Do-Over! Solids of Revolution

For my do-over blog, I thought about the prints that I have done this semester. Of all my prints, the one I liked the least was the first one. Due to my lack of experience with the software and 3D printing process, the resulting product was lackluster in my eyes. With this iteration, I wanted to take a look at the same functions I chose the first time around, and see if choosing a different domain would create an object that was more like the shape I desired. The goal was and is to create a bowl shaped object that looks good on my desk.

Previous Mistakes

My previous print was a visual representation of the approximate solid of revolution created by the two curves: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] The domain of the original approxmiation was from the y values of [0,1]. The shape of this graph was very wide and short making it look more like a saucer. To fix this, I took the domain from [0,11] which takes more of the shape of the region created by the curves. This will alter the ratio of the height of each washer to the inner and outer radii. There will be one additional washer in this design, and it will be much skinnier. The OnShape preview creates an object that matches the desired shape when compared to the original print. Here is a picture of the new version:

Caution in Printing

A semester's worth of experience has allowed me to pick up tricks and tips for using 3D printers. Taking advantage of an object's printing orientation has great potential to avoid using supports. This time around I will print my model upside down which will give it a steady base to print. Another thing to watch is the scaling done in the Prusa slicer. It will be important to make sure that the scale factor makes sense, and the dimensions scale correctly.

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The Approximation of a Solid of Revolution

Most math teachers I've had have been able to break down Calculus into two very broad categories: derivatives and integrals. What is truly amazing, is how much you can do with these two tools. By using integration, it is possible to approximate the shape of a 2-D function that is rotated around an axis. This solid created from the rotation is known as a solid of revolution. To explain this concept, we will take a look at the region bounded by the two functions: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] bounded at the line y = 1. This region is meant to represent a cross section of a small bowl. While it may not perfectly represent this practical object, the approximation will be quite textured, and will provide insight into how the process works. The region bounded by the two functions can be rotated around the y-axis to create a fully solid object. This is easy enough to talk about, but what exactly does this new solid look like? Is...

Do Over: Integration for Over Regions in the Plane

Introduction Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons. Why Double Integrals Again? As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue. The main cause of ...

Middle of Mass Measured from Moment

The center of mass of an object is the point that lies in the average position of mass in that object. For a normal polygon, that point is in the middle of the shape. However, if the shape is irregular, has concave angles, or has holes, the center of mass can be far from the middle of the object. Two things come into factor when determining the center of mass of an object. Weight and distance. Obviously, if one side of an object has more mass than the other, the center of mass will be on the heavier side. But, distance plays a role, too. If two sides of an object have the same mass, but one is farther from the middle, the center of mass will be on the side that has farther mass. Multiplying both of these factors together gives the moment of an object. Dividing the moment by the overall mass gives the location of the center of mass. The reason I chose my shape is to mimic how those toy balancing birds work. The toy is a model bird with its wings outstretched. The ...

MA 391 Composition and Communication

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