Designing for Mass Efficiency

ME 204: Advanced Mechanical Systems Design

Fall 2024, Stanford University

In Fall 2024, I took a class at Stanford called ME 204, which was quite possibly my favorite class ever. The two main projects of the class were mass-optimization mechanism design challenges. Through the class I learned a lot about characterizing load patterns and paths, calculating internal stresses and deflections, conducting FEA, and minimizing energy loss.

project 1

Design a mass-minimized mechanism that observes a linear force-deflection relationship, deflecting 10 cm at the max force of 100 N and 0 cm at 0 N.

Figure 1: FEA of final ring design

After brainstorming a variety of designs for this challenging, I eventually landed on a thin aluminum ring as the most mass-efficient and sufficiently linearly deflecting mechanism. To calculate deflection, I applied curved beam theory as shown in figure 2b to determine the relationship between deflection and force, in terms of the thickness and width of the cylinder. This equation (fig 2a) then provided me with the ideal thickness / width relationship for the cylinder.

Figure 2a (left) and 2b (above): Curved Beam Theory and deformation equation

In addition to dimensions, I also investigated materials. Aluminum was the most promising as it offered a high Young's Modulus and low density, but after calculating the material factor of this design, it seemed Aluminum 7075 T6 was the only alloy with a high enough yield stress to perform as desired within a reasonable size range.

With Aluminum 7075 chosen as the desired material, I then iterated the width and thickness in my CAD and ran FEA in Solidworks, with a force applied at a portion of the top of the cylinder and a virtual wall on the opposite side. After trying many iterations and ensuring the von Mises stress was below the yield stress by the desired safety factor of about 1.3, I settled on a design for my hoop.

During the actual test, the hoop deflected relatively linearly as applied force increased, but it achieved 10cm of deflection at around 80 N. Increasing the width of the hoop would have achieved this desired force-displacement relationship, but unfortunately I did not allow myself enough time for prototyping to discover this before the test date.

Figure 3: Final Hoop

project 2

Design a mass-minimized mechanism that maximizes the impulse applied on a surface as it displaces 10 cm over 30s.

Unlike the first project, our goal for the second project was to maximize the force output of our mechanism. We were given a DC motor with a fixed max rated power at 6V and, based on our mechanism design, I aimed to translate this power into a force over a steady velocity over the given time frame (30s). I began by choosing the motor setpoint to be max power, where the motor runs at half the no-load speed. Testing the motor at various voltages and load conditions, I determined that the maximum power was about 9.3 W and half the no-load speed was about 14000 RPM. Given these fixed specifications, my goal was to then create a mechanism with a transmission ratio that would translate the rotational speed at max power Smp to the linear speed of 10 cm over 30s, or 0.33 cm/s.

I began by setting the output radius of the motor shaft to be 5.5 mm. I determined this number by estimating the force in the pulley rope, finding a Kevlar rope to fit the specifications, and determining the minimum bend radius of this particular rope. For the synthetic rope of my choice, the max rated force is 200 lb = 890 N, and minimum bend ratio is 4:1 (it can bend at a radius of at least 4x the radius of the rope). With my spool radius choice of 5.5mm, this provides a safety factor of 4.5 for the tension force and 2.5 for the bend radius.

With this choice, I calculated that the "gear ratio" or transmission ratio of the system would need to be about 2400:1. We were given a gearbox with gear ratio 400:1, so my designed mechanism had to have a mechanical advantage of about 6:1. Looking through all potential mechanical devices that provide leverage, like levers, gears, and pulleys, I determined that the most efficient possible design would be a block and tackle mechanism, as shown in figure 5. With 3 pairs of pulleys, a block and tackle can provide a force multiplication of 6x; that is to say, the motor spool would provide a tension force of T, while the block and tackle mechanism would provide an output pushing force of 6T, thereby reaching the full transmission ratio of 2400:1.

Figure 4: Kevlar Rope and minimum Bend Radius

Figure 5: Block and Tackle

Figure 6a: Frame FEA

Figure 6b: Piston FEA

Given the force multiplication of the block and tackle system, I also had to design a frame for the pulleys that could withstand the same force I expected it to apply, which was about 1200 N. In designing for mass efficiency, I strove to minimize the frame dimensions as much as possible while also ensuring it could take the load in compression with some sufficient safety factor.

Finite Element Analysis of my designed frame and piston under the estimated compressive and tensile loads, respectively are shown in figure 6. Both have a safety factor of about 1.3.

Figure 7: Force Applied vs Displacement in final test

During the final test of the mechanism, there were a few issues I didn't account for. For example, I had tied a knot at the end of the rope on the spool, which rubbed against the frame and caused some energy loss. Additionally, about halfway through the test, the spring pin, which was the only connection between the motor shaft and spool, crumpled and failed, and essentially prevented the motor torque from translating to the spool. The maximum force applied was about 500 N, about half what I expected, as shown in figure 7. Though these failure points were small and fixable, testing for the first time truly demonstrated to me what the weakest points of my design were and gave me insight on how to redesign the mechanism to function as my theoretical calculations predicted.

A video of my mechanism in motion is shown on the right.

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