MEEN 368 – Solid Mechanics in Mechanical Design Fall 2016 Supplemental Problem 15 Recall that four prominent and useful methods for calculating deflections include double-integration, double-integration with singularity functions, Castigliano's theorem, and superposition. Consider the shaft of Problem 3-68 (shown below). Answer the question for each case described and briefly discuss the pros and cons of each deflection method for determining the desired quantity. (a.) The bearing at O is wearing out prematurely. What quantity, or quantities, would you want to calculate? (b.) The belt connecting to the pulley at B is loosening. What quantity, or quantities, would you want to calculate? (c.).The pulley at A is replaced by a spur gear, and the gear is chattering and wearing excessively. What quantity, or quantities, would you want to calculate? (d.) The shaft is vibrating too much, so you need to determine the maximum deflection and its location. (e.) Consult Table 7-2. What type of bearing is most robust (i.e., tolerates the most misalignment)? What type of bearing is most sensitive to misalignment? Is the amount of "deflection" permissible for spur gears generally higher or lower than for bearings?Solution (a.) The slope, or rotation, at O is what should be calculated (i.e., qO). For a single quantity like this, Castigliano's Theorem would be an attractive option (assuming no other deflection quantities are of interest at the time). However, it would also be permissible to use any of the other three methods. The likely order in terms of ease of calculation would be: superposition, double-integration with singularity functions, double-integration. (b.) The radial displacement at B is what should be calculated (i.e., dB), and perhaps the slope (qB) as well, as the latter may cause misalignment. This assumes that whatever is causing the loosening would not necessarily be the loads at B. As was discussed in class, I am interpreting the loosening to be due to other sources, i.e., not due to the belt itself being too tight. Obviously, a very tight belt would create "high" radial displacements at B, but this would not be consistent with loosening of the belt. Again, for a single quantity like this, Castigliano's Theorem would be an attractive option (assuming no other deflection quantities are of interest at the time). However, it would also be permissible to use any of the other three methods. The likely order in terms of ease of calculation would be: superposition, double-integration with singularity functions, double-integration. (c.) Based on the information in Table 7-2, it is apparent that proper functioning of meshing gears can be negatively affected by excessive "deflections" of both types: radial displacement and slope, or rotation. Thus, the quantities to calculate in this case are the radial displacement at A (dA) and the slope, or rotation, at A (qA). In this case, any of the four deflection methods would seemingly be of equal or similar benefit. (d.) In cases for which the location of the maximum deflection is not apparent, using Castigliano's Theorem is typically quite difficult to use. Thus, an expression for the deflection as a function of position needs to be generated first, and then that expression used to determine the location of the maximum displacement and its value. Consequently, the other three methods would be preferred (any but Castigliano's Theorem). (e.) To answer this, we consult Table 7-2. The most "robust" bearing will be the one that has the largest maximum deflection (slope, or rotation) value. Accordingly, there are actually two types of bearings that allow the most deflection: "spherical ball" bearings and "self-aligning ball" bearings. Here's a question…..(maybe a good BONUS question on some future assignment or exam) Noting that "deep-groove ball" bearings do not allow as much deflection as "spherical ball" bearings, why would anyone ever want to use deep-groove ball bearings? That is, what advantage does a deep-groove ball bearing offer compared to a spherical ball bearing. The one that is least robust (i.e., allows the least misalignment) is the one with the lowest value for the maximum deflection value. According to Table 7-2, this would be "tapered roller" bearings. Based on the information in Table 7-2 for "uncrowned spur" gears, the amount of deflection is generally lower for gears than for