Banked turn
Encyclopedia
A banked turn is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to its path.
is vertically upwards, so the only force keeping the vehicle turning on its path is friction
, or traction
. This must be large enough to provide the centripetal acceleration, a relationship which can be expressed as an inequality,
The expression on the right hand side is the centripetal acceleration multiplied by mass, so the force required to turn the vehicle. The left hand side is the maximum frictional force, which equals the coefficient of friction μ multiplied by the normal force. Rearranging the maximum cornering speed is
Note that μ can be the coefficient for static or dynamic friction. In the latter case, where the vehicle is skidding around a bend, the friction is at its limit and the inequalities becomes equations. This also ignores effects such as downforce
which can increase the normal force and cornering speed.
But once the car starts to move on a banked turn, it 'collides' with the turn itself, the turn feels this force, and returns it in the normal, causing the car to move circularly.
The Normal is thus greater than simply its gravitational component.
If the Normal were simply the component to gravity, one could not say mg = Ncosθ = mgcos2θ when the vertical acceleration is 0.
As opposed to a car riding along a flat circle, inclined edges add an additional force that keeps the car in its path and prevents it from being "dragged into" or "pushed out of" the circle. This force is the horizontal component of the car's normal force. In the absence of friction, the normal force is the only one acting on the car in the direction of the center of the circle. Therefore, as per Newton's second law, we can set the horizontal component of the normal force equal to mass multiplied by centripetal acceleration:
Because there is no motion in the vertical direction, the sum of all vertical forces acting on the system must be zero. Therefore we can set the vertical component of the car's normal force equal to its weight:
Solving the above equation for the normal force and substituting this value into our previous equation, we get:
Solving for velocity we have:
This provides the velocity that in the absence of friction and with a given angle of incline and radius of curvature, will ensure that the car will remain in its designated path. The magnitude of this velocity is also known as the "rated speed" of a turn or curve. Notice that the rated speed of the curve is the same for all massive objects, and a curve that is not inclined will have a rated speed of 0.
Once again, there is no motion in the vertical direction, allowing us to set all opposing vertical forces equal to one another. These forces include the vertical component of the normal force pointing upwards and both the car's weight and vertical component of friction pointing downwards:
By solving the above equation for mass and substituting this value into our previous equation we get:
Solving for v we get:
This equation provides the maximum velocity for the automobile with the given angle of incline, coefficient of static friction and radius of curvature. By a similar analysis of minimum velocity, the following equation is rendered:
The difference in the latter analysis comes when considering the direction of friction for the minimum velocity of the automobile (towards the outside of the circle). Consequently opposite operations are performed when inserting friction into equations for forces in the centripetal and vertical directions.
Improperly banked road curves increase the risk of run-off-road and head-on crashes. A 2% deficiency in superelevation (say, 4% superelevation on a curve that should have 6%) can be expected to increase crash frequency by 6%, and a 5% deficiency will increase it by 15%. Up until now, highway engineers have been without efficient tools to identify improperly banked curves and to design relevant mitigating road actions. A modern profilograph
can provide data of both road curvature
and cross slope
(angle of incline). A practical demonstration of how to evaluate improperly banked turns was developed in the EU Roadex III project, see the linked referenced document below.
When a fixed-wing aircraft
is making a turn (changing its direction) the aircraft must roll to a banked position so that its wing
s are angled towards the desired direction of the turn. When the turn has been completed the aircraft must roll back to the wings-level position in order to resume straight flight.
When any moving vehicle is making a turn, it is necessary for the forces acting on the vehicle to add up to a net inward force, to cause centripetal acceleration. In the case of an aircraft making a turn, the force causing centripetal acceleration is the horizontal component of the lift
acting on the aircraft.
In straight, level flight, the lift acting on the aircraft acts vertically upwards to counteract the weight of the aircraft which acts downwards. During a balanced turn where the angle of bank is θ the lift acts at an angle θ away from the vertical. It is useful to resolve the lift into a vertical component and a horizontal component. If the aircraft is to continue in level flight (i.e. at constant altitude), the vertical component must continue to equal the weight of the aircraft. The horizontal component is unbalanced, and is thus the net force
causing the aircraft to accelerate inward and execute the turn.
During a banked turn in level flight the lift on the aircraft must support the weight of the aircraft, as well as provide the necessary component of horizontal force to cause centripetal acceleration. Consequently, the lift required in a banked turn is greater than that one required in straight, level flight and can be achieved either by increasing the angle of attack
of the wing (typically by pulling on the elevator
control) or by deploying flaps
. The maneuver is usually complemented by an increase in power, in order to maintain airspeed.
