# 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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