\quad \phi_{1x}(L)=0\,,\cr
(\phi_1+\psi_{0x})(L)=0\,,\quad \psi_{1x}(L)=0\,,\cr
(\sigma_0(\phi_0+w_{0x})+{1\over 2}\sigma_2w_{2x})(L)=0\cr
(\phi_1+w_{1x})(L)=0\,,\cr
(2\sigma_2(\phi_0+w_{0x})+\nu_0w_{2x})(L)=0\,.
}$$
It is convenient to rewrite the equations using vector notation.
Towards this end, we introduce the column vector-valued function
$$v={\rm col}(\phi_0,\psi_1,w_0,w_2,\phi_1,\psi_0,w_1)$$
and the matrices
$${\cal E}_0=\left[\matrix{
0 & \mu & 0 & 0 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & 0 & 0 & 0\cr
1 & 0 & 0 & 0 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & 1 & 0 & 0\cr
0 & 0 & 0 & 0 & 1 & 0 & 0\cr}\right]\,,
\quad {\cal E}_1=\left[\matrix{
0 & 1 & 0 & 0 & 0 & 0 & 0\cr
0 & 1 & 0 & 2 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & 0 & 1 & 1\cr}\right]\,,$$
$$A=\left[\matrix{
K_0 & 0 & 0 & 0 & 0 & 0 & 0\cr
0 & \tau_2 & 0 & 0 & 0 & 0 & 0\cr
0 & 0 & \sigma_0 & \sigma_2 & 0 & 0 & 0\cr
0 & 0 & \sigma_2 & \sigma_4 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & K_2 & 0 & 0\cr
0 & 0 & 0 & 0 & 0 & \tau_0 & 0\cr
0 & 0 & 0 & 0 & 0 & 0 & \sigma_2\cr}\right]\,,
\quad C=\left[\matrix{
(1-\mu^2)K_0 & 0 & 0\cr
0 & \sigma_2 & 0\cr
0 & 0 & \sigma_0\cr}\right]\,.$$
Under the above assumptions we see that the matrix functions
$A(x)$ and $C(x)$ are positive definite for each $x\ {\rm in}\ [0,L]$.
\begin{lemma} \label{Lemma2.2} There exists a positive constant $\kappa_0$ such
that for any vector ${\bf u} \in {\mathbb R}^7$,
$${\bf u}^*A(x) {\bf u} \ge \kappa_0 {\bf u}^* {\bf u}$$
and
$${\bf u}^*C(x) {\bf u} \ge \kappa_0 {\bf u}^* {\bf u}\,.$$
\end{lemma}
\paragraph{Proof.} Clearly, by choosing $\kappa_0$ sufficiently small, but positive, the
condition holds for $C(x).$ From assumption (P), we have
$${\bf u}^*A(x) {\bf u} \ge \nu_0 \lbrace u^2_1+u^2_2+u^2_4+u^2_5+u^2_6
+u^2_7\rbrace
+\lbrack u_3,u_4\rbrack \left[\matrix{
\sigma_0 & \sigma_2\cr
\sigma_2 & \sigma_4\cr}\right]
\left[\matrix{u_3\cr
u_4\cr}\right]$$
But
$$\lbrack u_3,u_4\rbrack \left[\matrix{
\sigma_0 & \sigma_2\cr
\sigma_2 & \sigma_4\cr}\right]
\left[\matrix{u_3\cr
u_4\cr}\right] \ge \lambda_0(x)
(u^2_3+u^2_4)$$
where
$$\lambda_0(x)={1\over 2}\lbrace \sigma_0(x)+\sigma_4(x)-
\lbrack(\sigma_0(x)+\sigma_4(x))^2-4(\sigma_0(x)\sigma_4(x)-
\sigma^2_2(x))\rbrack^{1\over 2}\rbrace.$$
From assumption (P) we find that the function
$$x\mapsto \lambda_0(x)$$
is piecewise continuous on $[0,L]$ and there is a positive number $\nu_1$
such that $\lambda_0(x) \ge \nu_1$
for any $x\in [0,L]$ if
$\sigma_0(x)\sigma_4(x)-\sigma^2_2(x) > 0$
for all $x\in [0,L]$.
