Keywords: inverted tooth chain, link of the chain, topology optimization, shape optimization, minimum compliance, minimum stress.

The inverted tooth chain is also known as the silent chain. It is different from the widely used roller chain in many ways. The traditional roller chains are driven by sprockets. The contact between the sprocket and the chain occurs at the teeth of the sprocket and the rollers of the chain. Because the links of the chain are connected by rollers, the instantaneous chain velocity varies from the average velocity from time to time. This phenomenon is called chordal action [1]. The different velocity causes the links to move up and down with rotation through the angle of articulation and this in turn makes noise and impact between the chain and the sprocket. The links of inverted-tooth chain are also connected by pins. However the location of contact between the chain and the sprocket is different from that for roller chains. It occurs between the teeth of the sprocket and links of the chain. This reduces the chordal action and causes a quieter operation. The inverted-tooth chain is more expensive and can be used at higher speed. The purpose of this paper is to optimize the shape of the link of an inverted-tooth chain so that the optimized chain can transmit more power and have a longer life.

There are three levels of structural optimization. The first one is the topology optimization followed by shape and sizing optimizations. The topology optimization generates an approximate topology of the structure. The shape optimization produces the exact boundaries of the structure. The sizing optimization determines the optimum dimensions of the structure under a given topology. Since the link of the inverted-tooth chain is made of steel plate, the thickness of the plate cannot be changed. Therefore only two stages of optimization need to be done. The first stage is the topology optimization. The objective in this stage is to minimize the compliance of the link subject to the weight constraint. Because the link moves with time during operation, the forces acting on the link are different at different locations. Before doing the optimization the forces acting on the link at four critical locations are computed. The first location is the link at the tight side of the chain without contact with the sprocket. The second location is the place where the link is just in contact with the sprocket. The third and the fourth locations are links in full contact with the sprocket. The maximum forces acting on the link occur at these four locations. Because the chain is not a fixed structure, it is hard to set the boundary conditions in order to compute the stresses in the link using finite element method. To solve this problem some pseudo-springs are used to constrain the link. The Hertz contact theory is also used to compute the contact area between the pin and the link. The pseudo-springs are created within this area to simulate the constraints given by the pin. After knowing the forces acting on the link and the constraints on the link, the optimum topologies with different weight constraints at the four critical locations are generated. Based on engineering judgment the best four topologies at the four locations are chosen respectively from the generated topologies. The second stage optimization is the shape optimization. The initial shapes are the four best topologies from the first stage of optimization. The objective in this stage is to minimize the maximum stress in the link during operation so that the fatigue life of the link can be increased. The constraint is also imposed on the weight of the link. The MSC.Nastran optimization module is used to do the shape optimization. Four optimum shapes are found for the four critical locations. A cross examination of stresses of the four optimum shapes at the four critical locations is then performed. The best one of the four optimum shapes is chosen as the final design. The optimum design processes introduced previously are successfully used to design the link of a ANSI 3/4" pitch inverted tooth chain[2].

1

A.C. Ugural, "Mechanical Design: An Integrated Approach", The McGraw Hill Companies, Inc., New York, 2004.

2

"Inverted tooth (silent) chains and sprockets", American National Standard, ANSI B29.2M, 1982.

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