Tare weight of train is the weight of an empty train (without passengers).
The weight of passengers, baggage or goods conveyed by a train is called Pay or Net weight.
Total weight of train = Pay or Net weight of train + Tare weight of train
Train resistance is the force resisting the motion of train on track. When the tractive force developed by train is greater than the train resistance, then the train will start to accelerate. Specific resistance is equivalent to the train resistance per tonne i.e. train resistance / weight of train. On straight & level track, train resistance consist of following resistances :
1. Friction Resistance
2. Rolling Resistance
3. Air Resistance
Starting resistance, appears at the time of starting, is to overcome the non-elastic effects of track & wheel surface, inertia and bearing friction. Starting Resistance depend upon the type of bearings, weight per axle & temperature of the bearing and drops rapidly as the train speed increases. Generally, Starting Resistance are taken as 4 and 6 kg/tonne.
Running resistance of a train can be calculated by using empirical expression TR= A+B*V+C*V^{2} Kg/ton, where 'A', 'B' & 'C' are constants, V is speed in Kmph and TR is specific train resistance in Kg/ton.
Constant 'A' mainly depend upon the weight of train and independent of speed. Constant 'B' mainly depend upon the sliding friction between the flange of the wheel & the inner face of the rail. Constant 'C' depend upon the cross-section area of train, space in-between coaches, equipment under the coaches, air resistance of the sides, draft and turbulence created by the train etc. Contribution of term "C*V^{2}" is more visible with speed going more than 70 kmph
When the train travels upward, a component of its gravitational force works in a direction opposite to its motion and vice versa. Gradient Resistance is the force required to overcome train movement due to component of gravitational force acting on the train. If train is moving upward at a slope of θ, then the mass of the train (W) has two components; one perpendicular to the track (i.e. W*g*Cos θ) & the other along the track (i.e. W*g*Sin θ). The component along the track is known as gradient resistance.
Gradient Resistance = W*g*Sin θ Kilo Newton;
For a up gradient of 1 in 'G', value of Sin θ is equal to 1/(1+G^{2})^{1/2}.
For higher value of 'G', value of Sin θ may be considered equal to 1/'G'. Hence, Gradient Resistance = W*g/G Kilo Newton
An up grade produces a resistive force (+ve value) while a down grade produces an accelerating force (-ve value). Value of gradient resistance is independent of train speed.
When a train goes around a curve, a centrifugal force push the train outward which increase the sliding friction resistance between outer wheel flange & the inner face of the outer rail. In addition to this, the outer wheel rotates faster than the inner wheel causing transverse slip thus adding to sliding friction. To overcome curve resistance, outer rail is gradually lifted above the level of the inner rail and difference in height between the inner and outer rail on a curve is called superelevation or cant. The necessary cant in a curve depends on the expected speed of the trains and the radius. If a train goes around a curve at speed such that the component of centrifugal force in the lateral direction is equal to the component of gravitational force in the opposite direction there is very little curve resistance. This speed is called balancing speed of the curve. if train speed is either higher or lower from curve balancing speed, the curve resistance increases due to the unbalance in forces which tends to pull the vehicle sideways. Generally, curve resistance in Kg/ton is given by :
Curve Resistance = 0.4 * Degree of curvature Kg/ton;
Curve Resistance = 0.4 * Degree of curvature * g * weight of train Kilo Newton;
Degree of curvature = 1750 / Radius of curvature in meters;
Total train resistance = Train resistance + Gradient resistance + Curve Resistance
Rotating mass of train is equivalent to the rotational Inertia of the train, transformed by gear ratio and wheel diameter respectively. The equivalent non-rotating mass of train may be defined as a function of its true static mass and its rotational inertia. It is based on the principle that the total energy stored in the object is a combination of its translational kinetic energy and its rotational kinetic energy and mentioned below:
Equivalent non-rotating mass of train = Static mass of train + Rotational inertia of train * (Gear ratio / wheel radius)^{2} ;
Note:
Rotational inertia of traction motor rotor will be transformed by gear ratio & wheel diameter, however rotational inertia of wheels will be transformed by wheel diameter only.
Above relationship is independent of the vehicle speed, so the consideration is valid at any velocity, and is independent of vehicle power. Generally, rotating mass of a metro train is considered 6 to 8 % of the tare mass of train.
