Machine Component Design: an Analysis of Deflection of a Beam

Machine Component Design, or MCD, was one of my favorite classes due to the variety of projects we were tasked with. Under the academic course guide, MCD is described as "Applications of fundamentals of engineering mechanics in analysis and synthesis of machine components and systems. Special emphases placed on stress/strength analyses and fatigue failures. Design of mechanical components and systems including threaded fasteners, springs, bearings, gears, shafts, clutches, brakes, belts, chains, and couplings."

Focused on the fundamentals of design, the second project we were challenged with involved a redundantly supported beam, pictured above. This bridge was made out of a felled tree over a creek with a large rock as a middle support. The load P is a person walking slowly across said bridge. The "Basic" Level of this project was to determine the deflection of the beam at the load itself, P = 850 N, without a rock support and with variations of a.

By utilizing singularity functions to determine the deflection of the indeterminate beam, I was able to create an equation to later use in Matlab. A for loop was then written in order to iterate through different positions, or a, of the load P and create a plot of the deflection of the beam.

The "Medium" Level involved creating another plot that includes the redundant rock at b = 3 m for the same load P. In order to complete this task the integration relationship for deflection was calculated at all boundary conditions, a static analysis of the beam resulted in equations for the forces and moments, and finally a system of equations was solved for. My analysis of this problem is shown below:

The "Advanced" Level requires the optimization of the location of the rock, b, to limit the maximum bridge deflection to no more than 0.02 m, with the rock as close as possible to the left end. The reported value must be within 0.1 m. Using the singularity functions, logistics were created for the locations of a and b: a > b for when (a - b) and b > a for when (b - a). The iteration in Matlab would break if the deflection was greater than 0.02 m and had a step of 0:0.1:L/2. Two for loops were written since a and b are changing and the final result was b = 3.9 m.

This project focused on the foundations of calculating forces, stresses, and deflections of machine components using the tools and language of mechanics, as well as verifying the reasonableness or primary results. A more in-depth analysis and my step-by-step approach can be downloaded as a PDF on the main Portfolio page.