Animation:
An animation using PMKS was created to show how the pulley and string move as an input angular velocity is applied to the forearm. This simulation shows the user performing bicep curl using our concentric exercise device.
We created a fixed position at the top pin (which resembles the elbow of the user), and a fixed position at the bottom pin (which resembles the pulley with the fan). The horizontal link at the top resembles the users forearm, the horizontal link at the bottom resembles the radius of the pulley, and link combining the two is the chain that the user pulls upward.
PMKS text file:
Kinematic Analysis:
Kinematics analysis was performed on the mechanism to see how the system would react to the user pulling the chain at various speeds and forces. We first created a four bar mechanism using the forearm, chain, the radius of the pulley (treating it as a link), and a virtual link from the fixed elbow to the fixed pulley. Figure 1 displays how the arm, pulley, and chain function as a four bar mechanism. Figure 2 displays the set up for the kinematic analysis.
Figure 1: Diagram of the arm, pulley, and chain as a four bar
Figure 2: Diagram of the arm, pulley, and chain as a four bar mechanism used for kinematic analysis
Parameters that we deemed of interest to this analysis were the angular velocity of the chain as the user pulled the chain at different speeds. The position, velocity, and acceleration analysis for this mechanism is displayed in figures # through #.
Figure 3: Position Analysis of the Four Bar Mechanism
Figure 4: Velocity and Acceleration Analysis of the Four Bar Mechanism
Three different MATLAB codes were created for this analysis: a script to change the parameters of the function, a position function, and a velocity function. The three MATLAB scripts are below:
To see how the angular velocity of the string would react to and increase in the angular velocity of pulling the string, figures # and # display the angular velocity of the string as a function of theta 2 with angular velocities of the forearm at 22 and 30 radians per second, respectively. As you can see, there is a direct relationship because as the angular velocity of the forearm increased, the angular velocity of the string increased as well.
Figure 5: Low input Angular Velocity (22 rad/sec) Figure 6: High input Angular Velocity (30 rad/sec)
Another aspect of the mechanism we were interested in was how the angular acceleration, tangential acceleration, normal acceleration, and the output torque of the first pulley would change compared to the mechanism having a pulley without the fan (thus minimizing the resistance) when varying the force applied to the string. Figure 7 displays the sketch used to set up the analysis. The first step of this analysis was to calculate the moment of inertia of the pulley with the fan, shown in figure 8. The second step was to find the linear and angular velocities, shown in figure 9. The final calculations for the equations used in MATLAB for the angular acceleration, tangential acceleration, normal acceleration, and the output torque are displayed in figure 10.
Figure 7: Sketch of Concentric Exercise Device Design
Figure 8: Finding the Moment of Inertia of the pulley with the fan
Figure 9: Finding the Angular Velocity and Linear Velocity of the Pulley with the Fan
Figure 10: Finding the Angular, Tangential, and Normal Accelerations, and Output Torque of the Pulley with the Fan
After finding all of these equations, a MATLAB code was created to find all of the parameters of interest, displayed below.
The MATLAB plots for the angular acceleration, tangential acceleration, normal acceleration, and the output torque as a function of input force are displayed in figure 11 through 14. The fan causes the pulley to have a greater moment of inertia. This helps create resistance when the user performs a bicep curl. As you can see in figure 11 and 12, there is a direct relationship bewteen the angular and tangential acceleration and input force on the string for both cases. As the force pulled on the string increases, the angular and tangential acceleration increase. Additionally, the angular, tangential, and normal accelerations are greater without than fan versus with the fan. This is due to the greater moment of inertia on the pulley with the fan causing a greater resistance and therefore it accelerates slower. Figure 14 shows that the output toque on the is smaller for the pulley with the fan. This is the results that we wanted to produce a greater amount of resistance for our concentric exercise device.
Figure 11: Angular Acceleration vs Input Force Figure 12: Tangential Acceleration vs Input Force
Figure 13: Angular Acceleration vs Input Force Figure 14: Tangential Acceleration vs Input Force