EN3: Introduction to Engineering and Statics
Division of Engineering
Brown University
2. Forces
We start with a very basic question. What is a force?
Engineering design calculations always use classical (Newtonian) mechanics. In classical mechanics, the concept of a `force’ is based on experimental observations that everything in the universe seems to have a preferred configuration – masses appear to attract each other; objects with opposite charges attract one another; magnets can repel or attract one another; you are probably repelled by your professor. But we don’t really know why this is (except perhaps the last one).
The idea of a force is introduced to quantify the tendency of objects to move towards their preferred configuration. If objects accelerate very quickly towards their preferred configuration, then we say that there’s a big force acting on them. If they don’t move (or move at constant velocity), then there is no force. We can’t see a force; we can only deduce its existence by observing its effect.
Specifically, forces are defined through Newton’s laws of motion
0. A `particle’ is a small mass at some
position in space.
1. When the sum of the forces acting on a particle is zero, its velocity is constant;
2. The sum of forces acting on a particle of constant mass is equal to the product of the mass of the particle and its acceleration;
3. The forces exerted by two particles on each other are equal in magnitude and opposite in direction.
Isaac Newton on a bad hair day
The second law provides the definition of a force – if a mass m has acceleration a, the force F acting on it is
Of course, there is a big problem with Newton’s laws – what do we take as a fixed point (and orientation) in order to define acceleration? The general theory of relativity addresses this issue rigorously. But for engineering calculations we can usually take the earth to be fixed, and happily apply Newton’s laws. In rare cases where the earth’s motion is important, we take the stars far from the solar system to be fixed.
Forces may arise from a number of different effects, including
(i) Gravity;
(ii) Electromagnetism or electrostatics;
(iii) Pressure exerted by fluid or gas on part of a structure
(v) Wind or fluid induced drag or lift forces;
(vi) Contact forces, which act wherever a structure or component touches anything;
(vii) Friction forces, which also act at contacts.
Some of these forces can be described by universal laws. For example, gravity forces can be calculated using Newton’s law of gravitation; electrostatic forces acting between charged particles are governed by Coulomb’s law; electromagnetic forces acting between current carrying wires are governed by Ampere’s law; buoyancy forces are governed by laws describing hydrostatic forces in fluids. Some of these universal force laws are listed in Section 2.6.
Some forces have to be measured. For example, to determine friction forces acting in a machine, you may need to measure the coefficient of friction for the contacting surfaces. Similarly, to determine aerodynamic lift or drag forces acting on a structure, you would probably need to measure its lift and drag coefficient experimentally. Lift and drag forces are described in Section 2.6. Friction forces are discussed in Section 12.
Contact forces are pressures that act on the small area of contact between two objects. Contact forces can either be measured, or they can be calculated by analyzing forces and deformation in the system of interest. Contact forces are very complicated, and are discussed in more detail in Section 8.
2.3 Units of force and typical magnitudes
In SI units, the standard unit of force is the Newton, given the symbol N.
The Newton is a derived unit, defined through Newton’s second law of motion – a force of 1N causes a 1 kg mass to accelerate at 1 .
The fundamental unit of force in the SI convention is kg m/s2
In US units, the standard unit of force is the pound, given the symbol lb or lbf (the latter is an abbreviation for pound force, to distinguish it from pounds weight)
A force of 1 lbf causes a mass of 1 slug to accelerate at 1 ft/s2
US units have a frightfully confusing way of representing mass – often the mass of an object is reported as weight, in lb or lbm (the latter is an abbreviation for pound mass). The weight of an object in lb is not mass at all – it’s actually the gravitational force acting on the mass. Therefore, the mass of an object in slugs must be computed from its weight in pounds using the formula
where g=32.1740 ft/s2 is the acceleration due to gravity.
