I have a nonlinear ODE (second order) and exponential force in right side of equation ( q(t)=p0(1-t/t0)EXP[(-r*t)/t0] ) . p0, t0 and r are constant. I see in Nayfeh and Mook book only harmonic force. but i have a exponential force. How I can use from the Method of multiple scale to solve this problem?
A very rough idea: rather than solving in the time domain, you could try to solve in the frequency domain. Although the signal is not steady-state, one could use similar concepts as the Fourier transform. Perhaps the wavelet transform can help you. Before that, you should try to linearise your problems around certain equilibrium points. Not easy, but perhaps Newton-Raphson makes the job in a 2nd order ODE. By the way, it is one equation or a system of equations?
This looks like an impact on an object or the pressure from an explosion. The response function in the frequency domain is solveable analytically.
thank you Mr Maurizio Sciancalepore for your response
I have 3 PDE coupled equations for a plate. at the first, by use of some method, I changed 3 equations to 1 and then, by use of Galerkin method, I have 1 ODE equation that is a function of T. To solve free vibration, By use of the MMS method, I don't have any problem, but in force vibration, because of exponential force in the right hands of the equation, I don't know how I can solve it. this force is transient and after a little time, it's finished.
thank you Mr Anders Buen for your response.
yes, this is a sudden pressure wave that changes exponentially.
if I want to explain completely, I have 3 PDE-coupled equations for a plate. at the first, by use of some method, I changed 3 equations to 1, and then, by use of Galerkin method, I have 1 ODE equation that is a function of T. after use of the Galerkin method, the equation is in the frequency domain, But I do not know how to solve it when I have an exponential force.
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