Solution of a Non-Classical Integral Equation Modeling a Rotating Thin Rod Under Some Constraints and Technical Feasibility

In this paper, a mechanical system composed of a thin rotating rod under some constraints is considered. For this system, the total torque of the gravity forces is fixed and the unknown function to be determined is the mass density of the rod. This kind of problem is faced in several engineering applications as in aerospace. The resulting problem is formulated as a non classical integral equation, where the conventional methods of resolution do not apply. Therefore, a special treatment is required to solve the obtained integral equation. First, the obtained integral equation is transformed into a system of mixed integral and linear differential equations with two unknown functions. The latter transformation allows the inspiration of the general expression of the requested functions. Consequently, a highly non linear system with several unknowns is obtained. During the resolution of the latter system several mathematical technics are used. After applying all these technics an analytical solution of the studied integral equation is obtained. Finally, the technical feasibility from an engineering viewpoint of the production of a thin rod with the obtained mass density function is briefly discussed. In this context, the Functionally Graded Material is proposed as a material satisfying the obtained mass density function.