Tractive effort
Encyclopedia
As used in mechanical engineering
, the term tractive force is the pulling or pushing force exerted by a vehicle
on another vehicle or object. The term tractive effort is synonym
ous with tractive force, and is often used in railway engineering to describe the pulling or pushing capability of a locomotive
. The published tractive force value for any vehicle may be theoretical—that is, calculated from known or implied mechanical properties—or obtained via testing under controlled conditions. The discussion herein covers the term's usage in mechanical applications in which the final stage of the power transmission system is one or more wheel
s in friction
al contact with a roadway or railroad track.
to the driving wheels, the wheel diameter, coefficient of friction (μ) between the driving wheels and supporting surface, and the weight applied to the driving wheels (m). The product
of μ and m is the factor of adhesion
, which determines the maximum torque that can be applied before the onset of wheelspin
or wheelslip
.
Vehicles in which a hydrodynamic coupling
, hydrodynamic torque multiplier
or electric motor
is a component of the power transmission system may also have a maximum continuous tractive effort rating, which defines the highest tractive force that can be produced while in motion for a short period of time without causing component harm. The period of time in which the maximum continuous tractive effort may be safely generated is usually limited by thermal considerations. such as temperature rise in a traction motor
.
The basic shape of the graph is shown schematically (diagram right). The line AB shows the operation at the maximum tractive effort, the line BC shows the relationship of continuous tractive effort being inversely proportional to speed.
Tractive effort curves will often have graphs of rolling resistance
superimposed on them—the intersection of the rolling resistance graphThe graphs will typically show rolling resistance for standard train lengths or weights, both on the level or on an uphill gradient and tractive effort graph gives the maximum velocity (i.e. when the net tractive effort is zero).
, axle
bearing
friction
, the friction of the wheels on the rails (which is substantially greater on curved track than on tangent track), and the force of gravity if on a grade. Once in motion, the train will develop additional drag as it accelerates due to aerodynamic forces, which increase with the square of the speed. Drag may also be produced at speed due to truck (bogie) hunting
, which will increase the rolling friction between wheels and rails. If acceleration continues, the train will eventually attain a speed at which the available tractive force of the locomotive(s) will exactly offset the total drag, causing acceleration to cease. This top speed will be increased on a downgrade due to gravity assisting the motive power, and will be decreased on an upgrade due to gravity opposing the motive power.
Tractive effort can be theoretically calculated from a locomotive's mechanical characteristics (e.g., steam pressure, weight, etc.), or by actual testing with drawbar
strain sensor
s and a dynamometer car
. Power at rail is a railway term for the available power for traction, that is, the power that is available to propel the train.
of the piston.and the diameter of the wheel. It can be shown as a first approximation that half the stroke distance is approximately the same as the radial distance from the coupling of the driving rod to the centre of the driven wheel and the diameter of the wheel. The torque developed by the action of the linear motion of the piston depends on the angle that the driving rod makes with the tangent of the radius on the driving wheel.The relationship is simply: Torque = Forcepiston x R (the radial distance to the point of connection of the driving rod) x cos(A), where A is the angle the driving rod makes with the tangent to the radius from wheel centre to driving rod attachment For a more useful value an average value over the rotation of the wheel is used. The driving force is simply the torque divided by the wheel radius.
For a two cylinder locomotive the average force is twice that of a single cylinder locomotive.
Thus as an approximation the following equation can be obtained (for a 2 cylinder locomotive)As with any physical formula, consistent units of measurement
are required: pressure in psi and lengths in inches give tractive effort in lbf, while pressure in Pa and lengths in metres give tractive effort in N.:
where
The constant 0.85 was the Association of American Railroads
(AAR) standard for such calculations, and certainly over-estimated the efficiency of some locomotives and underestimated that of others. Modern locomotives were equipped with roller bearings were probably underestimated.