A force of 1 lb(f) causes a mass of 1 lb(m) to accelerate at 32.1740 ft/s2
The conversion factors from lb to N are
|
1 lb |
= |
4.448 N |
|
1 N |
= |
0.2248 lb |
(www.onlineconversion.com is a handy resource, as long as you can tolerate all the hideous advertisements…)
As a rough guide, a force of 1N is about equal to the weight of a medium sized apple. A few typical force magnitudes (from `The Sizesaurus’, by Stephen Strauss, Avon Books, NY, 1997) are listed in the table below
|
Force |
Newtons |
Pounds Force |
|
Gravitational Pull of the Sun on Earth |
||
|
Gravitational Pull of the Earth on the Moon |
||
|
Thrust of a Saturn V rocket engine |
||
|
Thrust of a large jet engine |
||
|
Pull of a large locomotive |
||
|
Force between two protons in a nucleus |
||
|
Gravitational pull of the earth on a person |
||
|
Maximum force exerted upwards by a forearm |
||
|
Gravitational pull of the earth on a 5 cent coin |
||
|
Force between an electron and the nucleus of a Hydrogen atom |
2.4 Classification of forces: External forces, constraint forces and internal forces.
When analyzing forces in a structure or machine, it is conventional to classify forces as external forces; constraint forces or internal forces.
External forces arise from interaction between the system of interest and its surroundings.
Examples of external forces include gravitational forces; lift or drag forces arising from wind loading; electrostatic and electromagnetic forces; and buoyancy forces; among others. Force laws governing these effects are listed later in this section.
Constraint forces are exerted by one part of a structure on another, through joints, connections or contacts between components. Constraint forces are very complex, and will be discussed in detail in Section 8.
Internal forces are forces that act inside a solid part of a structure or component. For example, a stretched rope has a tension force acting inside it, holding the rope together. Most solid objects contain very complex distributions of internal force. These internal forces ultimately lead to structural failure, and also cause the structure to deform. The purpose of calculating forces in a structure or component is usually to deduce the internal forces, so as to be able to design stiff, lightweight and strong components. We will not, unfortunately, be able to develop a full theory of internal forces in this course – a proper discussion requires understanding of partial differential equations, as well as vector and tensor calculus. However, a brief discussion of internal forces in slender members will be provided in Section 9.
2.5 Mathematical representation of a force.
Force is a vector – it has a magnitude (specified in Newtons, or lbf, or whatever), and a direction.
A force is therefore always expressed mathematically as a vector quantity. To do so, we follow the usual rules, which are described in more detail in the vector tutorial. The procedure is
1. Choose basis vectors or that establish three fixed (and usually perpendicular) directions in space;
2. Using geometry or trigonometry, calculate the force component along each of the three reference directions or ;
3. The vector force is then reported as
For calculations, you will also need to specify the point where the force acts on your system or structure. To do this, you need to report the position vector of the point where the force acts on the structure.
The procedure for representing a position vector is also described in detail in the vector tutorial. To do so, you need to:
1. Choose an origin
2. Choose basis vectors or that establish three fixed directions in space (usually we use the same basis for both force and position vectors)
3. Specify the distance you need to travel along each direction to get from the origin to the point of application of the force or
4. The position vector is then reported as
Engineers often need to measure forces. According to the definition, if we want to measure a force, we need to get hold of a 1 kg mass, have the force act on it somehow, and then measure the acceleration of the mass. The magnitude of the acceleration tells us the magnitude of the force; the direction of motion of the mass tells us the direction of the force. Fortunately, there are easier ways to measure forces.
In addition to causing acceleration, forces cause objects to deform – for example, a force will stretch or compress a spring; or bend a beam. The deformation can be measured, and the force can be deduced.
The simplest application of this phenomenon is a spring scale. The change in length of a spring is proportional to the magnitude of the force causing it to stretch (so long as the force is not too large!)– this relationship is known as Hooke’s law and can be expressed as an equation
where the spring stiffness depends on the material the spring is made from, and the shape of the spring. The spring stiffness can be measured experimentally to calibrate the spring.