Because centripetal acceleration is:
Newton's second law in the horizontal direction can be expressed mathematically as:
where:
In straight flight, lift is approximately equal to the aircraft weight. In turning flight the lift exceeds the aircraft weight, and is equal to the weight of the aircraft (mg) divided by the cosine of the angle of bank:
where g
is the gravitational field strength.
The radius of the turn can now be calculated:
This formula shows that the radius of turn is proportional to the square of the aircraft’s true airspeed
. With a higher airspeed the radius of turn is larger, and with a lower airspeed the radius is smaller.
This formula also shows that the radius of turn is inversely proportional to the angle of bank. With a higher angle of bank the radius of turn is smaller, and with a lower angle of bank the radius is greater.
The angle of bank is the sole determinant of the aircraft’s load factor during the turn.
Aeronautics
Turn on flat surfaces
If the bank angle is zero the surface is flat, simplifying the calculations, and the vehicle is just driving in a circle. The normal forceNormal force
In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...
is vertically upwards, so the only force keeping the vehicle turning on its path is 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:...
, or traction
Traction (engineering)
Traction refers to the maximum frictional force that can be produced between surfaces without slipping.The units of traction are those of force, or if expressed as a coefficient of traction a ratio.-Traction:...
. This must be large enough to provide the centripetal acceleration, a relationship which can be expressed as an inequality,
The expression on the right hand side is the centripetal acceleration multiplied by mass, so the force required to turn the vehicle. The left hand side is the maximum frictional force, which equals the coefficient of friction μ multiplied by the normal force. Rearranging the maximum cornering speed is
Note that μ can be the coefficient for static or dynamic friction. In the latter case, where the vehicle is skidding around a bend, the friction is at its limit and the inequalities becomes equations. This also ignores effects such as downforce
Downforce
Downforce is a downwards thrust created by the aerodynamic characteristics of a car. The purpose of downforce is to allow a car to travel faster through a corner by increasing the vertical force on the tires, thus creating more grip....
which can increase the normal force and cornering speed.
Normal reaction in a banked turn
In the case of a car being parked on a banked turn, the Normal force would simply be:But once the car starts to move on a banked turn, it 'collides' with the turn itself, the turn feels this force, and returns it in the normal, causing the car to move circularly.
The Normal is thus greater than simply its gravitational component.
If the Normal were simply the component to gravity, one could not say mg = Ncosθ = mgcos2θ when the vertical acceleration is 0.
Frictionless banked turn
Because there is no motion in the vertical direction, the sum of all vertical forces acting on the system must be zero. Therefore we can set the vertical component of the car's normal force equal to its weight:
Solving the above equation for the normal force and substituting this value into our previous equation, we get:
Solving for velocity we have:
This provides the velocity that in the absence of friction and with a given angle of incline and radius of curvature, will ensure that the car will remain in its designated path. The magnitude of this velocity is also known as the "rated speed" of a turn or curve. Notice that the rated speed of the curve is the same for all massive objects, and a curve that is not inclined will have a rated speed of 0.
Banked turn with friction
When considering the effects of friction on the system, once again we need to note which way the friction force is pointing. When calculating a maximum velocity for our automobile, friction will point down the incline and towards the center of the circle. Therefore we must add the horizontal component of friction to that of the normal force. The sum of these two forces is our new net force in the centripetal direction:Once again, there is no motion in the vertical direction, allowing us to set all opposing vertical forces equal to one another. These forces include the vertical component of the normal force pointing upwards and both the car's weight and vertical component of friction pointing downwards:
By solving the above equation for mass and substituting this value into our previous equation we get:
Solving for v we get:
This equation provides the maximum velocity for the automobile with the given angle of incline, coefficient of static friction and radius of curvature. By a similar analysis of minimum velocity, the following equation is rendered:
The difference in the latter analysis comes when considering the direction of friction for the minimum velocity of the automobile (towards the outside of the circle). Consequently opposite operations are performed when inserting friction into equations for forces in the centripetal and vertical directions.
Improperly banked road curves increase the risk of run-off-road and head-on crashes. A 2% deficiency in superelevation (say, 4% superelevation on a curve that should have 6%) can be expected to increase crash frequency by 6%, and a 5% deficiency will increase it by 15%. Up until now, highway engineers have been without efficient tools to identify improperly banked curves and to design relevant mitigating road actions. A modern profilograph
Profilograph
The profilograph is a device used to measure pavement surface roughness. In the early 20th century, Profilographs were low speed rolling devices. Today many Profilographs are advanced high speed systems with a laserbased height sensor in combination with a inertial system that creates a large...
can provide data of both road curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
and cross slope
Cross slope
Cross slope is a geometric feature of pavement surfaces; the transversal slope [%] with respect to the horizon. It is a very important safety factor. One task is to make water run off the surface to a drainage system, as Cross Slope is the main contributor to Pavement Drainage gradient. Very low...
