CEE 3604: Introduction to Transportation Engineering Fall 2011 Date Due: September 26, 2011 Assignment 4:Rail Analysis and Stopping/Passing Distances Instructor: Trani Problem 1 The basic resistance of the French TGV rail system can be modeled using a modified Davis equation of the form: R = k 1 + k 2 V + k 3 where the coefficients k 1, k 2 and k 3 are estimated by the manufacturer to be 7.1 (kn), 0.09940 (kn/s/m) and 0.01074 (kn s-s/ m-m) so that the units of the basic resistance equation above are kilonewtons. A six-car train weighing 415,000 kg is pulled by a single 14,000 HP locomotive. In solving this problem I recommend you can use Excel or the Matlab script in pages 8 and 12 of the Rail Resistance notes. a) Estimate and plot the basic resistance as a function of speed. Figure 1. Matlab Script to Estimate Resistance and Tractive Force. b) Estimate the tractive force of the complete rail unit if the locomotive has a efficiency (η ) of 0.8. Plot the tractive force vs. speed in the same plot as the resistance (part a). Since the tractive force will tend to infinity for small values of speed, the locomotive has an engine control to limit the tractive force to 800 kn below 23 mph. CEE 3604 A4 Trani Page 1 of 5
Figure 2. TGV Tractive Force and Resistance Diagram. c) Using the tractive force obtained in part (b) and the basic resistance obtained in part (a) calculate the acceleration function of the train as a function of speed. Plot and comment on the acceleration profile observed. Figure 3. Acceleration Function for TGV System. CEE 3604 A4 Trani Page 2 of 5
d) Estimate the time for the train to accelerate from a station to reach 80 mph. Repeat for 30 mph and 120 mph and comment on the trend observed for acceleration times. Problem 2 a) Calculate the speed (in mph) at which the braking and the reaction distances are equal. Observed from Table 3.1 in the handout Stopping and Passing distances that the speed where reaction and braking distances are equal lies between 35 and 40 mph. Solve for speed in the standard formula to estimate SSD (see formula below) and get a value of V =~ 36.5 mph. Vt r = a 254( 9.81 ) Figure 4. Reaction and Braking Distances for Various Speeds. b) Repeat part (a) if the vehicle travels uphill in a section of the highway with a slope of 6%. c) The physical evidence of an auto accident investigation on a secondary road reveals tire marks on the pavement with a length of 513 feet. The driver goes to court and challenges the police report by stating that he was driving close to the 45mph posted speed limit of the road. The weather at the time of the accident was clear skies and 80 degrees F. The section of the road with tire marks has an average local slope of 1.8% uphill. Perform the necessary calculations and provide evidence for a judge on whether the likelihood that the driver was in violation of the speed limit. Using the simple braking distance calculations, CEE 3604 A4 Trani Page 3 of 5
S b = a 254( 9.81 + G = 100 ) 254( 3.4 9.81 + 1.80 100 ) = 513 ft 3.28 ft / m Solve for speed and get V =~ 120 km/hr. This is 75 mph. Obviously the driver was speeding. d) Use the AASHTO equivalent friction coefficients presented on page 23 of the Stopping Distance handout to generate two curves in a single plot containing the braking and reaction distances vs. speed for a roads with 3% upgrade conditions. Compare the values obtained with those shown on page 21 of the same handout. Problem 3 a) Calculate the safe passing distance for a road where vehicles pass at an average speed of 90 km/hr. Use the AASHTO recommended standards. Use Exhibit 3-5 in the handout to get 1,918 feet (or 583 meters). b) Estimate how sensitive is the passing distance requirement for part (a) if the value of m (the difference in speed between the passing and overtaken vehicle) changes by 5 km/hr. Explain any consequences in highway design. Problem 4 A traffic engineer estimates the jam concentration of a highway to be 88 vehicles per kilometer per lane. The free flow speed is known to be 110 km/hr. Using traffic cameras, the engineer suspects Greenshield s model applies to her traffic situation. a) Find the maximum flow possible on this 4-lane divided highway (2 lanes each way) during the morning commute. Assume the morning commute means heavy traffic moves on two lanes of the highway towards the downtown area only. q max = u k f j 4 (110 km/hr)(88veh/km-la) q max = = 2,420veh/hr per lane 4 b) Find the speed of the traffic flow if one day cameras surveying the highway register an average of 40 meters between successive cars (front bumper to front bumper distance). Assume the critical design vehicle is a passenger car with a typical length of 19 feet. u = u f u f k j k u = 110 110 (25veh/km-la) = 78.5 km/hr 88 The speed would be around 78.5 km/hr. c) Find the travel time between two ramps spaced 2.6 km. away for conditions in part (b). 1.98 minutes. d) Plot (using Matlab or Excel) all traffic conditions for this highway. Plot density vs. speed and speed vs. flow. Label accordingly. CEE 3604 A4 Trani Page 4 of 5
Figure 5. Density-Flow Relationship. Greenshield Model. Figure 6. Flow-Speed Relationship. Greenshield Model. CEE 3604 A4 Trani Page 5 of 5