Optimal design of the limb in compound bows

Abstract

Most of the emphasis in research on compound bows was done on the kinetic analysis of the cam mechanics and their optimization. When the mechanics of the limbs are considered, many assumptions are made, and mostly experimental data are used. There is not an analytical model that provides a method for the design of the limb. A mathematical model is presented in this paper, which can accurately determine the deflection of the limb, the stored strain energy, and the stress in the material. While calculating these values, many parameters are considered which makes the model a realistic representation for the bow limb analysis. These parameters include the pre-strained condition of the limb in un-drawn state, the angle of the force acting on the tip of the limb, the characteristics of composite materials used in bows, and the variable cross section of the limb beam. The numerical model is programmed in Octave and the results are compared with a finite element analysis model in Ansys to show its accuracy. An optimization problem is set up to obtain the optimum design variables for the beam. The objective is to maximize the stored energy in the limb at its full-drawn position. It is shown that a beam with a variable cross section of a polynomial gives the maximum stored energy, and the optimum parameters are obtained. In another optimization problem, different composite materials are compared according to their characteristics to show which provides the best properties. While all materials will result in the same strain energy, the compared properties are cost and weight. The data used in the optimization are taken with respect to measurements on real bows.