Kinematics analysis: Head

To analyze the head I first had to simplify the motion into two dimensions to be compatible with the analyses and methods learned in this class.  Because the model has many linkages and is three-dimensional, I had to make many simplifications meaning that the analysis is a good approximation of the motion, but it is not a completely exact model.  I first decided to model the head similarly to a slider crank mechanism and assume that the head was moving vertically.

Based off of measurements I took of the joints while building, I first calculated the position of the head.

When the input crank is at θ2 = 0, the head moves upwards, hits it’s maximum height, then comes back down and settles back to its original position to be able to pop back up again.  Notably the head moves its full range of motion in one turn of the crank. 

I then took the derivative of the position of the head to find the velocity and said that for the purposes of this calculation that w2 = 10 deg/s and a2 = 0 (the angular speed and acceleration of the input crank, respectively) and then took the derivative of the velocity to get acceleration of the head.  Interestingly, the head moves faster when moving upwards, as shown by the steep slope of the velocity initially and the large acceleration. This acceleration gives the ‘popping’ of the head affect. When the bear’s head returns to its initial position it does so with a slower acceleration.  Furthermore, the acceleration in the first half of the motion is larger and changes faster while the second half when the bears head is going down is slower and the acceleration is also slower.