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Extensible beam with attached load

In the paper The effect of axial force on the vibration of hinged bars (J.Appl. Mech. 17, 35-36 (1950) ) S. Woinowski-Krieger introduced the following model:
\begin{eqnarray*}u_{tt}+\alpha u_{xxx}-\left(\beta+k\int\limits _0^\ell u_\xi^2(\xi,t)d\xi\right)u_{xx}=0\end{eqnarray*}
Here denote $\ell$ the length of the beam, $k$ the stretching force, $\alpha=E I/\rho$$\beta= H/\rho, k=EA_c/2\rho\ell, H=EA_C\Delta/\ell$, where $E$ is the Young modulus, I is the cross-sectional second moment of area, $\rho$ the density, $A_c$ the cross-sectional area and $\Delta$ the elongation of the beam due to extensibility. More recently a weakly damped vibration model is considered:
\begin{eqnarray*}&&u_{tt}++ 2\gamma u_t+\alpha u_{xxx}-\left(\beta+k\int\limits......nt\limits _0^\ell u_\xi^2(\xi,t)d\xi\right)u_x\right](\ell,t)=0.\end{eqnarray*}
Here $Q=mg$ is the load. Together with the initial conditions:
 
 
 
 
\begin{eqnarray*}u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad u(\ell,0)=\eta, \quad u_t(\ell,0)=\mu\end{eqnarray*}
we get an initial-boundary value problem. The non-linear terms describe the change in the tension of the beam due to its extensibility. The boundary condition $(Qu_{xx}+\rho g u_x)(\ell,t)=0$ can be seen as the restriction on the movement of the beam. Under this conditions this problem was formulated in 1973 by J.M. Ball (Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42, 61-90). Nowadays often formulations as an abstract evolution problem are considered.
 

A late reference: M. Grobbelaar-Van Dalsen, On the Initial-boundary-value Problem for the Extensible Beam with Attached Load, MMAS, 19, 943-957 (1996).


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collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg