Single Track Vehicle Design Final Report
ME 441, Winter 2012
March 8, 2012
As a long time bicycle enthusiast and newcomer to the freak bike building arena, I could not wait to take Single Track this quarter to further my knowledge of the inner workings of bicycle design. “Nessy” is the name of my new recumbent cargo bicycle and I could not be more happy with her. In this report, I chronicle my understanding of the handling qualities, loading, gearing, ergonomics, and manufacturing considerations that went into Nessy.
My original intent was to design and build a recumbent cargo bicycle suited for touring because of the following three reasons:
After riding more recumbent bicycles as well as realizing how much Nessy was going to weigh, I decided that the design would be best suited for cargo trips, and not touring. The design still features an integrated cargo frame, and is still optimized for maximum comfort to reduce any strain when putting in the extra effort to move the cargo. Thus, my target customer is the “do everything on a bike” type person who never wants to use a car to move around heavy items or to get groceries. A lower speed range will be selected for this bicycle and gearing will be accomplished with an internally geared hub for cleanliness and simplicity rather than a derailleur and cassette. Maximum comfort will be achieved by implementing a fully padded office chair as the seat, as well as positioning the bars such that the rider can just dangle their arms and relax. The estimated cost to build Nessy is $100, mostly going towards the ball joints for the remote steering, and the webbing for the cargo bay. See Appendix B for initial design sketches.
Carrying up to 100 pounds of cargo will severely change the handling when compared to a traditional recumbent bicycle. The bicycle must perform well when operating with such loads, in particular, it must have excellent low speed stability so as to avoid tipping over during starts and stops, and up steep hills. It will be important to keep the control spring small or negative at low speeds. At high speed, there must be adequate stability and a lower sensitivity. The control spring must remain small enough so that the rider does not become fatigued while steering during long rides. Though it is impossible that all handling criteria be ideally reached in one design, a good design will find the proper balance of parameters that meet the target goals without excessive compromise.
The model seen in Figure 1 was used to find pertinent geometric parameters to be used in the Patterson Control Model (PCM), determine the frame geometry for the finite element analysis (FEA), create drawings for manufacturing purposes, as well as determine how a comfortable rider fits on the bike. The rider used is 6 feet tall (my height) and has a distributed weight equal to 145 pounds (my weight). The cargo represents a distributed 40 pound weight. The bike, wheels, and components were said to be a total of 30 pounds. For detailed drawings of the bicycle and the ergonomic geometry, see Appendices C and D. Note that the only possible interference between the rider and the frame would be in the heel path. Appendix D illustrates that this clearance is adequate.
Figure 1: Solidworks Model
Stability and Handling Analysis
Initially, I had a longer wheelbase, a slacker head tube angle, and a higher center of mass than what is shown in Figure 1. I found the final design parameters (Figure 2) by changing 1 parameter at a time from my initial setup and analyzing the effect on handling with the PCM. A few of the changes and effects are tabulated in Table 1. I found that decreasing the wheelbase or the head tube angle would decrease both K1 and K2. Then, I could increase K2 by lowering the COM, and leave K1 relatively the same. Many design iterations were created, and each one was run through the PCM. I found that it was difficult to change K1 and K2 a lot simply because of other initial constraints I had set in stone such as 26 inch wheels in the front and rear. I really wanted to stick with my concept/layout and so I was pretty limited with how much I could change the parameters. The final design meets all constraints while balancing K1, K2, sensitivity, ergonomics, and aesthetics well.
Table 1: Parameter Changes and Effects on Control Spring
Lower COM to 25in
Tuck rear wheel in while keeping everything else in place (A decreases to 55in and B decreases to 10in)
Decrease Beta to 10 degrees
Figure 2: Final Geometric Parameters used in the PCM
The Control Sensitivity (Figure 3) plotted very similar to that of a Safety Bike (Figure A2), which is what I was shooting for. In the velocity sweet spot range which I define as between 5 and 10 m/s (11 and 22 mph), the control sensitivity stays between 6 and 12 rad/s/m which according to Patterson in “The Lords of the Chainring” is under the maximum control sensitivity of 12 rad/s/m that is comfortable for touring. This should allow for easy handling of the bike, ie: a small amont of force is required at the bars to steer the bicycle. I figured that the easiest way to adjust the Control Sensitivity is to change the handlebar radius which can be done last minute to the bike as well if necessary.
Figure 3: Control Sensitivity Plot
The Control Spring plot (Figure 4) shows a similar curve to that of the Safety Bike (Figure A3) but with smaller magnitudes. K1 is small, allowing for excellent low speed handling, because the bike will have a small intention to continue to steer with an input. It will help prevent the constant over-correction and readjustment of the bars when starting and stopping or just moving slow, something that I figured would be especially important because the remote steering setup is going to make the bike more difficult to handle by nature. I decided that the K2 value being smaller than the safety bike was also desirable because of the ease of handling it will provide during long rides at higher speeds (less force will be required to steer because the bike will not resist steering as much).
