CEE 3604: Introduction to Transportation Engineering Fall 2012 Date Due: September 28, 2012 Assignment 4:Rail Analysis and Stopping/Passing Distances Instructor: Trani Problem 1 You are ased to evaluate the suitability of a new bus produced by an American manufacturer. The bus parameters are provided below. Ca = 0.03; C1 = 7.2; C2 = 0.06; W = 130; A = 7.2; % Coefficient in SAE equation % Coefficient in SAE equation % Coefficient in SAE equation % Bus weight in N % Frontal area (square meters) D = 1.05; % Wheel diameter (m) u(1) = 4.3; u(2) = 3.4; u(3) = 2.5; u(4) = 1.8; u(5) = 1.3; J = 4.2; nu = 0.80; % efficiency of engine % gear ratios in explicit vector form % Differential reduction ratio P = [100 180 230 275 290 300 310]; % power output vector (HP) RPM = [800 1400 1800 2400 2800 3400 4200]; % RPM vector a) Find the total travel time between two adjacent bus stations located 300 meters apart. Assume the bus driver uses full throttle to accelerate the bus to the speed limit of the road. Assume the driver decelerates at 1 m/s 2 from cruise speed to zero speed at the final station. The road has a speed limit of 56 m/hr (35 mph) and the segment of road has zero gradient. Show me all your wor. In solving this problem, you can use the tractive force approximation demonstrated in class. b) Calculate the fraction of the time the bus is accelerating, cruising and decelerating. Comment on the observed trends. c) If there are 20 bus stops along this line and the average distance is 300 meters, estimate the travel time of the bus to complete one cycle (i.e., traveling from an initial station to the end station and then bac to the initial station). Consider 3 minutes of wait time at the end station. CEE 3604 A4 Trani Page 1 of 9
Figure 1. Tractive effort vs. Speed Diagram. Data Points in Circles are Selected Points to Approximate the TE Function. Figure 2. Acceleration Function Using a Fourth-Order Polynomial (segment from 0-20 m/hr) can be Assumed to be Constant Because the Tractive Effort is flat in that Speed Regime. CEE 3604 A4 Trani Page 2 of 9
The acceleration function is: dv dt = 2.79 0.222V + 0.0077V 2 0.000134V 3 + 8.64e 7V 4 if V > 5.5 else dv dt = 1.775 where V is in m/s and dv/dt is in m/s 2. Using this function we can now integrate numerically the equation for dv/dt and get the bus speed profile and the bus distance profile. The table that follows illustrates the procedure. In the solution presented we solved the following equations. I used a step size of 0.25 seconds. V t = V t Δt + dv dt Δt S t = S t Δt + ds dt Δt where : Δt = 0.25 seconds The bus accelerates quicly and reaches 15.6 m/s (35 mph) in 12.1 seconds. At this point the bus has traveled 107 meters (see Table 1). The deceleration distance from 15.6 m/s is calculated to be: S 3 = V 2 ( )2 2a = 15.6m/s 2(1m/s 2 ) = 122 meters t 3 = V a = 15.6m / s 1m / s 2 = 15.6 seconds The bus then is able to cruise for a small period of time because the sum of the acceleration and deceleration distances is 229 meters. The bus cruises for 71 meters or 4.6 seconds. The total distance traveled is 300 meters and the total travel time between bus stops is 32.3 seconds. CEE 3604 A4 Trani Page 3 of 9
Table 1. Numerical Integration Procedure to Study Bus Performance. First 10 seconds of the Trajectory Shown. CEE 3604 A4 Trani Page 4 of 9
Problem 2 a) Calculate the Stopping Sight Distance for a highway designed for 125 m/hr. Do not interpoloate from the AASHTO equations. Assume zero grade. Compare your result with the AASHTO tables provided in the class notes. Using the simple braing distance calculations, V2 125 2 = = 177.5 m a G 3.4 0 254( + ) 254( + ) 9.81 100 9.81 100 125m/hr Sr = Vt r = (2.5 seconds)=86.8 m 3.6 (m/hr)/(m/s) Sb = SSD = Sb + Sr = 264.3 m Be careful to convert the value of reaction distance to the correct units. In the calculation above, the value of speed was converted to m/s before doing the calculations.the AASHTO table shows that for 130 m/hr the value of SSD in flat terrain should be 284 meters. The value calculated above seems reasonable. Problem 3 a) Calculate the safe passing distance for a road where vehicles pass at an average speed of 110 m/hr. Use the AASHTO recommended standards. 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 overtaen vehicle) changes by 5 m/hr. Explain any consequences in highway design. Problem 4 A traffic engineer estimates the jam concentration of a highway to be 75 vehicles per ilometer per lane. Using Autoscope cameras, the free flow speed is nown to be 130 m/hr. Apply Greenshield s model equations and answer the following: a) Find the maximum flow for the 6-lane divided highway (3 lanes each way) during the morning commute. Assume the morning commute means heavy traffic moves on three lanes of the highway towards the downtown area only. uf j 4 (130 m/hr)(75veh/m-la) = = 2, 437veh/hr per lane 4 qmax = qmax CEE 3604 A4 Trani Page 5 of 9
for three lanes in the direction of traffic this translates into 7,311 vehicles per hour 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. When the spacing between successive cars is 40 meters, the lane density is 25 veh/la-m. Substitute the value of density into the Greenshield equation. u = u f u f u = 130 130 (25veh/m-la) = 86.7 m/hr 75 c) Estimate the speed of the traffic flow if one day cameras surveying the highway count 25 cars per lane per m. This produces the same answer as part (b). u = u f u f u = 130 130 (25veh/m-la) = 86.7 m/hr 75 d) Find the average spacing between cars for part (b). Assume the critical design vehicle is a passenger car with a typical length of 19 feet. e) Find the travel time between two highway ramps spaced 1.3 miles away for conditions in part (b). The travel time between two ramps is: t t = S / u t t = S / u = 2.09m = 0.0241hrs =1.45 minutes 86.7 m/hr f) 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 6 of 9
Figure 3. Flow vs. Speed Relationship. Figure 4. Density vs Flow Relationship. CEE 3604 A4 Trani Page 7 of 9
Figure 6. Density vs. Speed Diagram. Problem 5 a) For the Greenshield traffic flow model show (using differential calculus) that the maximum flow condition is found by differentiating the basic equation for flow (q) and solving for the values of density () and speed (u) that maximize q. u = u f u f q = u q = u f u f 2 dq d = 0 = u f 2 u f u f = 2 u f 1 = 2 = 2 Let the value of density found above be called m of the density for maximum flow condition. Then, m = 2 CEE 3604 A4 Trani Page 8 of 9
Repeat the process shown above starting with the equation that relates flow (q) and speed (u). = u f u q = u q = u u f u 2 Find the value of (u) that maximizes flow (q) taing the first derivative with respect to density (u). u m = u f 2 q = u Since flow is the product of density () and speed (u), q m = u m m = u f 2 q m = u f 4 2 = u f j 4 b) For the Greenberg model show (using differential calculus) that the condition for maximum flow is: Use the same principle for this part of the problem. CEE 3604 A4 Trani Page 9 of 9