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Definition

The total force necessary to maintain a rail vehicle in motion is the rail vehicle resistance (or simply resistance). This force depends on a number of variables and is of crucial importance for the energy efficiency of the vehicle as it is proportional to the tractive effort ^[1]. For the speed of the vehicle to remian the same, the locomotive must express the proper tractive force, otherwise the speed of the vehicle will change until this condition is met ^[2].

Davis equation

A number of experimental measurements ^[3] ^[4] ^[5] of the train resistance have shown that this force can be expressed as a quadratic equation with respect to speed as shown below:

$R=A+BV+CV^{2}$

Where $R$ is the resistance, $V$ is the speed of the rail vehicle and $A$ , $B$ , and $C$ are experimentally determined coefficients. The most well-known of these relations was proposed by Davis^[3] and is named after him.

Speed-independent term

The first term in Davis' ( $A$ ) equation accounts for the contributions to the resistance that are independent from speed. Track gradient and acceleration contribute to this term, since they cause the work produced by the locoomtive to be converted to potential and kinetic energy respectively. Other contributions are bearing friction and rolling friction due to the the local deformation of the rail at the point of contact with the wheels, these latter quantities are independent from vehicle speed but they are be affected by number of axles and axle loads ^[6]. If the tracks curve, an additional component of this term needs to be added to account for the excess friction of the wheels pressed against the rails as a result of centrifugal force (CITAZIONE SCHITO?).

Speed-linear term

The coefficient in the second term of Davis' equation ( $B$ ) relates to the terms linearly dependent on speed and is sometimes omitted becuase it is negligible compared to the other terms^[7].

This term is as easy to account with a direct physical explanation as the others, but it is typically though of as a mass-related viscous term^[8]

Speed-quadratic term

The coefficient in the third term of Davis' equation ( $C$ ) accounts for the aerodynamic forces acting on the vehicle, it is explained by the fact that as the train moves through the air, it sets some of the air surrounding it in motion (this is called slipstream). To maintain constant speed, the continous transfer of momentum to the air needs to be compensated by an additional tractive force by the locomotive, this is accounted for by thid term. As train speed increases, this contribution grows in importance, and is accounts for most of the resistance for high-speed trains (CITAZIONE SCHITO?) and for freight trains^[9]. This term is highlyl dependent on the geometry of the

Measurement of tractive force

External links

References

^ "Power - physics". Encyclopedia Britannica. Retrieved July 8, 2024.
^ "Newton's laws of motion". Encyclopedia Britannica. Retrieved July 8, 2024.
^ ^a ^b Davis, W. J. (1926). "The Tractive Resistance of Electric Locomotives and Cars,". General Electric Review. p. 3.
^ Schmidt, E. C. (1910). ""Freight Train Resistance; Its Relation to Average Car Weight". University of Illinois Engineering Experiment Station.
^ Tuthil, J. K. (1938). "High-Speed Freight Train Resistance: Its Relation to Average Car Weight". University of Illinois Engineering Bulletin: 376.
^ Lukaszewicz, P (2007-03-01). "Running resistance - results and analysis of full-scale tests with passenger and freight trains in Sweden". Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit. 221 (2): 183–193. doi:10.1243/0954409JRRT89. ISSN 0954-4097.
^ Gielow, M. A.; Furlong, G. F. "Results of wind tunnel and full scale tests conducted from 1983 to 1987 in support of the Association of American Railroad's Energy program". Retrieved July 9, 2024.
^ Rochard, B. P.; Schmid, F. (2000). "A review of methods to measure and calculate train resistances". Proceedings of the Institution of Mechanical Engineers: 187.
^ Li, Chao; Burton, David; Kost, Michael; Sheridan, John; Thompson, Marc C. (2017). "Flow topology of a container train wagon subjected to varying local loading configurations". Journal of Wind Engineering and Industrial Aerodynamics. 169: 12.

