ME 2016 Sections A-B-C-D Fall Semester 2005 Computing Techniques 3-0-3 Homework Assignment 2 Modeling a Drivetrain Model Accuracy Due: Friday, September 16, 2005 Description and Outcomes In this assignment, you will create a first model for a complete drivetrain. The model combines the engine model from last week with an automatic transmission, a differential and the wheels. In addition, we will model the load experienced by the car due to air and tire resistance. We will use these models to determine the gear in which the car runs most efficiently. In addition to learning about drive train models, you will refresh your Matlab knowledge about functions and control structures. The learning objectives of this assignment are: to learn how to formulate, solve, and interpret an engineering problem using models to learn about the impact of model accuracy on solution results to learn how to use Matlab to solve engineering problems to continue developing as an independent learner of Matlab features to learn the basics of Matlab functions to learn typical control statements in Matlab: conditions, loops, etc. to learn how to use structures in Matlab Background In this homework assignment, we focus on the drivetrain of a car, specifically the drivetrain of a Ford Taurus V6 3L sedan. The drivetrain consists of the engine, torque converter, transmission, differential, and wheels. Several of these components are pretty complex devices and modeling them accurately is a challenging problem. However, a simple model will be sufficient for the approximate simulations we perform in this class. Next, we discuss each drivetrain component in more detail. Engine Transmission Tires Car Body Torque Converter Differential Engine: The engine is modeled in the same fashion as in assignment 1, namely, a map of specific fuel consumption values as a function of engine speed and torque and bounded by the maximum torque curve. Given an engine speed and torque, the engine map provides us with the Page 1
specific fuel consumption. For instance, we determined that at 3000 rpm and 100 Nm the engine consumes 290g/kWh. In this assignment, we are interested in the actual fuel consumption (in g/km), which can be computed using the following equation: bsfc P fc = 3600 v where fc is the fuel consumption (in g/km), bsfc is the brake specific fuel consumption (in g/kwh), P is the power provided by the engine (in W; remember P = τω ), and v is the forward velocity of the car (in m/s). Torque Converter: In this assignment, we assume that the car is cruising at constant velocity it is in steady-state. That means that the torque converter really serves no purpose and can be omitted from the drivetrain model. Transmission: The model of the transmission is a lossless speed reduction with a ratio of n:1. The gear ratio, n, can take on 4 different values corresponding to 1 st through 4 th gear. (We are not modeling the reverse.) An ideal gear reduction is modeled by the following two equations for the rotation speed and torque: ω τout out = ω / n in = nτ in where the subscript in refers to the torques and velocities for the shaft closest to the engine, and n is the gear ratio. For a Ford Taurus, the gear ratios are: gear ratio 1 st gear: 2.77 gear ratio 2 nd gear: 1.54 gear ratio 3 rd gear: 1.00 gear ratio 4 th gear: 0.69 Differential: Since we assume that the car is driving in a straight line, we can model the differential using the exact same equations it simply serves as another gear reduction with a gear ratio of 4.266 to 1. Wheels: The wheels are modeled as a rigid cylinder rolling without slip over a flat road surface (notice the similarity with the equations for the transmission above): v F car car = Rω = τ wheel wheel / R where R is the radius of the wheel. The wheels of a Ford Taurus have a standard diameter of 16 inches or 0.4064m. Air and Tire Resistance: To complete the model of the motion of the car, we also need to model the external forces that are acting on the car the forces the engine needs to overcome to keep the car moving. The tire or 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 = mgµ r where m is the mass of the car (1650kg), g is the gravity constant (9.81m/s 2 ), and friction coefficient (0.009). Finally, the air or drag resistance is given by the equation: 1 Fdrag = CDρ Av 2 2 Page 2 µ r is the rolling