Using the definitions from the equations (2.5), we find that for each
$x\in[0,L]$
$$\sigma_0(x) \sigma_4(x) -\sigma^2_2(x) = \int^{k(x)}_{-k(x)} h(x,y) dy
\int^{k(x)}_{-k(x)}y^4 h(x,y) dy-
(\int^{k(x)}_{-k(x)}y^2 h(x,y) dy)^2.$$
Thus, $\sigma_0(x) \sigma_4(x)-\sigma^2_2(x)$ is positive on $[0,L]$
by the Cauchy-Schwarz inequality. It follows that $\lambda_0(x)$ is
real-valued and bounded away from zero. Selecting $\kappa_0$ to be less than
$\nu_0$ or $\nu_1$ now yields the result.
With the above definitions, we may express the
strain potential energy functional (2.6) as
$${\cal V}(t)={1\over 2}\int^L_0\{(v_x+{\cal E}_0 v)^*A(v_x+{\cal E}_0 v)+v^*({\cal
E}^*_1 C{\cal E}_1)v\}dx\,.\eqno(2.12)$$
The form of the kinetic energy is obtained by introducing the matrix
$$M=\left[\matrix{
I_{\rho,0} & 0 & 0 & 0 & 0 & 0 & 0\cr
0 & I_{\rho,2} & 0 & 0 & 0 & 0 & 0\cr
0 & 0 & \rho_0 & \rho_2 & 0 & 0 & 0\cr
0 & 0 & \rho_2 & \rho_4 & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & I_{\rho,2} & 0 & 0\cr
0 & 0 & 0 & 0 & 0 & I_{\rho,0} & 0\cr
0 & 0 & 0 & 0 & 0 & 0 & \rho_2\cr}\right].$$
The kinetic energy functional is now expressed as
$${\cal T}(t)={1\over 2}\int^L_0 {v_t}^* Mv_t dx.\eqno(2.13)$$
Finally, defining the column vector
$$F={\rm col}(0,0,F_0,F_2,0,0,F_1),$$
the linear functional expressing the work done by external forces is
$${\cal W}(t)=\int^L_0 F^* v dx.$$
The system of partial differential equations becomes
$$Mv_{tt}-(A(v_x+{\cal E}_0v))_x+{\cal E}^*_0 A(v_x+{\cal E}_0v)+{\cal
E}^*_1 C {\cal E}_1v=F$$
with cantilevered boundary conditions
$$v(0)=0\quad {\rm and} \quad (v_x+{\cal E}_0 v)(L)=0\,.$$
Of course, initial conditions $v(0)=v_0$ and $v_t(0)=v_1$
must be specified as well. \smallskip
We next discuss the weak formulation for our problems.
Towards this end, we designate the Sobolev space
$${\bf V}=H^1(0,L;{\mathbb R}^7)
= \lbrace (v_1,v_2,v_3,v_4,v_5,v_6,v_7): v_i\in H^1(0,L)\ i=1,...,7\rbrace$$
with norm
$$\Vert v \Vert_{\bf V}\ =\ (\sum^7_{i=1} \Vert v_i \Vert^2_{H^1(0,L)})
^{1/2}$$
and the Hilbert space ${\bf H}=L^2(0,L;{\mathbb R}^7)$.
