We can solve for the six unknowns including m 13 1 m 24 1 and f i 1 i \u00bc 1 2 3 4

We can solve for the six unknowns including m 13 1 m

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We can solve for the six unknowns, including m 13 1 , m 24 1 and f i 1 , i ¼ 1 ; 2 ; 3 ; 4, from screw Eq. ( 8.40 ). However, how to obtain m 1 1 , m 3 1 and m 2 1 , m 4 1 from m 13 1 , m 24 1 will be discussed in next section. Since f i 1 $ f i 1 is obtained from Eq. ( 8.40 ), the active forces f A i 1 $ A i 1 along the axis of the prismatic pair and produced by the forces acting only on the platform can be solved z F F $ y x c 1 a 4 a 3 a 2 a 13 h h 1 11 m m $ 24 1 21 m m $ 1 1 11 f f $ 2 1 21 f f $ 3 f f $ 4 1 31 1 41 f f $ Fig. 8.3 Free-body diagram of platform 302 8 Dynamic Problems of Parallel Mechanisms
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from the force equilibrium equations of link t , where the unit vector of active force $ A i 1 of limb i acting on link t is parallel to $ f i 1 and f A i 1 is the magnitude of the active force. 8.2.1.2 Main-Pair Reactions Produced by Limb Applied Force F 1 t $ F 1t Without the loss of generality, we suppose that there is only an applied force F 1 t $ F 1 t acting on link t of the limb. To calculate the unknown main-pair reaction force caused by F 1 t $ F 1 t , the link t is first analyzed. (1) The equilibrium of link t in limb 1 under limb applied force F 1 t $ F 1 t The free-body diagram of the two-pair link t is shown in Fig. 8.4a . We can analyze the reactions of U and P pairs in link t . 1. Constraint forces in U pair at a 1 The U pair also belongs to the sub-mechanism shown in Fig. 8.4b . In order to determine the reactions of the U pair, we first need to determine the mobility of the right part of the sub-mechanism, which is a 3-UPU mechanism. From the Modified G-K Criterion Eq. ( 3.5 ), its mobility is M ¼ 6 ð 8 ± 9 ± 1 Þ þ 15 þ 1 ¼ 4 (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 the one-DOF mechanism, there should be five constraints to the platform in total. These are shown in Fig. 8.4b : f i 2 $ f i 2 ; i ¼ 2 ; 3 ; 4, is the constraint force of the U pair at a 2 , a 3 or a 4 only caused by force F 1 t $ F 1 t along the axis of the prismatic pair of limb i ; m 3 2 $ m 32 is the constraint couple of the U pair at a 3 and is only caused by the force F 1 t $ F 1 t ; m 24 2 $ m 22 is the resultant constraint couple of the U pair at a 2 and a 4 , where $ m 22 is parallel to $ m 42 and also caused by the force F 1 t $ F 1 t . Let m 2 2 and m 4 2 be the magnitudes of the constraint couples of the U pair at a 2 and a 4 , respectively. Then, m 24 2 is the resultant couple of m 2 2 and m 4 2 . m 2 2 , m 3 2 , and m 4 2 are three couples which are exerted on the platform by the three different limbs 2, 3, 4. f i 2 ; i ¼ 2 ; 3 ; 4 is the magnitude of the constraint force of U-pair at a i . For the 3-UPU mechanism, the five constraints acting on the platform can be expressed into screw coordinates in a c - xyz system as follows $ f 22 ¼ s y c a ± c y c a ± s a ; ð c = 2 Þ s a ð c = 2 Þ s a ð c = 2 Þ c a ð s y ± c y Þ ð Þ $ f 32 ¼ s y c a c y c a ± s a ; ±ð c = 2 Þ s a ð c = 2 Þ s a ð c = 2 Þ c a ð c y ± s y Þ ð Þ $ f 42 ¼ ± s y c a c y c a ± s a ; ±ð c = 2 Þ s a ±ð c = 2 Þ s a ð c = 2 Þ c a ð s y ± c y Þ ð Þ $ m 32
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