So i have this problem. centrifugal governor has three different factors: x1,x2,x3 - they are nondimensional angle, angular rate, and output shaft rotational speed respectively.

I have the following three equations:

dx1/dt = X2

dx2/dt = X3^2(sinX1cosX1) - sinX1 - yX2

dx3/dt= k(cosX1-p)
k,p are costants

I’m trying to put these equations into Jacobian matrix in order to get a transfer function for the system. Does anyone know how to do that?

I understand the Jacobian matrix. What throws me off is that the variables are the derivatives, such as

x1=x2
x2’=x1’’=X3^2(sinX1cosX1) - sinX1 - yX2

( ’ stands for prime)

so all of these are interdependent?
would it be an accurate assumption to treat all of these as functions and perform partial differentiations with respect to (x1,x2,x3) for these particular:

dx1/dt = X2

dx2/dt = X3^2(sinX1cosX1) - sinX1 - yX2

dx3/dt= k(cosX1-p)

(nondimensional angle, angular rate, and output shaft rotational speed respectively)
in order to get a jacobian matrix?

so that J =
( 0 | 1 | 0; X3^2 (cos2X1)-cosX1 | y | X3sin(2X1); -ksinx1 | 0 | 0)
ps did the trig identity for sinxcosx = (sin2x)/2