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June 24th, 2017, 07:39 AM  #1 
Newbie Joined: Jun 2017 From: Paris Posts: 1 Thanks: 0  Infinite beam under moving mass
I am studying a paper which is about a point mass $m$ moving at constant velocity $v$ along an infinite beam resting on an elastoviscous support. The motion of the mass consists of a motion with constant velocity $v$ along the $Ox$ axis, and a motion along the $Oy$ axis together with the beam, without separating from it. The behaviour of the system is studied with reference to the $O \xi x$ system of coordinates moving along the $Ox$ axis with velocity $v$, $\xi=xvt$. The equation of the beam flexure and the conditions for matching the solution at the position where the point mass appears, after changing to dimensionless coordinates, have the following form: \begin{equation} \frac{\partial^2 W}{\partial t^2} 2v\frac{\partial^2 W}{\partial t \partial \xi}+ v^2\frac{\partial^2 W}{\partial \xi^2}+\frac{\partial^4 W}{\partial \xi^4}+h\Bigg( \frac{\partial W}{\partial t} \frac{\partial W}{\partial \xi} \Bigg)+\frac{1}{4}W=0 \end{equation} \begin{equation} W^+(0,t)=W^(0,t) \end{equation} \begin{equation} \frac{\partial W^+(0,t)}{\partial \xi}=\frac{\partial W^(0,t)}{\partial \xi} \end{equation} \begin{equation} \frac{\partial^2 W^+(0,t)}{\partial \xi^2}=\frac{\partial^2 W^(0,t)}{\partial \xi^2} \end{equation} \begin{equation}\label{(3)} \frac{\partial^3 W^(0,t)}{\partial \xi^3}\frac{\partial^3 W^+(0,t)}{\partial \xi^3}=P+M\frac{\partial^2 W(0,t)}{\partial t^2}. \end{equation} In addition, the solution of the problem must satisfy the condition of boundedness when $x \in (\infty,+\infty)$. In the coordinate system attached to the beam, the equations of motions and the matching conditions are: \begin{equation} \frac{\partial^2 W}{\partial t^2} +\frac{\partial^4 W}{\partial x^4}+h \frac{\partial W}{\partial t} +\frac{1}{4}W=0 \end{equation} \begin{equation} W^+(vt,t)=W^(vt,t) \end{equation} \begin{equation} \frac{\partial W^+(vt,t)}{\partial x}=\frac{\partial W^(vt,t)}{\partial x} \end{equation} \begin{equation} \frac{\partial^2 W^+(vt,t)}{\partial x^2}=\frac{\partial^2 W^(vt,t)}{\partial x^2} \end{equation} \begin{equation} \frac{\partial^3 W^(vt,t)}{\partial x^3}\frac{\partial^3 W^+(vt,t)}{\partial x^3}=M\Bigg( \frac{\partial^2 W(vt,t)}{\partial t^2} +2v\frac{\partial^2 W(vt,t)}{\partial x \partial t}+v^2\frac{\partial^2 W(vt,t)}{\partial x^2}\Bigg). \end{equation} I don't know how to obtain the condition on the third derivative. I tried to write the Lagrangian and use Hamilton's principle, but I got just the equation of motion and the conditions on the first and second derivative. Last edited by skipjack; June 24th, 2017 at 02:25 PM. 

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beam, infinite, mass, moving 
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