5.1 Spatial concurrent force systemTExample 1 An object with a weight ofPis supported by aαErodAB and ropes AC andADlocatedonthe sameαhorizontalplane,as shown in the figure.P=1oo0N,AβCE-ED-12cm,EA=24cm, β =45regardlessofrod weightIPFind the tension of the rope and the force on the rod.BSolution: Taking hinge A as the research object,ZDdthe stress is as shown in the figure.αAEZX=0:T sin α-T, sin α=0JaZY=O:-T.cosα-T, cosα-Ssin β=0SnβZZ=0:-Scosβ-P=0IpB224Bygeometric relationscosα:V12+2425Get the solution:Tc = Tp = 559NS=-1414N
A B C D E P A B C D E P TD TC S x y z Example 1 An object with a weight of P is supported by a rod AB and ropes AC and AD located on the same horizontal plane, as shown in the figure. P=1000N, CE=ED=12cm, EA=24cm, regardless of rod weight; Find the tension of the rope and the force on the rod. = 45 Solution: Taking hinge A as the research object, the stress is as shown in the figure. = 0: sin − sin = 0 X TC TD Y = 0: −T cos −T cos − S sin = 0 C D Z = 0 : −S cos − P = 0 By geometric relations 5 2 12 24 24 cos 2 2 = + = Get the solution： S = −1414N TC = TD = 559N 5.1 Spatial concurrent force system
5.2 Moment of force on axis and moment of force on point二、 Moment of force to pointThe moment of force to point is expressed by vector--Moment vectorBThree elementsF(1) Magnified: the product of force Fand arm一(2) Direction: the direction of rotation(3) Action surface: moment action surface.Mo(F)=r×FThat is, the moment vector of the spatial force to the point is equal tothevectorproduct of the vector diameter from the moment centertothe action point of the force
The moment of force to point is expressed by vector-Moment vector Three elements (1) Magnified: the product of force Fand arm (2) Direction: the direction of rotation (3) Action surface: moment action surface. 5.2 Moment of force on axis and moment of force on point 二、Moment of force to point That is, the moment vector of the spatial force to the point is equal to the vector product of the vector diameter from the moment center to the action point of the force
5.2 Moment of force on axis and moment of force on pointSet up as shown in figure coordinates, there areZBr=xi+yi+zkmo(F)FKF=Fi+Fj+F.k(x,y,zAySo:aijkmo(F)=rxF=1xZyFF.FxL=(yF, -zF,)i +(zF-xF)j+(xF,-yF)kThe above formula is the analytical expression of themoment of force to point
Set up as shown in figure coordinates, there are r xi yj zk = + + F F i F j F k = + + x y z So： ( ) ( ) ( ) ( ) O x y z z y x z y x i j k m F r F x y z F F F yF zF i zF xF j xF yF k = = = − + − + − The above formula is the analytical expression of the moment of force to point. O x y z F A( x, y,z) B r d m (F) O i j k 5.2 Moment of force on axis and moment of force on point
5.2 Moment of force on axis and moment of force on pointProjection of moment of force on Opoint on three coordinate axes;zM。(F). = yF,- zF)HMo(F)kM.(F))=zF.-xFA(XZ)J1M(F))=xF,-yF
Projection of moment of force on O point on three coordinate axes; 5.2 Moment of force on axis and moment of force on point
5.2 Moment of force on axis and moment of force on point1zMoment of force to axisFBThe moment of a force to any axis is equal to theAmoment of the projection of the force on a planeperpendicular to the axis to the intersection of theaxis and the plane.B'AThe moment of forceF to z axis is defined as:A'X1m.(F) = ±F..d = ±2△OA'B' areaSymbolregulation:lookingfrom the positivedirection ofZ axis,if the forcemakes the rigid body turn counterclockwise,it will take positive, otherwiseitwill takenegative.The moment offorce on the axis is an algebraic quantitySimilarly, there is a resultant moment theorem for the moment of force to axis:the moment of resultantforce to any axis is equalto the algebraic sum ofmoments ofeach componentto the same axis.Thatis:m.(R)=Zm,(F)
一、Moment of force to axis d A B F O A B z xy Fxy The moment of a force to any axis is equal to the moment of the projection of the force on a plane perpendicular to the axis to the intersection of the axis and the plane. mz (F) = Fxyd = 2OAB area Symbol regulation: looking from the positive direction of Z axis, if the force makes the rigid body turn counterclockwise, it will take positive, otherwise it will take negative. The moment of force on the axis is an algebraic quantity. Similarly, there is a resultant moment theorem for the moment of force to axis: the moment of resultant force to any axis is equal to the algebraic sum of moments of each component to the same axis. That is: m (R) m (F) z z = F The moment of force to z axis is defined as: 5.2 Moment of force on axis and moment of force on point