Spring scales are not exactly precision instruments, of course. But the same principle is used in more sophisticated instruments too. Forces can be measured precisely using a `force transducer’ or `load cell’ (A search for `force transducer’ on any search engine will bring up a huge variety of these – a few are shown in the picture). The simplest load cell works much like a spring scale – you can load it in one direction, and it will provide an electrical signal proportional to the magnitude of the force. Sophisticated load cells can measure a force vector, and will record all three force components. Really fancy load cells measure both force vectors, and torque or moment vectors.
Simple force transducers capable of measuring a single force component. The instrument on the right is called a `proving ring’ – there’s a short article describing how it works at http://www.mel.nist.gov/div822/proving_ring.htm
A sophisticated force transducer produced by MTS systems, which is capable of measuring forces and moments acting on a car’s wheel in-situ. The spec for this device can be downloaded at www.mts.com/downloads/SWIF2002_100-023-513.pdf.pdf
The basic design of all these load cells is the same – they measure (very precisely) the deformation in a part of the cell that acts like a very stiff spring. One example (from http://www.sandia.gov/isrc/Load_Cell/load_cell.html ) is shown on the right. In this case the `spring’ is actually a tubular piece of high-strength steel. When a force acts on the cylinder, its length decreases slightly. The deformation is detected using `strain gages’ attached to the cylinder. A strain gage is really just a thin piece of wire, which deforms with the cylinder. When the wire gets shorter, its electrical resistance decreases – this resistance change can be measured, and can be used to work out the force. It is possible to derive a formula relating the force to the change in resistance, the load cell geometry, and the material properties of steel, but the calculations involved are well beyond the scope of this course.
The
most sensitive load cell currently available is the atomic force microscope
(AFM) – which as the name suggests, is intended to measure forces between small
numbers of atoms. This device consists
of a very thin (about 1 ) cantilever beam, clamped at one end, with a
sharp tip mounted at the other. When
the tip is brought near a sample, atomic interactions exert a force on the tip
and cause the cantilever to bend. The
bending is detected by a laser-mirror system.
The device is capable of measuring forces of about 1 pN (that’s N!!), and is used to explore the properties
of surfaces, and biological materials such as DNA strands and cell
membranes. A nice article on the AFM
can be found at http://www.di.com
Selecting a load cell
As an engineer, you may need to purchase a load cell to measure a force. Here are a few considerations that will guide your purchase.
1. How many force (and maybe moment) components do you need to measure? Instruments that measure several force components are more expensive…
2. Load capacity – what is the maximum force you need to measure?
3. Load range – what is the minimum force you need to measure?
4. Accuracy
5. Temperature stability – how much will the reading on the cell change if the temperature changes?
6. Creep stability – if a load is applied to the cell for a long time, does the reading drift?
7. Frequency response – how rapidly will the cell respond to time varying loads? What is the maximum frequency of loading that can be measured?
8. Reliability
9. Cost
In this section, we list equations that can be used to calculate forces associated with
(i) Gravity
(ii) Forces exerted by linear springs
(iii) Electrostatic forces
(iv) Electromagnetic forces
(v) Hydrostatic forces and buoyancy
(vi) Aero- and hydro-dynamic lift and drag forces
Gravitation
Gravity forces acting on masses that are a large distance apart
Consider two masses and that are a distance d apart. Newton’s law of gravitation states that mass will experience a force
Here, the symbol indicates that this is the force exerted by mass (2) on mass (1) (it’s not hard to remember that (2/1) means two on one). Also, is a unit vector pointing from mass to mass , and G is the Gravitation constant. Mass will experience a force of equal magnitude, acting in the opposite direction.
In SI units,
The law is strictly only valid if the masses are very small (infinitely small, in fact) compared with d – so the formula works best for calculating the force exerted by one planet or another; or the force exerted by the earth on a satellite.