Figure 4: Control Spring Plot
A structural analysis in Mechanica was completed to find the weakest areas of the frame. I chose all tubing to have a 1 inch O.D. and wall thickness of 0.05 inches and be made of plain carbon steeI. Constraints are placed on the front and rear dropouts as well as the seat tube. Forces were positioned for rider weight, cargo weight, steering column force, pedaling force and moment, and chain forces on the bottom bracket and jackshaft. Figure 5 shows the diagram of the frame in Mechanica.
Figure 5: Forces, Moments, and Constraints on the Frame
Figure 6 shows an exaggerated deformed state of the frame. It can be seen that the fork bends outward the most, followed by the front half of the cargo bay bending down. Under static loading, the max deflection of the fork is 0.058 inches, a negligible value. If I hit a hypothetical pothole that caused an impact of 5 times the static loading, the fork deflection is still small at 0.29 inches. The frame displaces a maximum of 0.16 inches in the pothole case, which should cause no problems with the rest of the mechanics of the bike.
Figure 6: FEA Displacement Plot
The force diagram of Figure 7 shows that the beams under the most tension are the down tube followed by the tubes under the cargo bay. The static loading case causes a maximum tensile force in the down tube of 75 pounds and in the pothole case, 375 pounds, a relatively small magnitude for the steel tubing to be used. The beams with the highest compressive forces are the seat stays, followed by the top tube. The highest compressive force in the seat stay during the hypothetical pothole is 750 pounds. Using Euler buckling analysis, the max allowable force in these beams is 18.3 kips, yielding a factor of safety against buckling greater than 24. Phew!
Figure 7: FEA Frame Forces
Figure 8 shows that the highest principle stress is in the fork. As I am not designing a fork, I will assume that whatever fork I use is going to be strong enough for my application. However, the highest stresses in the frame are located at the joints. Because my welding skills are not up to commercial manufacturing par, these areas could be critical. In the pothole case, the max stress in the frame is 23.3 ksi. With the yield strength of steel being 32 ksi, the factor of safety against yielding at the joints is 1.37. Though this is small, I think the pothole conditions are an absolute worst case scenario.
Figure 8: FEA Frame Stresses
The entire bike was constructed at my house with a gas-less MIG welder. All materials were from the bicycle graveyard. A headset, top and down tube combo with the correct geometry was selected for the front half of the bike and a rear triangle was used for the rear half. 1” chromoly tubing was used to join the front and rear halves on the bottom of the frame. 3/4” galvanized tubing was used for the rest of the cargo framing. Tubes were cut and mitered to fit together as tight as possible with a grinder and then laid out on a flat surface to be welded together. Angles were measured with a protractor and re-measured after tacking to ensure accuracy. Though I had limited jigging capabilities, I was able to get the entire frame squared up and accurate to my dimensions.
For the production of this design, it will be best to build the cargo bay first with a nice jig. Then add the front and rear halves, again with a jig, to ensure proper alignment of the dropouts, seat tube, steering axis, and bottom bracket. Decent steel of the same alloy will be used throughout the frame to keep costs low but strength high. Based on the FEA, 1” tubing with 0.05” wall thicknesses will be used throughout the frame. This is because the factor of safety guarding against yielding was just in the comfort range for the hypothetical pothole case of 5 times the static loading, and almost all of the members had stress concentrations at the joints (Figure 8). For the less loaded members, the larger tubes are still desired for an overall robustness of the bike and to prevent damage from loading cargo in and out.
This bicycle design is primarily intended to be used at lower speeds, and not exceed 15 mph unless traveling downhill. The design will also have to be able to hill climb while full of cargo. Due to this, a lower gearing range is optimal, and because of the complexity of the chain line, an internal 7 speed hub will be used for simplicity.
The maximum grade that this bicycle should be able to climb is 7% and it should be able to do it at a speed of 4mph. The gradability redux calculation in Appendix E shows that 89 Watts of power would be required for this climb at 100% efficiency which is completely possible for a human. At minimum and maximum speeds of 3 and 15 mph, the gear ratio needs to change from 1.57 for low gear and 0.314 for high gear. A 34 tooth front chainring and 28 tooth rear cog, in combination with a 34 x 34 and 34 x 76 overdrive 2 speed on the left side cranks will allow for the internal hub to give a ratio range of 2.08 to 0.35. This is adequate as the rider can simply increase cadence in high gear to reach the 15mph mark. See Appendix E for calculations.
The maximum braking calculation as seen in Appendix F show that stopping will always be limited by friction forces, and not the risk of overturning. This is due to the long wheelbase and the low center of mass height. The maximum deceleration in wet road conditions is 0.1 g’s and in dry road conditions it is 0.8 g’s.
This project was a lot of fun. I mean, c’mon, when else do I get to build a bicycle for homework! It was much more work than I had anticipated but because of this, I learned a whole lot. Nessy handles great at low and high speeds, reinforcing my faith in the PCM and in the class as a whole. I’m really happy with my end product and hope to make more bicycles in the future with the knowledge I gained in this class.
Figure A1: Safety Bike Parameters
Figure A2: Safety Bike Control Sensitivity
Figure A3: Safety Bike Control Spring
Table A1: Safety Bike Values