Hi, I am working on modeling a simple rigid body dynamic system through a certain range of motion. Here is the picture of the system.

I am trying to solve the system for the variables x1'', y1'', x2'',y2'', the acceleration variables for points A and B respectively, the angular acceleration which we will call t'' corresponding to the angle seen in the diagram, and the force in the rod which is given as Fab.

I could have used a typical approach of basing coordinates and equations around the center of mass but I decided I wanted to try it this way. ALSO, the rod connecting the two masses is considered massless. The masses are given value 1, so is r the rod length for mathematical simplification.

Here are the general equations of motion used. SUM OF FORCES AT A(x1,y1)

x1'' = -F + Fab*sin(t)

y1'' = Fab*cos(t) - Na = 0

SUM OF FORCES AT B(x2,y2)

x2'' = Nb - Fab*sin(t) = 0

y2'' = -Fab*cos(t)

So far we have 5 variables and 4 equations, we will proceed to the moment equation about B

Mb = Rab x Nab - Rab x F = I*t''

The "x" here stands for the cross product of the two, simplifying we get

Mb = Narsin(t) - Frcos(t) = It'' = m1r^2*t''

As stated earlier, r and m are equal to 1 for mathematical simplification, so we get

Nasin(t) - Fcos(t) = t''

The last equation is the relative acceleration for point A

Ra = Rb + Rab

Ra'' = Rb'' + t'' x Rab - (w^2)*Rab

Ra'' = [0i - Fab*cos(t)j] + [t''k x (-rsin(t)i - rcos(t)j)] - t'^2[rsin(t)i - rcos(t)j]

Ra'' = [0i - Fab*cos(t)j] + [rt''cos(t)i - rt''sin(t)j] + [rt'^2sin(t)i + rt'^2cos(t)j]

Setting r = 1 as given from before and organizing terms under the correct components we get

Ra'' = [t''cos(t) + t'^2sin(t)]i + [t'^2cos(t) - t''sin(t) - Fab*cos(t)]j

Comparing these components to the components for point A

x1'' = t''cos(t) + t'^2sin(t)

y1'' = 0 = t'^2cos(t) - t''sin(t) - Fab*cos(t)

Solving for t'^2 in the second equation we have

t'^2 = t''tan(t) + Fab

Plugging this into the new equation for x1'' we get

x1'' = t''cos(t) + Fabsin(t) + t''tan(t)sin(t)

x1'' = Fab*sin(t) + t''[tan(t)sin(t) + cos(t)]

At this point, I have enough equations to solve for my 6 variables(x1'',y1'',x2'',y2'',Fab,t''). I put these into a matrix form and got a nonsense answer, I was hoping to get some help. I understand I can formulate the problem in a different way around the center of mass but thats not what I was aiming to do. I want to formulate it with these coordinates. I feel like I got a sign wrong somewhere but I'm stuck trying to find it.