Gravity forces acting on a small object close to the earth’s surface
For engineering purposes, we can usually assume that
1. The earth is spherical, with a radius R
2. The object of interest is small compared with R
3. The object’s height h above the earths surface is small compared to R
If the first two assumptions are valid, then one can show that Newton’s law of gravitation implies that a mass m at a height h above the earth’s surface experiences a force
where M is the mass of the earth; m is the mass of the object; R is the earth’s radius, G is the gravitation constant and is a unit vector pointing from the center of the earth to the mass m. (Why do we have to show this? Well, the mass m actually experiences a force of attraction towards every point inside the earth. One might guess that points close to the earth’s surface under the mass would attract the mass more than those far away, so the earth would exert a larger gravitational force than a very small object with the same mass located at the earth’s center. But this turns out not to be the case, as long as the earth is perfectly uniform and spherical).
If the third assumption (h<<R) is valid, then we can simplify the force law by setting
where g is a constant, and j is a `vertical’ unit vector (i.e. perpendicular to the earth’s surface).
In SI units .
The force of gravity acts at the center of gravity of an object. For most engineering calculations the center of gravity of an object can be assumed to be the same as its center of mass. For example, gravity would exert a force at the center of the sphere that Mickey is holding. The location of the center of mass for several other common shapes is shown below. The procedure for calculating center of mass of a complex shaped object is discussed in more detail in section 6.3.
Some subtleties about gravitational interactions
There are some situations where the simple equations in the preceding section don’t work. Surveyors know perfectly well that the earth is no-where near spherical; its density is also not uniform. The earth’s gravitational field can be quite severely distorted near large mountains, for example. So using the simple gravitational formulas in surveying applications (e.g. to find the `vertical’ direction) can lead to large errors.
Also, the center of gravity of an object is not the same as its center of mass. Gravity is actually a distributed force. When two nearby objects exert a gravitational force on each other, every point in one body is attracted towards every point inside its neighbor. The distributed force can be replaced by a single, statically equivalent force, but the point where the equivalent force acts depends on the relative positions of the two objects, and is not generally a fixed point in either solid. One consequence of this behavior is that gravity can cause rotational accelerations, as well as linear accelerations. For example, the resultant force of gravity exerted on the earth by the sun and moon does not act at the center of mass of the earth. As a result, the earth precesses – that is to say, its axis of rotation changes with time.
Forces exerted by springs
A solid object (e.g. a rubber band) can be made to exert forces by stretching it. The forces exerted by a solid that is subjected to a given deformation depend on the shape of the component, the materials it is made from, and how it is connected to its surroundings. Solid objects can also exert moments, or torques – we will define these shortly. Forces exerted by solid components in a machine or structure are complicated, and will be discussed in detail separately. Here, we restrict attention to the simplest case: forces exerted by linear springs.
A spring scale is a good example of a linear spring. You can attach it to something at both ends. If you stretch or compress the spring, it will exert forces on whatever you connected to.
The forces exerted by the ends of the spring always act along the line of the spring. The magnitude of the force is (so long as you don’t stretch the spring too much) given by the formula
where a is the un-stretched spring length; L is the stretched length, and k is the spring stiffness.
In the SI system, k has units of N/m.
In Imperial units k has units of lb/ft (or sometimes lb/in)
Note that when you draw a picture showing the forces exerted by a spring, you must always assume that the spring is stretched, so that the forces exerted by the spring are attractive. If you don’t do this, your sign convention will be inconsistent with the formula , which assumes that a compressed spring (L<a) exerts a negative force.
Electrostatic forces
As an engineer, you will need to be able to design structures and machines that manage forces. Controlling gravity is, alas, beyond the capabilities of today’s engineers. It’s also difficult (but not impossible) to design a spring with a variable stiffness or unstretched length. But there are forces that you can easily control. Electrostatic and electromagnetic forces are among the most important ones.
