A slender cantilever beam has the following properties: length $L$, rigidity $EI$,
cross sectional area $A$ and density $\rho$. It undergoes free lateral vibrations over
time $t$. The boundary conditions are as follows:
$y(0) = 0$,
$\frac{dy}{dx}(0) = 0$,
$EI\frac{d^2y}{dx^2}(L) = 0$,
$EI\frac{d^3y}{dx^3}(L) = 0$,
where $x$ is the parameter $0 \le x \le L$ and $y(x)$ is the deflection.
(a) State what each of the boundary conditions means.
(b) Using the relation $EI\frac{d^4y}{dx^4} = \rho A \frac{d^2y}{dt^2}$ show by direct integration, that the
deflection
$y(x) = C_1 \cos \beta x + C_2 \sin \beta x + C_3 \cosh \beta x + C_4 \sinh \beta x$.
And state what $\beta$ is in terms of the beam's properties.
(c) Determine the constants of integration and show that
$\cosh \beta L \cos \beta L = -1.$
