# Cant deficiency

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The term "cant deficiency" is defined in the context of travel of a rail vehicle at constant speed on a constant radius curve. Cant itself is a British synonym for the superelevation of the curve, that is, the elevation of the outside rail minus the elevation of the inside rail. Cant deficiency is present when a vehicle's speed on a curve is greater than the speed at which the components of wheel to rail force are normal to the plane of the track. In that case, the resultant force (aggregated force of gravitational force and centrifugal force) exerts the outside rail more than the inside rails, in which it creates lateral acceleration toward outside of the curve. In order to reduce cant deficiency, the speed can be reduced or the superelevation can be increased. The amount of cant deficiency is expressed in term of required superelevation to be added in order to bring the resultant force into balance between the two rails. In the contrary, it is said to be "cant excess" if the resultant force exerts more against the inside rail than the outside rail, for instance, a high superelevation curve with a train traveling at a low speed. [1]

Rail transport is a means of transferring of passengers and goods on wheeled vehicles running on rails, also known as tracks. It is also commonly referred to as train transport. In contrast to road transport, where vehicles run on a prepared flat surface, rail vehicles are directionally guided by the tracks on which they run. Tracks usually consist of steel rails, installed on ties (sleepers) and ballast, on which the rolling stock, usually fitted with metal wheels, moves. Other variations are also possible, such as slab track, where the rails are fastened to a concrete foundation resting on a prepared subsurface.

The cant of a railway track or camber of a road is the rate of change in elevation (height) between the two rails or edges. This is normally greater where the railway or road is curved; raising the outer rail or the outer edge of the road providing a banked turn, thus allowing vehicles to maneuver through the curve at higher speeds than would otherwise be possible if the surface is flat or level.

In Newtonian mechanics, the centrifugal force is an inertial force that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis passing through the coordinate system's origin and parallel to the axis of rotation. If the axis of rotation passes through the coordinate system's origin, the centrifugal force is directed radially outwards from that axis. The concept of centrifugal force can be applied in rotating devices, such as centrifuges, centrifugal pumps, centrifugal governors, and centrifugal clutches, and in centrifugal railways, planetary orbits and banked curves, when they are analyzed in a rotating coordinate system. The term has sometimes also been used for the reactive centrifugal force that is a reaction to a centripetal force.

## Forces

The forces that bear on the vehicle in this context are illustrated in the following figure.

A vehicle's motion at speed v along a circular path embodies centripetal acceleration of magnitude v2/R toward the center of the circle, the curvature of that path being 1/R where R is the radius of the circle. This centripetal acceleration is produced by horizontal forces applied by the rails to the wheels of the vehicle, directed toward the center, and having sum equal to Mv2/R where M is the mass of the vehicle.

The net horizontal force producing the centripetal acceleration is generally separated into components that are respectively in the plane of the superelevated (i.e., banked) track and normal thereto.

The component normal to the track acts together with the much larger component of gravitational force normal to the track and is generally neglected. It can slightly increase the vertical load seen by the vehicle suspension but it does not create lateral acceleration as perceived by passengers or and does not cause lateral deflection of the vehicle suspension.

The track is superelevated so that the component of the acceleration of gravity in the plane of the track will provide some fraction of the horizontal acceleration in the plane of the track due to the circular motion. Referring to the figure above, it can be seen that the components of gravitational and centripetal acceleration in the plane of the track will be equal when the following balance equation is satisfied, where α is the bank angle.

In physics, acceleration is the rate of change of velocity of an object with respect to time. An object's acceleration is the net result of all forces acting on the object, as described by Newton's Second Law. The SI unit for acceleration is metre per second squared (m⋅s−2). Accelerations are vector quantities and add according to the parallelogram law. The vector of the net force acting on a body has the same direction as the vector of the body's acceleration, and its magnitude is proportional to the magnitude of the acceleration, with the object's mass as proportionality constant.

${\displaystyle {v^{2} \over R}\cos \alpha =g\sin \alpha }$

For a given curve radius and bank angle (i.e., superelevation) the speed V that satisfies the balance equation is called the balancing speed and is given by

${\displaystyle V_{bal}=\left({Rg\tan \alpha }\right)^{\tfrac {1}{2}}}$

For reasons that will be mentioned below, passenger vehicles usually traverse a curve at a speed higher than the balance speed. The amount by which the actual speed exceeds the balance speed is conveniently expressed via the so-called cant deficiency, i.e., by the amount by which the superelevation would need to be increased to raise the balance speed to the speed at which the vehicles actually travel. Letting gaugese denote the rail gauge from low rail gauge side corner to high rail field side corner, letting super_el denote the actual superelevation, and letting Vact denote the actual speed, it follows from the definition that the cant deficiency, CD, is given by the formula

${\displaystyle CD={gauge_{se} \over {\left(1+{R^{2}g^{2} \over {V_{act}^{4}}}\right)^{\tfrac {1}{2}}}}-super\_el}$

