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Timoshenko's Beam Equations
Timoshenko's theory of beams constitutes an improvement over the Euler-Bernoulli theory, in that it incorporates shear and rotational inertia effects [77]. This is one of the few cases in which a more refined modeling approach allows more tractable numerical simulation; the reason for this is that Timoshenko's theory gives rise to a hyperbolic system, unlike the Euler-Bernoulli system, for which propagation velocity is unbounded. It is this partially parabolic character of the Euler-Bernoulli system which engenders severe restrictions on the maximum allowable time step (at least in the case of explicit methods, of which type are all the scattering-based methods included in this work). For a physical derivation of Timoshenko's system, we refer the reader to [77,146,152,187,188], and simply present it here:
![$\displaystyle \begin{eqnarray}\rho A\frac{\partial^{2}w}{\partial t^{2}} &=& \f...
...\right)+A\kappa G\left(\frac{\partial w}{\partial x}-\psi\right) \end{eqnarray}$](img2363.png) |
(5.17a) |
As before,
represents the transverse displacement of the beam from an equilibrium state, and the new dependent variable
is the angle of deflection of the cross-section of the beam with respect to the vertical direction. Here, the quantities
,
,
and
are as for the Euler-Bernoulli Equation (5.1).
is the shear modulus (usually called
in other contexts) and
is a constant which depends on the geometry of the beam. For generality, we assume that all these material parameters are functions of
. Losses or sources are not modeled.
Nitsche [131], in his MDWD network-based approach preferred to use the more fundamental set of four first order PDEs from which system (5.16) is condensed:
![$\displaystyle \begin{eqnarray}\rho A \frac{\partial v}{\partial t} &=& \frac{\p...
...{\partial q}{\partial t} &=& \frac{\partial v}{\partial x}-\omega\end{eqnarray}$](img2365.png) |
(5.18a) |
|
![$\displaystyle \begin{eqnarray}\rho I \frac{\partial \omega}{\partial t} &=& \fr...
...ac{\partial m}{\partial t} &=& \frac{\partial \omega}{\partial x}\end{eqnarray}$](img2366.png) |
(5.19a) |
|
We have introduced here the quantities
is interpreted as transverse velocity,
as an angular velocity,
as the bending moment, and
as the shear force on the cross-section. Each of the subsystems (5.17) and (5.18) has the form of a lossless (1+1)D transmission line system; they are coupled by constant-proportional terms, and it is this coupling that gives the Timoshenko system its dispersive character. The Euler-Bernoulli system (5.4) is recovered in the limit as
and
[131].
This is a symmetric hyperbolic system of the form given in (3.1), with
,
and
Subsections
Next: Dispersion
Up: Applications in Vibrational Mechanics
Previous: Free End
Stefan Bilbao
2002-01-22