where C D is the drag coefficient (0.32), ρ is the density of air (1.29 kg/m3 ), A is the frontal area of the car (2.2 m 2 ), and v is the velocity of the car (in m/s). Important: Verify that the units of both sides of the equation match. Putting it all together: The Entire Car Model. In this assignment, the goal is to determine how much gas the car uses in a specific gear and at a specific speed. How do we combine all the models discussed above to solve that problem? Well, we know that for a given engine speed and torque, we can determine the fuel consumption from the engine model. The question then remains: what is the torque that the engine needs to provide to run the car at a constant speed corresponding to a given engine speed. The way to compute this is to move through all the drivetrain models from the engine through the transmission, differential and wheels, and back. Additional details are provided in the description of Task 3. One last twist to the story: The models are inaccurate. Remember what we discussed in class: All models are wrong, some are useful. Here too we have made quite a few assumptions and idealizations so that we can be pretty sure that the models are not 100% accurate. If we want to draw conclusions from these models we should take the inaccuracy into account. Assume that both the engine model (bsfc as a function of speed and torque) and the resistance model have an error of at most ±5%. For instance, at 3000 rpm and 100 Nm the specific fuel consumption is not exactly 290g/kWh, but somewhere in the range [275.5, 304.5] g/kwh. Similarly, the total force determined by the resistance model, Ftotal = Frolling + Fdrag, is really within the range [0.95 F,1.05 F ]. This inaccuracy must be taken into account in Task 3 below. total total NOTES: Make sure to use compatible units throughout your computations. It is good practice to use SI units throughout. Since this is only the second homework assignment, we will still provide you with significant scaffolding. That is, we will lead you through the problem step by step. Each assignment, some of the scaffolding will be removed until you can solve the entire assignment by yourself. For instance, in this assignment we will no longer specify that you need to include a header in each of your m-files; you learned that in the previous assignment and are expected to carry that over in all future assignments (and beyond even when you write programs for your coop or full-time job later, you should always include a header). Tasks The main task of this assignment is to create a program that generates the following graph: 130 120 Fuel Consumption as a Function of Car Velocity for Different Gears (Chris Paredis) 110 fuel consumption in [g/km] 100 90 80 70 60 gear 1 gear 2 50 gear 3 gear 4 40 0 20 40 60 80 100 120 140 160 car speed in [km/h] Page 3
Task 1: Create the function get_engine_data_xyz (where XYZ are your initials) The function get_engine_data_xyz re-uses some of your program from HW1 to extract an engine model from an Excel file. The function should have as an input the name of the Excel file and as an output a structure containing the following 5 fields: o speed_max_torque: vector of engine speeds in [rpm] (index for max_torque data) o max_torque: corresponding vector of maximum torque values in [Nm] o speed_bsfc: vector of engine speeds in [rpm] (row index for bsfc data) o torque_bsfc: vector of torques in [Nm] (column index for bsfc data) o bsfc: matrix of brake specific fuel consumptions in [g/kwh] These are the same variables as appear in the solution to HW1, but now they appear as fields of a structure (e.g., bsfc becomes engine_map.bsfc). Using a structure allow us to pass a lot of information to a function with only one variable. Yet, all the information is still easily accessible and is stored in an easily comprehensible format. If you are not familiar with Matlab structures, learn more about them at: http://www.mathworks.com/access/helpdesk/help/techdoc/matlab_prog/ch02_d27.html. Task 2: Create the function resistance_model_xyz (where XYZ are your initials) The function resistance_model_xyz implements the model that computes the total resistance force, F total = F rolling + F drag, as a function of the car velocity, v. The input should thus be the car velocity (in m/s) and the output should be the total resistance force (in Nm). Make sure to vectorize your function so that it can accept a vector of velocities as an input and produce a corresponding vector of resistance forces. Task 3: Create the script compute_fuel_efficiency_xyz (where XYZ are your initials) This is the main script. It should create the plot above. For each of the 4 gears, the plot shows a nominal fuel consumption in a solid line and the accuracy bounds in dotted lines. Let s first discuss how the nominal fuel consumption can be plotted. To start, we recognize that the independent variable (along the x-axis) is the car velocity. However, the valid range of velocity values is different for every gear because the valid velocities correspond to the range of valid engine speeds (from 500 to 6000 rpm). Therefore, it is easiest to consider the engine speed as the independent variable and compute both the car velocity and the fuel consumption as a function of engine speed, as is shown in the figure on the right. For a given engine speed, use the transmission model to determine the speed of the shaft between the transmission and differential, then use the differential model to determine the rotational speed of the wheels, and finally the wheel model to determine the forward velocity of the car. From the car velocity, we can compute the total resistance force (load force) resulting from tire and air resistance. This load force can