calculate moment of inertia pulley with multiple shapes

#Calculate moment of inertia pulley with multiple shapes how to#

In particular, method 'b' seems a bit more efficient, since it employs one less subarea, compared to method 'c'. For each one of their subareas, there are easily available analytical formulas, we can use to calculate their moments of inertia around a parallel to global axis. Methods 'b' and 'c' on the other hand, do not present such a problem. Surely, it can be done this way, but it is not very efficient calculation wise. Firstly, we have to find the moments of inertia I x and I y around the inclined centroidal axes of the rectangle, and then transform them around a horizontal axis, using the formulas for axis rotation. No formula is available for this particular angle of inclination of the rectangle that gives its moment of inertia around an axis parallel to the global one. Method 'a' seems more simple, and employs the least number of subareas, but the calculation of the moment of inertia of the central inclined rectangle is rather hard. Three alternative ways to divide the top-left gray area to subareas, in order to find its moment of inertia around a horizontal axis.

With method 'a', the number of subareas is 4, with method 'b' it is 5 and and with method 'c', 6. The global axis of rotation is indicated with red dashed line. The following picture demonstrates a case of a composite area, that is decomposed to smaller subareas, using three different methods (among many others possible). Efficiency is important, because sometimes there are many ways to decompose an area, but not all of them are equally easy for calculations. While doing so, we must ensure that we can efficiently obtain the moment inertia of each subarea, around a parallel axis. With step 1 we aim to divide the complex area under investigation to smaller and more manageable subareas. Flowchart for the calculation of the moment of inertia of a composite area Add (or subtract for negative subareas, see examples) the moments of inertia from the last step.įigure 1.Apply the Parallel Axes Theorem to find the moment of inertia of each subarea around the global axis.Determine the moment of inertia of each subarea, around a parallel axis, passing through subarea centroid.Determine the distance from global axis of the centroid of each one of the subareas.Identify simply shaped subareas the composite area can be decomposed to.

In general, the steps for the calculation of the moment of inertia of a composite area, around an axis (called global axis hereafter), are summarized to the following:

#Calculate moment of inertia pulley with multiple shapes how to#

In this article, it is demonstrated how to calculate the moment of inertia of complex shapes, using the Parallel Axes Theorem. The possible shape geometries one may encounter however, are unlimited, but most of the times, these complex areas can be decomposed to more simple subareas. The given analytical formulas for the calculation of moments of inertia usually cover, just a handful of rather simple cases.