University of Puerto Rico
Mayagüez Campus
Department of Mechanical Engineering
Machine Component Design 1, 2007-I
INME 4011
Shaft Design for Transmission Application
Irmydel Lugo Velazquez
843-03-3851
Victor Molina Vargas
802-02-4285
Eladio Pereira Troche
843-00-5500
May 8, 2007
Objective
Design a shaft that transmits 40 HP with a fluctuating rotation of 100 to 3000 RPM. This shaft will be designed for an infinite live with a safety factor. Other requirements of the design are minimize the weight of the shaft and appropriate material selection.
Description
Design a shaft that will be used in some type of transmission application. This shaft is used to transmit a maximum amount of power of 40 hp at 3000 rpm. This element is subjected to forces applied when the gears it’s rotating. We have a restriction of length of 2 ft and a minimal thickness of .25 inches. Another important aspect of this design is that we need choose a material that holds all the stresses applied to it and must have a very high safety factor to assure that the shaft will have infinite life. The material in addition must be as light weight as possible and durable, endure other types of fatigue, although our design will take into consideration the critical points of the shaft and optimize them.
The data given in this design submits the shaft to fluctuating stresses. The main idea is for the shaft to withstand the change in loads, for example, rising from 1000 to 3000 RPM and back again. The following figure is a description of the shaft that we must design.
Design Details
· First we calculate the torque applied to the shaft at 3000 RPM:
· Now we calculate the force due to the torque:
· For the next step, we made a diagram show how the forces are applied at the point of the spur gear, we assume the angle of attack for the teeth of the spur gear is 20º:
· Diagrams:
o X-Y Plane:
o X-Z plane:
o Stresses Obtained:
· Moment of Inertia for each diameter:
· First Moment of Area:
· Stresses:
o Section with diameter of 2 in:
o Section with diameter of 4.8 in:
o Section with diameter of 6 in:
· Principal Stresses:
o Section with diameter of 2 in:
o Section with diameter of 4.8 in:
o Section with diameter of 6 in:
Material Selection:
· Boundary Conditions:
o
o
· Substituting Constants:
o
· Now we determine the material indices that minimize the mass of the shaft, so it will be as light as possible:
To minimize the mass we have to minimize the relation . So we maximize the inverse to find appropriate materials in the Elasticity modulus-density graph. The graph tells us that the best material is B4C, but a ceramic is not recommendable because it’s a very brittle and small cracks will propagate very fast. Amongst the best materials to be used are aluminum alloys, titanium alloys and steel. Also the ceramic and carbon fiber alloys, but due to the fact that the forces and stresses are so high, it is not recommendable to use them. With the relation mentioned before we obtain that aluminum has the lowest ratio with 1.68 followed by titanium with 2.70 and finally steel with 2.85. We found some recommend materials to be used for this application; which are the following; Aluminum 2024-T4, 1020 Steel and Beta Titanium Alloy.
For the fluctuating stresses we start out with the following graph to show Force vs RPM;
Deflection:
The maximum deflection occurs at ; where a and b are the distances between the applied force and the supports, where b > a. In our design a is 9 in and b is 19 in.
At this point the maximum deflection is obtained by the following equation; where P is 168.064lb, I is .78233 in^4 and L is 28 in.
We obtained the value of E for Al-2024 T-4 (E = 10602 KSI), ASI-1020 (E = 29732 KSI) and Ti-15Mo5Zr3Al (β titanium alloy) (E = 10878 KSI).
We can compare these values and find that the least probable to have deflection is steel followed by titanium and finally aluminum.
Calculating Safety Factor:
In each of the previous tables, which we develop in excel, we can find all the calculations done to obtain the safety factor for each material and different point on the shaft. The safety factor was calculated by taken into consideration that we use infinite life as a goal of the design. The highest safety factor was obtained by titanium, followed by steel and aluminum respectively.
Discussion
After compiling all the data, from the initial dimensions to the safety factor for each individual material, we found that between zero and 100 rpm the forces are so great due to the fact that they tend to infinity, with this it would not transfer the 40 hp required by the design. So we decided to evaluate the design with fluctuating stresses relating from 100 rpm to 3000 rpm. In the material selection we obtained various materials, but ceramics and carbon fibers were discarded due to the fact that they would not withstand the forces exerted. The remaining materials were analyzed to reduce weight and increase the safety factors. Steel is the best option, even though being the heaviest of the materials; the reasons outweigh the weight reduction. For example, there is a big difference between the deflection of the steel was of. 002736 in, and this is approximately 3.5 smaller times that that of titanium and that of aluminum. This is a determinant factor for design a shaft since helps to reduce the fatigue and obtaining infinite life. Also cost comes into effect because of the fact that both the titanium and aluminum considered are fairly expensive and rare.
A disadvantage that we confronted designing the shaft was the constants changes in diameters. Because we have no radius to deal with, it became a challenge.
Conclusion
The shaft that we designed works for velocities between 100rpm to 3000rpm, generating a constant power of 40 HP. This shaft was designed to minimize the weight and the cost of manufacturing. To minimize the weight we selected diameters that will give small cross sections and, that way, we use less material.
Our main objective was to make a shaft that will have infinite life. To achieve that, we searched for materials that resist deflection, cracks and fatigue. We selected and analyzed three preliminary materials: an aluminum alloy, a titanium alloy and a stainless steel. Our best option was the steel AISI 1020, since it filled better the requisites. This material gave the lowest possible deflection of the shaft, in comparison with the other two materials analyzed. The safety factors were 2.43 for one critical section, and 6.78 for the other critical section studied. These safety factors are good to make the shaft useful for infinite life. We calculated with the Von Mises (equivalent), amplitude and mean stresses of each section. We found on tables the fatigue limit stress (Sf) and the ultimate stress (Sut) of the materials, and with these quantities we calculated the safety factor, using the Goodman model.
Appendix