Mass moment of inertia evaluation using the Monte-Carlo method and the parallel axis theorem

The Monte-Carlo method is a powerful tool for analysing a wide range of problems in mechanical engineering and physics. This paper considers how to introduce the Monte-Carlo method to undergraduate engineering students. It is proposed to use the Monte-Carlo method to evaluate the mass moment of inertia as an example application. This is an ideal application area as there is a hierarchy of complexity of implementation, starting with a one-dimensional shape, followed by a three-dimensional shape and finally a composite shape. A number of variants of the Monte-Carlo method are considered, with different complexities of implementation and numerical accuracy. The Monte-Carlo method that uses the parallel axis theorem as part of its basis is the most efficient method, with a maximum speed-up of 58,500 compared to the Monte-Carlo method that is the easiest to implement when reduced run-time is factored into the analysis. If the parallel axis theorem is not part of the Monte-Carlo method basis, then the maximum speed-up parameter is reduced to 79.4. The Monte-Carlo method that uses the parallel axis theorem uses proportionate stratified sampling to allocate function evaluations to the shapes that make up the composite shape.