Pointwise Comparison Principle for clamped Timoshenko beam

Grzegorz Bartuzel, Andrzej Fryszkowski

Abstract


We present the properties of three Green functions for:
\roster
\item"1." general complex ``clamped beam''
$$
\gather
D_{\alpha ,\beta }[y] \equiv y^{\prime \prime \prime \prime}
-(\alpha ^{2}+\beta ^{2}) y^{\prime \prime }+\alpha ^{2}\beta^{2}y=f,
\\
y(0) =y(1) =y^{\prime }(0) =y^{\prime}(1) =0.
\tag{BC}
\endgather
$$
\item"2." Timoshenko clamped beam $D_{\alpha ,\overline{\alpha }}[y] \equiv f$
with (BC).
\item"3." Euler-Bernoulli clamped beam $D_{k(1+i) ,k(1-i)} [ y] \equiv f$
with (BC).
\endroster
In case 1. we represent solution via a Green operator expressed in terms of
Kourensky type system of fundamental solutions for homogeneous case. This
condense form is, up-to our knowledge, new even for the Euler-Bernoulli
clamped beam and it allows to recognize the set of $\alpha ^{\prime }s$ for
which the Pointwise Comparison Principle for the Timoshenko beam holds. The
presented approach to positivity of the Green function is much
straightforward then ones known in the literature for the case 3
(see \cite{12}).

Keywords


Timoshenko beam; Euler-Bernoulli clamped beam; Pointwise Comparison Principle; nonnegativity of Green function; fourth order ODE

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