rail_resistance

rail_resistance - CEE 3604 Rail Transportation: Addendum...

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Unformatted text preview: CEE 3604 Rail Transportation: Addendum Rail Resistance Equations Transportation Engineering (A.A. Trani) Fundamental Formula • A quadratic formula has been used for over 80 years to approximate rail vehicle resistance • von Borries Formel, Leitzmann Formel, Barbier and Davis worked on this equation R = A + BV + CV 2 • where R is the rail vehicle resistance (N), V is the velocity of the vehicle (m/s), and A (N), B (N s/m) and C ( Ns 2 / m 2 ) are regression coefficients obtained by fitting test data to the Davis equation Transportation Engineering (A.A. Trani) 2 Observations • The coefficients A and B in the Davis equation account for mass and mechanical resistance • The coefficient C accounts for air resistance (proportional to the square of the speed) • The Davis equation has been modified over the years for various rail systems and configurations . A few examples are shown in the following pages. Transportation Engineering (A.A. Trani) 3 Davis Equation - Committee 16 of AREA (American Railway Engineering Association) 20 KV 2 Ru = 0.6 + + 0.01V + w wn • • where: • Values of K are 0.07 for conventional equipment, 0.0935 for containers and 0.16 for trailers on flatcars Ru is the resistance in lb/ton, w is the weight per axle (W/n), n is the number of axles, W is the total car weight on rails (tons), V is the speed in miles per hour and K is a drag coefficient Transportation Engineering (A.A. Trani) 4 Additional Terms to the Davis Equation (Gradient Forces) Mg RG ( kN ) = X • • where: RG is the resistance (kN) due to gradients, M is the mass of the train in metric tons, g is the acceleration due to gravity (m/s2) and X is the gradient in the form 1 in X (for example: a grade of 3% is expressed as X = 1/0.03 = 33.33 in the formula above) Transportation Engineering (A.A. Trani) 5 Additional Terms to the Davis Equation (Resistance due to Curvature) k rc ( kN / t ) = 0.01 Rc • • where: rc is the resistance due to curvature (kN/ton), k is dimensionless parameter depending upon the train (varies from 500 to 1200), RC is the curve radius in a horizontal plane (meters). Transportation Engineering (A.A. Trani) 6 Application of Davis Equation to a High-Speed Rail System (Japan Shinkansen Series 200) per Rochard and Schmid1 R = 8.202 + 0.10656V + 0.01193V 2 • • where: R is the total resistance (kN), V is the speed of the train (m/s) train 1A review of Methods to Measure and Calculate Train Resistances (Proceedings of the Institute of Mechanical Engineers,Vol. 214 Part F) Transportation Engineering (A.A. Trani) 7 Matlab Script to Calculate Resistance Forces (Shinkansen Series 200) • % Script to estimate the total resistance of a Series 200 train % Equations provided by Rochard and Schmid (2000) % Coefficients of Davis equation applied to Japanese Shinkansen system % Series 200 A = 8.202; % units are kN B = 0.10656; % units are kN s/m C = 0.0119322; % units are kN s-s/m-m % Create a speed vector V = 0:1:90; % speed in meters/second % Calculate Resistance (in KiloNewtons) according to modified Davis equation R = A + B * V + C * V.^2; % Make a plot of total resistance vs speed plot(V,R,'o--') xlabel(' Speed (m/s)') ylabel('Resistance (kN)') title('Reisistance of Series 200 Shinkansen Rail System') Transportation Engineering (A.A. Trani) 8 Shinkansen Series 200 Tractive Effort Curve • The tractive effort can derived from knowledge of the shaft horsepower delivered by the rail engine(s) • Literature on the Shinkansen indicates that the series 200 locomotives deliver 15,900 HP of power • Lets assume that a single locomotive pulls a 6-car train unit Transportation Engineering (A.A. Trani) 9 Tractive Effort vs Power • A fundamental equation to convert power to tractive force (or effort) is shown below • This equation can be modified to convert units correctly (from HP to Newtons) VT P= η • where: P is the power output delivered by the engine, T is the tractive force or effort, η is the efficiency in converting power output to tractive force and V is the velocity of the vehicle Transportation Engineering (A.A. Trani) 10 Tractive Force or Effort in Typical Units ηP T = 2650 V • T in Newtons • P in horsepower • V in km/hr Transportation Engineering (A.A. Trani) 11 Matlab Script to Calculate Tractive Effort (Shinkansen Series 200) % Coefficients of Davis equation applied to Japanese Shinkansen system % Series 200 plot(V,R,'o--') xlabel(' Speed (m/s)') ylabel('Resistance (kN) or T (kN)') title('Reisistance of Series 200 Shinkansen Rail System') grid hold on % Calculate the Tractive Effort (T) profile P = 15900; % horsepower (hp) Vkmhr = V*3.6; % velocity in km/hr (needed in the TE equation) nu = 0.7; % efficiency T = 2650 * nu * P ./ Vkmhr / 1000; % in kN plot(Vkmhr/3.6,T,'^-r') grid Transportation Engineering (A.A. Trani) 12 Resistance or Tractive Effort (kN) Plot of Resistance and Tractive Force vs Speed Tractive Force η = 0.7 Resistance Force Speed (m/s) Transportation Engineering (A.A. Trani) 13 Observations • According to these plots, the high-speed rail system will reach its maximum velocity at 82.8 m/s (298 km/hr) • The value of efficiency has been assumed to be 0.7 (conservative) • The plot applies to level ground (zero gradient) Transportation Engineering (A.A. Trani) 14 ...
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This note was uploaded on 12/31/2011 for the course CEE 3604 taught by Professor Katz during the Fall '08 term at Virginia Tech.

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