European designers used a constant of 0.6 instead of 0.85, so the two cannot be directly compared without a conversion factor. In Britain, the main-line railways generally used a constant of 0.85 but builders of industrial locomotives often used a lower figure, typically 0.75.
The value of the constant c also depends on the cylinder dimensions and the time at which the steam inlet valves are open; if the steam inlet valves are closed immediately after obtaining full cylinder pressure the piston force can be expected to have dropped to less than half the initial force.See Gas laws
for an explanation. giving a low c value. If the cylinder valves are left open for longer the value of c will rise nearer to 1.
For other numbers and combinations of cylinders, including double and triple expansion engines the tractive effort can be estimated by adding the tractive efforts due to the individual cylinders at their respective pressures and cylinder strokes.The value of the constant c for a low-pressure cylinder is taken to be 0.80 when the value for a high pressure cylinder is taken to be 0.85
's 2-8-8-8-4 Triplex locomotive, which in simple expansion mode had a starting T.E. of 199,560 lbf (887.7 kN) — but this did not translate into power, for the boiler was undersized and could not produce enough steam to haul at speeds over 5 mph (8 km/h).
Of more successful large steam locomotives, those with the highest rated starting tractive effort were the Virginian Railway AE-class 2-10-10-2
s, at 176,000 lbf (783 kN) in simple-expansion mode. The Union Pacific's famous Big Boys
had a starting T.E. of 135,375 lbf (602 kN); the Norfolk & Western's Y5, Y6, Y6a, and Y6b class 2-8-8-2
s had a starting T.E. of 152,206 lbf (677 kN) in simple expansion mode (later modified, resulting in a claimed T.E. of 170,000 lbf (756 kN)); and the Pennsylvania Railroad
's freight Duplex Q2
attained 114,860 lbf (510.9 kN, including booster) — the highest for a rigid framed locomotive. Later two cylinder passenger locomotives were generally 70,000 to 80,000 lbf (300 to 350 kN) of T.E.
or a Diesel-electric locomotive, starting tractive effort can be calculated from the amount of weight on the driving wheels (which may be less than the total locomotive weight in some cases), combined stall torque
of the traction motor
s, the gear ratio between the traction motors and axles, and driving wheel diameter
. For a Diesel-hydraulic locomotive, the starting tractive effort is affected by the stall torque of the torque converter
, as well as gearing, wheel diameter and locomotive weight.
Freight locomotives are designed to produce higher maximum tractive effort at any given speed than passenger units of equivalent power, necessitated by the much higher weight that is typical of a freight train. In modern locomotives, the gearing between the traction motors and axles is selected to suit the type of service in which the unit will be operated. As traction motors have a maximum speed at which they can rotate without incurring damage, gearing for higher tractive effort is at the expense of top speed. Conversely, the gearing used with passenger locomotives favors speed over maximum tractive effort.
Mechanical engineering
Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the...
, the term tractive force is the pulling or pushing force exerted by a vehicle
Vehicle
A vehicle is a device that is designed or used to transport people or cargo. Most often vehicles are manufactured, such as bicycles, cars, motorcycles, trains, ships, boats, and aircraft....
on another vehicle or object. The term tractive effort is synonym
Synonym
Synonyms are different words with almost identical or similar meanings. Words that are synonyms are said to be synonymous, and the state of being a synonym is called synonymy. The word comes from Ancient Greek syn and onoma . The words car and automobile are synonyms...
ous with tractive force, and is often used in railway engineering to describe the pulling or pushing capability of a locomotive
Locomotive
A locomotive is a railway vehicle that provides the motive power for a train. The word originates from the Latin loco – "from a place", ablative of locus, "place" + Medieval Latin motivus, "causing motion", and is a shortened form of the term locomotive engine, first used in the early 19th...
