13.2 Kinematic Analysis for the Egg Cracker
Kinematic Analysis for the slider-crank subsystem and the subsequent four-bar mechanism is required in order to insure proper speed and torque translation between the systems. This analysis section is broken down by subsystem, with analysis of the slider-crank presented first followed by the four-bar mechanism.
Slider-Crank Analysis
The slider-crank subsystem is driven by the DC motor. It is oriented vertically in order to output a vertical sliding motion that is desired for the rack and pinion system that drives the four-bar mechanism. A schematic of the slider-crank is shown below in Figure 1.
Based on the desired rotation for the four-bar mechanism (explained later) and based on the desired bearing ratios and mechanical advantages, link lengths of 1.01 inches and 4 inches were defined for L2 and L3, respectively.
Mobility Calculation
The Gruebler equation for this basic slider-crank mechanism is rather simple and is shown below, resulting in a 1 degree of freedom subsystem:
Gruebler Equation: M = 3(L - 1) - 2J1 - J2
L = 4 links, J1 = 4 full joints, J2 = 0 half joints
M = 3(4 - 1) - 2(4) - 0 = 9 - 8 = 1 Degree of Freedom
Because this system is a slider-crank, the Grashof condition does not apply since Link 4 is in essence infinite. The maximum translation of the slider is 2.02 inches from its maximum height to its minimum height.
Position and Velocity Analysis
The position and velocity of the slider at different input angles is of interest because it directly impacts the position and velocity of the four-bar mechanism. The position curve of the basic slider-crank is presented in Figure 2 and the subsequent velocity curve is presented in Figure 3.
The velocity of the slider is of particular interest because the output linear velocity of the slider-crank is directly converted by the rack and pinion into the input angular velocity of the four-bar mechanism. This calculation is shown below in the four-bar analysis section based on the pitch radius of the gear.
Mechanical Advantage
The mechanical advantage of the slider-crank is of interest because it is needed in order to know the torque requirements of the DC motor. Figure 4 below depicts the mechanical advantage of the slider-crank mechanism. This mechanical advantage was calculated by taking the ratio between input and output linear velocities.
It is important to note that the range of input angles only goes from 0 to 180 degrees for this graph because that corresponds to the desired range of motion of the four-bar mechanism. Once the slider starts to slide back up (when the input angle is larger than 180 degrees), the four-bar will re-trace its path back to its original position. Thus, the mechanical advantage curve shown above is repeated for the range from 180 to 360 degrees.
Also, this mechanical advantage profile is deemed acceptable since the most mechanical advantage is needed at the beginning to initially push the rack down onto the egg in order to crack it. After this initial force on the egg, significantly less force is needed to further translate the rack down and rotate the gears.
Finally, acceleration of the slider-crank mechanism was not analyzed because it is the velocity profile that is most crucial for subsequent analysis of the four-bar mechanism.
Four-Bar Mechanism Analysis
The four-bar subsystem is driven by an input link that is directly attached to the gear being driven by the rack, which is driven by the slider-crank mechanism. The purpose of this four-bar is to push the egg shells out of the holders in order to properly dispose of them. For this reason, the position profile of the point of interest P was the most important part of this analysis. A kinematic diagram of the four-bar mechanism overlayed on the Solidworks model is shown below in Figure 5.
As alluded to earlier when discussing link lengths for the slider-crank, the desired range of motion for this mechanism is only about 100 degrees. This rotation range is achieved through the use of the rack and pinion. Based on the specified gear pitch diameter and the fact that a 100 degree turn of the gear was desired, a linear translation of the rack of about 2.02 inches was calculated. Hence, Link 2 for the slider-crank mechanism is 1.01 inches long.
Since L3 is the shortest link, the configuration of the four-bar is determined by if L2 and L4 cross each other when L2 is in quadrant 1. With this in mind, it is clear to see that this mechanism is in the crossed configuration. Note that Figure 5 is the backside of the mechanism to better visualize all four links. The link lengths are laid out below:
L1 = 1.50 inches
L2 = 1.94 inches
L3 = 1.30 inches
L4 = 2.43 inches
Based on these link lengths, mobility calculations can be performed and are shown in the next section.