We now define the bilinear form $a(\cdot,\cdot)$ on ${\bf V}$ by
$$a(u,v)\ =\ \int^L_0 \lbrace (u_x+{\cal E}_0 u)^* A (v_x+{\cal E}_0
v)+u^*{\cal E}_1^* C {\cal E}_1 v\rbrace dx.\eqno(2.14)$$
Note from Lemma 2.2 that for any $u\in {\bf V}$
$$a(u,u)\ge \kappa_0 \int^L_0\lbrace \vert u_x +{\cal E}_0 u\vert^2+\vert
{\cal E}_1 u\vert^2\rbrace dx\,.$$
\begin{remark} \label{Rem2.3} From the positive definiteness of $a(u,u)$ and from the
Cauchy-Schwarz inequality, it follows that there are positive numbers
$\gamma_0$ and $\gamma_1$ such that
$$a(u,u)+\gamma_0\Vert u \Vert^2_{\bf H} \ge
\gamma_1\Vert u \Vert^2_{\bf V}.\eqno(2.15)$$
By the Cauchy Schwarz inequality it follows that there is a
positive constant $\gamma_2$ such that
$$\vert a(u,v)\vert \le \gamma_2
\Vert u \Vert_{\bf V} \Vert v \Vert_{\bf V}.\eqno(2.16)$$
\end{remark}
\begin{proposition} \label{Prop2.4} Suppose that ${\bf V}_0$ is a closed
subspace of ${\bf V}$ with the property
$$ {\rm if}\ u\in {\bf V}_0\ {\rm is\ such\ that}\ a(u,u)=0,\
{\rm then}\ u=0,\eqno(2.17)$$
then there exists a positive number $\gamma$
such that for any $u\in {\bf V}_0$
$$a(u,u)\ge \gamma \Vert u \Vert^2_{\bf V}.$$
\end{proposition}
\paragraph{Proof.} We show there is a positive constant $\alpha'$ such that
$$a(u,u)\ge \alpha' \Vert u\Vert^2_{\bf H}$$
for any $u\in {\bf V}_0.$
If this were not the case, then for each n
there exists $u_n$ with $\Vert u_n\Vert_{\bf H}=1$ and such that
$$0 \le a(u_n,u_n) \le 1/n.$$
It follows from (2.15) that
$$1/n + \gamma_0 \ge \gamma_1 \Vert u_n\Vert^2_{\bf V}.$$
Thus, there is a subsequence again denoted by
$\lbrace u_n\rbrace_{n=1}^{\infty}$
such that
$$ \displaylines{
u_n \rightarrow u\quad {\rm weakly\ in}\ {\bf V}\,,\cr
u_n \rightarrow u\quad {\rm strongly\ in}\ {\bf H}
}$$
since ${\bf V}$ embeds compactly in ${\bf H}$.
The limit satisfies $\Vert u\Vert_{\bf H} = 1$. The
continuity condition (2.16) implies the weak lower
semicontinuity of the bilinear form $a(\cdot,\cdot)$ on ${\bf V}$,
i. e.,
$$\liminf a(u_n,u_n) \ge a(u,u).$$
Moreover, since ${\bf V}_0$ is a closed subspace of ${\bf V},$ it follows
that $u\in {\bf V}_0$.
Thus, we conclude that $a(u,u)=0$ and therefore, $u=0,$ contradicting
$\Vert u\Vert_{\bf H}=1.$ The results follow by setting
$\gamma = 1/(1+\mu \alpha')$,
\begin{remark} \label{Rem2.5}
The previous proposition applies to those subspaces of ${\bf V}$ for which
$$u_x+{\cal E}_0 u = 0\eqno(2.18)(i)$$
and
$${\cal E}_1 u = 0 \eqno(2.18)(ii)$$
imply that $u=0$.
\end{remark}
Note that the two conditions (2.18)(i) and (2.18)(ii) imply that
$$
\psi_1= w_2 =\phi_1= \phi_{0x}=\psi_{0x}= w_{1x} =
w_{0x}+\phi_0=0\,.\eqno(2.19)$$
Suppose ${\bf V}_0$ is specified as above and that
${\bf V}_0$ is dense in ${\bf H}$.
Designating ${\bf H}$ as the pivot space, let ${\bf V}^\prime_0$ be the
dual of ${\bf V}_0.$
Let $F$ be in ${\bf V}^\prime_0.$ The weak formulation of
the static problem is then given as follows:
Find $ u\in {\bf V}_0$ such that
$$ a(u,v)=(F,v)\ {\rm for\ any}\ v\in {\bf V}_0\eqno(2.20)$$
where $(\cdot,\cdot)$ expresses the duality pairing
between ${\bf V}_0$ and ${\bf V}^{\prime}_0.$
The existence of a unique solution of (2.20) is classical and follows
from Proposition 2.4.
