I'm researching about a Timoshenko Beam with a Transverse vibration. The displacements are very small.
Equation of moving load is:
Fm=-(N+m*d^2v/dt^2) δ(x-ut)
N=mg+f , mg is load weight and f is a constant force , V is the transverse displacement
When velocity is constant, with the change of a variable the equation can be solved, but when the velocity is variable, how can I solve the equation of beam?
Hmm, not sure I understand your problem. You state an ODE showing Fm=-(N+m*d^2v/dt^2) δ(x-ut) where v = velocity, correct? When v = a constant, no problem you can solve it, true? So your problem comes when v = a non-constant for example v = 3.21 * Sin( pi * 120 * t), right so far? I use the freeware program FortranCalculus, http://fortrancalculus.info/apps/fc-compiler.html , to solve my math problems and would right the math model as ...
v = 3.21 * Sin( pi * 120 * t) ... or whatever the velocity function is.
Fm=-(N+m*d^2v/dt^2) δ(x-ut)
Now integrate the above equations ... I feel like I missed something. Hope these comments help.
The total acceleration of the moving mass is given by
\frac{d^2}{dt^2} = \frac{\partial^2 w}{\partial t^2} + 2 v \frac{\partial^2 w}{\partial x \partial t} + \frac{\partial^2 w}{\partial x^2} v^2 + \frac{\partial w}{\partial x} a
where v denotes the velocity of the moving mass and a the acceleration of the moving mass. For a constant velocity, the last term vanishes.
Sorry, can't understand your typing ... \frac{ ... please state in text only.
e.g. d2v/dt2 = 2v * d2w/dxdt + d2w / dx2 * v^2 + dw / dx * a ???
a = dv/dt ???
If so, that is a partial differential equation (PDE). My compiler, alpha version, only handles ODEs. But, do you know Method of Lines? I don't. If so, one can solve it on my level compiler. Your equation should be simple to solve with the right equation if solved using Method of Lines or PDE solver ... (coming in next release).
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