8.1 The concept of absolute motion relative motioninvolving motionExample1: CAMjacking mechanismMoving point: center O of the CAMBDynamicfastening:consolidatedonrodABabsolutemotion:horizontal straightnessVrelative motion:An arc withA as the centerand OAas theradiustransport motion: The vertical translationtransportpoint: Where AB overlaps with O
8.1 The concept of absolute motion relative motion involving motion 0 v A B O Moving point: center O of the CAM Dynamic fastening: consolidated on rod AB x y x’ y’ Example1：CAM jacking mechanism absolute motion： relative motion： transport motion： transport point： horizontal straightness An arc with A as the center and OA as the radius The vertical translation Where AB overlaps with O e v r v a v
8.1 The concept of absolute motion relative motioninvolving motionExample2Moving point: center O of the CAMBDynamic system:consolidatedonrodABrelativemotion:beparalleltothelineABVtransportpoint:WhereABoverlapswithC
8.1 The concept of absolute motion relative motion involving motion 0 v A B O Moving point: center O of the CAM Dynamic system: consolidated on rod AB relative motion： transport point： be parallel to the line AB Where AB overlaps with O e v Example2
8.1 The concept of absolute motion relative motioninvolving motionMExample3 As shown in thefigure, the bar haslength L and rotates at angular velocity @ aroundi福ae2Vaxis O, while the radius of the disk is r and rotates at?0oélMaeangular velocity ' around the axis o'. Find theM,involved velocity and acceleration ofthe edges andO0points M, and M, of the disk.Solution: The static fastening is taken on the ground, whilethe dynamic fastening is taken on the poleae1 =(l -r)o3(l-r)0Vel =Ve2 = Vi? +r0ae2 = Vl? +r??
o o M1 M2 Solution：The static fastening is taken on the ground, while the dynamic fastening is taken on the pole ve1 = (l − r) e1 v 2 1 ae = (l − r) e1 a 2 2 2 v l r e = + e2 v 2 2 2 ae2 = l + r e2 a M1 M2 o Example3 As shown in the figure, the bar has length L and rotates at angular velocity around axis O, while the radius of the disk is r and rotates at angular velocity around the axis . Find the involved velocity and acceleration of the edges and points and of the disk. 8.1 The concept of absolute motion relative motion involving motion
8.2 The velocity synthesis theorem of pointBB'M moves along the arc of AB and ABM2moves to the position at time t+△ t.M'12Vectorrelations in the figureVVMM'=MM, +M,M1MiDivide both sides of the above equation by t.Mand let △t-→0. Take the limit, and get:AAMMMMM,M'limlimJim△tAtAt4-01-0At->0MMMM,Defined by the speed,limVV?limAtAt020M.M'MM2vlimlim△t△tAt0At->0
M moves along the arc of AB and AB moves to the position at time t+∆ t. MM = MM + M M 1 1 t M M t MM t MM t t t + = → → → 1 0 1 0 0 lim lim lim a t v t MM = → lim 0 e t v t MM = → 1 0 lim r t t v t MM t M M = = → → 2 0 1 0 lim lim Ｂ’ Ｍ２ Ｍ Ａ Ｂ Ａ’ Ｍ’ Ｍ１ a v e v r v 8.2 The velocity synthesis theorem of point Divide both sides of the above equation by ∆t, and let ∆t→0. Take the limit, and get: Vector relations in the figure Defined by the speed
8.2 The velocity synthesis theorem of point,=v+i,Thus it can be obtained that:CThat is, the absolute velocity of a moving point at a certain instant is equal tothe vector sum of its associated velocity and relative velocity at that instant.That's thevelocity compositiontheoremfor pointsillustrate:★ Instantaneous relation★ Applicable to the resultant motion involving any form of point motion★ The vector relationship v, must be on the diagonal; If you know 4 ofthem, you can figure out 2 ofthem
Thus it can be obtained that: a e r v v v = + That is, the absolute velocity of a moving point at a certain instant is equal to the vector sum of its associated velocity and relative velocity at that instant. That's the velocity composition theorem for points. Instantaneous relation Applicable to the resultant motion involving any form of point motion The vector relationship must be on the diagonal; If you know 4 of them, you can figure out 2 of them illustrate: a v 8.2 The velocity synthesis theorem of point