7.5. FOURBAR AND SLIDERCRANK COGNATES ROBERTS  CHEBYCHEV THEOREM When our concern is only the path traced by the coupler point of a fourbar or a slidercrank mechanism, we can determine other fourbar or slidercrank mechanism proportions that generate identically the same coupler point curve. The fourbar mechanisms that generate the identical coupler point curve are known as the cognates of a fourbar. Let us state a theorem and show how the cognates of a fourbar and slider crank mechanism can be found. RobertsChebychev Theorem Consider the fourbar mechanism shown (A_{0}B_{0}=a_{1}, A_{0}A=a_{2}, AB=a_{3}, B_{0}B=a_{4}, AP=b_{3}, BP=c_{3}). The coupler point P traces a curve as shown. For this mechanism the loop equation is:
The construction for determining the cognates can be given by the following steps (Fig.7.34):
4. Draw C_{1}C_{0} //PC_{2} and C_{2}C_{0}// C_{1}P to locate C_{0}. C_{0}C_{1}PC_{2} forms a parallelogram therefore:
5. Four bar mechanisms that trace the same coupler curve are:
C_{0}C_{2}B_{1}B_{0} (P on C_{2}B_{1} ) ( c) Now, the location of C _{0} can be written as : Writing these equations in complex numbers:
One can join any two or all three of the cognates by a revolute joint at P and the mechanism thus obtained will still be movable (provided that the cognates are movable). The mechanism will be of permanent critical form (the general degreeoffreedom equation will not apply). Another important use of the cognates of fourbar mechanisms is to obtain a rigid body motion which is in a translation (e.g. lines taken on this body remain parallel to their original positions) and every point on this body describe paths which are the same as the path of coupler point of the original fourbar. Referring to figures, note that the cranks A_{0}A and C_{0}C_{2} or B_{0}B and C_{0}C_{1} or B_{0}B_{1} and A_{0}A_{1} rotate by the same amount although these angles differ by a constant (a or pb). Let us move one of the cognates relative to the other such that the centers of rotation of the two cranks rotating at the same speed are coincident. For example, if C_{0} and A_{0} are to be made coincident every point of the cognate C_{0}C_{2}B_{1}B_{0} must be displaced parallel to A_{0}C_{0} by an amount A_{0}C_{0} . The relative positions of all the links of the cognate with respect to each other are kept, e.g. since the cranks C_{0}C_{2} and A_{0}A rotate at the same speed, their relative positions will not change and hence they can be connected to each other. Now consider points P and P’. Both of these points will always describe the same path and the distance PP’ will not change and will always be parallel to the initial line drawn. Therefore, we can attach a link in between the two points PP’ and connect them by revolute joints to the coupler links of the two fourbars. The seven link mechanism thus obtained is an overclosed mechanism with link PP’ in a translation. We can eliminate this overclosure by removing one of the cranks B_{0}B or C_{0}C_{2} and obtain a sixlink mechanism as shown. The mechanism can still perform a constrained motion with PP’ in translation. Note that the six link mechanism for the required motion is not unique (there are other sixlink mechanisms to generate a translation of a link as that of point P.
Example 4.9. Mechanism shown below is known as Chebyshev lambda or Hoecken straight line mechanism. The coupler point P describes an approximate straightline. The link length proportions are given as: AB = BP = B_{0}B = 2.5 A_{0}A, A_{0}B_{0} = 2 A_{0}A. We would like to obtain a rigid body in an approximate rectilinear translation guided by a sixlink mechanism. Figure 7.39(a) The construction of the cognates and the design of the six link mechanism is shown in the following flash file
This mechanism has been used as the suspension system for an offroad vehicle.
One can also extend A_{0}AB on a straight line and obtain the link length dimensions as shown..
