06.3 Kinematics Analysis

Kinematic and Force Analysis

Figure 1 describes the mechanism, a four-bar slider-crank mechanism designed to drive a cutting blade vertically through cylindrical vegetables, that we propose for the vegetable cutter. The values for the link L2, L3, and eccentricity E are found with an optimization study of the system described in the section “Optimization of the link lengths”. To scale up for improved performance, these dimensions were doubled to L2 = 60 mm, L3 = 240 mm and e = 40 mm, enough to slice through larger vegetables.

Figure 2 shows the trajectory of the linkages and Figure 3 shows the animation of the linkage.

Figure 1: Initial Drawing of Mechanism and Linkage Setup (angles are not to scale)

Figure 2: Vertical vs horizontal cut movement; vertical displacement dominates, confirming minimal ‘wasted’ lateral motion

Figure 3: Linkage Animation

Optimization of the link lengths

To find the optimal lengths of the links we computed the mean transmission angle in the vegetable (thickness of 30 mm) and its standard deviation, and we look for a combination of e and L3 that aims to:

Maximize the mean of the transmission angle in the vegetable
Minimize the standard deviation of the transmission angle in the in the vegetable

The best configuration requires high value of e, and L3 to maximize the mean transmission angle and to reduce the standard deviation.

Moreover, we computed the speed at the surface of the vegetable, again as a function of e and L3. In this case to maximize the speed we would need shorter links but the difference would be less than 10%.

Note: the length of link 2, L2 is fixed during the optimization process to 30 mm.

Note: these simulations were run with code written in python.

Figure 4: Mean transmission angle; shows mean of 90° (1.5708 radians) at a length close to 120 mm

Figure 5: Standard deviation of transmission angle in repeated cuts; shows consistent kinematic behavior with deviations below 3°

Figure 6: Mean cut surface speed in 30 mm vegetables, demonstrating uniformity of speed during major cutting phases

Figures 7 – 9: Transmission angle, blade velocity, and acceleration plotted against blade/motor separation. These validate efficient force delivery and stable, smooth cutting motion.

Figure 7: Transmission angle plotted as a function of link L1 (i.e. the distance blade/center of motor).

Figure 8: Blade Velocity plotted as a function of link L1 (i.e. the distance blade/center of motor).

Figure 9: Blade Acceleration plotted as a function of link L1 (i.e. the distance blade/center of motor).

Analyzing the transmission angle between L3 and L4, we optimized the linkage so that the transmission angle approaches 90° at both the start and end of the cutting stroke. This maximizes delivered force when initiating and finishing the cut, ensuring penetration through tough outer layers with high efficiency (Figure 6). Throughout the motion, the transmission angle remains greater than 85°, maintaining a favorable force profile.

Velocity and acceleration analyses reveal peak blade velocity at mid-stroke, coinciding with the period of lowest acceleration (minimal wasted effort due to outside movement). Plots of blade velocity and acceleration versus crank/blade separation (Figures 7 and 8) confirm that velocity is highest when force transmission is most efficient.

Final comments

Kinematic mobility analysis gives one degree of freedom controlled by input crank rotation. Python simulations yielded profiles for all major linkage points, indicating peak torque and velocity during the critical cutting phases. Calculated mechanical advantage exceeds 1.6 when the blade contacts the vegetable, providing sufficient force for smooth cutting through typical kitchen vegetables.