Equivalent Systems of Forces
Equivalent Systems of Forces
ContentsIntroduction（绪论）VectorProducts of TwoVectors（矢量积）Moment ofaForceAbouta Point（力对点的矩）Moment of a Force About a Given Axis（力对轴的矩）MomentofaCouple（力偶矩）CouplesCanBeRepresentedByVectors（力偶矩矢量）Resolution ofaForce Into aForce and a Couple（力的平移与分解)System ofForces: Reduction to a Force and a Couple （力 系 :简化为单个力与单个矩）2
Contents • Introduction（绪论） • Vector Products of Two Vectors（矢量积） • Moment of a Force About a Point（力对点的矩） • Moment of a Force About a Given Axis（力对轴的矩） • Moment of a Couple（力偶矩） • Couples Can Be Represented By Vectors（力偶矩矢量） • Resolution of a Force Into a Force and a Couple（力的平移与 分解） • System of Forces: Reduction to a Force and a Couple（力系： 简化为单个力与单个矩） 2
Introduction: Treatment of a body as a single particle is not always possible. Ingeneral, the size of the body and the specific points of application of theforces must be considered.: Most bodies in elementary mechanics are assumed to be rigid, i.e., theactual deformations are small and do not affect the conditions ofequilibrium or motion of the body: Current chapter describes the effect of forces exerted on a rigid body andhow to replace a given system of forces with a simpler equivalent system. moment ofa force about a pointmomentofaforceaboutanaxis: moment due to a couple: Any system of forces acting on a rigid body can be replaced by anequivalent system consisting of one force acting at a given point and onecouple.3
Introduction • Treatment of a body as a single particle is not always possible. In general, the size of the body and the specific points of application of the forces must be considered. • Most bodies in elementary mechanics are assumed to be rigid, i.e., the actual deformations are small and do not affect the conditions of equilibrium or motion of the body. • Current chapter describes the effect of forces exerted on a rigid body and how to replace a given system of forces with a simpler equivalent system. • moment of a force about a point • moment of a force about an axis • moment due to a couple • Any system of forces acting on a rigid body can be replaced by an equivalent system consisting of one force acting at a given point and one couple. 3
Vector Product of Two Vectors: Concept of the moment of a force about a point isV=PxQmore easily understood through applications ofthe vector product or cross product.Vector product of two vectors P and Q is definedas the vector V which satisfies the following(a)conditions:1. Line of action of V is perpendicular to planecontaining P and Q2.Magnitude of Vis V = POsin0(b)3.Direction of V is obtained from the right-handrule.. Vector products:- are not commutative, Q× P=-(P×Q)P×(Q1 +02)= P×Q1 + P×Q2- are distributive,(P×Q)xS+ P×(Q×S)arenotassociative,4
Vector Product of Two Vectors • Concept of the moment of a force about a point is more easily understood through applications of the vector product or cross product. • Vector product of two vectors P and Q is defined as the vector V which satisfies the following conditions: 1.Line of action of V is perpendicular to plane containing P and Q. 2. Magnitude of V is 3. Direction of V is obtained from the right-hand rule. V PQsin • Vector products: - are not commutative, - are distributive, - are not associative, Q P PQ P Q1 Q2 PQ1 PQ2 PQ S PQ S 4
Vector Products: Rectangular ComponentsVectorproductsofCartesianunitvectors,ixi=0jxi=-k kxi=jixi--ixj=kjxj=0kxj=-ikxk=0ixk=-j jxk=iixj=kVectorproductsintermsofrectangularcoordinates = (Pi + P,j + P,k)x(Qxi +O,j+Q.k)(P,Q, - P.0,) +(P.x - PxO.))+(PxOy - PyOx)kjkiP2PsPyOxy Q5
Vector Products: Rectangular Components • Vector products of Cartesian unit vectors, 0 0 0 i k j j k i k k i j k j j k j i i i j i k k i j • Vector products in terms of rectangular coordinates V P i P j P k Q i Q j Q k x y z x y z P Q P Q k P Q P Q i P Q P Q j x y y x y z z y z x x z x y z x y z Q Q Q P P P i j k 5