### Example

Taking an example, a curve with curvature 1.0 degree per 100 ft chord (radius 1,746.40 m = 5,729.65 ft), gauge_se = 1511.3 mm (59.5 inches), and super_el = 152.4 mm (6.0 inches) will have

${\displaystyle V_{bal}={\sqrt {1746.4\cdot 9.80665\cdot \tan(\arcsin(152.4/1511.3))}}}$:
${\displaystyle =41.6638\,\mathrm {m/s} =149.99\,\mathrm {km/h} =93.20\,\mathrm {miles/h} }$

If a vehicle traverses that curve at a speed of 55.880 m/s (= 201.17 km/h = 125 mph), then the cant deficiency will be

${\displaystyle CD={\frac {1511.3}{\sqrt {1+(1746.4^{2}\cdot 9.8066^{2}/55.8^{4})}}}-152.4}$
${\displaystyle =118.7\,\mathrm {mm} (=4.67\,\mathrm {inches} )}$

On routes that carry freight traffic in cars with the maximum allowed axle loads it will be desirable to set superelevations so that the balancing speed of each curve is close to the speed at which most such traffic runs. This is to lessen the tendency of heavy wheel loads to crush the head of either rail.

## Limit values

For passenger traffic superelevations and authorized speeds can be set so that trains run with as much cant deficiency as is allowed, based on safety, on relevant regulations and on passenger comfort. As of 2007 the US Federal Railroad Administration regulations limit CD to 7 in (178 mm) for tilting passenger vehicles, 3 in (76 mm) for conventional vehicles. [ citation needed ] This FRA regulation is based on AAR standards based on a single study in the 1950s on a rail line in Connecticut. [ citation needed ] In Germany, where axle loads are typically lower than those in the USA, tilting trains are allowed to operate with 12 in (305 mm) CD in some cases[ citation needed ]. CD above 6 in (152 mm) can be considered too uncomfortable for passengers (e.g. things on tables might slide off), except for tilting trains.

The Federal Railroad Administration (FRA) is an agency in the United States Department of Transportation (DOT). The agency was created by the Department of Transportation Act of 1966. The purpose of FRA is to promulgate and enforce rail safety regulations, administer railroad assistance programs, conduct research and development in support of improved railroad safety and national rail transportation policy, provide for the rehabilitation of Northeast Corridor rail passenger service, and consolidate government support of rail transportation activities.

A tilting train is a train that has a mechanism enabling increased speed on regular rail tracks. As a train rounds a curve at speed, objects inside the train experience centrifugal force. This can cause packages to slide about or seated passengers to feel squashed by the outboard armrest, and standing passengers to lose their balance. Tilting trains are designed to counteract this; by tilting the carriages towards the inside of the curve it compensates for the g-force. The train may be constructed such that inertial forces cause the tilting, or it may have a computer-controlled powered mechanism.

The FRA issued new information on cant deficiency in 2009 under FRA-2009-0036-0003. [2] Due to the circumstances outlined, the federal regulations on cant deficiency were amended such that any rail vehicle may operate with up to 3 inches of cant deficiency and any vehicle that is to be operated above this number must be approved by the FRA for such operations. Approval is governed by conditions outlined in CFR chapter 49 section 213.329 part (d) [3] and based on the idea that the car cannot unload the inside wheel on a curve by more than 60% of static loading.

Allowed CD is set below the value that would be allowed based on safety in order to reduce wheel and rail wear and to reduce the rate of degradation of geometry of ballasted track. Choice of design CD will be less constrained by passenger comfort in the case of vehicles that have tilting capability. One historical approach to determining safe cant deficiency was the requirement that the projection to the plane of the track of the resultant of the inertial and gravitational forces acting on a vehicle fall within the middle third of the track gauge. Contemporary engineering studies would likely use vehicle motion simulation including cross wind conditions to determine margins relative to derailment and rollover.

If the superelevation determined for a dedicated passenger route curve on regulatory and safety bases is below 6 in (152.4 mm) it may be desirable to increase the superelevation and reduced the cant deficiency. However, if on such a curve some trains regularly travel at low speeds, then raising the superelevation may be inadvisable for passenger comfort reasons.

On a mixed traffic route owned by a freight rail company, freight considerations are likely to prevail. On a mixed traffic route owned by a passenger rail company some kind of compromise may be needed.

Cant deficiency is generally looked at with respect to ideal track geometry. As geometry of real track is never perfect it may be desirable to supplement the static considerations laid out above with simulations of vehicle motion over measured geometries of actual tracks. Simulations are also desirable for understanding vehicle behaviour traversing spirals, turnouts, and other track segments where curvature changes with distance by design. Where simulations or measurements show non-ideal behaviour traversing traditional linear spirals, results can be improved by using advanced spirals. Good track geometry including advanced spirals is likely to foster passenger acceptance of higher CD values.