Further, it satisfies the estimate
$$\gamma \Vert u \Vert_{{\bf V}_0} \le \Vert F \Vert_{{\bf V}_0}.$$
Furthermore, results for the hyperbolic system and the
associated eigenvalue problem are classical as well [3,8].
\smallskip
The weak formulation of the dynamic problem is obtained by introducing
the bilinear form
$$m(u,v)=\int^L_0 u^* M v dx$$
on ${\bf H}.$ Observe, that under the assumptions above, there are
positive constants $\beta_0\ {\rm and}\ \beta_1$ such that for
any $u$ and $v$ in ${\bf H}$,
$$ m(u,u)\geq \beta_0 \Vert u\Vert^2_{\bf H}$$
and
$$m(u,v)\leq \beta_1 \Vert u\Vert_{\bf H} \Vert v \Vert_{\bf H}.$$
The weak form of the dynamic problem is stated as follows:
Find $u\in L^2(0,T;{\bf V}_0)$ such that for any $v\in {\bf V}_0$
$$m(u_{tt}(t),v) + a(u(t),v)\ =\ (F(t),v)_{\bf H}\eqno(2.21)$$
and
$$(u(0),v)_{\bf H} = (u_0,v)_{\bf H}$$
$$(u_t(0),v)_{\bf H} = (u_1,v)_{\bf H}$$
for any $v\in {\bf V}_0$. \smallskip
The associated eigenvalue problem is posed as follows:
Find those numbers $\lambda^2$ such that there exist
nontrivial solutions of the equation
$$a(u,v)= \lambda^2 m(u,v).\eqno(2.22)$$
\section{Numerical approximation and model validation}
In this section, we give a finite element formulation of the above
problems. The starting point is the weak formulation. Once we have
obtained the system of approximating equations, we present results that
constitute a validation of the model by comparing calculated natural
frequencies for our model to compare with those observed
experimentally in the laboratory.
To obtain the finite dimensional analogue, we specify a set of
linearly independent real valued functions, $\{b_i\}^M_{i=1}$ defined on
the interval $(0,L)$ contained in the Sobolev space, $V$.
We define the $M$-row vector-valued function
$$b(x)=(b_1(x),\ldots,b_M(x))$$
and the
$7\times 7M$ matrix-valued function
$$B(x)=\left[\matrix{
b(x) & 0 & 0 & 0 & 0 & 0 & 0\cr
0 & b(x) & 0 & 0 & 0 & 0 & 0\cr
0 & 0 & b(x) & 0 & 0 & 0 & 0\cr
0 & 0 & 0 & b(x) & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & b(x) & 0 & 0\cr
0 & 0 & 0 & 0 & 0 & b(x) & 0\cr
0 & 0 & 0 & 0 & 0 & 0 & b(x)\cr}\right]\eqno(3.1)$$
where here, $0$ represents a $M-{\rm row\ vector}$ whose components are
all zeros. Further, let
$$c={\rm col}(c_1,\ldots,c_{7M}).$$
For this approximation we look for vector-valued functions expressed as
$$u^M(x)=B(x) c.$$
In equation (2.20) setting $v(x)=B(x) d,$
we obtain using (2.14)
$$c^* \lbrace \int^L_0 \lbrack (B_x +{\cal E}_0 B)^* A(B_x+{\cal E}_0
B)+B^*{\cal E}^*_1 C {\cal E}_1 B\rbrack dx \rbrace d = \lbrace \int^L_0 F^* B
dx\rbrace d.$$
It follows that we seek the solution of the linear system
$$\lbrace \int^L_0 \lbrack (B_x +{\cal E}_0 B)^* A(B_x+{\cal E}_0
B)+B^*{\cal E}^*_1 C {\cal E}_1 B\rbrack dx \rbrace c = \lbrace \int^L_0 F^* B
dx\rbrace .\eqno(3.2)$$
In a similar manner
for the dynamic system (2.21), we set
$u(t)=B c(t)$
to obtain the initial value problem
$$\displaylines{
\lbrace \int^L_0 B^*M Bdx\rbrace c_{tt}+
\lbrace \int^L_0 \lbrack (B_x +{\cal E}_0 B)^* A(B_x+{\cal E}_0
B)+B^*{\cal E}^*_1 C {\cal E}_1 B\rbrack dx \rbrace c \cr
\hfill = \lbrace \int^L_0 F(t)^* B\,dx\rbrace \hfill\llap{(3.3)}
}$$
with initial conditions
$$\lbrace \int^L_0 B^* B dx\rbrace c(0) = \int^L_0 B^* u_0 dx$$