. The published tractive force value for any vehicle may be theoretical—that is, calculated from known or implied mechanical properties—or obtained via testing under controlled conditions. The discussion herein covers the term's usage in mechanical applications in which the final stage of the power transmission system is one or more wheel
Wheel
A wheel is a device that allows heavy objects to be moved easily through rotating on an axle through its center, facilitating movement or transportation while supporting a load, or performing labor in machines. Common examples found in transport applications. A wheel, together with an axle,...
s in friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
al contact with a roadway or railroad track.
Defining tractive effort
The term tractive effort is often qualified as starting tractive effort, continuous tractive effort and maximum tractive effort. These terms apply to different operating conditions, but are related by common mechanical factors: input torqueTorque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
to the driving wheels, the wheel diameter, coefficient of friction (μ) between the driving wheels and supporting surface, and the weight applied to the driving wheels (m). The product
Product (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
of μ and m is the factor of adhesion
Factor of adhesion
In railroad engineering, the factor of adhesion of a locomotive is the weight on the driving wheels divided by the starting tractive effort.A common rule is that for a steam locomotive a good factor of adhesion equals or exceeds 4, but not by too much...
, which determines the maximum torque that can be applied before the onset of wheelspin
Wheelspin
A wheelspin, occurs when the force delivered to the tire tread exceeds that of available tread-to-surface friction and one or more tires lose traction.Standard differentials always apply equal torque to each wheel...
or wheelslip
Locomotive wheelslip
Locomotive wheelslip is an event that affects railway motive power when starting from stationary.The greatest effort is required from a locomotive when starting...
.
- Starting tractive effort: Starting tractive effort is defined as the tractive force that can be generated at a standstill. This figure is of great importance in railway engineering because it implicitly limits the maximum train weight that a locomotive can set into motion from a dead stop.
- Maximum tractive effort: Maximum tractive effort is defined as the highest tractive force that can be generated under any condition that is not injurious to the vehicle or machine. In most cases, maximum tractive effort is developed at low speedSpeedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
and may be the same as the starting tractive effort value.
- Continuous tractive effort: Continuous tractive effort is the tractive force that can be generated at any given speed. Due to the relationship between powerPower (physics)In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...
(P), velocity (v) and force (F), described as:
- P = vF or P/v = F
- continuous tractive effort will inversely vary with speed at any given level of available power. Continuous tractive effort is often represented in graph form at a range of speeds as part of a tractive effort curve.
Vehicles in which a hydrodynamic coupling
Fluid coupling
A fluid coupling is a hydrodynamic device used to transmit rotating mechanical power. It has been used in automobile transmissions as an alternative to a mechanical clutch...
, hydrodynamic torque multiplier
Torque converter
In modern usage, a torque converter is generally a type of hydrodynamic fluid coupling that is used to transfer rotating power from a prime mover, such as an internal combustion engine or electric motor, to a rotating driven load...
or electric motor
Electric motor
An electric motor converts electrical energy into mechanical energy.Most electric motors operate through the interaction of magnetic fields and current-carrying conductors to generate force...
is a component of the power transmission system may also have a maximum continuous tractive effort rating, which defines the highest tractive force that can be produced while in motion for a short period of time without causing component harm. The period of time in which the maximum continuous tractive effort may be safely generated is usually limited by thermal considerations. such as temperature rise in a traction motor
Traction motor
Traction motor refers to an electric motor providing the primary rotational torque of a machine, usually for conversion into linear motion ....
.
Tractive effort curves
Technical specifications of locomotives often include tractive effort curves, which show the relationship between tractive effort and velocity.The basic shape of the graph is shown schematically (diagram right). The line AB shows the operation at the maximum tractive effort, the line BC shows the relationship of continuous tractive effort being inversely proportional to speed.
Tractive effort curves will often have graphs of rolling resistance
Rolling resistance
Rolling resistance, sometimes called rolling friction or rolling drag, is the resistance that occurs when a round object such as a ball or tire rolls on a flat surface, in steady velocity straight line motion. It is caused mainly by the deformation of the object, the deformation of the surface, or...
superimposed on them—the intersection of the rolling resistance graphThe graphs will typically show rolling resistance for standard train lengths or weights, both on the level or on an uphill gradient and tractive effort graph gives the maximum velocity (i.e. when the net tractive effort is zero).
