Presentation on theme: "Force, Moment, Couple and Resultants"— Presentation transcript:
1 Force, Moment, Couple and Resultants 2 Force SystemsForce, Moment, Couple and Resultants
2 Objectives Students must be able to #1 Course ObjectiveDescribe the characteristics and properties of forces and moments, analyse the force system, obtain the resultant and equivalent force systems.Chapter ObjectivesUse mathematical formulae to manipulate physical quantitiesObtain position vectors with appropriate representation.Use and manipulate force vectorsUse and manipulate moment vectorsAnalyse the force system resultantsDescribe and obtain equivalent systems
3 Force Definition Force is a vector quantity (why?) Force is the action of one body on another. [Statics]Force is an action that tends to cause acceleration of an object. [Dynamics]The SI unit of force magnitude is the newton (N).One Newton is equivalent to one kilogram-meter per second squared (kg·m/s2 or kg·m · s – 2)Examples of mechanical force include the thrust of a rocket engine, the impetus that causes a car to speed up when you step on the accelerator, and the pull of gravity on your body.Force can result from the action of electric fields, magnetic fields, and various other phenomena.33
4 FORCE SYSTEMS Force is a vector Line of action is a straight line colinear with the forceForce System:concurrent if the lines of action intersect at a pointparallel if the lines of action are parallelycoplanar if the lines of action lie on the same planex
5 Writing Convention Hand Print Scalar Vector Unit Vector Magnitude of VectorsamesymbolIn this course, you have to write in this convention.Recommended Style
13 Component Resolution of a Vector A vector may be resolvedinto two components.
14 Basic relations of Triangle (C/6, law of cosine, sine) Law of sineb
15 a c b Hint Given V, and , find Law of cosine Law of sine 2 1 b b qbcbGiven V, and , findLaw of cosine(Law of sine)Law of sine
16 Vector Component and Projection b: vector components of (along axis a and b)a: projections of(onto axis a and b)=bspecial case:projection vectors are orthogonal to each othera: orthogonal projections & vector components
19 EXAMPLE 2-1 y T ANS x kN Given the magnitude of the tension in the cable, T = 9 kN, express T interms of unit vector i and jTxy3 S.F.Correct?kNANS
20 (a) parallel and perpendicular to arm AB We are using robot arm to put the cylindrical part into a hole.Determine the components of the force which the cylindrical part exerts on the robot along axes(a) parallel and perpendicular to arm AB(b) parallel and perpendicular to arm BCP = 90 NparperDefiningdirectionP = 90 Nperpararm ABANSarm BCANS
21 Vector Component (Algebraic) 2/2 Combine the two forces P and T, which act on the fixed structure at B, into a single equivalent force RGraphicsP=800 N (8cm)RT=600 N (6cm)GeometricPTRVector Component (Algebraic)Correct?Point of application is B
22 Two forces is not acting Example Hibbeler Ex 2-1 #1Determine the magnitude and direction of the resultant force.Two forces is not actingat the same point.Geometric
23 Vector Component (Algebraic) GeometricGood?(get full score?)- more explanation- mark answer- 5S.F. Then 3S.F.
28 Reference axis (very very important) Many problems do not come with ref. axis.Assignment based on convenience/experienceOriginally pass through OVector summation (addition)Three ways to be masteredyxoF1yF1xF2yF2x1. GraphicallyRyRx2. Geometrically3. Vector component (algebraically)The calculations do not reveal the point of application of the resultant force.In case where forces do not apply at the same point of application, you have to find it too!
31 Three Dimensional Coordinate System Real-life Coordinate System is 3D.Introduce rule for defining the 3rd axis- “right-hand rule”: x-y-z- for consistency in math calculation(cross vector)zxHow does 2D differs from 3D?y2Dzx
32 Rectangular Components (3D) projection& componentzyx- cos(x), cos(y), cos(z) : “directional cosines” of(maybe +/-)- cos2(x)+cos2(y)+cos2(z) = 1is a unit vector in the direction of- If you known the magnitude and all directional cosines, you can write force in the form ofdirectional cosine Method
33 Example Hibbeler Ex 2-8Find Cartesian components of Fzxy
34 Given the cable tension T = 2 kN. Write the vector expression of 1) directional cosine methodxyzABRealdirectionalcosineBdirectionl cosine = -0.92A
36 Directional Cosines by Graphics cos2(x)+cos2(y)+cos2(z) = 1
37 (a) Two points on the line of action of force is given (F also given). - Usually, the direction of force is not given using the directional cosines. Need some calculation.- Two examples(a) Two points on the line of action of force is given (F also given).zB (x2, y2, z2)PositionvectorTwo-Point MethodA (x1, y1, z1)yx
45 Example Hibbeler Ex 2-15 #1ForceThe roof is supported by cables as shown. If the cables exert forces FAB = 100 N and FAC = 120 N on the wall hook at A as shown, determine the magnitude of the resultant force acting at A.