Electrostatic forces are exerted on, and by, charged objects. The concepts of electrical potential, current and charge are based on experiments. A detailed discussion of these topics is beyond the scope of this course (it will be covered in detail in EN51), but electromagnetic and electrostatic forces are so important in the design of engines and machines that the main rules governing forces in these systems will be summarized here.
Electrostatic forces acting on two small charged objects that are a large distance apart
Coulomb’s Law states that if like charges and are induced on two particles that are a distance d apart, then particle 1 will experience a force
(acting away from particle 2), where is a fundamental physical constant known as the Permittivity of the medium surrounding the particles (like the Gravitational constant, its value must be determined by experiment).
In SI units, are specified in Coulombs, d is in meters, and is the permittivity of free space, with fundamental units . Permittivity is more usually specified using derived units, in Farads per meter. The Farad is the unit of capacitance.
The value of for air is very close to that of a vacuum. The permittivity of a vacuum is denoted by . In SI units its value is approximately
Like gravitational forces, the electrostatic forces acting on 3D objects with a general distribution of charge must be determined using complicated calculations. It’s worth giving results for two cases that arise frequently in engineering designs:
Forces acting between charged flat parallel plates
Two parallel plates, which have equal and opposite charges and are separated by a distance , experience an attractive force with magnitude
The force can be thought of as acting at the center of gravity of the plates.
Two parallel plates, which have area A, are separated by a distance d, and are connected to a power-supply that imposes an electrical potential difference V across the plates, experience an attractive force with magnitude
The force can be thought of as acting at the center of gravity of the plates.
Applications of electrostatic forces:
Electrostatic forces are small, and don’t have many applications in conventional mechanical systems. However, they are often used to construct tiny motors for micro-electro-mechanical systems (MEMS). The basic idea is to construct a parallel-plate capacitor, and then to apply force to the machine by connecting the plates to a power-supply. The pictures below show examples of a comb drive motor.
An experimental comb drive MEMS actuator developed at Sandia National Labs, http://mems.sandia.gov/scripts/index.asp
A rotary comb drive actuator developed at iolon inc. Its purpose is to rotate the mirror at the center, which acts as an optical switch.
The configurations used in practice are basically large numbers of parallel plate capacitors. A detailed discussion of forces in these systems will be deferred to future courses.
Electrostatic forces are also exploited in the design of oscilloscopes, television monitors, and electron microscopes. These systems generate charged particles (electrons), for example by heating a tungsten wire. The electrons are emitted into a strong electrostatic field, and so are subjected to a large force. The force then causes the particles to accelerate – but we can’t talk about accelerations in this course so you’ll have to take EN4 to find out what happens next…
Electromagnetic forces
Electromagnetic forces are exploited more widely than electrostatic forces, in the design of electric motors, generators, and electromagnets.
Ampere’s Law states that two long parallel wires which have length L, carry electric currents and , and are a small distance d apart, will experience an attractive force with magnitude
where is a constant known as the permeability of free space.
In SI units, has fundamental units of , but is usually specified in derived units of Henry per meter. The Henry is the unit of inductance.
The value of is exactly H/m
Electromagnetic forces between more generally shaped current carrying wires and magnets are governed by a complex set of equations. A full discussion of these physical laws is beyond the scope of this course, and will be covered in EN51.
Applications of electromagnetic forces
Electromagnetic forces are widely exploited in the design of electric motors; force actuators; solenoids; and electromagnets. All these applications are based upon the principle that a current-carrying wire in a magnetic field is subject to a force. The magnetic field can either be induced by a permanent magnet (as in a DC motor); or can be induced by passing a current through a second wire (used in some DC motors, and all AC motors). The general trends of forces in electric motors follow Ampere’s law: the force exerted by the motor increases linearly with electric current in the armature; increases roughly in proportion to the length of wire used to wind the armature, and depends on the geometry of the motor.