Adhesion is caused by friction. Available adhesion between the driving wheels and the track depends on the weight per wheel, irregularities of contact surfaces, environmental conditions etc. and determine the force that can be applied before the wheels begin to slip or slide during traction or braking respectively. Adhesion value between rail wheel and track is observed between 35% to 50%, whilst under extreme conditions (i.e. slippery condition ) it can fall to as low as 5%. Generally, useable adhesion limit is kept below the maximum available adhesion limit and its general range is 12 to 22 % for metro trains. Hence, tractive effort per motor axle can be increased by increasing the motor torque but only upto a certain point, which design adhesion limit allows.
Tractive effort of the train is equivalent to the force developed by the traction units at the wheel rims for moving the train. Tractive effort is generally higher than tractive force by the amount of rolling resistance present. Tractive effort develops necessary linear and angular acceleration to the train mass, overcome the gravity component of the weight of the train, overcome the wind, frictional and curve resistance of the train.
Tractive effort vs Speed graph may be classified as follow :
1. Constant Tractive Effort / Maximum Tractive Effort Zone
2. Quadrant portion of TE vs Speed graph; wherein, generally TE is inversely proportional of the speed
3. Quadrant portion of TE vs Speed graph; wherein, generally TE is Inverse Quadratic of the speed
Starting tractive effort is the tractive effort that can be generated at a standstill. It is important on railways because it determines the maximum train weight that a locomotive can set into motion. Tractive effort equivalent to constant power varies inversely with the speed. In this zone, relationship between power (P), velocity (v) and tractive effort (F) is described as below:
Power (kW) = Speed (m/s) * Tractive effort (kN) Similar to tractive effort graph, regenerative braking effort graph may be classified as follow :
1. Constant Regenerative Braking Effort Zone
2. Quadrant portion of Reg. BE vs Speed graph; wherein, generally Reg. BE is inversely proportional of the speed
3. Quadrant portion of Reg. BE vs Speed graph; wherein, generally Reg. BE is Inverse Quadratic of the speed
Regenerative braking effort is sub-set of total braking effort graph (Regenerative Braking Effort graph + Pneumatic Braking Effort graph)
Newton’s second law is applied to calculate the acceleration of the train.
Train acceleration at given speed = (Tractive effort of train - Train resistance) / Effective mass of train including rotating mass,
Starting resistance of the train should also be included in train resistance for calculation of starting acceleration of the train.
The residual acceleration of train is equivalent to the value of acceleration at maximum service speed. Trains which have low value of the residual acceleration cover longer distance to achieve the maximum service speed.
Balancing speed of train corresponds to the speed at which tractive effort of the train is equal to train resistance. At this speed, acceleration of train is supposed to zero. Balancing speed of metro train should be higher than its service speed / test speed. Higher value of the residual acceleration leads to higher value of the balancing speed as compared to maximum service speed of the train.
Service brake is used to apply and release of the brakes in a controlled way during normal operations of train. Generally, service brake deceleration of train is kept lower than the emergency brake deceleration of train.
Emergency brake is used for emergency stop of the train in the event of a brake pipe failure, control electronics failure or an emergency application by the train operator as a last resort. A completely separate mechanism from the service braking system is used to stop the train as quickly as possible. The emergency brake applies considerably more braking force than the full-service brake.
SEC of the train is the energy consumed (in KWh from the supply system) per ton mass per km length of the run.
SEC = (Energy consumed from the supply system during traction - Reg. energy fed to the supply system during braking) / (Weight of train * travelled distance),
The specific energy consumption of a train is influenced by
1. Distance between stations
2. Gross weight of the train
3. No. of speed restrictions in sections
4.
Acceleration & Deceleration
5. Maximum service speed
6. Auxiliary load of train
7. Shape of regenerated braking effort vs speed graph
8. Efficiency of propulsion equipment
Generally, percentage regeneration of a train for a given driving cycle is the ratio of regenerated electrical energy to consumed electrical
energy measured at pantograph in presence of specified auxiliary load.
Train operation modes may be classified as follow :
1. Full Traction mode : In this mode, master controller is kept in full traction position.
2. Driving in Allout mode : means full traction upto maximum service speed, operation at that speed followed by full service brake.
3. Cruising mode : In this mode, train is operated with constant speed and develop tractive effort equal to train resistance, hence value of acceleration is zero.
4. Coasting Mode : In this mode, no electric energy is supplied to traction motors and energy required to overcome train resistance is supplied from its kinetic energy, which result gradual reduction in train speed.
5. Full service brake : In this mode, master controller is kept in full service brake position.