49 Fx = Fxy cos() = F cos() cos() Fy = Fxy sin() = F cos() sin() (b) Two Angles orienting the line of action of force are given (, )Othorgonal projection MethodResolve into two components at a timezyFz = F sin()Fxy = F cos()Fx = Fxy cos() = F cos() cos()Fy = Fxy sin() = F cos() sin()x
52 2/110 A force F is applied to the surface of the sphere as shown. The 2 angles (zeta, phi) locate Point P, and point M is themidpoint of ON. Express F in vector form, using the givenx-,y- z-coordinates.
72 Homepage URLsStatics official HP (User: Prince Password: Caspian)Session 1 HP(after the end of registration period)
73 FORCE SYSTEMS 2-D Force Systems 3-D Force Systems Moment Moment Couple VectorBasic Concept2-D Force Systems3-D Force SystemsMomentCoupleResultantsMomentCoupleResultants
74 Force Definition Force is a vector quantity (why?) Force is the action of one body on another. [Statics]Force is an action that tends to cause acceleration of an object. [Dynamics]The SI unit of force magnitude is the newton (N).One newton is equivalent to one kilogram-meter per second squared (kg·m/s2 or kg·m · s – 2)Examples of mechanical force include the thrust of a rocket engine, the impetus that causes a car to speed up when you step on the accelerator, and the pull of gravity on your body.Force can result from the action of electric fields, magnetic fields, and various other phenomena.
75 Force Representation Use different colours in diagrams Body outline blueLoad redMiscellaneous black(dimension, angle, etc.)Vector quantityMagnitudeDirectionPoint of application10 N
76 Type of ForcesApplied forceExternal forceReactive forceForceStressInternal forceStrainConcentratedContact forceForceForceDistributedBody force
78 2/2 Combine the two forces P and T, which act on the fixed structure at B, into a single equivalent force RP=800 N (8cm)GraphicalRT=600 N (6cm)GeometricPTRAlgebraicCorrect?Point of application is B
79 How to add sliding vectors (forces)? Principle ofTransmissibilityis applied at point AAPoint of applicationNot OK. !Still OK.Point of Applicationis wrongAA
80 Special case: Addition of Parallel Sliding Force Point of applicationline ofactionR2RR1R2R1RRThis graphical method can be used to find Line of actionThe better and efficient way will be discussed later, when we learn the concept of “moment”, “couple”, and “resultant force”80
81 T VD1 Ty Tx x y 60 Ans Move all forces to that concurrent point Point of application,But no physical meaningAnsApplication Point
82 How to add sliding vectors (forces)? is applied at point APoint of applicationThere is better way to find the point of application(or line of action), but you have to learn the concept ofmoment and couples first.
84 MomentIn addition to the tendency to move a body, force may also tend to rotate a body about an axis(magnitude)summationFrom experience (experiment)magnitude depends only on “F” and “d”momentaxisDirectionMoment is a vector
85 Moment Definition Moment is a vector quantity. Magnitude Direction xyzOMoment is a vector quantity.MagnitudeDirectionAxis of RotationThe unit of moment is N·mThe moment-arm d (perpendicular distance)The right-hand ruledetermined by vector cross productSign convention: 2D +k or CCW is positive.Moment of a force or torque
86 Mathematical Definition (3D) from A to point of application of the forceMoment about point A :-Magnitude:a-Direction:right-hand ruleXr-Point of application: point AAd(Unit: newton-meters, N-m)2D- 2D, need sign convention and be consistent; e.g. + for counter- clockwise and – for clockwiseM=Fdd+
87 sum of moment (of each force) = moment of sum (of all force) Varignon’s Theorem (Principle of Moment)can be used withmore than2 componentsSame?The moment of a force about any point is equal to the sum of the moments of the components of the force about that pointsum of moment (of each force) = moment of sum (of all force)Useful with rectangular componentsd2Mo = -Fxd2+Fyd1yOd1+x
88 Principle of Transmissibility & Moment Principle of Transmissibility is based on the fact that“moving force along the line of action causes no effect in changing moment”position vector:from A to any point online of action of the force.OconvenientaXrAdYZ- direction: same- magnitude:M = Fr sin a = FdSliding force has the same momentO
89 Sample 2/5 Calculate the magnitude of the moment about the base point O of 600N force in five different ways.2mA4002m4m600NAd400yxO600NSolution II: 3D Vector Approach4mSolution I: 2D Scalar ApproachOCW or CCW?CWCorrect?CW
91 EXAMPLE 2.8In raising the flagpole, the tension T in the cable must supply aMoment about O of 72 kN-m. Determine T.o12 m30 md15 mANS
92 Example Hibbeler Ex 4-7 #1 Correct? MomentDetermine the moment of the force about point O.Correct?
93 Scalar Approach (Varignon’s theorem) Example Hibbeler Ex 4-7 #2Moment3D Vector ApproachScalar Approach (Varignon’s theorem)
94 Couple- Couple is a summed moment produced by two force of equal magnitude but opposite in direction.adM = F(a+d) – Fa = Fdmagnitude does not depend on distance a (point O),i.e. any point on the body has the same magnitude.Effect of Pure Rotation+O- tendency to rotate the “whole” object.- no effect on moving object as translation.2D representations: (Couples)CCCcouple is a free vector
95 Moment Couple Definition #2 Moment of a CoupleBAOA couple moment is a free vectorIt can act at any point since M depends only upon the position vector r directed between the forces.
