[![enter image description here][1]][1]


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 = Na*r*sin(t) - F*r*cos(t) = I*t'' = m1*r^2*t''

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

Na*sin(t) - F*cos(t) = t''

**The last equation is the relative acceleration for point A**

Ra = Rb + Rab

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

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

Ra'' = [0**i** - 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) + Fab*sin(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.**


  [1]: https://i.sstatic.net/eUmUh.png