ME217 Fall 2017 Calibration Assignment

Transcription

1 ME217 Fall 2017 Calibration Assignment Jane Doe November 6,

2 1 Summary of problem statement Prof. Mammoli seeks to estimate the mechanical energy needed to propel himself and his roadbike from his address at 775 Windsong Lane in Corrales to UNM main campus, return trip. The preference is to ride along designated bike paths, possibly well away from roads. Details on the bike characteristics and rider mass are provided. 2 Roadmap of analysis We will first determine a route from home to UNM and back, and attempt to accurately describe the path in terms of distance and elevation change. We will then identify the various sources of mechanical energy expenditure. We will then attempt to quantify the mechanical energy expenditure along the route, using appropriate equations inserted in a spreadsheet. For simplicity, we will assume calm wind conditions, and constant speed. 3 Analysis Since the path from the home address to UNM is not a straight line, we will need a reasonably detailed path description to allow an accurate calculation. To obtain this, we can use Google maps. The output is shown in Fig. 1. We choose the longer route, along the river trail, because it is much more pleasant, reflecting the good professor s desire to be away from roads. We also note that the Google time calculator allocates a 1h 22m travel time, for the 25.3 km ride. Based on his average speed, the professor should do the ride in less than an hour, but the Google time calculator probably accounts for things such as slowing down for peds on the trail, stoplights, intersections and so on. 2

3 Figure 1: Google maps output of bike route from home address to office on UNM main campus 3

4 To a first approximation, there are three components to energy expenditure on a bike: 1. rolling resistance 2. aerodynamic drag 3. gravity (i.e. going up hills) OK, now we need to prepare for the calculation itself. To start with, we can assume that rolling resistance results from a constant force F r, that in turn results from deformation of the tires as they roll on the ground. There is also additional resistance from wheel bearings, crank bearings and from the chain. We make an informed decision that tire deformation is the dominant contributor to rolling resistance. So, the energy dissipated by rolling resistance for riding along a distance d, independently of speed, is given by: The aerodynamic drag force F d is given by E r = F r d. (1) F d = 1 2 C daρv 2, (2) where C d is the drag coefficient, A is the cross-sectional area of the rider in the direction of travel, ρ is the density of air and v is the speed. Energy dissipated by aerodynamic drag for a distance d is then E d = F d d. (3) For gravity, the energy required to go up a hill is given by E g = mg h, (4) where m is the combined mass of rider and bike, g is the acceleration of gravity, and h is the change in elevation experienced while traveling along distance d [3]. When going uphill, the change in potential energy is added to the dissipation from rolling resistance and drag. When going downhill, we assume that loss in potential energy offsets rolling resistance and drag (i.e. the rider does not have to pedal as hard). However, if loss in potential energy is greater than rolling resistance and drag combined, we assume that the rider applies the brakes to maintain constant speed, with zero net energy expenditure on the part of the rider. With this, we are ready to get some numbers. All we need is the sequence of rectilinear paths that approximates the total trip from home to UNM. Sadly, the Google map is not enough information! Fortunately, it is possible to obtain a detailed route with GPS coordinates and other derived information by feeding the Google Maps route to the GPS Visualizer website. 4

5 Figure 2: Sample output of GPS Visualizer website. 5

6 The output of GPS visualizer is a text file that looks like the sample in Fig. 2. The information fields are self-explanatory: latitude, longitude, altitude, distance, and distance interval. From this, we will use the distance interval and the altitude, which provide us with the quantities d and h respectively in Equations 1, 3 and 4. Before attempting the calculation, we need a few more bits of data. First, we need the rolling resistance of the tire. This can be obtained from the Bicycle rolling resistance website [1]. We choose a mid-range road bike tire, namely the Schwalbe Ultremo ZX, that has a rolling resistance of 15 W at a speed of 29 km/h and a load of 42.5 kg weight. Given that rolling power P r is related to rolling resistance by: P r = F r v, (5) we can calculate the constant rolling resistance, F r = 1.87 N per tire. For the aerodynamic drag, we use the CyclingPowerLab website [2]. For a bike with drop bars, the typical C d is The frontal area for a typical cyclist using drop bars is 0.32 m 2. Finally, the density of air in Albuquerque, for typical conditions, is kg/m 3. That s it! All that remains is to plug the formulas in the spreadsheet, and add it all up. The outcome, for the route from home to UNM, is an energy expenditure of 323,247 J, or kwh, or 77 calories (compare this with a typical 2000 calorie energy intake from food). Also note, however, that the chemical energy consumed by the body to produce this mechanical energy is likely much higher, probably something to the tune of 450 calories (the human body is not the most efficient energy converter). The power as a function of distance for the home to UNM trip is shown in Fig. 3. Figure 3: Power output of rider as a function of distance. This looks pretty reasonable, with a power of just under 100 W during the flat part, and higher power in the uphill section (from the river valley to UNM). One should also note that the 800 W power spikes are not realistic - given that pro riders can sustain a power output of 500 W for a few minutes, and the really good sprinters maybe 1000 W for 30 seconds! A more realistic sustained power output for the uphill section for the aging professor is about 200W (according to the treadmill at the gym), meaning that he would 6

7 have to slow down to a few km/h on the really steep sections. Nevertheless, the energy expenditure is almost the same - there would be a small reduction in aerodynamic drag, but gravity and rolling resistance would be the same. Finally, the way back is a little harder - with a total energy expenditure of 333,503 J, as a result of the asymmetrical layout of the uphill and downhill segments of the ride. 7

8 References [1] Bicycle Rolling Resistance. Accessed: [2] CyclingPowerLab. Accessed: [3] Richard A Dunlap. Sustainable energy. Cengage Learning,