Rail vehicles
In order to start a train and accelerate it to a given speed, the locomotive(s) must develop sufficient tractive force to overcome the train's drag (resistance to motion), which is a combination of inertiaInertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
, axle
Axle
An axle is a central shaft for a rotating wheel or gear. On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to its surroundings, with the wheels rotating around the axle. In the former case, bearings or bushings are provided at the mounting points where the axle...
bearing
Bearing (mechanical)
A bearing is a device to allow constrained relative motion between two or more parts, typically rotation or linear movement. Bearings may be classified broadly according to the motions they allow and according to their principle of operation as well as by the directions of applied loads they can...
friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
, the friction of the wheels on the rails (which is substantially greater on curved track than on tangent track), and the force of gravity if on a grade. Once in motion, the train will develop additional drag as it accelerates due to aerodynamic forces, which increase with the square of the speed. Drag may also be produced at speed due to truck (bogie) hunting
Hunting oscillation
Hunting oscillation is an oscillation, usually unwanted, about an equilibrium. The expression came into use in the 19th century and describes how a systems 'hunts' for equilibrium...
, which will increase the rolling friction between wheels and rails. If acceleration continues, the train will eventually attain a speed at which the available tractive force of the locomotive(s) will exactly offset the total drag, causing acceleration to cease. This top speed will be increased on a downgrade due to gravity assisting the motive power, and will be decreased on an upgrade due to gravity opposing the motive power.
Tractive effort can be theoretically calculated from a locomotive's mechanical characteristics (e.g., steam pressure, weight, etc.), or by actual testing with drawbar
Drawbar (haulage)
A drawbar is a solid coupling between a hauling vehicle and its hauled load. Drawbars are in common use with rail transport, road trailers, both large and small, industrial and recreational, and with agricultural equipment.-Agriculture:...
strain sensor
Strain gauge
A strain gauge is a device used to measure the strain of an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern. The gauge is attached to the object by a suitable...
s and a dynamometer car
Dynamometer car
A dynamometer car is a railroad maintenance of way car used for measuring various aspects of a locomotive's performance. Measurements include tractive effort , power, top speed, etc.-History:...
. Power at rail is a railway term for the available power for traction, that is, the power that is available to propel the train.
Steam locomotives
An approximate theoretical value for the tractive effort of a single cylinder steam locomotive can be obtained by considering the cylinder pressure, cylinder area, strokeStroke (engines)
Reciprocating motion, used in reciprocating engines and other mechanisms, is back-and-forth motion. Each cycle of reciprocation consists of two opposite motions: there is a motion in one direction, and then a motion back in the opposite direction. Each of these is called a stroke...
of the piston.and the diameter of the wheel. It can be shown as a first approximation that half the stroke distance is approximately the same as the radial distance from the coupling of the driving rod to the centre of the driven wheel and the diameter of the wheel. The torque developed by the action of the linear motion of the piston depends on the angle that the driving rod makes with the tangent of the radius on the driving wheel.The relationship is simply: Torque = Forcepiston x R (the radial distance to the point of connection of the driving rod) x cos(A), where A is the angle the driving rod makes with the tangent to the radius from wheel centre to driving rod attachment For a more useful value an average value over the rotation of the wheel is used. The driving force is simply the torque divided by the wheel radius.
For a two cylinder locomotive the average force is twice that of a single cylinder locomotive.
