Results to published data for Chevrolet Corvette

  This mini project is designed to give you a real-world application of some of the content we have covered in this course. As part of this project, you will be analyzing the properties of a Chevrolet Corvette, and will compare your results to published data. You will be forced to make some assumptions, and we will explore the implication of these assumptions at the end of the project. This assignment will require you to use relationships of power to force and acceleration, model dynamic resistive forces, and find displacement and time associated with non-constant acceleration. You can work in groups of two or three on this project. Model description For this project, we will be analyzing the performance of a Chevrolet Corvette Coupe accelerating as it drives in a straight line. Information about performance that you will be asked to find include: 1/8 mile elapsed time 1/8 mile speed (miles per hour) 1/4 mile elapsed time 1/4 mile speed (miles per hour) 0-60 foot elapsed time Maximum speed Free-body diagram Treat the car as a particle, where the vehicle is subjected to rolling resistance and wind resistance. Basic system properties Assume that the vehicle has a: Weight of 3241 lb Wheel diameter of 26.88 inches Engine data The vehicle used in this study is the C6-05. For this vehicle, the following table relating engine speed and wheel horsepower is provided. Data was pulled from the Rototest Research Institute (http://www.rototestinstitute.org/). Full-throttle operation is assumed. The tests conducted on this engine determine wheel horsepower, which is the power available at the wheel to propel the vehicle forward. By using wheel horsepower, drivetrain inefficiencies are accounted for. Engine speed Wheel power n (rpm) P (hp) 1751 100.6 1993 118.3 2493 151.3 2994 186.7 3496 219.9 3999 267.8 4214 281.3 4401 295.6 4603 309.0 5006 331.3 5507 346.0 5810 353.9 6010 353.8 6212 350.2 6429 346.6 Transmission shift schedule Cars need transmissions because of the physics associated with the internal combustion engine. Each engine has a redline, the maximum rpm value sustained by the engine before it fails. The transmission allows the gear ratio between the engine and the drive wheels to change as the car speeds up. You shift gears so the engine can stay below the redline and near the rpm band of its best performance. Figure 1. Representation of a transmission Further detail on transmissions can be found at: http://auto.howstuffworks.com/transmission.htm The ratio for the gears in the transmission is the transmission input (engine output) speed to the transmission output speed. The following table describes the transmission shift-up schedule for this vehicle: Gear Ratio Start rpm Shift rpm 1 3.27 1700 5500 2 2.20 3700 5500 3 1.56 3900 5500 4 1.22 4300 5500 5 1.00 4500 5500 6 0.82 4500 - Differential ratio In aiming the engine power to the wheels, a differential is also responsible for acting as the final gear reduction in the vehicle. This allows the rotational speed of the transmission to be slowed one final time before it hits the wheels. The differential also allows power to be transmitted to the wheels in such a way that they can rotate at different speeds when turning. This eases the stress on the components, but is not considered in this project. Further detail on differentials can be found at: http://auto.howstuffworks.com/differential.htm The differential ratio is the differential input (transmission output) speed to differential output (wheel input) speed. The ratio of this vehicle is 3.08. Figure 2. Diagram of a differential Rolling resistance Rolling resistance can be simply modeled with a coefficient of rolling friction fr. For pneumatic rubber tires on a hard surface (like a drag strip), fr has an assumed value of 0.012. Wind resistance Wind resistance can be modeled as a quadratic loss or a velocity-squared loss. Assume the following relationship: F_W=((C_d ρA)/2) v^2 where: Fw = wind resistance C = drag coefficient (assume 0.35 for a sport car)  = air density (assume a value of 0.0807 lbn/ft3) A = frontal area (assume 24 ft2) v = velocity of the car Simplifying assumptions Engine performance data, although produced during steady-state conditions, applies to acceleration conditions. Neglect shift times Torque converter locked up at all times (no torque multiplication) http://auto.howstuffworks.com/auto-parts/towing/towing-capacity/information/torque-converter.htm Neglect mass moments of inertia of wheels, drivetrain, and engine Sufficient tire grip is present to prevent slip Neglect tire growth Use a start velocity of 13.5 mph (engine at 1700 rpm with transmission in first gear). The actual practice of drag racing involves running the engine to about 4000 rpm with the brakes locked and the torque converter stalled. The brakes are then released, which impacts acceleration. This is beyond the scope of what I expect you to model. Engine speed n and car velocity v are directly proportional, with the proportionality depending on the transmission gear. Procedure for analysis Having defined the model you will use, the procedure for analysis is discussed in this section. It is important to highlight that transmission shifts are based on engine speed, not car velocity. Therefore, engine speed will be the independent variable the analysis is constructed around. To complete this project, you will need to finish the following tasks: Derive the appropriate free-body diagram and equations of motion. Develop an equation for horsepower that is a function of engine speed. Using engine speed as the independent variable, create a spreadsheet showing start rpm to shift rpm for each transmission gear. Use engine speed increments of 200 rpm. Find the wheel horsepower for each engine speed in your spreadsheet. Determine the vehicle’s velocity as a function of engine speed in your spreadsheet. Calculate the force generated by wheel horsepower and the resistive forces from your free-body diagram as a function of engine speed. Calculate the acceleration of the vehicle as a function of engine speed. Use the trapezoidal rule to integrate and find values of elapsed time, t. Use the trapezoidal rule to integrate and find values of displacement, s. Determine the required performance parameters listed above. Reporting Submitted reports should be typed and clearly presented. For your report: Document the results of your work from steps 1-9 above, describing the procedure and equations used. Calculate the percent difference between your results and the results presented below that were taken from a real vehicle. 1/8 mile elapsed time 7.210 seconds 1/8 mile speed 96.980 mph 1/4 mile elapsed time 11.270 seconds 1/4 mile speed 121.620 mph 0-60 foot elapsed time 1.660 seconds www.dragtimes.com Provide plots of acceleration, velocity and displacement as a function of time. Discuss each figure briefly. Discuss the maximum speed result you found for your vehicle. How did you determine where this occurred? Examine the acceleration of the vehicle as a function of engine speed in terms of g’s. Given the behavior here, what behavior do you expect to see out of the vehicle? How does weight distribution and center of gravity height influence this? Discuss the significance of wind resistance at low and high speeds Spreadsheet must be turned in via blackboard