Mobility Calculations
Beginning with the Gruebler equation for degrees of freedom, it is clear that this four-bar mechanism has one degree of freedom, as evidenced by the work shown here:
Gruebler Equation: M = 3(L - 1) - 2J1 - J2
L = 4 links, J1 = 4 full joints, J2 = 0 half joints
M = 3(4 - 1) - 2(4) - 0 = 9 - 8 = 1 Degree of Freedom
Next, the Grashof condition may be used to determine which class mechanism this particular four-bar is. This calculation is presented below:
Grashof Condition: S + L < P + Q
S = shortest link = 1.30 inches
L = longest link = 2.43 inches
P and Q are the other links lengths: 1.50 inches and 1.94 inches.
Thus: 1.30 + 2.43 < 1.50 + 1.94 → 3.73 < 3.44
This four-bar mechanism does not meet the Grashof condition and is thus a Class II Kinematic Chain. In the context of this project, the mechanism being non-Grashof is not an issue since only about a 100 degree range of motion is desired.
Animation
An animation of the four-bar mechanism is shown below in Figure 6. This animation was produced in MATLAB.
The point of interest Point P is the magenta point in the animation. The oscillating motion shown is the motion of the four-bar mechanism for one full rotation of the motor that is connected to the slider-crank (one full oscillation for one full rotation of the motor).
Position, Velocity, Acceleration Analysis
The position, velocity, and acceleration profiles of Point P of the four-bar mechanism are presented below in Figures 7, 8 and 9. Point P is the point of interest because this is the point on the four-bar in charge of discarding the empty egg shells.
The theory behind the position profile of point P was that a path was needed that would minimize interference with the gear and egg holder while in motion but also curve tight enough to push the egg shell out of the holder at the end of its motion. This desired profile is shown below in Figure 7.
In context of the overall system, it is worth mentioning that this curve is traced by point P for the slider-crank input angle range from 0 to 180 degrees. This path is then re-traced back over itself for the range from 180 to 360 degrees. For the four-bar mechanism itself, this path is accomplished with an input angle range of 134 to 231 degrees. Once again, this is made possible by the rack and pinion subsystem that converts the linear motion of the slider into rotational motion of the gear and link 2.
For the velocity magnitude profile, the input angular velocity of link 2 was calculated by taking the linear velocity of the slider/rack and dividing that by the pitch radius of the pinion, in this case 1.125 inches. With this vector of angular velocities as input for the four-bar mechanism, the velocity profile of point P shown in Figure 8 was calculated using the principle of velocity for a point on a link. Similarly, the accelerations were calculated and are presented in Figure 9.
Note that since the input link is rotating clockwise, these graphs should be read from “right to left”, where the initial velocities and accelerations are shown on the right side. The high velocities in the middle portion are needed so that Point P can swing around the top of its path quickly to get in a position to push the egg shell out. As can be seen, the velocity at the end of the path of Point P is low. This is desirable to insure a controlled, repeatable ejection of the egg shell.
Also note that these graphs depict the magnitudes of the velocities and accelerations. It is evident that with a decreasing velocity, acceleration should be in the opposite direction of velocity but for the sake of simplicity, the magnitudes of each were graphed.
Mechanical Advantage
Finally, the mechanical advantage of the four-bar mechanism was calculated, using Point P as the output point. This is shown below in Figure 10.
The specific mechanical advantages of interest are on the left side of the graph, because this is where Point P will need to apply a small force to eject the egg shell. Although the mechanical advantage in this range is only slightly greater than 1, not much force will be required to push the shell out given the design of the arm holding the egg. Also, the corresponding mechanical advantages of the slider-crank in this angle range are higher, in the range of 10-15. Since mechanical advantages from subsystems propagate through the system, we anticipate no issues with a reasonably small DC motor being able to supply enough torque to drive the four-bar and point P to eject the egg shell.