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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 HighSpeed
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 ss/mm
% 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
6car 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 highspeed 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.
 Fall '08
 KATZ

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