**Nächste Seite:**Quasi-static
Bingham fluid**Aufwärts:**Models
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type equations

##
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:
Here denote
the length of the beam,
the stretching force, , ,
where
is the Young modulus, I is the cross-sectional second moment of area,
the density,
the cross-sectional area and
the elongation of the beam due to extensibility. More recently a weakly
damped vibration model is considered:
Here
is the load. Together with the initial conditions:

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
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).

**Nächste Seite:**Quasi-static
Bingham fluid**Aufwärts:**Models
of Mathematical Physics**Vorherige Seite:**Sobolev-Galpern
type equations
collected and worked out by Wolfgang Sprößig, TU-Bergakademie
Freiberg