@@ Line 15: / Line 15: @@
 ===Speed-independent term ===
+[[File:Track gradient.svg|thumb|Illustrative scheme of tracks on a gradient]]
-The first term of the equation accounts for the contributions to the resistance that are independent from speed. Track gradient and acceleration contribute to this term, since they cause the work produced by the locoomtive to be converted to potential and kinetic energy  respectively. Other contributions are bearing friction and rolling friction due to the the local deformation of the rail at the point of contact with the wheels, these latter quantities are independent from vehicle speed but they are be affected by number of axles and axle loads <ref>{{Cite journal |last=Lukaszewicz |first=P |date=2007-03-01 |title=Running resistance - results and analysis of full-scale tests with passenger and freight trains in Sweden |url=http://journals.sagepub.com/doi/10.1243/0954409JRRT89 |journal=Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit |language=en |volume=221 |issue=2 |pages=183–193 |doi=10.1243/0954409JRRT89 |issn=0954-4097}}</ref>.
+The first term in Davis' (<math>A</math>) equation accounts for the contributions to the resistance that are independent from speed. Track gradient and acceleration contribute to this term, since they cause the work produced by the locoomtive to be converted to potential and kinetic energy  respectively. Other contributions are bearing friction and rolling friction due to the the local deformation of the rail at the point of contact with the wheels, these latter quantities are independent from vehicle speed but they are be affected by number of axles and axle loads <ref>{{Cite journal |last=Lukaszewicz |first=P |date=2007-03-01 |title=Running resistance - results and analysis of full-scale tests with passenger and freight trains in Sweden |url=http://journals.sagepub.com/doi/10.1243/0954409JRRT89 |journal=Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit |language=en |volume=221 |issue=2 |pages=183–193 |doi=10.1243/0954409JRRT89 |issn=0954-4097}}</ref>.
-If the tracks curve, an additional component of this term needs to be added to account for the excess friction of the wheels pressed against the rails as a result of centrifugal force (CITAZIONE SCHITO).
+If the tracks curve, an additional component of this term needs to be added to account for the excess friction of the wheels pressed against the rails as a result of centrifugal force (CITAZIONE SCHITO?).
 ===Speed-linear term ===
-The second term of Davis' equation relates to the terms linearly dependent on speed and is sometimes omitted becuase it is negligible compared to the other terms<ref>{{cite web |last1=Gielow |first1=M. A. |last2=Furlong |first2=G. F. |title=Results of wind tunnel and full scale tests conducted from 1983 to 1987 in support of the Association of American Railroad's Energy program |url=https://trid.trb.org/View/293260 |access-date=July 9, 2024}}</ref>.
-This term is as easy to account with a direct physical explanation, but typically it is though of as a mass-related viscous term<ref>{{cite journal |last1=Rochard |first1=B. P. |last2=Schmid |first2=F. |date=2000 |title=A review of methods to measure and calculate train resistances |journal=Proceedings of the Institution of Mechanical Engineers |pages=187}}</ref>
+The coefficient in the second term of Davis' equation (<math>B</math>) relates to the terms linearly dependent on speed and is sometimes omitted becuase it is negligible compared to the other terms<ref>{{cite web |last1=Gielow |first1=M. A. |last2=Furlong |first2=G. F. |title=Results of wind tunnel and full scale tests conducted from 1983 to 1987 in support of the Association of American Railroad's Energy program |url=https://trid.trb.org/View/293260 |access-date=July 9, 2024}}</ref>.
+This term is as easy to account with a direct physical explanation as the others, but it is typically though of as a mass-related viscous term<ref>{{cite journal |last1=Rochard |first1=B. P. |last2=Schmid |first2=F. |date=2000 |title=A review of methods to measure and calculate train resistances |journal=Proceedings of the Institution of Mechanical Engineers |pages=187}}</ref>
 ===Speed-quadratic term ===
-Aerodynamic forces.
+The coefficient in the third term of Davis' equation (<math>C</math>) accounts for the aerodynamic forces acting on the vehicle, it is explained by the fact that as the train moves through the air, it sets some of the air surrounding it in motion (this is called slipstream). To maintain constant speed, the continous transfer of momentum to the air needs to be compensated by an additional tractive force by the locomotive, this is accounted for by thid term. As train speed increases, this contribution grows in importance, and is accounts for most of the resistance for high-speed trains (CITAZIONE SCHITO?) and for freight trains<ref>{{Cite journal |last=Li |first=Chao |last2=Burton |first2=David |last3=Kost |first3=Michael |last4=Sheridan |first4=John |last5=Thompson |first5=Marc C. |year=2017 |title=Flow topology of a container train wagon subjected to varying local loading configurations |url=https://www.sciencedirect.com/science/article/pii/S0167610516303385?via%3Dihub |journal=Journal of Wind Engineering and Industrial Aerodynamics |volume=169 |pages=12}}</ref>.
+This term is highlyl dependent on the geometry of the
 == Measurement of tractive force==