## Related Research Articles

A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits.

In physics, the Coriolis force is an inertial or fictitious force that seems to act on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.

In physics, jerk is the rate of change of acceleration; that is, the time derivative of acceleration, and as such the second derivative of velocity, or the third time derivative of position. According to the result of dimensional analysis of jerk, [length/time3], the SI units for its magnitude are m/s3 ; this can also be expressed in standard gravity per second (g/s).

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

In both road and rail vehicles, the wheelbase is the distance between the centers of the front and rear wheels. For road vehicles with more than two axles, the wheelbase is the distance between the steering (front) axle and the centerpoint of the driving axle group. In the case of a tri-axle truck, the wheelbase would be the distance between the steering axle and a point midway between the two rear axles.

A fictitious force is an apparent force that acts on all masses whose motion is described using a non-inertial frame of reference, such as a rotating reference frame. Examples are the forces that act on passengers in an accelerating or braking automobile, and the force that pushes objects toward the rim of a centrifuge.

A derailment occurs when a vehicle such as a train runs off its rails. This does not necessarily mean that it leaves its track. Although many derailments are minor, all result in temporary disruption of the proper operation of the railway system, and they are potentially seriously hazardous to human health and safety. Usually, the derailment of a train can be caused by a collision with another object, an operational error, the mechanical failure of tracks, such as broken rails, or the mechanical failure of the wheels. In emergency situations, deliberate derailment with derails or catch points is sometimes used to prevent a more serious accident.

Rolling resistance, sometimes called rolling friction or rolling drag, is the force resisting the motion when a body rolls on a surface. It is mainly caused by non-elastic effects; that is, not all the energy needed for deformation of the wheel, roadbed, etc. is recovered when the pressure is removed. Two forms of this are hysteresis losses, and permanent (plastic) deformation of the object or the surface. Another cause of rolling resistance lies in the slippage between the wheel and the surface, which dissipates energy. Note that only the last of these effects involves friction, therefore the name "rolling friction" is to an extent a misnomer.

A banked turn is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the inside of the curve. The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to the horizontal.

A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral acceleration. In plane, the start of the transition of the horizontal curve is at infinite radius, and at the end of the transition, it has the same radius as the curve itself and so forms a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of bank is reached.

A track geometry car is an automated track inspection vehicle on a rail transport system used to test several geometric parameters of the track without obstructing normal railroad operations. Some of the parameters generally measured include position, curvature, alignment of the track, smoothness, and the crosslevel of the two rails. The cars use a variety of sensors, measuring systems, and data management systems to create a profile of the track being inspected.

In railroad engineering, curve resistance is a part of train resistance, namely the additional rolling resistance a train must overcome when travelling on a curved section of track. Curve resistance is typically measured in per mille, with the correct physical unit being Newton per kilo-Newton or N/kN. Older texts still use the wrong unit of kilogram-force per tonne or kgf/t, which mixes an (outdated) unit of force and a unit of mass. Sometimes also kg/t was used, which confused the resisting force with a mass.

An Euler spiral is a curve whose curvature changes linearly with its curve length. Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals.

The minimum railway curve radius is the shortest allowable design radius for the center line of railway tracks under a particular set of conditions. It has an important bearing on constructions costs and operating costs and, in combination with superelevation in the case of train tracks, determines the maximum safe speed of a curve. Minimum radius of curve is one parameter in the design of railway vehicles as well as trams. Monorails and guideways are also subject to minimum radii.

Rail speed limits in the United States are regulated by the Federal Railroad Administration. Railroads also implement their own limits and enforce speed limits. Speed restrictions are based on a number of factors including curvature, signaling, track condition, the physical condition of a train, and the presence of grade crossings. Like road speed limits in the United States, speed limits for rail tracks and the trains that run on them use miles per hour (mph).

A roller coaster is a machine that uses gravity and inertia to send a train of cars along a winding track. This combination of gravity and inertia, along with g-forces and centripetal acceleration give the body certain sensations as the coaster moves up, down, and around the track. The forces experienced by the rider are constantly changing, leading to feelings of joy in some riders and nausea in others. The basic principles of roller coaster mechanics have been known since 1865, and since then roller coasters have become a popular diversion.

Track geometry is three-dimensional geometry of track layouts and associated measurements used in design, construction and maintenance of railroad tracks. The subject is used in the context of standards, speed limits and other regulations in the areas of track gauge, alignment, elevation, curvature and track surface. Although, the geometry of the tracks is three-dimensional by nature, the standards are usually expressed in two separate layouts for horizontal and vertical.

## References

1. Marquis, Brian. Cant Deficiency, Curving Speeds and Tilt (PDF). US DOT Volpe National Transportation Systems Center. Retrieved 29 September 2015.
2. "Regulations.gov". www.regulations.gov.