EGR 181 Centroid-Related Assignments & Notes, Fall 1997 (Dr. Cutchins):
The text has short but thorough descriptions of the topics of center of mass, centroids, and the related Appendices that deal with area moments of inertia. Appendix B refers to mass moments of inertia in the Dynamics book, a very related topic.
Once these area moments of inertia are determined, principal values of them and the axes to which they refer, can be found. These special properties are often called eigenvalues and eigenvectors. Similarly, these properties exist for mass moment of inertia problems. Many shapes are tabulated; those of area on pps. 511-2, and those of mass on pps. 513-6.
The same subjects (as those described above) applied to composite bodies appear on pps. 263-4 & p. 471. Since we will not have time to address all of these topics completely this quarter, and some of the topics require integral concepts, it is this composite body topic that will be emphasized in EGR 181. The theorems of Pappus (pps.274-5) can be utilized to solve problems involving bodies of revolution if you understand the concept of centroids and develop a good 3-D perspective.
In the space of a few days, you should learn to do the following:
a) Be able to find a centroid location of a composite area, and the center of mass location of simple mass arrangements.
b) Apply the Theorems of Pappus to simple cases to find surface areas and volumes of bodies of revolution.
c) Use a software package on the Shop Bldg. PC's called EigTool that allows you to do ALL the computations for the topics above (except the Pappus Theorems), for both composite point-area problems and systems of masses even in three dimensions. There are restrictions, however, that the parts making up the composite body have to be point-areas or point-masses, or lines (area or mass) and there cannot be more than 99 of each. At times you may want to use lines with zero mass or zero area to better define a 3-D shape that involves point-areas or point-masses. [The software will also do principal stress problems in 3-D, problems that are covered in a course on Strength of Materials. EigTool was developed by Dr. Cutchins and one of his grad students and is in the process of further improvement.]
d) Determine (by hand) area moments of inertia and mass moments of inertia for the simplest composite bodies usually using just one of the parallel axis theorems on p. 471 and some of the simplest tabulated values on pps. 511-516.
A very simple example:
|||||||------------------ The shape to the left is comprised of a point mass, m, and a rod of mass, 2m and length, L. We choose the origin for x at the bar's right end, positive to the left.
X to the center of mass is determined by applying the first of Eqs. 5/7, p. 263.
X = [2m(L/2) + mL] / [2m + m] = 2mL / 3m = 2L/3. This would be the balance point, located from the right end.
The mass moment of inertia about a vertical line at the right end of the bar would be
1/3 (2m) L^2 (see bottom case, p. 514) + m L^2 (a mass term just like the last term of one of the area equations near the bottom of p. 471).
The mass moment of inertia about a vertical line through the mass center would be the mass at the left times (L/3)^2 + 1/12 (2m) L^2 + (2m) [2/3 L -1/2 L]^2.
Assignments, Due Wed. 12/3: 5/44, p. 267 (Treat the circle as "negative area".) 5/52, p. 269 A/36, p. 473
5/57, p. 270 using EigTool. Let a = 6" and model the curved rod with 3 straight lines of 30 degrees arc each. You should determine the (x,y,z) values of each end of these straight-line arc lengths before getting on the computer since they will be required input in EigTool. The software will be further described in class.
____________________________________
For the eager, but not required for this quarter:
The text has descriptions of the topics of center of mass (pps. 243-5), centroids (pps. 245-9), and the related Appendices (App. A, pps. 455-9) that deal with area moments of inertia. Page 491 (Appendix B) refers to mass moments of inertia in the Dynamics book., a very related topic.
Once these area moments of inertia are determined, principal values of them and the axes to which they refer, can be found (see pps. 478-top of p. 480). These special properties are often called eigenvalues and eigenvectors. Similarly, these properties exist for mass moment of inertia problems (The equations are in the companion Dynamics text). At first glance it may appear to be many, many equations. But note the similarity of so many of them and you will not have to stretch your memory so much (Ex. p. 456, p. 244 & p. 263). Also, many shapes are tabulated like those of area on pps. 511-2, and those of mass on pps. 513-6.