2.11Resultant and Equilibrium Condition of Planar Concurrent ForceSystems2. EquilibriumF3FsRConclusion:The necessary and sufficient conditions for geometricequilibrium of planar concurrent force system are:The force polygon formed by each force vector in the force systemis self-closed;Orthevector sum of each force vectoris equal to zero.Expressedasavector::R=0 或 ZF=0
2. Equilibrium Conclusion: The necessary and sufficient conditions for geometric equilibrium of planar concurrent force system are: The force polygon formed by each force vector in the force system is self-closed; Or the vector sum of each force vector is equal to zero. Expressed as a vector： ： R 0 或 F 0 F2 F3 F1 F F4 5 R 2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems
2.1 Resultant and Equilibrium Condition of Planar Concurrent ForceSystems二. Analytical Method1, The projection and components of a forceThe components ofaforceR=F+FAccording to the parallelogram law of force, the resultant force of twoconcurrent forces is unigue. If a force is decomposed into twocomponents, the solution is not unique without sufficient conditions.There are six elements in the above formula, four of which must beknown to determine the remaining two.yOrthogonal decomposition offorce:RFF. =FcosαβaF, =Fcosβ=FsinαFx
1、The projection and components of a force. The components of a force According to the parallelogram law of force, the resultant force of two concurrent forces is unique. If a force is decomposed into two components, the solution is not unique without sufficient conditions. R F1 F2 x y F Fx Fy Orthogonal decomposition of force： cos sin cos F F F F F y x There are six elements in the above formula, four of which must be known to determine the remaining two. 二. Analytical Method 2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems
2.1 Resultant and Equilibrium Condition of Planar Concurrent ForceSystemsTheprojectionofaforceBFQbaXxX = FcosαThat is, the projection of force on an axis is equal to the forcemultiplied by the cosine of the angle between the force and theforward direction of the projection axis
The projection of a force A B F x a b X X F cos That is, the projection of force on an axis is equal to the force multiplied by the cosine of the angle between the force and the forward direction of the projection axis. 2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems
2.1 Resultant and Equilibrium Condition of Planar Concurrent ForceSystemsProjection offorceon rectangularAnalytical expression offorcecoordinateaxisX = F.i = Fcos(F,i)atBY = F.j= Fcos(F,j)FYThe magnitude and direction ofthe known projection force areAFF=VX?+y?OXxYXcos(F,i)=F,cosFFIn a rectangular coordinate systemF= F+ F,= Xi +YiThis formula is the analytical expression of force
Analytical expression of force x y A B O F Fx Fy X Y i j Projection of force on rectangular coordinate axis cos( , ) cos( , ) Y F j F F j X F i F F i The magnitude and direction of the known projection force are F Y F j F X F i F X Y cos( , ) , cos( , ) 2 2 In a rectangular coordinate system F F F Xi Yj x y This formula is the analytical expression of force. 2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems
2.1 Resultant and Equilibrium Condition of Planar Concurrent ForceSystems2,ResultantforceprojectiontheoremFrom the geometric method of planar concurrent force systemR=ZFsynthesis:If rectangular coordinate system is used, the analytical formula ofresultantforceandcomponentforceisR=R,i+R,jF,=X,i+YJSubstitute into the above formula to getRi+R,j=Z(Xi+Yj) =(ZX)i +(ZY))R, =ZXFrom the concepts of vector equalitywe can getR, =ZY,That is, the projection of the resultant force of the planar concurrent forcesystem on a certain axis is equal to the algebraic sum of the projection of eachcomponent force in the force system on the same axis. This is the resultantforceprojectiontheorem
2、Resultant force projection theorem From the geometric method of planar concurrent force system synthesis： R Fi If rectangular coordinate system is used, the analytical formula of resultant force and component force is Substitute into the above formula to get R i R j X i Y j X i Y j x y i i i i ( ) ( ) ( ) From the concepts of vector equality , we can get y i x i R Y R X That is, the projection of the resultant force of the planar concurrent force system on a certain axis is equal to the algebraic sum of the projection of each component force in the force system on the same axis. This is the resultant force projection theorem. R R i R j x y F X i Y j i i i 2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems