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Using scaled equations to go from $rho u_{tt}(x, t) + Ek^2u_{xxxx}(x, t) = 0$ to $v_{tau tau} + Jv_{zeta zeta zeta zeta} = 0$

Mathematics Asked by The Pointer on December 16, 2020

I am given the partial differential equation $rho u_{tt}(x, t) + Ek^2u_{xxxx}(x, t) = 0$ for modelling a beam/rod, where

$0 < x < l$, where $l$ is the length of the rod,
$u(0, t) = a sin(omega t)$,
$u_x(0, t) = 0$,
$u_{xx}(l, t) = 0$,
$u_{xxx}(l ,t) = 0$.

I am then provided with the scaled equations

$$x = l zeta, tau = omega t, u = av,$$

and told that we therefore have that

$$v_{tau tau} + Jv_{zeta zeta zeta zeta} = 0,$$

where $J = dfrac{E k^2}{rho omega^2 l^4}$, where $E$ is Young’s modulus, $k$ is the radius of gyration, $rho$ is the density, $omega$ is the frequency, and $l$ is the length of the rod.

However, there is no explanation for how the scaled equations are used to go from $rho u_{tt}(x, t) + Ek^2u_{xxxx}(x, t) = 0$ to $v_{tau tau} + Jv_{zeta zeta zeta zeta} = 0$. And this is the first exposure to such material, so there is no basis for expecting someone to know what happened here. I would greatly appreciate it if people would please take the time to explain this.

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