and Results Discussion
This section presents the results obtained through the analysis described above for the geared five-bar linkage. Three different designs of the geared five-bar linkage have been considered. For case one, the radius of gear 2 is 2 inches and the radius of gear 3 is 3 inches. For case two, the radius of gear 2 is 3 inches and the radius of gear 3 is 3 inches. For case three, the radius of gear 2 is 4 inches and the radius of gear 3 is 2 inches. As stated above, the angular velocity of input link 2 is a constant 10 rad/sec. This section will present the results for case two, when the gear ratio is one-to-one. The results for case one is shown in Appendix B and the results for case three can be found in Appendix C. For case two, the input link began at zero degrees and rotated counterclockwise 360 degrees in 5 degree increments. Using the Newton-Raphson method described the position variables for links 3, 4, and 5 were determined. The plot of the position variables of links 4 and 5 is shown in Figure 4a while the plot of the position variables for link 3 is shown in Figure 4b. The numerical values for the position variables of links 3, 4, and 5 are shown in Table 3 found in Appendix A.
Figure 4a: Position Variables Links 4 and 5 against Input position
Figure 4b: Position Variable Link 3 against Input Position
It can be seen in Figure 4a that both links make a complete turn. This conclusion is made because the plot shows that both links return to their initial position. Figure 4a not only gives the position analysis for links 4 and 5 but also confirms that the conditions of the input link are acceptable for this analysis. Equation 9 shows that the singular configurations occur when the angle of link 4 differs the angle of link 5 by 180 degrees or when the two angles are equal. It is clear from Figure 4a that neither of these conditions is satisfied and thus a singular position does not when the radius of gear 2 is 3 inches and the radius of gear 3 is 3 inches. To verify that a singular position does not occur for case two, a plot of the value of the determinate versus the input link is shown in Figure 2. It is clear from Figure 5 that the determinate value at no point obtains a value of zero. It was found that no singular position occurred for any of the three different gear ratios. The closest any of the different gear ratios had to becoming singular was for Case 1 at an input position of approximately 390 degrees.
Figure 5: Determinate versus Input Position
Next, the first-order kinematic coefficients were found using equations (16a) and (16b). The numerical values found for the first-order kinematic coefficients for case two can be found in Table 3 located in Appendix A. Figure 6 below plots the first-order kinematic coefficients versus the input position.
Figure 6: First-Order Kinematic Coefficients Versus Input Position
The second-order kinematic coefficients were found using equations (19a) and (19b). The numerical values obtained are given in Table 3 located in Appendix A. The plot of the secondorder kinematic coefficients against the input angle is shown in Figure 7. In the analysis section, it was determined that the second-order kinematic coefficient for link 3 would be zero. Figure 7 below depicts that this is indeed the case. The first-order kinematic coefficients are used to determine the angular velocity and the angular acceleration. Unlike Figure 6 above, Case 1 shows that the kinematic coefficients for links 4 and 5 are almost the same for most of the cycle.
Figure 7: Second-Order Kinematic Coefficients versus Input
The values obtained for the first- and second-order kinematic coefficients are further validated in Appendix D and Appendix E. Appendix D uses the method of instant centers. Appendix E uses the method of finite differences in order to verify the results obtained. Next, the angular velocity was analyzed. This analysis was done using equations (20a) , (20b) , and (20c) for links 3, 4, and 5. Note that the value for the first-order kinematic coefficient for link 3 is -1. With an input of 10 rad/s, equation (20a) gives a value of -10 rad/s for the angular velocity of link 3. The numerical values obtained for the angular velocity is shown in Table 4 located in Appendix A. Figure 8 shows the plot obtained for the angular velocity of links 3, 4, and 5 versus the input position angle. The angular velocity and angular acceleration seem to mirror the first-and second-order kinematic coefficients for all three cases. This demonstrates the principle that when the output velocity or acceleration is needed to be changed, rather than changing the entire program based off of the desired output using angular velocity and angular acceleration, the first-and secondorder kinematic coefficients can be evaluated. This also demonstrates the principle that the angular velocity and angular acceleration although are time dependent variables, are actually a function of position. This is an important concept that can be carried over for any linkage / mechanism.
Figure 8: Angular Velocity of Links 3, 4, and 5 versus Input Angle
Using equations (21a), (21b) and (21c) the angular acceleration for links 3, 4, and 5 was found. The plot of the angular acceleration for links 3, 4, and 5 against the input position angle is shown in Figure 6. The numerical values obtained for the angular acceleration for links 3, 4, and 5 can be found in Table 4 located in Appendix A.
