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Chapter 2Review of Forces and Moments2.1 ForcesIn this chapter we review the basic concepts of forces, and force laws. Most of this material is identicalto material covered in EN030, and is provided here as a review. There are a few additional sections – forexample forces exerted by a damper or dashpot, an inerter, and interatomic forces are discussed inSection 2.1.7.2.1.1 Definition of a forceEngineering design calculations nearly always use classical (Newtonian) mechanics. In classicalmechanics, the concept of a force’ is based on experimental observations that everything in the universeseems to have a preferred configuration – masses appear to attract each other; objects with oppositecharges attract one another; magnets can repel or attract one another; you are probably repelled by yourprofessor. 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 preferredconfiguration. If objects accelerate very quickly towards their preferred configuration, then we say thatthere’s a big force acting on them. If they don’t move (or move at constant velocity), then there is noforce. 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 motion0. 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 isconstant;2. The sum of forces acting on a particle of constant mass is equal to theproduct of the mass of the particle and its acceleration;3. The forces exerted by two particles on each other are equal in magnitudeand 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 itisF maOf 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 forengineering calculations we can usually take the earth to be fixed, and happily apply Newton’s laws. Inrare cases where the earth’s motion is important, we take the stars far from the solar system to be fixed.2.1.2 Causes of forceForces 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 calculatedusing Newton’s law of gravitation; electrostatic forces acting between charged particles are governed byCoulomb’s law; electromagnetic forces acting between current carrying wires are governed by Ampere’slaw; buoyancy forces are governed by laws describing hydrostatic forces in fluids. Some of theseuniversal 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 mayneed to measure the coefficient of friction for the contacting surfaces. Similarly, to determineaerodynamic lift or drag forces acting on a structure, you would probably need to measure its lift and dragcoefficient experimentally. Lift and drag forces are described in Section 2.6. Friction forces arediscussed in Section 12.Contact forces are pressures that act on the small area of contact between two objects. Contact forces caneither be measured, or they can be calculated by analyzing forces and deformation in the system ofinterest. Contact forces are very complicated, and are discussed in more detail in Section 8.2.1.3 Units of force and typical magnitudesIn 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 1kg mass to accelerate at 1 ms 2 .The fundamental unit of force in the SI convention is kg m/s2In US units, the standard unit of force is the pound, given the symbol lb or lbf (the latter is an abbreviationfor 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 reportedas weight, in lb or lbm (the latter is an abbreviation for pound mass). The weight of an object in lb is notmass at all – it’s actually the gravitational force acting on the mass. Therefore, the mass of an object inslugs must be computed from its weight in pounds using the formulam (slugs) W (lb)g (ft/s 2 )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/s2The conversion factors from lb to N are1 lb1N 4.448 N0.2248 lb(www.onlineconversion.com is a handy resource, as long as you can tolerate all the hideousadvertisements )As a rough guide, a force of 1N is about equal to the weight of a medium sized apple. A few typical forcemagnitudes (from The Sizesaurus’, by Stephen Strauss, Avon Books, NY, 1997) are listed in the tablebelowForceGravitational Pull of the Sun on EarthNewtons3.5 10Gravitational Pull of the Earth on the Moon2 1020Thrust of a Saturn V rocket engine3.3 107Thrust of a large jet engine7.7 105Pull of a large locomotive5 105Force between two protons in a nucleus104Gravitational pull of the earth on a person7.3 102Maximum force exerted upwards by a forearm2.7 102Gravitational pull of the earth on a 5 cent coin5.1 10 2Force between an electron and the nucleus of a Hydrogen atom 8 10 622Pounds Force7.9 10214.5 10197.4 1061.7 1051.1 1051031.6 102601.1 10 21.8 10 8

