My Math Forum  

Go Back   My Math Forum > College Math Forum > Differential Equations

Differential Equations Ordinary and Partial Differential Equations Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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 $v$, $\xi=x-vt$.

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.
geenatrombetta is offline  
 
Reply

  My Math Forum > College Math Forum > Differential Equations

Tags
beam, infinite, mass, moving



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
a problem on a moving searchlight beam; implicit diff. Lambin Calculus 2 March 7th, 2013 03:14 PM
Question in CANTILEVER BEAM r-soy Physics 0 December 29th, 2011 04:07 AM
A double beam oscilloscope r-soy Physics 9 July 7th, 2011 08:50 AM
Solving For Mass, Given mass as constant and 1 unknown DesolateOne Elementary Math 1 December 16th, 2010 08:37 PM
Beam Equation LariRudi Algebra 0 March 18th, 2010 03:09 AM





Copyright © 2017 My Math Forum. All rights reserved.