Figure 9: Angular Acceleration of Links 3, 4, and 5 versus Input Angle
The subsequent material of the Results section will show results obtained for the coupler point P. The first task was to determine the position of the coupler point P. The position of the coupler point P was found using equations (26a) and (26b). The tabulated data for the position of the coupler point P can be found in Table 8 in Appendix A. The first-order kinematic coefficients for the coupler point P were found using equations (27a) (27b). and The second-order kinematic coefficients of the coupler point P were found next. The Velocity of the coupler point P is shown below in Figure 11 and the acceleration of the coupler point P is shown in Figure 12. Both plots are against the input position. It is clear that the velocity of the coupler point P for Case 2 when the two radii are equal takes a much smother path. For both Case 1 and Case 3 the velocity of the coupler point P tended to contain abrupt changes in the velocity. The same is true for the acceleration of the coupler point P.
Figure 11: Velocity of Coupler Point P against Input Position
Figure 12: Acceleration of Point P against Input Position
The position of the coupler point P is shown in Figure 13. The maximum position of the coupler point P in the x-direction is given as 6.9738 inches. This maximum position of the coupler point
P in the x-direction occurs when the input, 2 is at 75 degrees. The minimum position of the coupler point P in the x-direction is -6.8272 inches. This minimum position of the coupler point P in the x-direction occurs when the input, 2 is at 75 degrees 240 degrees. The maxim um value of the position of the coupler point P in the y-direction is 8.588 inches. When the input is at 75 degrees, the maximum value of the position of the coupler point P occurs. This agrees and confirms with the results for the maximum displacement of the coupler point P in the x-direction. The minimum value of the position of the coupler point P in the y-direction is 2.9412. The minimum value of the coupler point P occurs when the input is at an angle of 265 degrees. The maximum displacement of point P, that is, the difference between the maximum value of the position of point P and the minimum value of point P, in the x-direction is 13.8011. The maximum displacement of point P in the y-direction is 5.9176. These values are shown in table 3 below. Table 7: Coupler Point P Displacement Coupler Point P Value (inches) Max Displacement Coupler Point (x 6.97388" direction) Min Displacement Coupler Point (x-6.8272" direction) Max Displacement Coupler Point (y8.588" direction) Min Displacement Coupler Point (y2.9412" direction) Max Displacement (x-direction) 13.8011" Max Displacement (y-direction) 5.9176" 2 75 degrees 240 degrees 75 degrees 265 degrees -
Figure 13: Position of Coupler Point P
Figure 13 depicts that the position of the coupler point P is manageable and useful for Case 2. The path of the coupler point is not as near as unpredictable as it is for Case 1 and 3. The plots of the position of the coupler point P are not smooth as shown in Figure 13. It is clear that having a smooth position of the coupler point is desirable rather than having a plot with abrupt changes in the position. The next analysis that was done was to determine if any values for the initial position of link three that would allow point P to travel on a path that does not cross over itself. This was a tedious process as there is no formula to determine this. The process was to go into Matlab, and for each case, change the initial position as well as the initial positions of the other links. Using the position analysis done, the corresponding positions were able to be inputted into Matlab. The initial position of link 3 was varied in 10 degree increments to analyze the coupler position. Once a position was found, the increments were decreased by 5 degrees to determine the minimum and maximum values for the initial position of link 3. It was determined that point P will always cross over its path for case 1 and 3. However, for case 2, a range of initial positions of link 3 was determined that allowed the path of point P to not cross over itself. Table 7 below shows what was determined as the minimum and maximum allowed values of the initial position of link 3 to permit the path of point P to not cross over itself. The plot of the coupler point when the initial value of the angle for link 3 is -90 degrees is shown in Figure 14. Table 7: Range of Initial Values that Permit Coupler Point to not Cross Over Itself [Degrees]
120 140
[Degrees]
-90.000 -110.000
[Degrees]
1.940 10.070
[Degrees]
102.057 97.763
Figure 14: Initial Value of link 3 (-90 Degrees)
Conclusion
The geared five-bar mechanism has been analyzed for the three different cases that has been described above. During this analysis important insight was gained as to the influence the firstand second-order kinematic coefficients have on the angular velocity and acceleration for all three cases. The analysis also showed how the plot of the coupler point P is affected by different gear ratios. It is assumed that the plot of the coupler path point P depends on the first- and second-order kinematic coefficients for the quality of the curve. This assumption is based on the analysis of the angular velocity and angular acceleration analysis done above. When looking at the graphs of the position for each case, they are much different for Case 2. For Cases 1 and 3, the position
of the links 3, 4, and 5 seems somewhat chaotic. This leads to the first- and second-order kinematic coefficients being in a hectic state along with the angular velocity and acceleration. Then when analyzing the path of the coupler point P, as might be predicted, the paths for Case 1 and Case 3 tend to be confused while in Case 2 the path seems much more "friendly." It has been concluded that a change in the radius of the gears has an impact on not just the position of the other links, but also on the first-and second-order kinematic coefficients, the angular velocity and acceleration and finally the path of the coupler point. Future work that could be attempted is to consider a force analysis of the mechanism. The geared five-bar linkage is an important mechanism to understand as it is useful by having mobility of 1 and having the ability to provide more complex motions than the more common planar four-bar linkage.