Our analysis was split into two parts, the kinematics of the four-bar and the dynamics introduced by the spring. 

• Kinematics

We began analyzing our four-bar mechanism with a simple vector loop.  We then went ahead and took the first derivative and second derivative for the velocity and acceleration analysis of the mechanism. The following images will show the sequence of steps done to derive our functions to plug in later on in Matlab.  

• Dynamics

Finally for our dynamics, we began with calculating the first link's torque. We amplified our motor's torque by using a 2:1 gear ratio system.  The driven gear had 32 teeth, while the driver had 16 teeth.  Once the Torque was known, we knew that stepping the torque up would step the angular velocity down by the same factor.  We then use this information to calculate the force of the sliding block that is compressing the spring.  We then calculate the spring's force pushing back on the slider block with Hooke's law.  Once those two forces have been found as a function of theta2, we are able to sum the forces to find the resultant force being exerted by the slider block. We had to make sure our force out by the four-bar mechanism was greater than the force of the spring in order to fully compress it.  We are able to finally calculate the shaft's linear velocity by equating the spring's potential energy to its kinetic energy. and solve for velocity.  All of this can be seen in the following image.