10.2 - Project Prototype

Kinematic Analysis

Ideal Motion/Force Profiles

The ideal motion profile remains the same as described in the Project Proposal page. We want a zip-zag pattern type motion created by the end effector, which in this mechanism is the whisk. The motion can be seen in the figure below. It should be noted that the end effector motion should fit within the dimensions of the bowl in order to properly mix within the bowl.

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Mobility Calculations

Gruebler’s equation:

For the mobility calculation, the mechanism was modeled as a planar system consisting of the ground link, the input gear, the compound middle gear pair mounted on a common shaft, the output gear, the first crank, the first coupler, the slider block, the second crank, and the second coupler, giving a total of L=9 links. The lower pairs were counted as all revolute and prismatic joints in the mechanism: one revolute joint for the input gear to ground, one revolute joint for the compound middle gear shaft to ground, one revolute joint for the output gear to ground, one revolute joint for the first crank to ground, one revolute joint between the first crank and first coupler, one revolute joint between the first coupler and the slider, one prismatic joint between the slider and ground, one revolute joint for the second crank to ground, one revolute joint between the second crank and second coupler, and one revolute joint between the second coupler and the slider. This gives J1=10 lower pairs. The higher pairs were the two gear meshes: one between the input gear and the middle gear, and one between the second gear on the compound shaft and the output gear, so J2=2.

Using Gruebler’s equation for planar mechanisms,

M = 3(L-1) - 2J₁ - J₂

M = 24 - 20 - 2 = 2

This raw count gives M=2, but one degree of freedom is removed by the gear train because the two crank motions are not independent. The gear train kinematically constrains the angular motion of the second crank to the first through the fixed gear ratio, so only one independent input is required to drive the entire mechanism. Therefore, the actual mobility of the mechanism is M = 1.

Grashof’s equation:

Each side of the mechanism was treated as an offset slider-crank. Using the condition l≥r+e, where r is the crank length, l is the connecting rod length, and e is the offset between the crank center and slider axis, both loops were checked for full crank rotatability. The following numbers were based off of the link lengths generated by MotionGen. For the top loop, r=0.382 in, l=2.962 in, and e=1.697 in, giving r+e=2.079 < l. For the bottom loop, r=0.632 in, l=3.272 in, and e=2.114 in, giving r+e=2.7466 in <l. Therefore, both slider-cranks satisfy the rotation condition and can operate with continuous crank motion.

Position, Velocity, and Acceleration Analysis:

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The mechanism was modeled as two crank-coupler loops connected to the same whisk end effector point, which was approximated as a slider that only moves vertically along a line x. The top loop consists of the upper crank and coupler, and the bottom loop is the other crank and coupler pair, this one connected to the motor. The lower crank angle θ1 was taken as the input and the upper crank angle related to it through the fixed gear ratio, g = 2.9 with the equation θ2 = gθ1. For each input angle, the coordinates of the two crank endpoints were first found using standard trigonometric position relations. Then, using the fact that each coupler has a fixed length and the whisk lies on the vertical slider line, a vertical position for the whisk was found.

Since both loops act on the same output point, the final whisk position was taken as the average of the two vertical position values. After the position was found over the full cycle, velocity and acceleration were obtained numerically by differentiating the position with respect to time. This gave position, velocity, and acceleration plots for the whisk over one full rotation of the input crank. The approach is a simplified kinematic model, but it is useful for showing how the geared dual-crank arrangement affects the motion of the whisk throughout the cycle.

Force & Torque Analysis:

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Figure 3: Force Analysis	Figure 4: Torque Analysis

Force and Torque Analysis

The dynamic behavior of the mechanism was analyzed by calculating the inertial loads generated as the whisk follows its the intended "S-motion" path. Because the whisk has a physical mass, it resists the rapid changes in speed and direction dictated by the dual-crank mechanism that we have. By multiplying the whisk's mass by the acceleration data found in the kinematic study, we determined the total inertial force acting on the linkages. The results show that the highest forces occur during the sharpest turns in the gear cycle, representing the points where the mechanical components and pins are under the most structural stress.

Motor Justification

The selection of a 20 RPM motor is specifically justified by the high torque peaks identified in this analysis. While the average torque is manageable, significant "spikes" occur when the whisk must overcome its own inertia at the extremes of the motion path. A standard high-speed motor would likely stall or skip at these points, but a 20 RPM motor utilizes internal gear reduction to multiply its twisting force, or torque. Because the internal gearbox of the Greartisan 12V Gearbox Motor provides a 250:1 reduction ratio, this converts the 5000 RPM motor's raw speed into the massive mechanical advantage needed to handle the maximum 5 N·m loads given by the torque analysis done. The motor ensures the whisk has more than the necessary torque needed to maintain a steady, powerful motion throughout the non-linear path without the risk of mechanical failure due to friction between gears or linkages or failure due to a motor burnout.

