- #1

Tomder

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- TL;DR Summary
- I have a non linear system to which I implement a PD controller, but when applying the kinetic-work theorem I can't solve an integral.

The integral is (dx/dt)^2 dt, where x=x(t) so it can't be just x + C.

The non linear system for whom wants to know how did I get to that point is:

d(dx/dt)/dt = sqrt(a^2+b^2)*sin(x+alfa+phi) - Kd*(dx/dt); where alfa = atan(a/b), phi = constant angle, Kd = constant coefficient.

After applying the kinetic work theorem by multiplying both sides by dx/dt I get:

d(dx/dt)/dt *dx/dt = sqrt(a^2+b^2)*sin(x+alfa+phi)*dx/dt - Kd*(dx/dt)*dx/dt ; So, by integration by dt I get to:

1/2*(dx/dt)^2 = - sqrt(a^2+b^2)*cos(x+alfa`phi) - INTEGRAL[(Kd*(dx/dt)^2]dt + C ;

Rearranging terms:

1/2*(dx/dt)^2 + sqrt(a^2+b^2)*cos(x+alfa`phi) = C - INTEGRAL[(Kd*(dx/dt)^2]dt ;

And by this without the Kd term i could get the total energy of the system and the velocity at every point but I don't know how to proceed with the Kd term.

I'm sure maybe some of the theory may be wrong explained so I say sorry in advance.

For further explanation, my system doesn't lose energy when Kd = 0 because the total energy would be constant but with Kd term I assume it is like a frictional component that takes the energy out and the system would slowly stop oscillating in the equilibrium point.

The non linear system for whom wants to know how did I get to that point is:

d(dx/dt)/dt = sqrt(a^2+b^2)*sin(x+alfa+phi) - Kd*(dx/dt); where alfa = atan(a/b), phi = constant angle, Kd = constant coefficient.

After applying the kinetic work theorem by multiplying both sides by dx/dt I get:

d(dx/dt)/dt *dx/dt = sqrt(a^2+b^2)*sin(x+alfa+phi)*dx/dt - Kd*(dx/dt)*dx/dt ; So, by integration by dt I get to:

1/2*(dx/dt)^2 = - sqrt(a^2+b^2)*cos(x+alfa`phi) - INTEGRAL[(Kd*(dx/dt)^2]dt + C ;

Rearranging terms:

1/2*(dx/dt)^2 + sqrt(a^2+b^2)*cos(x+alfa`phi) = C - INTEGRAL[(Kd*(dx/dt)^2]dt ;

And by this without the Kd term i could get the total energy of the system and the velocity at every point but I don't know how to proceed with the Kd term.

I'm sure maybe some of the theory may be wrong explained so I say sorry in advance.

For further explanation, my system doesn't lose energy when Kd = 0 because the total energy would be constant but with Kd term I assume it is like a frictional component that takes the energy out and the system would slowly stop oscillating in the equilibrium point.