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. F is the input force that is held constant.

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)

$$x1'' = -F + F_{AB}\sin(t)$$

$$y1'' = F_{AB}\cos(t) - N_{A} = 0$$

SUM OF FORCES AT B(x2,y2)

$$x2'' = Nb - F_{AB}\sin(t) = 0$$

$$y2'' = -F_{AB}cos(t)$$

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

$$Mb = R_{BA} \times N_{ab} - R_{BA} \times F = I_{A}t''$$

$$Mb = rN_{A}sin(t) - rFcos(t) = I_{A}t'' = m_{1}r^2t''$$

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

$$N_{A}*sin(t) - F*cos(t) = t''$$

The last equation is the relative acceleration for point A

$$R_{A} = R_{B} + R_{BA}$$

$$R_{A}'' = R_{B}'' + t'' \times R_{BA} - w^2R_{BA}$$

$$R_{A}'' = [0\hat i - F_{AB}cos(t)\hat j] + [-t''\hat k \times (-rsin(t)\hat i - rcos(t)\hat j)] - t'^2[-rsin(t)\hat i - rcos(t)\hat j]$$

$$R_{A}'' = [0\hat i - F_{AB}cos(t)\hat j] + [-rt''cos(t)\hat i + rt''sin(t)\hat j] + [rt'^2sin(t)\hat i + rt'^2cos(t)\hat j]$$

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

$$R_{A}'' = [-t''cos(t) + t'^2sin(t)]\hat i + [t'^2cos(t) + t''sin(t) - F_{AB}cos(t)]\hat j$$

Comparing these components to the components for point A

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

$$y1'' = 0 = t'^2cos(t) + t''sin(t) - F_{AB}cos(t)$$

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

$$t'^2 = -t''tan(t) + F_{AB}$$

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

$$x1'' = -t''cos(t) + F_{AB}sin(t) - t''tan(t)sin(t)$$

$$x1'' = F_{AB}sin(t) + t''[-tan(t)sin(t) - cos(t)]$$

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. Edit: a picture of the matlab of coefficients and output vector B were added.