and
$$\lbrace \int^L_0 B^* B dx\rbrace c_t(0) = \int^L_0 B^* u_1 dx.$$
Finally, the generalized eigenvalue problem is
given by
$$ \lbrace \int^L_0 \lbrack (B_x +{\cal E}_0 B)^* A(B_x+{\cal E}_0
B)+B^*{\cal E}^*_1 C {\cal E}_1 B\rbrack dx \rbrace c =
\lambda^2 \lbrace \int^L_0 B^*M Bdx\rbrace c.\eqno(3.4)$$
\begin{remark} \label{Rem3.1}
The error analysis for the above approximations is standard and
discussions may be found for example in \cite{s2}.
\end{remark}
To test the model, we measured the natural
frequencies for an aluminum structure in the shape of a paddle composed
of 2 rectangles with the larger atop the smaller. The dimensions of the
structure are as follows:
\begin{center}
\begin{tabular}{ll}
Total length & 35 in \\
Length of the lower rectangle & 14 in\\
Length of the upper rectangle & 21 in \\
Thickness & 0.125 in\\
Width of the lower rectangle & 8 in \\
Width of the upper rectangle & 22 in \\
\end{tabular}\end{center}
Note that the width $k$ is a piecewise constant function of $x$ given by
$$k(x)=\cases{
4\,, & $x\in (0,14)$\cr
11\,, & $x\in [14,35]\,.$\cr}$$
The observed frequencies are:
$$2.81\quad 9.69\quad 20.94\quad 28.12\quad 44.37\quad
85.31\quad 95.62\quad 99.06\quad 115.94\, .$$
Because the shearing and inplane motion is small compared to the
motion normal to the plane, we set
$$\phi_0=\phi_1=\psi_0=\psi_1=0\,.$$
We may also obtain affine motions with only linear cross-sectional
deformations admissible by neglecting $w_2$ as well. We use a uniform mesh
with 14 subdivisions on which the functions $b_i$ are taken to be
``hat'' functions with boundary condition $b_i(0)=0$ imposed to reflect the
clamped boundary conditions at $x = 0$.
We calculate the following frequencies
$$3.16 \quad 9.42 \quad 19.7 \quad 26.0\quad 45.1\quad 84.7
\quad 95.6\quad 99.1\quad 118.4\,.
$$
By calculating the frequencies in the affine case, we find that
$$3.16\quad 9.42\quad 19.7\quad 26.0\quad 95.6\quad 99.1 $$d
appear to be related to the affine motion of the structure while the
frequencies
$$45.1\quad 84.7\quad 95.62\quad 118.4$$
appear to be associated with bowed motions.
\section{Coupling of Bowed Plates}
In this section, we present models for structures that may be viewed as
coupled narrow plates. Our approach is to designate the plates by
assigning local coordinate systems. In a manner similar to that of
the previous sections, we then determine the potential energy functional
for each of the separate plates. The sum of these
functionals forms the total potential energy of the
deformations of the structure. Coupling and boundary constraints are
imposed to determine the class of admissible deformations.
The resulting constraints are then included in the
functional by means of penalization. Inclusion of the constraints
by penalization in the potential energy functional amounts to inclusion
of certain potential energy functionals with large elastic constants.
We present our approach for several cases. Examples of mode shapes
resulting from our formulation are given.
For convenience, we assume that the plates are of the same length and
constant thickness and width. Thus, in all cases the functions h and k
are considered constants. We first explicitly
give stiffness matrices corresponding to those in Section 3.
