3. Kinematic Analysis

Position Analysis

The first task in my analysis was to determine the position of three points on the dinosaur head:

• P1 = tip of bottom jaw
• P2 = tip of top jaw
• H = hinge of jaw

As described previously, the base of the extender mechanism was modeled by a four-bar vector loop (R2 + R3 - R4 - R1 = 0), and the other linkages could be calculated based on geometric relationships to the vectors in the initial vector loop. The fourbarpos.m MATLAB function was used to determine the angles of each of the vectors (Θ_A, Θ_B, Θ_C, Θ_D). From here, the appropriate vectors were split into their real and imaginary components,

ae^jΘ_A + be^jΘ_B - ce^jΘ_C - de^jΘ_D = 0

a(cosΘ_A + jsinΘ_A) + b(cosΘ_B + jsinΘ_B) - c(cosΘ_C + jsinΘ_C) - d(cosΘ_D + jsinΘ_D) = 0

then summed to determine the X and Y coordinate for points P1, P2, and P3. The results of the position analysis are shown in Figures C and D.

Figure C. Position plot of dinosaur head based on the crank angle

Figure D. Animation of dinosaur head movement

The crank angle starts at 90 degrees and completes a full rotation in the figures shown above. Figure C shows the path that points P1, P2, and H took as the input crank rotated a full 360 degrees. All three points complete a closed path which means that one rotation of the input crank results in one full cycle of the dinosaur head. In addition, from Figure D we see that points P1 and P2, which represent the tips of the dinosaur jaw, start by growing closer together, then drift further apart, then come back together at the end of the loop. The movement seen here results from the angular relationship of the linkages that are connected to the base of the mechanism. When the crank is pointed opposite from the other ground pin, the dinosaur neck is fully contracted and open. In contrast, when the crank is pointed directly towards the other ground pin, the dinosaur neck is fully extended and closed. 

Velocity Analysis

Using the values from the position analysis, as well as assuming the crank moves at an angular velocity of 10 rad/sec CCW, the angular velocities of the jaw pieces were found. The following equation was used to calculate the angular velocities: 

ω_P1 = (a/m)*sin(Θ_C - Θ_A)/sin(Θ_B - Θ_C)

ω_P2 = (a/n)*sin(Θ_A - Θ_B)/sin(Θ_C - Θ_B)

Figure E. Plot of angular velocity of P1 and P2 in relation to angle of crank

From Figure E we see that P1 and P2 follow the same general angular velocity pattern. However, P2 seems to lead P1 slightly. As the crank turns, P1 continues to increase its angular velocity in the CW direction. P2 exhibits CCW rotation at first but quickly decelerates to move CW. After the crank reaches an angle of 225 degrees, both cranks accelerate towards CCW rotation. P2 continues accelerating until the end of the cycle, while P1 decelerates around 350 degrees. The dinosaur jaws are closing in the regions where P2 experiences positive angular velocity at the same time P1 experiences CW rotation. Supported by the position analysis in Figure D, the dinosaur begins the cycle by closing its mouth until the crank reaches about 200 degrees. After this point, both P1 and P2 grow apart, until coming back together at the end. 

Force Analysis

Finally, I wanted to determine the "bite force" that the dinosaur is capable of. To calculate the force of the bite, I assumed a 1 pound force was exerted on the crank. Using the following relationship, I determined the force for both the top and bottom pieces of the jaw:

Fout/Fin = (ω_inr_in)/(ω_outr_out)

Further simplified to solve for force at P1 and P2

F_out = (ω_inr_in)/(ω_outr_out)*F_in

Where F_in is assumed to be 1 pound, angular velocities are known, and r_out is the distance from the tip of the jaw to the hinge. The total bite force was determined by summing the force of the top and bottom jaw. Positive force for P1 occurs when angular velocity of P1 is negative, while positive force of P2 occurs when angular velocity of P2 is positive. Figure F shows the force analysis. 

Figure F. (Top) Force of top jaw. (Middle) Force of bottom jaw. (Bottom) Total bite force of dinosaur jaw.

One thing to notice about the force plots shown above are the asymptotes which occur when the angular velocity of the jaw equals zero. At these points, force is calculated as infinity. 

I also wished to calculate the mechanical advantage of the jaws based on the force applied to the crank. Using the force-angular velocity relationship stated above, I calculated the mechanical advantage plotted in Figure G. 

Figure G. Mechanical advantage of dinosaur jaw based on force on input crank

Again, we see very noticeable asymptotes that are the result of angular velocity going to zero.