96 Force-couple systems - Line of action of a force on a body may be changed if a couple is added to compensated for the change in the tendency to rotate of that body.No changes in net external effectABABBdPrinciple of transmissibilityAForce-couple systemThe direction and magnitude of Force can not be changed, only line of action(i.e. only change to other pararell line)Procedure may be reversed to combine a force with a couple
97 F A B B F C A C A B F B A F F F A B A B from new location (B) to old location (A)CACABFBAFFNo Moment:Principle of TransmissibilityFABAPrinciple of Transmissibilityis based on the fact thatmoving force alongthe line of action causes no effect in changing momentB
98 Why using equivalent system? BBPrinciple of transmissibilityAForce-couple systemAll force systems are equal.real (physical)systemIn the viewpoint of Mechanics,Result of force to these systemsare equalequivalent systemequivalent system
99 Understanding Force-Couple system Moment about point B of force F= tendency of force F to rotate the object at point B couple occurs when moving Force F from A to B( couple occurs when moving Force F parallel toits line of action to the point B)Equivalent SystemABBDDA
100 Be careful of the direction of moment PVector DiagramF12mPAns
101 2/11 Replace the force F by an equivalent force-couple system at point O. xy50 kN0.25 m0.1m50 kNMCouple occurred when moving F to O= Moment of F about OCCWCorrect?Ans
102 Engine number 3 fails. Determine the force-couple system on the body about point o. Moving all 3 forces to point O(direction: left)couples occuring when moving forces.xysum of moments?+(CW)ANSSum of couplesGot the meaning?
103 Example Hibbeler Ex 4-14 #1ResultantReplace the current system by an equivalent resultant force and couple moment acting A.
107 2/6 Simplest ResultantResultant of many forces-couple is the simplest force-couple combination which can replace the original forces/couples without changing the external effects on the body they act onyxqPoint of application-Add two at a time get line of action of-Add many do not get line of action of
108 Easier way to get a resultant + its location F1d12) Replace each force with a force at point O + a coupleF2d2F3d33) Add forces and momentsOMo=(Fidi)Oarbitraryd1d2d31) Pick a point (easy to find moment arms)(forces + couples : same procedures)resultantd=Mo/R2Dany forces + couples systemO single-force system (no-couple)Mo=Rdor single-couple system4) Replace force-couple system with a single force3Dany forces + couples system single-force + special single-couple (wrench)
109 2/87 Determine the resultant and its line of action of the following three loads. why?Move 3 forces to point O,Sums their force and couplesM(force-couple system)ORNote: M depends on the location where we move the force to+M = -2.4*0.2cos *0.12cos20-3.6*0.3cos20 kN-mNote: R is the same regardless with the location point we move the force toR = ( 2.4cos sin cos20 ) i+( -2.4sin cos cos20 ) j kNmove the R to point X whereResultant Moment is zerofind the point X where the Resultant Moment is zeroROXM+N= 0But we want to find the line of action of the “pure” resultant force(the one which has no additional couple)
110 M R O (0,0) P (x,y) M O N R P R M = -1.635 kN-m + Sys 1 couples At point O (0,0)Sys 1MORM = kN-mPAt point X (x,y)O (0,0)P (x,y)couplescancelledCorrect?Sys 2MTwo equivalent systemsMoment at any pointmust be the same on both systemOPick Point ONRPR( line of action )
111 Manually Canceling Couples At point O (0,0)+MORPM = kN-mdManually Canceling CouplesHow to locate Point PHow to find line of action ?O (0,0)dOPdorP
112 Equivalent System Definition RPRTwo force-couple systems are equivalent
113 A car stuck in the snow. Three students attempt to free the car by exert forces on the car at point A, B and C while the driver’s actions result in a forward thrust of 200 N as shown in picture.Determine1) the equivalent force-couple system at the car center of mass G2) locate the point on x-axis where the resultant passes.xyG
114 ANS For line of action of resultant y x y b x G G Sys IISys ICouple CancellationAt y = 0; x = m.ANSTwo equivalent systemsMoment at point Gmust be the same on both systemIf you want to find only b (not line of action itself)Two equivalent systems(2D)+ or - , you have to find out manually
115 Determine the resultant (vector) and the point on x and y axes which must pass. G
116 ANS y y x x For line of action of resultant O O If y = 0; x = 7.42 m. x = 0; y = m.ANS