Animation of Linkage:

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The animation of the linkage mechanism illustrates the simultaneous operation of the geared dual-crank system as it drives the whisk through its non-linear "S-motion" path. The simulation visualizes the horizontal movement of the slotted link or frame, driven by the red primary linkage, while the blue secondary linkage simultaneously moves the whisk's vertical position within that sliding track. By tracing the whisk’s end-effector in real-time, the animation confirms that the geometry remains within the intended physical limits that we want, and maintaining constant coupler lengths produces the complex, repeating oscillation of the “S-motion” required for effective matcha whisking.

Gear Analysis:

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We designed a 32/11 (2.909) gear ratio to make sure the whisk moves fast enough to actually froth as the bridge slotted link moves left and right, while the input angular speed given by the motor moves slow enough to cover the whole bowl without the path just repeating over itself in a loop. The initial aim for this design was to have a gear ratio of 3, but due to the sizing and spacing of the gears for the prototype we landed on having a gear ratio of around 2.9 as it was still sufficient to meet our goals. For the final project, we will attempt to reach a gear ratio of 3 with our compound gear train.

Iteration Documentation:

Iteration Step 1: Slider-Crank Mechanism

As stated in the proposal, our end goal is for the matcha whisk to move in a zigzag pattern across the bowl. The first stage of our design process was to determine what type of mechanism we should incorporate to achieve this shape. We brainstormed a couple different ideas including one with a cam follower that is outlined in the proposal. Ultimately, we decided against that idea because of the complexity of the cam follower system. We went with a system of two slider-cranks that work together to produce two-axis motion. Each converts rotary input into linear motion: one controls the whisk’s left-to-right movement, while the other controls its up-and-down motion within the first slider. One crank rotates at three times the angular speed of the other, allowing for the zig-zag path needed for effective whisking.

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An animation was created using MotionGen to visualize the motion of the dual slider-crank system. This simulation shows both cranks operating simultaneously at the 1:3 frequency ratio. It's important to note that we decided to have the vertical slider moving at three times the speed of its horizontal counterpart, whereas the animation shows the opposite. Regardless, the simulation indicates that our system will output the zigzag motion we desire.

Iteration Step 2: Ideal Link Lengths and Gear Size

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The next iteration of our prototype involved deciding on ideal link lengths and how we would power our system. Initially we thought about having two motors to power each of the cranks separately. We eventually decided to make it more complex by having a motor power one crank that was connected to the other by a gear train. This allowed us to power the whole system with only one motor and control the 1:3 angular speed ratio through the size of the gears. We calculated maximums and minimums to determine the link lengths, gear dimensions, and number of teeth we would use in the physical prototype. We then used SolidWorks to CAD the system of slider-cranks, the gear train, and the body on which it would rest, and created an assembly to visualize all of those parts together.

Iteration Step 3: Experimenting with Laser Cut Links and Gears

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Our next step was to laser cut the links and gears to make sure they worked with each other and could press fit onto the shafts and bearings we were provided. This is where we had to take a lot of measurements and reiterate many versions of our links and gears. We were working with ball bearings of many different outer diameters, so we had to account for which bearing we were using for which joint. We also took into account the materials we were working with. It made sense for us to use 3mm plywood for the links, because of the thickness of the bearings, and 6mm plywood for the gears, to maximize the contact between the pinion and the gear.

Iteration Step 4: Physical Prototype

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After we finalized the links and gears we were going to use in the physical prototype, we began to build. We ran into challenges with cutting the shafts down to the size we needed. Initially we were sawing the stainless-steel shafts by hand which was taking a lot of time and effort. Then we moved to having the machine shop cut them for us which was much more efficient. Overall, the gear train and dual slider-cranks work to generate the motion we need to whisk the matcha. However, there are facets of the design we can improve to make the motion smoother before we add the motor.

Key Takeaways and Design Updates:

• The horizontal slider needs to be a little longer to accommodate the full range of motion of the whisk.

• Need to figure out how to screw the horizontal slider to the linear bearing housing. Right now, the screws are too short to fit a nut on the end. Previously we tried longer screws, but they hit the side of the bowl and interfered with the motion.

• The 3D printed elements should be made out of PETG instead of PLA so they will be stronger.

• The platform that the gear train is attached to needs to be 12 mm taller.

• The shaft that the vertical crank is grounded to needs to be 1 inch upwards in the y-direction.

• The links and gears should be laser cut out of acrylic instead of plywood so that they are smoother.

• The press fit of the gears and the links needs to be more precise so that the motor doesn’t pull anything loose.

• Need to identify exact placement of links, joints, and gears in the z-axis so as to eliminate any motion in that direction and leave only planar motion.

• Overall, ensuring that the end motion is incredibly smooth before we add the motor.

Bill of Materials: