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a3_cee3604_2011 sol

# a3_cee3604_2011 sol - CEE 3604 Introduction to...

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Unformatted text preview: CEE 3604: Introduction to Transportation Engineering Fall 2011 Assignment 3: Vehicle Forces and Kinematics Date Due: September 16, 2011 Instructor: Trani Problem 1 a) A commuter train with a mass of 305,000 kg. travels downhill at a grade of -2% and 50 mph when the brakes fail. Estimate the maximum speed the train will achieve downhill if the following parameters apply: S = 14 m 2 , f roll = 0.0018, C d = 0.75 (dim) and C l = 0.03 (dim) Here we do not have any tractive effort since the train is un runaway mode. Balance of the forces as the train travels downhill is the gravity force accelerating the train and the friction and drag forces opposing the motion. At equilibrium: G f = F f + D mg sin( θ ) = ( mg cos( θ ) − L ) f roll + D mg sin( θ ) = ( mg cos( θ ) − 1 2 ρ V 2 SC l ) f roll + 1 2 ρ V 2 SC d where: G f = gravity force pushing the train F f = friction force D = drag force all in Newtons Solve for V in the equation to obtain the balancing speed of the train at equilibrium. A numerical solution can be obtained using a simple Matlab script to estimate the sum all three forces as a function of speed and then watch for the value of speed that makes zero the sum of all forces acting on the train. The diagram is shown in the figure below. CEE 3604 A3 Trani Page 1 of 11 Figure 1. Diagram of Forces for a Runaway Train (No Tractive Force). The “balancing” speed is estimated at 91.35 m/s (204 mph). b) During normal operations, the train described in part (a) has a tractive force as shown in Table 1. The tractive force was obtained from actual testing of the vehicle. Table 1. Tractive Force for Train in Problem 1. Speed (m/s) Tractive Force (N) 180000 20 155000 40 95000 60 65000 80 50000 100 35000 c) Plot the speed vs tractive force data shown in Table 1 using Matlab. CEE 3604 A3 Trani Page 2 of 11 Figure 2. Tractive Force and Resistance Function for Rail System. d) Using Matlab’s basic fitting capabilities, obtain a third degree polynomial (cubic equation) that relates the tractive force as a function of speed. Write down the equation of the cubic model. The equation is of the form: T ( N ) = a + bV + cV 2 + dV 3 where V is in m/s and coefficients a to d are correct to produce T in Newtons Using the Matlab basic fitting function we find the values of the model coefficients. T ( N ) = a + bV + cV 2 + dV 3 a = 1.8317 e 5 b = − 1.7308 e 3 c = − 12.302 d = 0.1505 e) Under maximum tractive force conditions, what is the maximum speed the train system could reach in flat terrain?...
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a3_cee3604_2011 sol - CEE 3604 Introduction to...

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