Thus as an approximation the following equation can be obtained (for a 2 cylinder locomotive)As with any physical formula, consistent units of measurement
Units of measurement
A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...
are required: pressure in psi and lengths in inches give tractive effort in lbf, while pressure in Pa and lengths in metres give tractive effort in N.:
where
- t is tractive effort
- c is a constant representing losses in pressure and friction; normally 0.85 is usedFor a 'perfect' locomotive with cylinder piston pressure equal to boiler pressure (independent of stroke) and with no frictional losses the constant c can be taken as 1
- P is the boilerBoilerA boiler is a closed vessel in which water or other fluid is heated. The heated or vaporized fluid exits the boiler for use in various processes or heating applications.-Materials:...
pressurenote that the boiler pressure may be greater than the cylinder pressure - d is the pistonPistonA piston is a component of reciprocating engines, reciprocating pumps, gas compressors and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-tight by piston rings. In an engine, its purpose is to transfer force from...
diameter (bore) - s is the piston stroke
- D is the driving wheelDriving wheelOn a steam locomotive, a driving wheel is a powered wheel which is driven by the locomotive's pistons...
diameter
The constant 0.85 was the Association of American Railroads
Association of American Railroads
The Association of American Railroads is an industry trade group representing primarily the major freight railroads of North America . Amtrak and some regional commuter railroads are also members...
(AAR) standard for such calculations, and certainly over-estimated the efficiency of some locomotives and underestimated that of others. Modern locomotives were equipped with roller bearings were probably underestimated.
European designers used a constant of 0.6 instead of 0.85, so the two cannot be directly compared without a conversion factor. In Britain, the main-line railways generally used a constant of 0.85 but builders of industrial locomotives often used a lower figure, typically 0.75.
The value of the constant c also depends on the cylinder dimensions and the time at which the steam inlet valves are open; if the steam inlet valves are closed immediately after obtaining full cylinder pressure the piston force can be expected to have dropped to less than half the initial force.See Gas laws
Gas laws
The early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases...
for an explanation. giving a low c value. If the cylinder valves are left open for longer the value of c will rise nearer to 1.
For other numbers and combinations of cylinders, including double and triple expansion engines the tractive effort can be estimated by adding the tractive efforts due to the individual cylinders at their respective pressures and cylinder strokes.The value of the constant c for a low-pressure cylinder is taken to be 0.80 when the value for a high pressure cylinder is taken to be 0.85
Values and comparisons for steam locomotives
Tractive effort is the figure most often quoted when comparing the power of different steam locomotives, but its use can be misleading, because tractive effort shows the ability to start a train, not the ability haul it. Possibly the highest figure for starting tractive effort ever recorded was for the Virginian RailwayVirginian Railway
The Virginian Railway was a Class I railroad located in Virginia and West Virginia in the United States. The VGN was created to transport high quality "smokeless" bituminous coal from southern West Virginia to port at Hampton Roads....
's 2-8-8-8-4 Triplex locomotive, which in simple expansion mode had a starting T.E. of 199,560 lbf (887.7 kN) — but this did not translate into power, for the boiler was undersized and could not produce enough steam to haul at speeds over 5 mph (8 km/h).
Of more successful large steam locomotives, those with the highest rated starting tractive effort were the Virginian Railway AE-class 2-10-10-2
2-10-10-2
Under the Whyte notation for the classification of steam locomotive wheel arrangements, a 2-10-10-2 is a locomotive with two leading wheels, two sets of ten driving wheels, and a pair of trailing wheels.Other equivalent classifications are:...
s, at 176,000 lbf (783 kN) in simple-expansion mode. The Union Pacific's famous Big Boys
Union Pacific Big Boy
Big Boy was the name of the Union Pacific Railroad's 4000-class 4-8-8-4 articulated steam locomotives, built between 1941 and 1944 by American Locomotive Company...
had a starting T.E. of 135,375 lbf (602 kN); the Norfolk & Western's Y5, Y6, Y6a, and Y6b class 2-8-8-2
2-8-8-2
.A 2-8-8-2, in the Whyte notation for describing steam locomotive wheel arrangements, is an articulated locomotive with a two-wheel leading truck, two sets of eight driving wheels, and a two-wheel trailing truck. The equivalent UIC classification is, refined to Mallet locomotives, D1...
s had a starting T.E. of 152,206 lbf (677 kN) in simple expansion mode (later modified, resulting in a claimed T.E. of 170,000 lbf (756 kN)); and the Pennsylvania Railroad
Pennsylvania Railroad
The Pennsylvania Railroad was an American Class I railroad, founded in 1846. Commonly referred to as the "Pennsy", the PRR was headquartered in Philadelphia, Pennsylvania....