The material functions given in the equations (2.5) are the constants
$$\displaylines{
K_0={{4h^3kE}\over{3(1-\mu^2)}}\,,\quad
K_2={{4h^3k^3E}\over{9(1-\mu^2)}}\,,\cr
\sigma_0=hkG\,,\quad \sigma_2={{hk^3G}\over 3}\,,\quad
\sigma_4={{hk^5G}\over 5}\,,\cr
\tau_0={{h^3kG}\over 3}\,,\quad \tau_2={{h^3k^3 G}\over 9}\,.
}$$
Let the matrices ${\cal E}_0,\ {\cal E}_1,\ A,\ {\rm and}\ C$
be as defined in Section 2.
The matrix valued function $B$ is defined with a full basis
without regard to boundary conditions so that essential boundary
conditions are not imposed directly on the basis elements.
Hence, dividing the interval $(0,L)$ into N subintervals and using
piecewise linear elements with respect to the resulting mesh yields
vector functions $b(x)$ with $M=N+1$ terms.
The essential boundary conditions are imposed by penalization in
the potential energy functional. With these assignments
$$B_x(x)+{\cal E}_0 B_(x)=\left[\matrix{
b_x(x) & \mu b & 0 & 0 & 0 & 0 & 0\cr
0 & b_x(x) & 0 & 0 & 0 & 0 & 0\cr
b(x) & 0 & b_x(x) & 0 & 0 & 0 & 0\cr
0 & 0 & 0 & b_x(x) & 0 & 0 & 0\cr
0 & 0 & 0 & 0 & b_x(x) & 0 & 0\cr
0 & 0 & 0 & 0 & b(x) & b_x(x) & 0\cr
0 & 0 & 0 & 0 & b(x) & 0 & b_x(x)\cr}\right].$$
Define the $M \times M$ matrices
$$\displaylines{
g_0 = \int^L_0 b^*(x) b(x)\, dx \,,\quad
g_1 = \int^L_0 b_x^*(x) b(x)\, dx\,,\quad
g_2 = \int^L_x b_x^*(x) b_x(x)\,dx\,.
}$$
We now obtain
\begin{eqnarray*}
G_2 &=& \int^L_0 \lbrack(B_x(x) + {\cal E}_0 B(x)*A (B_x(x) +
{\cal E}_0 B(x)) + B(x)^* {\cal E}^*_1 C {\cal E}_1 B(x)\rbrack dx \\
&=&\left[\matrix{G_2(1,1)&0\cr
0&G_2(2,2)\cr}\right]
\end{eqnarray*}
$$G_2(1,1)=\left[\matrix{
K_0 g_2+\sigma_0 g_0 & \mu K_0 g_1& \sigma_0 g_1^* &\sigma_2 g_1^*\cr
\mu K_0 g_1^*& \tau_2 g_2+(\sigma_2+\mu^2 K_0) g_0 &0 & 2\sigma_2g_0\cr
\sigma_0 g_1& 0 & \sigma_0 g_2 & \sigma_2 g_2 \cr
\sigma_2 g_1 & 2\sigma_2g_0 & \sigma_2 g_2 & \sigma_4 g_2 +4\sigma_2g_0\cr}\right]$$
$$G_2(2,2)=\left[\matrix{
K_2 g_2+(\tau_0+\sigma_2)g_0 & \tau_0 g_1^*& \sigma_2g_1^*\cr
\tau_0 g_1& \tau_0 g_2+\sigma_0g_0 & \sigma_0g_0\cr
\sigma_2 g_1 & \sigma_0g_0 & \sigma_2 g_2+\sigma_0g_0\cr}\right]$$
To formulate equations describing the motion of two narrow plates 1 and 2
that are coupled along one side, we view the plates as
being situated in such a way that they lie in the x-y plane. The
junction between the two plates lies along the x-axis. Local coordinate
systems for 1 and 2 are given by $x_1,\ y_1,\ {\rm and}\ z_1$ and
$x_2,\ y_2,\ {\rm and}\ z_2,$ respectively, where
$0