2.1.4 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 theseeffects are listed later in this section.Constraint forces are exerted by one part of a structure on another, through joints, connections or contactsbetween 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 stretchedrope has a tension force acting inside it, holding the rope together. Most solid objects contain verycomplex distributions of internal force. These internal forces ultimately lead to structural failure, and alsocause the structure to deform. The purpose of calculating forces in a structure or component is usually todeduce the internal forces, so as to be able to design stiff, lightweight and strong components. We willnot, unfortunately, be able to develop a full theory of internal forces in this course – a proper discussionrequires understanding of partial differential equations, as well as vector and tensor calculus. However, abrief discussion of internal forces in slender members will be provided in Section 9.2.1.5 Mathematical representation of a force.Force is a vector – it has a magnitude (specified in Newtons, or lbf, orwhatever), and a direction.kFzjFyA force is therefore always expressed mathematically as a vectorFxzquantity. To do so, we follow the usual rules, which are described inyOmore detail in the vector tutorial. The procedure isx1. Choose basis vectors {i, j, k} or {e1 , e 2 , e 2 } that establish threeifixed (and usually perpendicular) directions in space;2. Using geometry or trigonometry, calculate the force component along each of the three referencedirections ( Fx , Fy , Fz ) or ( F1 , F2 , F3 ) ;3. The vector force is then reported asF Fx i Fy j Fz k F1e1 F2e 2 F3e3 (appropriate units)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 origin2. Choose basis vectors {i, j, k} or {e1 , e 2 , e 2 } that establish three fixed directions in space (usuallywe 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 ofapplication of the force (rx , ry , rz ) or (r1 , r2 , r3 )4. The position vector is then reported asr rx i ry j rz k r1e1 r2e 2 r3e3 (appropriate units)2.1.6 Measuring forcesEngineers often need to measure forces. According to the definition, if we want to measure aforce, we need to get hold of a 1 kg mass, have the force act on it somehow, and then measurethe acceleration of the mass. The magnitude of the acceleration tells us the magnitude of theforce; the direction of motion of the mass tells us the direction of the force. Fortunately, thereare easier ways to measure forces.In addition to causing acceleration, forces cause objects to deform – for example, a force willstretch or compress a spring; or bend a beam. The deformation can be measured, and the forcecan be deduced.The simplest application of this phenomenon is a spring scale. The change in length of a spring isproportional to the magnitude of the force causing it to stretch (so long as the force is not too large!)– thisrelationship is known as Hooke’s law and can be expressed as an equationkδ Fwhere the spring stiffness k 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 moresophisticated 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 areshown in the picture). The simplest load cell works much like a spring scale – you can load it in onedirection, and it will provide an electrical signal proportional to the magnitude of the force. Sophisticatedload cells can measure a force vector, and will record all three force components. Really fancy load cellsmeasure both force vectors, and torque or moment vectors.Simple force transducers capable of measuring a single force component. The instrument on the right iscalled a proving ring’ – there’s a short article describing how it works athttp://www.mel.nist.gov/div822/proving ring.htm

A sophisticated force transducer produced by MTS systems, which is capable of measuring forces andmoments acting on a car’s wheel in-situ. The spec for this device can be downloaded atwww.mts.com/downloads/SWIF2002 100-023-513.pdf.pdfThe basic design of all these load cells is the same – they measure (very precisely) the deformation in apart of the cell that acts like a very stiff spring. One example (fromhttp://www.sandia.gov/isrc/Load Cell/load cell.html ) is shown on theright. In this case the spring’ is actually a tubular piece of high-strengthsteel. When a force acts on the cylinder, its length decreases slightly. Thedeformation is detected using strain gages’ attached to the cylinder. Astrain gage is really just a thin piece of wire, which deforms with thecylinder. When the wire gets shorter, its electrical resistance decreases –this resistance change can be measured, and can be used to work out theforce. It is possible to derive a formula relating the force to the change inresistance, the load cell geometry, and the material properties of steel, butthe calculations involved are well beyond the scope of this course.The most sensitive load cell currently available is the atomic forcemicroscope (AFM) – which as the name suggests, isintended to measure forces between small numbersof atoms. This device consists of a very thin (about1 μ m ) cantilever beam, clamped at one end, with asharp tip mounted at the other. When the tip isbrought near a sample, atomic interactions exert aforce 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 about1 pN (that’s 10 12 N!!), and is used to explore theproperties of surfaces, and biological materials suchas DNA strands and cell membranes. A nice articleon the AFM can be found at http://www.di.com

Selecting a load cellAs an engineer, you may need to purchase a load cell to measure a force. Here are a few considerationsthat will guide your purchase.1. How many force (and maybe moment) components do you need to measure? Instruments thatmeasure 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. Accuracy5. 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 themaximum frequency of loading that can be measured?8. Reliability9. Cost2.1.7 Force LawsIn 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 forcesGravitationGravity forces acting on masses that are a large distance apartConsider two masses m1 and m2 that are a distance dapart. Newton’s law of gravitation states that massm1 will experience a forcem2e12m1FdGm1m2F e12d2where e12 is a unit vector pointing from mass m1 to mass m2 , and G is the Gravitation constant. Massm2 will experience a force of equal magnitude, acting in the opposite direction.In SI units, G 6.673 10 11 m3 kg -1s -2The law is strictly only valid if the masses are very small (infinitely small, in fact) compared with d – sothe formula works best for calculating the force exerted by one planet or another; or the force exerted bythe earth on a satellite.

Gravity forces acting on a small object close to theearth’s surfaceFor engineering purposes, we can usually assumethat1. The earth is spherical, with a radius R2. The object of interest is small comparedwith R3. The object’s height h above the earthssurface is small compared to RIf the first two assumptions are valid, then one canshow that Newton’s law o

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