's freight Duplex Q2
PRR Q2
The Pennsylvania Railroad's class Q2 comprised one prototype and twenty-five production duplex steam locomotives of 4-4-6-4 wheel arrangement....
attained 114,860 lbf (510.9 kN, including booster) — the highest for a rigid framed locomotive. Later two cylinder passenger locomotives were generally 70,000 to 80,000 lbf (300 to 350 kN) of T.E.
Diesel and electric locomotives
For an electric locomotiveElectric locomotive
An electric locomotive is a locomotive powered by electricity from overhead lines, a third rail or an on-board energy storage device...
or a Diesel-electric locomotive, starting tractive effort can be calculated from the amount of weight on the driving wheels (which may be less than the total locomotive weight in some cases), combined stall torque
Stall torque
Stall torque is the torque which is produced by a device when the output rotational speed is zero. It may also mean the torque load that causes the output rotational speed of a device to become zero - i.e...
of the traction motor
Traction motor
Traction motor refers to an electric motor providing the primary rotational torque of a machine, usually for conversion into linear motion ....
s, the gear ratio between the traction motors and axles, and driving wheel diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
. For a Diesel-hydraulic locomotive, the starting tractive effort is affected by the stall torque of the torque converter
Torque converter
In modern usage, a torque converter is generally a type of hydrodynamic fluid coupling that is used to transfer rotating power from a prime mover, such as an internal combustion engine or electric motor, to a rotating driven load...
, as well as gearing, wheel diameter and locomotive weight.
Freight locomotives are designed to produce higher maximum tractive effort at any given speed than passenger units of equivalent power, necessitated by the much higher weight that is typical of a freight train. In modern locomotives, the gearing between the traction motors and axles is selected to suit the type of service in which the unit will be operated. As traction motors have a maximum speed at which they can rotate without incurring damage, gearing for higher tractive effort is at the expense of top speed. Conversely, the gearing used with passenger locomotives favors speed over maximum tractive effort.
See also
- Factor of adhesionFactor of adhesionIn railroad engineering, the factor of adhesion of a locomotive is the weight on the driving wheels divided by the starting tractive effort.A common rule is that for a steam locomotive a good factor of adhesion equals or exceeds 4, but not by too much...
, which is simply the weight on the locomotive's driving wheels divided by the starting tractive effort - Tractor pullingTractor pullingTruck and Tractor pulling, also known as power pulling, is a motorsport competition, popular in America, Europe , Australia and Brazil, which requires modified tractors to pull a heavy sledge along a 35ft. wide and length of 100 metre or 300ft+ track, with the winner being the tractor that pulls...
, bollard pullBollard pullBollard pull is a value that allows the comparison of the pulling force of watercraft, particularly tugboats. A mooring bollard may be used as a point of attachment for measuring the force, or pull of the craft.-Background:...
- articles relating to tractive effort for other forms of vehicle - Rail adhesionRail adhesionThe term adhesion railway or adhesion traction describes the most common type of railway, where power is applied by driving some or all of the wheels of the locomotive. Thus, it relies on the friction between a steel wheel and a steel rail. Note that steam locomotives of old were driven only by...
- Power classification - British Railways and London, Midland and Scottish railwayLondon, Midland and Scottish RailwayThe London Midland and Scottish Railway was a British railway company. It was formed on 1 January 1923 under the Railways Act of 1921, which required the grouping of over 120 separate railway companies into just four...
classification scheme - Drag equationDrag equationIn fluid dynamics, the drag equation is a practical formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid....