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The Renaudot approach vs. the Yakushev approach

The Renaudot approach

\delta(x-vt)\frac{\mbox{d}}{\mbox{d}t}\left[m\frac{\mbox{d}w(vt,t)}{\mbox{d}t}\right]=\delta(x-vt)m\frac{\mbox{d}^2w(vt,t)}{\mbox{d}t^2}\. $$

The Yakushev approach

\frac{\mbox{d}}{\mbox{d}t}\left[\delta(x-vt)m\frac{\mbox{d}w(vt,t)}{\mbox{d}t}\right]=-\delta^\prime(x-vt)mv\frac{\mbox{d}w(vt,t)}{\mbox{d}t}+\delta(x-vt)m\frac{\mbox{d}^2w(vt,t)}{\mbox{d}t^2}\. $$

Massless string under moving inertial load

Let us consider a massless string, which is a particular case of moving inertial load problem. The first solve the problem Smith \cite{smith64}. The analysis will follow the solution of Fryba \cite{fryba}. Assuming $ρ$=0, the equation of motion of a string under a moving mass can be put into the following form

-N\frac{\partial^2w(x,t)}{\partial x^2}=\delta(x-vt)P-\delta(x-vt)\,m\frac{\mbox{d}^2w(vt,t)}{\mbox{d}t^2}\. $$ We impose simply-supported boundary conditions and zero initial conditions. To solve this equation we use the convolution property. We assume dimensionless displacements of the string $y$ and dimensionless time $τ$ :

y(\tau)=\frac{w(vt,t)}{w_{st}}\ ,\ \ \ \ \tau\ =\ \frac{vt}{l}\ , $$ where $w$st is the static deflection in the middle of the string. The solution is given by a sum

y(\tau)=\frac{4\,\alpha}{\alpha\,-\,1}\,\tau\,(\tau-1)\,\sum_{k=1}^\infty\,\prod_{i=1}^k\frac{(a+i-1)(b+i-1)}{c+i-1}\;\frac{\tau^k}{k!}\ , $$ where $α$ is the dimensionless parameters :

\alpha=\frac{Nl}{2mv^2}\,>\,0\ \ \ \wedge\ \ \ \alpha\,\neq\,1\. $$ Parameters $a$, $b$ and $c$ are given below

a_{1,2}=\frac{3\,\pm\,\sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ b_{1,2}=\frac{3\,\mp\,\sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ c=2\. $$ In the case of $α$=1 the considered problem has a closed solution

y(\tau )=\left[\frac{4}{3}\tau (1-\tau) -\frac{4}{3}\tau \left( 1+2 \tau\ln (1-\tau )+2\ln (1-\tau )\right)\right]\. $$

References:
 * @Article{smith64,

author={C. E. Smith},title={Motion of a stretched string carrying a moving mass particle},journal={J. Appl. Mech.},year={1964},volume={31},number={1},pages={29-37}}