Homework Assignment 4 Root Finding Algorithms The Maximum Velocity of a Car Due: Friday, February 19, 2010 at 12noon

Transcription

1 ME 2016 A Spring Semester 2010 Computing Techniques Homework Assignment 4 Root Finding Algorithms The Maximum Velocity of a Car Due: Friday, February 19, 2010 at 12noon Description and Outcomes In this assignment, you will use a root-finding algorithm in Matlab to solve an engineering problem. The engineering problem is to find the maximum velocity that a Ford Taurus will reach in each of its gears while driving at full throttle up roads with slopes varying between -5% and +20%. The learning objectives of this assignment are: to increase your understanding of root finding methods to learn how to implement root finding methods in Matlab to learn how to solve engineering problems using root finding methods to continue developing as an independent learner of Matlab features to improve your debugging skills Background The overall goal of this assignment is to find the top speed of a car under different combinations of slope and gear. The top speed is reached when the forward force produced by the car engine is equal but opposite to the resistance force due to wind resistance, rolling resistance, and gravity. According to Newton s law, we can then conclude that the acceleration, which is equal to the sum of the forces, is zero. Solving this problem therefore requires that we use a root finding method to determine the point where the engine force is equal to the resistance force. As is shown in the figure below, both of these forces depend on the velocity of the car. In this case, the maximum steady-state velocity in 4 th gear is approximately 63 m/s (227 km/h or 141mph) not bad for a Ford Taurus Engine Force and Resistance Force for a slope of 0 degrees resistance force engine force (1st) engine force (2nd) engine force (3rd) engine force (4th) Force in [N] Velocity in [m/s] Page 1

2 You may recognize the shape of the engine torque-rpm characteristic that we have used previously. Since the drivetrain is converting the engine torque into a forward force, and the drivetrain has a different reduction ratio depending on the gear we are in, you see that the maximum force generated by the engine becomes smaller in higher gears. You have probably experienced this while driving: when the required force is large (e.g., when you are pulling a heavy trailer or when you are going uphill, you need to downshift to allow your car to generate a larger force). The load force looks like a parabola. It is a combination of three forces: a gravitational force component, rolling resistance, and wind resistance. The gravitational force component results from the projection of the gravitational force onto the forward direction of the car. It depends thus on the slope of the road, but is independent of the velocity of the car. F = mg sin( α) with grav slope in % tan( α ) = 100 where m is the mass of the car and g is the gravity constant. The rolling resistance is due to the deformation of the tire some of the energy is converted into heat inside the rubber of the tires (have you ever noticed how your tires get hot after driving on the highway?). The rolling resistance is typically modeled as a constant force proportional to the weight of the car: F rolling = µ mg cos( α) r where µ r is the rolling coefficient. Finally, the wind resistance is the only component of our load model that depends on the velocity of the car. It is given by the formula: 1 F = C ρ Av 2 2 drag D car where C D is the drag coefficient, ρ is the density of air, A is the frontal area of the car, and v is the velocity of the car. The parabolic shape of the load curve in the figure on the previous page is due to quadratic dependence of the drag force on the car velocity. As in assignment HW2, The model for the transmission and differential is an ideal (i.e., no friction losses) gear reduction: ω τ out out = ω / n in = nτ in where the subscript in refers to the torques and velocities closest to the engine, and n is the gear ratio. The wheels are modeled in a similar fashion, as a rigid cylinder rolling without slip over a flat road surface: v F car car = Rω = τ wheel wheel / R where R is the radius of the wheel. A final note: all equations above assume SI units. Program Structure To solve the problem of finding the maximum velocity of the car for each gear, we suggest breaking the problem down into the following Matlab script and functions: Page 2

3 1) The main script, called plot_max_velocities_xyz should adopt the same structure that we have used for all previous assignments: set up the problem, solve the problem, and present the result in a format that is easy to interpret. In the solution portion, the script should iterate through each of combination of gear selection and slope, each time solving the root finding problem and storing the result in a matrix. In the interpretation portion, the script should plot the maximum velocity for each of the gears as a function of slope, as shown below Maximum Steady-State Speed as a Function of Road Surface Slope (ME Chris Paredis) 1st gear 2nd gear 3rd gear 4th gear Maximum Speed [km/h] Slope [%] We recommend that you use the built-in Matlab function for root finding: fzero. It is faster and more robust than any of the methods we have studied in class. Make sure to check the appropriate syntax for the fzero function in the help pages: X = FZERO(FUN,X0). where FUN is the function handle to the function we want to solver and X0 is an initial guess or initial bracket. 2) The model for the engine, called engine_force_xyz: This function should compute the forward force provided by the engine, given the velocity of the car. For a given car velocity, you should use the models provided above to compute the engine RPMs (similar to what you did in HW2). Once you have determined the engine RPMs, you can use the Matlab function that you created for HW2 to compute the corresponding torques and power (NOTE: this Matlab function is available on the course web-site in the solution section of HW2 feel free to use it as is). Once you have computed the engine torque, you again use the appropriate conversions to compute how the engine torque is converted into a forward force acting on the car. This forward force is the output of the function. Since this force depends on both the car velocity and the gear number, these two should be inputs to the function. 3) The load model, called resistance_force_xyz: This function should compute the forces due to wind resistance, rolling resistance, and gravitation, based on the equations provided on page 2. Page 3

