How to consider the vector direction in a mechanism?
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3.4 Vector Loops of a Mechanism
The main difference between freely moving bodies and the moving links in a mechanism is that they have a constrained motion due to the joints in between the links. The links connected by joints form closed polygons that we shall call a loop. The motion analysis of mechanisms is based on expressing these loops mathematically.
In kinematic analysis we shall assume that all the necessary dimensions of each link is given and link length dimensions (i.e. the distance between the joints or the angles) can be determined from the given dimensions using the geometry of the link.
In Section 2.1.2, we have seen that it is sufficient to represent the position of each link (rigid body) by describing the position of any two points on that link. One way of selecting these two points on a link is to use the permanently coincident points. It is obvious that in such a procedure, the origin of a vector will be defined by the previous vector and thus the number of parameters to define the link positions will be decreased.

Let us consider a four-bar mechanism as shown above as a simple example. In this mechanism A0 ,is a permanently coincident point between links 1 and 2, A is peranently coincident point between links 2 and 3, B between 3 and 4 and B0 between 1 and 4. Let us disconnect joint B. In such a case we will obtain two open kinematic chains A0 AB (links 2,3) with two degrees of freedom and A0 B0B (links 1,4) with one degree of freedom (Fig.2.7b). To determine the positions of the links we must have a reference frame. One obvious choice is to select the fixed pivots A0, B0 as one of the co-ordinate axes and select A0or B0 as the origin. Next,in order to define the position of link 2, we must define angle 12 , which is related with the degree of freedom of the joint between links 1 and 2. To determine the position of link 3, since the location of the permanently coincident point A between 2 and 3 can be determined when 12 defined, we must now define 13 , which is related to the freedom of the joint between links 2 and 3. Similarly 14 must be defined to determine the position of link 4. Hence we need 3 parameters (12 , 13 and 14 ) which are all related to the joint freedoms for the open kinematic chains obtained when we disconnect a joint to eliminate a loop.
The main difference between freely moving bodies and the moving links in a mechanism is that they have a constrained motion due to the joints in between the links. The links connected by joints form closed polygons that we shall call a loop. The motion analysis of mechanisms is based on expressing these loops mathematically.
In kinematic analysis we shall assume that all the necessary dimensions of each link is given and link length dimensions (i.e. the distance between the joints or the angles) can be determined from the given dimensions using the geometry of the link.
In Section 2.1.2, we have seen that it is sufficient to represent the position of each link (rigid body) by describing the position of any two points on that link. One way of selecting these two points on a link is to use the permanently coincident points. It is obvious that in such a procedure, the origin of a vector will be defined by the previous vector and thus the number of parameters to define the link positions will be decreased.

Let us consider a four-bar mechanism as shown above as a simple example. In this mechanism A0 ,is a permanently coincident point between links 1 and 2, A is peranently coincident point between links 2 and 3, B between 3 and 4 and B0 between 1 and 4. Let us disconnect joint B. In such a case we will obtain two open kinematic chains A0 AB (links 2,3) with two degrees of freedom and A0 B0B (links 1,4) with one degree of freedom (Fig.2.7b). To determine the positions of the links we must have a reference frame. One obvious choice is to select the fixed pivots A0, B0 as one of the co-ordinate axes and select A0or B0 as the origin. Next,in order to define the position of link 2, we must define angle 12 , which is related with the degree of freedom of the joint between links 1 and 2. To determine the position of link 3, since the location of the permanently coincident point A between 2 and 3 can be determined when 12 defined, we must now define 13 , which is related to the freedom of the joint between links 2 and 3. Similarly 14 must be defined to determine the position of link 4. Hence we need 3 parameters (12 , 13 and 14 ) which are all related to the joint freedoms for the open kinematic chains obtained when we disconnect a joint to eliminate a loop.
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