Two examples of DC motors – the picture on the right is cut open to show the windings. You can find more information on motors at http://my.execpc.com/~rhoadley/magmotor.htm
Hydrostatic and buoyancy forces
When an object is immersed in a stationary fluid, its surface is subjected to a pressure. The pressure is actually induced in the fluid by gravity: the pressure at any depth is effectively supporting the weight of fluid above that depth.
A pressure is a distributed force. If a pressure p acts on a surface, a small piece of the surface with area is subjected to a force
where n is a unit vector perpendicular to the surface. The total force on a surface must be calculated by integration. We will show how this is done shortly.
The pressure in a stationary fluid varies linearly with depth below the fluid surface
where is atmospheric pressure (often neglected as it’s generally small compared with the second term); is the fluid density; g is the acceleration due to gravity; and d is depth below the fluid surface.
Archimedes’ principle gives a simple way to calculate the resultant force exerted by fluid pressure on an immersed object.
The magnitude of the resultant force is equal to the weight of water displaced by the object. The direction is perpendicular to the fluid surface. Thus, if the fluid has mass density , and a volume of the object lies below the surface of the fluid, the resultant force due to fluid pressure is
The force acts at the center of buoyancy of the immersed object. The center of buoyancy can be calculated by finding the center of mass of the displaced fluid (i.e. the center of mass of the portion of the immersed object that lies below the fluid surface).
The buoyancy force acts in addition to gravity loading. If the object floats, the gravitational force is equal and opposite to the buoyancy force. The force of gravity acts (as usual) at the center of mass of the entire object.
Aerodynamic lift and drag forces
Engineers who design large bridges, buildings, or fast-moving terrestrial vehicles, spend much time and effort in managing aero- or hydro-dynamic forces. Hydrodynamic forces are also of great interest to engineers who design bearings and car tires, since hydrodynamic forces can cause one surface to float above another, so reducing friction to very low levels.
In general, when air or fluid flow past an object (or equivalently, if the object moves through stationary fluid or gas), the object is subjected to two forces:
1. A Drag force, which acts parallel to the direction of air or fluid flow
2. A Lift force, which acts perpendicular to the direction of air or fluid flow.
The forces act at a point known as the center of lift of the object – but there’s no simple way to predict where this point is.
The lift force is present only if airflow past the object is unsymmetrical (i.e. faster above or below the object). This asymmetry can result from the shape of the object itself (this effect is exploited in the design of airplane wings); or because the object is spinning (this effect is exploited by people who throw, kick, or hit balls for a living).
Two effects contribute to drag:
(1) Friction between the object’s surface and the fluid or air. The friction force depends on the object’s shape and size; on the speed of the flow; and on the viscosity of the fluid, which is a measure of the shear resistance of the fluid. Air has a low viscosity; ketchup has a high viscosity. Viscosity is often given the symbol , and has the rather strange units in the SI system of . In `American’ units viscosity has units of `Poise’ (or sometimes centipoises – that’s Poise). The conversion factor is . (Just to be confusing, there’s another measure of viscosity, called kinematic viscosity, or specific viscosity, which is , where is the mass density of the material. In this course we’ll avoid using kinematic viscosity, but you should be aware that it exists!) Typical numbers are: Air: for a standard atmosphere (see http://users.wpi.edu/~ierardi/PDF/air_nu_plot.PDF for a more accurate number) ; Water, ; SAE40 motor oil , ketchup (It’s hard to give a value for the viscosity of ketchup, because it’s thixotropic. See if you can find out what this cool word means – it’s a handy thing to bring up if you work in a fast food restaurant.)
(2) Pressure acting on the objects surface. The pressure arises because the air accelerates as it flows around the object. The pressure acting on the front of the object is usually bigger than the pressure behind it, so there’s a resultant drag force. The pressure drag force depends on the objects shape and size, the speed of the flow, and the fluid’s mass density .