(angle of incline). A practical demonstration of how to evaluate improperly banked turns was developed in the EU Roadex III project, see the linked referenced document below.
Banked turn in aeronautics
Fixed-wing aircraft
A fixed-wing aircraft is an aircraft capable of flight using wings that generate lift due to the vehicle's forward airspeed. Fixed-wing aircraft are distinct from rotary-wing aircraft in which wings rotate about a fixed mast and ornithopters in which lift is generated by flapping wings.A powered...
is making a turn (changing its direction) the aircraft must roll to a banked position so that its wing
Wing
A wing is an appendage with a surface that produces lift for flight or propulsion through the atmosphere, or through another gaseous or liquid fluid...
s are angled towards the desired direction of the turn. When the turn has been completed the aircraft must roll back to the wings-level position in order to resume straight flight.
When any moving vehicle is making a turn, it is necessary for the forces acting on the vehicle to add up to a net inward force, to cause centripetal acceleration. In the case of an aircraft making a turn, the force causing centripetal acceleration is the horizontal component of the lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...
acting on the aircraft.
In straight, level flight, the lift acting on the aircraft acts vertically upwards to counteract the weight of the aircraft which acts downwards. During a balanced turn where the angle of bank is θ the lift acts at an angle θ away from the vertical. It is useful to resolve the lift into a vertical component and a horizontal component. If the aircraft is to continue in level flight (i.e. at constant altitude), the vertical component must continue to equal the weight of the aircraft. The horizontal component is unbalanced, and is thus the net force
Net force
In physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together...
causing the aircraft to accelerate inward and execute the turn.
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
of the wing (typically by pulling on the elevator
Elevator (aircraft)
Elevators are flight control surfaces, usually at the rear of an aircraft, which control the aircraft's orientation by changing the pitch of the aircraft, and so also the angle of attack of the wing. In simplified terms, they make the aircraft nose-up or nose-down...
control) or by deploying flaps
Flap (aircraft)
Flaps are normally hinged surfaces mounted on the trailing edges of the wings of a fixed-wing aircraft to reduce the speed an aircraft can be safely flown at and to increase the angle of descent for landing without increasing air speed. They shorten takeoff and landing distances as well as...
. The maneuver is usually complemented by an increase in power, in order to maintain airspeed.
Because centripetal acceleration is:
Newton's second law in the horizontal direction can be expressed mathematically as:
where:
- L is the lift acting on the aircraft
- θ is the angle of bank of the aircraft
- m is the massMassMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of the aircraft - v is the true airspeedTrue airspeedTrue airspeed of an aircraft is the speed of the aircraft relative to the airmass in which it is flying. True airspeed is important information for accurate navigation of an aircraft.-Performance:...
of the aircraft - r is the radius of the turn
In straight flight, lift is approximately equal to the aircraft weight. In turning flight the lift exceeds the aircraft weight, and is equal to the weight of the aircraft (mg) divided by the cosine of the angle of bank:
where g
Earth's gravity
The gravity of Earth, denoted g, refers to the acceleration that the Earth imparts to objects on or near its surface. In SI units this acceleration is measured in metres per second per second or equivalently in newtons per kilogram...
is the gravitational field strength.
The radius of the turn can now be calculated:
This formula shows that the radius of turn is proportional to the square of the aircraft’s true airspeed
True airspeed
True airspeed of an aircraft is the speed of the aircraft relative to the airmass in which it is flying. True airspeed is important information for accurate navigation of an aircraft.-Performance:...
. With a higher airspeed the radius of turn is larger, and with a lower airspeed the radius is smaller.
This formula also shows that the radius of turn is inversely proportional to the angle of bank. With a higher angle of bank the radius of turn is smaller, and with a lower angle of bank the radius is greater.
The angle of bank is the sole determinant of the aircraft’s load factor during the turn.
See also
- Camber angleCamber anglethumb|100px|From the front of the car, a right wheel with a negative camber angleCamber angle is the angle made by the wheels of a vehicle; specifically, it is the angle between the vertical axis of the wheels used for steering and the vertical axis of the vehicle when viewed from the front or...
- Coriolis force (perception)
- Centripetal force
- Cant (road/rail)Cant (road/rail)The cant of a railway track or a road is the difference in elevation between the two edges...
- g-forceG-forceThe g-force associated with an object is its acceleration relative to free-fall. This acceleration experienced by an object is due to the vector sum of non-gravitational forces acting on an object free to move. The accelerations that are not produced by gravity are termed proper accelerations, and...
- Superelevation
External links
Surface vehicles- http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/imgmech/carbank.gif
- http://whitts.alioth.net
- http://www.batesville.k12.in.us/physics/PHYNET/Mechanics/Circular%20Motion/banked_no_friction.htm
Aeronautics