【20160321】Special Cases of the Mobility Equation
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* @- G8 S* b7 l; r" l/ a5 V vMobility is an extremely important property of amechanism. Among other facets, itgives insight into the number of actuators required to operate a mechanism. However, to obtain correct results, special care must be taken in using the Gruebler's equation. Some specialconditions are presented next.5 E8 Q$ r( s: j
自由度計算中有些特殊情況需要格外注意。. b$ [. }5 W- L! }; n1 o
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Coincident Joints
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Some mechanisms have three links that are all connected at a common pin joint, as shown in Figure 1.28. This situation brings some confusion to kinematic modeling. Physically, one pin may be used to connect all three links. However, by definition, a pin joint connects two links.
1 M; \& a% L; w! O8 E5 p同一銷釘連接三個桿件的情況，需要算作兩個銷釘。因為按照定義，一個銷釘連接兩個桿件。
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Exceptions to the Gruebler's Equation
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Another special mobility situation must be mentioned. Because the Gruebler's equation doesnot account for link geometry, in rare instances it can lead to misleading results. One such instance is shown in Figure 1.31.! D: \% N" g% v' A& _& J
自由度計算公式也有例外，比如由幾何因素引起的錯誤結(jié)果，一個例子如圖所示。
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' j) p1 ^! Q, d' \- jNotice that this linkage contains five links and six pin joints. Using Gruebler's equation, thislinkage has zero degrees of freedom. Of course, this suggests that the mechanism is locked. However, if all pivoted links were the same size, and the distance between thejoints on the frame and coupler were identical, this mechanism would be capable of motion, with one degree of freedom. The center link is redundant, and because it is identical in length to the other two links attached to the frame, it does not alter the action of the linkage.* b* p" T" b& _5 P, s# W8 i4 d; X+ j
5個桿件6個銷釘，按自由度計算公式其自由度為零。這意味著機構(gòu)鎖定，然而若所有轉(zhuǎn)軸等長，并且與機架相連的銷釘?shù)竭B桿的距離相等，這個機構(gòu)是可以運動的，有1個自由度。中間的曲柄是多余的，并不影響機構(gòu)的運動。
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Idle Degrees of Freedom4 X# A+ R. H: V' q
# U' X3 }" L" {In some mechanisms, links exhibit motion which does not influence the input and output relationship of the mechanism. These idle degrees of freedom present another situation where Gruebler's equation givesmisleading results. An example is a cam with a roller follower as shown in Figure 1.32. Gruebler's equationspecifies two degrees of freedom (4 links, 3 pins, 1 higher-order joint). With an actuated cam rotation, the pivoted link oscillates while the roller follower rotates about its center. Yet, only the motion of the pivoted link serves as the output of the mechanism. The roller rotation is an idle degree of freedom and not intended to affect the output motion of the mechanism. It is a design feature which reduces friction and wear on the surface of the cam. While Gruebler's equation specifies that a cam mechanism with a roller follower has a mobility of two, the designer is typically only interested in a single degree of freedom. Several other mechanisms contain similar idle degrees of freedom.& ~9 p0 u) D5 M. H
多余自由度，有的機構(gòu)中含有這樣的桿件，其運動對輸出運動不產(chǎn)生影響， 則其為多余自由度。如圖機構(gòu)含4個桿件，3個銷釘，1個高副約束，輸入端凸輪轉(zhuǎn)動，輸出端桿件擺動，中間滾輪從動件繞其中心轉(zhuǎn)動。然而只有桿件的擺動是機構(gòu)的輸出，滾輪的滾動是多余自由度，不影響機構(gòu)輸出。這樣設計是為了減小凸輪表面的摩擦。自由度計算公式給出，該機構(gòu)有2個自由度，然而事實上我們只對一個感興趣。還有其他例子，不作贅述。
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