05-03-2013, 02:38 PM
STRUCTURAL ASPECT OF SYMMETRY IN PLANETARY GEAR TRAINS
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ABSTRACT: -
Gear Trains are typically used in various mechanisms including robots to transmit specified motion and/or torque between two or more shafts and synthesis of Planetary Gear Trains (PGT) is creation of a Geared Kinematic Chains (GKC) to satisfy a desired functional requirement. In addition to generation of PGTs, it will be useful for the designer if other structural characteristics like symmetry in PGTs are known. Symmetry is important both from architectural beauty of the structure and generation of structures. Generation process is simplified if symmetry aspect is incorporated into synthesis process. Better balance can easily be achieved by having more symmetry in the PGT. Higher the number of symmetric pairs in a PGT greater is the symmetry.
INTRODUCTION
Structural synthesis and analysis of mechanisms has always been an important area of investigation for researchers in the field of mechanisms. A design methodology must ensure the identification of all the design alternatives and result in an optimum design. The structure of a kinematic chain is basic to an understanding of their function. An understanding of this structure requires a systematic development of the method of enumeration, identification, classification and isomorphism of kinematic chains. Gear Trains are typically used to transmit motion at desired velocity ratio and/or torque between two or more shafts like in applications automobile gear boxes, helicopter mechanisms, differentials, gas turbine engines, machine tool gear boxes, robot actuator mechanisms etc.
SYMMETRY IN PGTs:
From the aesthetics point of view and generation of structures of kinematic chains, symmetry plays a vital role. By knowing the symmetry in a graph the number of isomorphic structures enumerated is reduced to a larger extent. Symmetry in GKCs simplifies the enumeration process and automation can be achieved using digital computers. Symmetrical placement of identical links in a structure is known as symmetry in the structure. Identical location of identical links with reference to another link is a measure of symmetry about that link. Hamming number method is very useful in detecting symmetry in the kinematic chains with geared elements.
SYMMETRY:
Symmetry about a link in a GKC is known by identifying the identical links in the GKC. For identical position of identical links other than the links identical to a link about which symmetry is sought gives symmetry. Identification of identical links in a GKC is done by comparing the hamming strings for various links in the hamming matrix. Comparing the hamming strings for the five links of the graph in figure (2), the hamming strings for the three links 3, 4, & 5 are identical. To know about symmetry about a link in the GKC, search for identical location of these three links about the other two links. Thus links 3, 4&5 are symmetric with respect to link 1, since in the hamming matrix, the hamming values for these links 3, 4&5 are same in the hamming string for the first link, i.e. first row of the hamming matrix for this GKC. Similar is the case with hamming string of the second link. Hence the links 3, 4 & 5 are symmetric about the links 1 & 2.
CONCLUSIONS:
Symmetry plays a vital role from the aesthetics point of view and in the generation of geared kinematic chains. Knowing the symmetry in a graph of a GKC, the number of isomorphic graphs/Structures enumerated can be reduced to a greater extent. A GKC with higher symmetry is preferred to another isomorphic GKC with lesser symmetry.