then be translated back to a corresponding engine torque by using the wheel model, the differential model, and the transmission model (traversing the models in the opposite order as for the speeds). Once we have the engine speed and torque, then the engine model provides us with the fuel consumption. Finally, the specific fuel consumption (bsfc) can be converted into the fuel consumption (fc in g/km), which can be plotted as a function of the car velocity (an intermediate result of the computations outlined above). Task 4: Including Uncertainty Bounds Page 4 ωengine Engine bsfc = get_bsfc( ω, τ ) engine τ engine Transmission engine ωdiff = ω / n and τ = τ / n engine trans engine wheel trans ω diff Differential Wheel τ diff ω = ω / n and τ = τ / n wheel diff diff diff wheel diff v = ω R and τ = F R car wheel wheel wheel load wheel Air and Rolling Resistance F ω wheel load v car τ wheel F load = resistance_model( v ) car
The computations in Task 3 result in the nominal results the solid lines in the graph. To add the uncertainty bounds (the dotted lines), we repeat the same computations but considering the uncertainty bounds for both the resistance_model() and the get_bsfc() model. We need to consider the worst and best case combination of errors in these two models. When is the fuel consumption the largest? Clearly, the error will be large when the error is at the upper accuracy bound of the bsfc model (i.e, nominal bsfc + maximum error). But for which resistance force is the fuel consumption the largest for the nominal force plus or minus the maximum error? Think about it and verify your answer by testing it in Matlab. Remember that we assume a maximum error of ±5%. Once you have determined the maximum and minimum error on the fuel consumption, add the upper and lower bounds to your plot. Task 5: Interpretation of the Plot Answer each of the following 4 questions, and make sure to justify your answer. Question 1: Assuming that the nominal models are accurate, rank the gears from most to least efficient for a car velocity of 20km/h. Question 2: Does your answer to question 1 change when considering the accuracy bounds? Question 3: If one wants to drive as efficiently as possible, which gear should one use, and at which velocity should one drive? Answer the question twice: 1) considering the nominal model only; 2) considering the error bounds. Question 4: Some politicians have suggested lowering the speed limit again from 70mph (112km/h) to 55mph (88km/h) with the claim that this will save energy. Is this claim justified? Report Turn in a brief report in MS Word containing the following: 1. All your Matlab scripts and functions (Cut and Paste them into MS Word) 2. A copy of the figures (use the Edit / Copy Figure in the figure window, and paste into your Word document). 3. The answers to the questions in Task 5 4. A statement of collaboration (see below) 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 to the front of your submission. Submission Double Submission: 1. Hardcopy in class: At the beginning of class, hand in a hardcopy that includes the grade sheet and your report. 2. WebCT: zip all your files in a folder called HW2.FamilyName.FirstName.zip (replace FamilyName and FirstName with YOUR names) and submit this file in the Homework section of WebCT. The zip-file should contain: your Matlab script and your report. The deadline for electronic submission is 12 noon. Late Submission Remember that the deadline on this assignment will be strictly enforced. After 12 noon on Friday September 16, you will no longer be able to upload your work to WebCT, and you will receive a penalty. Don t wait until the last minute to get started! We repeat here the policy that is included in the syllabus: Late Submission WebCT will be set up such that late submissions are not accepted. We strongly encourage you to submit homework on time. We will accept late homework, but with the following penalties. Submit by noon Saturday and receive 20% off of your grade. Submit by noon Sunday and receive Page 5
40% off. Submissions after noon on Sunday receive a zero. If you are submitting late, and are in section A or C, bring a hardcopy to Dr. Paredis office (MARC 256), and email your zip file to chris.paredis@me.gatech.edu. If you are in section B, bring your hardcopy to Dr. Rosen s office (MARC 252) and e-mail your zip file to david.rosen@me.gatech.edu. If you are in section D, bring your hardcopy to Dr. Hahn s office (Love 123) and e-mail your zip file to steven.hahn@me.gatech.edu. The time-stamp on your e-mail message will be considered as your submission time. 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, and it definitely does not include solving it together and then copying the solution from each other. 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. To avoid any confusion, please, add to your report a short section identifying with whom you collaborated and what the extent of the collaboration was. Any copying on homework 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