《理论力学》课程教学资源（PPT课件，英文）Chapter 02 Planar Concurrent Force System and Plane Couples

IHEORYMECHANICS优Electronic teaching planChapter2 Planar Concurrent ForceSystem and Plane CouplesCollege of Mechanicaland Vehicle Engineering王晓君

College of Mechanical and Vehicle Engineering 王晓君 THEORY MECHANICS Electronic teaching plan

Chapter2PlanarConcurrentForce System andPlane CouplesIntroductionForce system is divided into:planar force system ;space force system@Planar concurrentplanarforce systemPlane specialforce②Plane Couplesforce systemPlanarparallelforcesystemsystem@Plane general forceCoplanar forcesystemsystemPlanar ConcurrentForce System:A force system in which the lines of action of all forces are in thesame plane and converge at a point。Example:CranehookResearch methods:Geometric methodT2analytical method

Planar Concurrent Force System ： A force system in which the lines of action of all forces are in the same plane and converge at a point。 Introduction Research methods: Geometric method, analytical method. Example：Crane hook Force system is divided into :planar force system ;space force system ①Planar concurrent force system ②Plane Couples ③Planar parallel force system ④Plane general force system planar force system Plane special force system Coplanar force system Chapter2 Planar Concurrent Force System and Plane Couples

Chapter2PlanarConcurrentForce System and Plane Couples≥ 2.1 Resultant and Equilibrium Condition ofPlanar Concurrent Force Systems> 2.2 Concept and Calculation of the Moment ofForce about a Co-planarPoint> 2.3Plane Couples

Ø 2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems Ø 2.2 Concept and Calculation of the Moment of Force about a Co-planar Point Ø 2.3 Plane Couples Chapter2 Planar Concurrent Force System and Plane Couples

2.1 ResultantandEquilibriumCondition of Planar Concurrent ForceSystems,GeometricalMethod1.ResultantThetriangleruleofforceFRRFAabFFTriangle abc is called force triangle; The above drawingmethod is called trianglerule offorce

The triangle rule of force Triangle abc is called force triangle; The above drawing method is called triangle rule of force. a b F1 c F2 R   一. Geometrical Method A F1 F2 2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems 1.Resultant R 

2.1 Resultant and Equilibrium Condition of Planar Concurrent ForceSystemsPolygonal ruleofforceFdF2Fc1E2RF4baThe polygon formed by each force vector and the resultant force vector is called aforce polygon. The drawing rule for finding the resultant force by force polygon iscalled the polygon rule of force. The edge representing the resultant force vector inthe force polygon is called the closed edge of the force polygonConclusion:The final result of planar concurrent force system resultantis aforce (called resultant force). Its action line passes through the concurrent pointof force system, and its size and direction are determined by the closed edge offorce polygon, which is equal to the vector sum of each force vector. Expressed asavector:R=F+F,+...+F,=ZF

Polygonal rule of force A F1 F2 F3 F4 R  a b F1 c F2 d F3 4 e F  The polygon formed by each force vector and the resultant force vector is called a force polygon. The drawing rule for finding the resultant force by force polygon is called the polygon rule of force. The edge representing the resultant force vector in the force polygon is called the closed edge of the force polygon. Conclusion: The final result of planar concurrent force system resultant is a force (called resultant force). Its action line passes through the concurrent point of force system, and its size and direction are determined by the closed edge of force polygon, which is equal to the vector sum of each force vector. Expressed as a vector： R F F Fn F       1  2     2.1 Resultant and Equilibrium Condition of Planar Concurrent Force Systems

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