3.2 Simplified results for plane general force system一、Analysis of simplified results1, The principalvector and the principalmoment are equal tozeroThe simplification result is(R'= 0, M。=0)Nowtheplaneforceindepe ndent of the location-ofsystemis in equilibriumthe simplification center.2、 The principal vectoris equal to zero,(R' = O, M。 ± O)the principalmomentis not equalto zeroThe plane force system is simplified to a couple, M = Zmo(FThe simplification resultdepends on the location of thesimplification center3Theprincipalvectoris notzero,(R'+0, M。=0)the principal momentis zeroAt this point, the plane force system is simplified to a net.force acting onR-ZFthe simplification center,and(R' ±0, M。 ±0)4,Theprincipalvectorandtheprincipalmoment are not equal to zeroAt this point,it can be further simplifiedto a net force
The simplification result depends on the location of the simplification center. 3.2 Simplified results for plane general force system 1、The principal vector and the principal moment are equal to zero ( = 0, = 0) R Mo Now the plane force system is in equilibrium. 2、The principal vector is equal to zero, the principal moment is not equal to zero ( = 0, 0) R MO 3、The principal vector is not zero, the principal moment is zero ( 0, = 0) R MO At this point, the plane force system is simplified to a net force acting on the simplification center, and R F = The plane force system is simplified to a couple， M m (F) O = 一、Analysis of simplified results 4、The principal vector and the principal moment are not equal to zero ( 0, 0) R MO At this point, it can be further simplified to a net force. The simplification result is independent of the location of the simplification center
3.25Simplifiedresults for plane general force systemR'RR!RMoO三OPModMo = mo(R)= Rd = R'dSOR'Bythe definition ofthepivotmoment:M。=Emo(F)So:mo(R) =Zmo(F)Sincethe simplificationcenteris arbitrarilyselected,theformulahasuniversalsignificanceConclusion: The moment of plane general force system is equal to thealgebraic sum of the moments of all forces on the same point in thesystem . It's the theorem of the moment of a resultant of plane generalforcesystem
O O MO R O O R R R d O O R d MO = mO (R) = Rd = Rd so R M d O = By the definition of the pivot moment: ( ) MO mO Fi = So： ( ) ( ) mO R mO Fi = Conclusion: The moment of plane general force system is equal to the algebraic sum of the moments of all forces on the same point in the system . It’s the theorem of the moment of a resultant of plane general force system. = = Since the simplification center is arbitrarily selected, the formula has universal significance. 3.2 Simplified results for plane general force system
3.3Equilibrium conditions and equations of plane generalforce system.Equilibrium conditions and equilibrium equations1Equilibriumconditions:R' = O Balance of plane intersection force systemM。 = OThe plane couple system is also balanced.The necessaryand sufficient condition for the equilibriumof planegeneral force systemis that the principalvectorof the force systemand the principalmoment of any point are equal to zero. That is.R'= /(ZX)? +(EY)~= 0M。=Zmo(F)=0
3.3 Equilibrium conditions and equations of plane general force system 一. Equilibrium conditions and equilibrium equations ' ( ) ( ) 0 2 2 R = X + Y = MO =mO (Fi ) = 0 R = 0 MO = 0 Balance of plane intersection force system The plane couple system is also balanced. The necessary and sufficient condition for the equilibrium of plane general force system is that the principal vector of the force system and the principal moment of any point are equal to zero. That is, 1、Equilibrium conditions ：
3.3Equilibrium conditions and equations of plane generalforce system2、Equilibriumequations:Due to theR'= /(X)? +(ZY)?, Mo=Zmo(F)Therefore, the analytic equation of the equilibrium condition is:ZX=0GeneralformulaZY=0Zmo(F)=0Thatis to say, the analytical condition ofequilibrium of plane general forcesystem is that the algebraic sum of all forces projected on two optionalcoordinate axes in the actionplane of force system is equal to zero, and thealgebraic sum ofmoments of all forces to any point is equalto zero. Theaboveformula is called thegeneral formula ofequilibrium equation ofplanegeneralforcesystem
Due to the ( ) ( ) , 2 2 R = X + Y ( ) MO mO Fi = Therefore, the analytic equation of the equilibrium condition is: That is to say, the analytical condition of equilibrium of plane general force system is that the algebraic sum of all forces projected on two optional coordinate axes in the action plane of force system is equal to zero, and the algebraic sum of moments of all forces to any point is equal to zero. The above formula is called the general formula of equilibrium equation of plane general force system. 2、Equilibrium equations： X = 0 Y = 0 mO (Fi ) = 0 General formula 3.3 Equilibrium conditions and equations of plane general force system
3.3Equilibrium conditions and equations of plane generalforce system二.Other forms of the equilibrium equationZX=0Two-moment typeNotice:Zm(F)=0There are threeindependent(Zm(F)=0equationsinthethreeformsofequilibriumConditions:the connectinglineAB between pointsequations of planeA and B cannot be perpendicular to the x axisgeneral force systemThree-moment typeand only three1Zm,(F)= 0unknowns can besolved.Zm(F)= 0Zmc(F)=0Conditions:PointsA,B and C cannot be on thesame straight line
二. Other forms of the equilibrium equation Two-moment type = = = ( ) 0 ( ) 0 0 B i A i m F m F X Conditions: the connecting line AB between points A and B cannot be perpendicular to the x axis. Three-moment type = = = ( ) 0 ( ) 0 ( ) 0 C i B i A i m F m F m F Conditions: Points A, B and C cannot be on the same straight line. Notice： There are three independent equations in the three forms of equilibrium equations of plane general force system, and only three unknowns can be solved. 3.3 Equilibrium conditions and equations of plane general force system