johanby

Hmmm, I wonder if it is applicable to treat the valve as a pure double differentiator? My argument is that the valve has so small interval where it is linear due to the construction with a tiny hole that opens and closes. I would rather see the valve like something of bang-bang type which is either open or closed. However, I agree that the piston is acting like an integrator. Note that I call the valve a double differentiator since (as you have explained) P=C1*m/A*d2x/dt2 and opposite (double integrator) with the piston. Moreover, multiplication with 's' usually means differentiation since if L{f(t)}=F(s), then L{f'(t)}=sF(s)-f(0-) (however, we can ignore the initial value), i.e. multiplication of the Laplace transform F(s) with s. In the same way, multiplication with '1/s' usually means integration. The mathematical details are anyway not important, what is more important is if the valve of your CVP is lying so "on the edge" that it is only operating in the linear part of the valve characteristics?

As for the differential equation, I think the thing is that the equation is non-homogeneous due to the harmonic excitation from the dithering process. The homogenous part of the solution will only give transient solutions (the exponential functions mentioned before with negative exponents), but the non-homogeneous part will give a stationary solution similar to the external excitation. That means that if the excitation is harmonic (i.e. trigonometric), the stationary response will also be harmonic. (This is usually known as the method of homogeneous and particular solutions). Anyway, this explains your observations.

Do I understand this correctly that the CVT is used mainly to change the timing of the two driving cylinders and therefore be able to regulate speed and direction? I am certain that you are aware of that this is also possible to achieve with a differential, like in http://www.brickshelf.com/cgi-bin/gallery.cgi?f=45669 or http://lego.roerei.nl/steam-engine/steam-engine.htm Anyway, you are on to something really nice and sophisticated. Congratulations! PS. A second order (homogeneous) differential equation with constant coefficients that contains A) function y and its second derivative y'' will have trigonometric solutions B) derivative y' and second derivative y'' will have a sum of a constant and an exponential function as solution C) function y, derivative y' and second derivative y'' will have either sum of two exponential functions, product of exponential function with trigonometric function or product of exponential function with linear function, depending on the roots of the characteristic polynomial. The way you describe the situation, I would guess that the differential equation is of type B or C, i.e. no pure trigonometric solutions.