Lift forces defy a simple explanation, despite the efforts of various authors to provide one. If you want to watch a fight, ask two airplane pilots to discuss the origin of lift in your presence. (Of course, you may not actually know two airplane pilots. If this is the case, and you still want to watch a fight, you could try http://www.wwe.com/, or go to a British soccer match). Lift is caused by a difference in pressure acting at the top and bottom of the object, but there’s no simple way to explain the origin of this pressure difference. Two correct explanations of the origin of lift forces can be found at http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html (this site has some neat Java applets that calculate pressure and flow past airfoils), and http://www.monmouth.com/~jsd/how/htm/airfoils.html. Unfortunately there are thousands more books and websites with incorrect explanations of lift, but you can find those for yourself (check out the explanation from the FAA!)
Lift and drag forces are usually quantified by defining a coefficient of lift and a coefficient of drag for the object, and then using the formulas
Here, is the air or fluid density, V is the speed of the fluid, and and are measures of the area of the object. Various measures of area are used in practice – when you look up values for drag coefficients you have to check what’s been used. The object’s total surface area could be used. Vehicle manufacturers usually use the projected frontal area (equal to car height x car width for practical purposes) when reporting drag coefficient. and are dimensionless, so they have no units.
The drag and lift coefficients are not constant, but depend on a number of factors, including:
1. The shape of the object
2. The object’s orientation relative to the flow (aerodynamicists refer to this as the `angle of attack’)
3. The fluid’s viscosity , mass density , flow speed V and the object’s size. Size can be quantified by or ; other numbers are often used too. For example, to quantify the drag force acting on a sphere we use its diameter D. Dimensional analysis shows that and can only depend on these factors through a dimensionless constant known as `Reynold’s number’, defined as
For example, the graph below shows the variation of drag coefficient with Reynolds number for a smooth sphere, with diameter D. The projected area was used to define the drag coefficient
The variation of drag coefficient with Reynolds number Re for a smooth sphere.
Many engineering structures and vehicles operate with Reynolds numbers in the range , where drag coefficients are fairly constant (of order 0.01 - 0.5 or so). Lift coefficients for most airfoils are of order 1 or 2, but can be raised as high as 10 by special techniques such as blowing air over the wing)
Lift and drag coefficients can be calculated approximately (you can buy software to do this for you, e.g. at http://www.hanleyinnovations.com/walite.html . Another useful resource is www.desktopaero.com/appliedaero ). They usually have to be measured to get really accurate numbers.
Tables of approximate values for lift and drag coefficients can be found at http://aerodyn.org/Resources/database.html
Lift and drag forces are of great interest to aircraft designers. Lift and drag forces on an airfoil are computed using the usual formula
The wing area where c is the chord of the wing (see the picture) and L is its length, is used in defining both the lift and drag coefficient.
The variation of and with angle of attack are crucial in the design of aircraft. For reasonable values of (below stall - say less than 10 degrees) the behavior can be approximated by
where , and are more or less constant for any given airfoil shape, for practical ranges of Reynolds number. The first term in the drag coefficient, , represents parasite drag – due to viscous drag and some pressure drag. The second term is called induced drag, and is an undesirable by-product of lift.
The graphs below, (taken from `Aerodynamics for Naval Aviators, H.H. Hurt, U.S. Naval Air Systems Command reprint’ shows some experimental data for lift coefficient as a function of AOA (that’s angle of attack, but you’re engineers now so you have to talk in code to maximize your nerd factor. That’s NF). The data suggest that , and in fact a simple model known as `thin airfoil theory’ predicts that lift coefficient should vary by per radian (that works out as 0.1096/degree)
The induced drag coefficient can be estimated from the formula
where , L is the length of the wing and c is its width; while e is a constant known as the `Oswald efficiency factor.’ The constant e is always less than 1 and is of order 0.9 for a high performance wing (eg a jet aircraft or glider) and of order 0.7 for el cheapo wings.
The parasite drag coefficient is of order 0.05 for the wing of a small general aviation aircraft, and of order 0.005 or lower for a commercial airliner.