This preview shows page 311 - 313 out of 429 pages.
We can solve for the six unknowns, includingm131,m241andfi1,i¼1;2;3;4, fromscrew Eq. (8.40). However, how to obtainm11,m31andm21,m41fromm131,m241will bediscussed in next section.Sincefi1$fi1is obtained from Eq. (8.40), the active forcesfAi1$Ai1along the axis of theprismatic pair and produced by the forces acting only on the platform can be solvedzFF$yxc1a4a3a2a13hh111mm$24121mm$1111ff$2121ff$3ff$4131141ff$Fig. 8.3Free-body diagramof platform3028Dynamic Problems of Parallel Mechanisms
from the force equilibrium equations of linkt, where the unit vector of active force$Ai1of limbiacting on linktis parallel to$fi1andfAi1is the magnitude of the active force.126.96.36.199Main-Pair Reactions Produced by Limb Applied Force F1t$F1tWithout the loss of generality, we suppose that there is only an applied forceF1t$F1tacting on link t of the limb. To calculate the unknown main-pair reaction forcecaused byF1t$F1t, the link t is first analyzed.(1) The equilibrium of linktin limb 1 under limb applied forceF1t$F1tThe free-body diagram of the two-pair linktis shown in Fig.8.4a. We can analyzethe reactions of U and P pairs in linkt.1.Constraint forces in U pair at a1The U pair also belongs to the sub-mechanism shown in Fig.8.4b. In order todetermine the reactions of the U pair, we first need to determine the mobility ofthe right part of the sub-mechanism, which is a 3-UPU mechanism. From theModified G-K Criterion Eq. (3.5), its mobility isM¼6ð8±9±1Þ þ15þ1¼4(8.41)Equation (8.41) indicates that the mobility of the sub-mechanism is also four.When its three prismatic pairs are locked, it still has one freedom. Then, for theone-DOF mechanism, there should be five constraints to the platform in total.These are shown in Fig.8.4b:fi2$fi2;i¼2;3;4, is the constraint force of theU pair ata2,a3ora4only caused by forceF1t$F1talong the axis of the prismaticpair of limbi;m32$m32is the constraint couple of the U pair ata3and is only causedby the forceF1t$F1t;m242$m22is the resultant constraint couple of the U pair ata2anda4, where$m22is parallel to$m42and also caused by the forceF1t$F1t. Letm22andm42bethe magnitudes of the constraint couples of the U pair ata2anda4, respectively.Then,m242is the resultant couple ofm22andm42.m22,m32, andm42are three coupleswhich are exerted on the platform by the three different limbs 2, 3, 4.fi2;i¼2;3;4 is the magnitude of the constraint force of U-pair atai.For the 3-UPU mechanism, the five constraints acting on the platform can beexpressed into screw coordinates in ac-xyzsystem as follows$f22¼syca±cyca±sa;ðc=2Þsaðc=2Þsaðc=2Þcaðsy±cyÞðÞ$f32¼sycacyca±sa;±ðc=2Þsaðc=2Þsaðc=2Þcaðcy±syÞðÞ$f42¼ ±sycacyca±sa;±ðc=2Þsa±ðc=2Þsaðc=2Þcaðsy±cyÞðÞ$m32