4 4) Finally, you need to combine engine_force_xyz and resistance_force_xyz into a function for which the root occurs at the maximum velocity of the car. Since the maximum velocity of the car depends on the slope and gear_number, you should include these in the definition of you function. I suggest using the anonymous functions we discussed in class: This program structure is only a suggestion. Other solution approaches are possible. If you decide to deviate from the structure above, make sure to justify in your report why the structure that you have chosen makes sense. In general, you should try to leverage as much as possible the code that you have developed in the past. Additional Hints: - Avoid copying and pasting lines of codes when you can easily accomplish the same functionality using a for-loop - warning off can get rid of annoying warnings that are generated by the engine_model function. Read the help file for the warning command for more information. - Leverage your previous efforts. Portions of this assignment are similar to portions of assignment HW2. Feel free to build on your past efforts or even the HW2 solutions posted on the course web-site. Parameter Values The same engine and drivetrain parameters as in HW2. In addition, use the following values: gravity constant: 9.81 m/s 2 air density: 1.29 kg/m 3 car mass: 1650 kg (includes occupants and cargo) drag coefficient 0.32 frontal area of the car: 2.2 m 2 rolling coefficient Tasks Task 1: Create the function engine_force_xyz (where XYZ are your initials) The function engine_force_xyz should compute the forward force provided by the engine. Its inputs should be the velocity of the car in [m/s] and the current gear_number (e.g., 1 for first gear, 2 for second gear, etc.). It may be useful to test quickly whether you obtain the same results as are shown in the figure on page 1 of this handout (but this is not a requirement for this HW). Task 2: Create the function resistance_force_xyz (where XYZ are your initials) The function resistance_force_xyz should compute the resistance force experienced by the car due to wind resistance, rolling resistance, and gravity. Its inputs should be the velocity of the car in [m/s] and the slope of the road surface in percent. Again, it may be useful to test quickly whether you obtain the same results as are shown in the figure on page 1 of this handout, but be careful, that curve is for a slope of zero percent. Task 3: Create the script plot_max_velocities_xyz (where XYZ are your initials) The scrip plot_max_velocities_xyz should result in a plot of the maximum velocity that a Ford Taurus will reach in each of its gears while driving up roads with slopes between negative 5% (downhill) and positive 20% (uphill) in 0.5% increments. To implement this functionality, you will need to define an additional function for which the root occurs at the maximum velocity of the car. (see comments on the previous page). Include the resulting graph in your report it should look like the graph on the previous page. Page 4

5 Task 4: Interpretation In your report, provide a short interpretation of your results. Question 1: How does the maximum velocity change as the slope increases? Question 2: At which slope should you down-shift from 4 th to 3 rd gear to maintain the maximum speed? At which slope should you down-shift even further from 3 rd to 2 nd gear? Explain why. Question 3: What happens when you drive up a 15% hill in 4 th gear? How does this manifest itself mathematically in the root finding problem? Report Turn in a brief report in MS Word containing the following: 1. A copy of all your Matlab code. 2. A copy of all your figures you generated. 3. Your answers to all the questions in task A statement of collaboration (see below) include this even if you did not collaborate with anybody Evaluation Criteria A detailed grade sheet for this assignment is provided as a separate file on the course web-site. Print out this sheet, fill in your name, and staple it to the front of your submission. Submission 1. Hardcopy in class: At the beginning of class, hand in a hardcopy that includes the grade sheet and your report. Make sure you staple everything together so that we don t lose anything. (no staple = incorrect submission) 2. T-square: save all your files in a zip-file called HW4.FamilyName.FirstName.zip (replace FamilyName and FirstName with YOUR names) and submit this file in the Assignments section of T-square. The zip-file should contain: your Matlab function and script, and your report. The deadline for electronic submission is at 12noon. Late Submission Remember that the deadline for this assignment will be strictly enforced. After the deadline has passed you will receive a late-penalty. Don t wait until the last minute to get started! We repeat here the policy that is included in the syllabus: Late Submission T-square is set up such that late submissions are accepted until 48 hours after the due date. We will accept late homework but with the following penalties. Submit before 24 hours late and receive 20% off of your grade. Submit before 48 hours late and receive 40% off. For submission more than 48 hours late, no more partial credit will be awarded. If you are submitting late, bring a hardcopy to the office of your instructor. If you plan on submitting late, a quick by the deadline would be appreciated so that we can make appropriate plans for grading your assignment. Collaboration We would like to re-emphasize the policy on collaboration. Collaboration is encouraged. Discussing the assignments with your peers will help you to develop a deeper understanding of the material. However, discussing the assignment does not mean solving it together; it does not mean asking your friend to debug your code for you. I encourage you to discuss how to approach the problem, which Matlab functions to use, or how to interpret the results, but I do expect each student to turn in a report and Matlab functions that reflect the student s individual work. Do not copy code from another student. Do not copy parts of other electronic documents. In general, an activity is acceptable if it promotes learning by you and your peers. For example, Page 5

6 you learn from discussing alternate solution approaches with your friend, but you don't learn from blindly copying your friend's code. To avoid any confusion, each homework solution should explicitly identify the students with whom you collaborated and what the extent of the collaboration was. Any copying on homework and/or exams will be dealt with severely and reported to the Dean of Students No exceptions. If you have questions about this collaboration policy, do not hesitate to ask your instructor. Page 6