Machine Design II - Lab (ME 308)
Project # 1 – Design of Shaft and Selection of Bearings
Summary of Class # 3
Step # 1: Calculation of Deflections under the gear # 3 and gear # 4
Moment Equation :
Slope Equation :
Deflection Equation :
Deflection equations for our Shaft in the y- axis:
Equations 1 through 6 above involve 6 unknowns (C1, C2 , C3 , C4 , C5 , C6 ) and hence, 6 boundary conditions are needed to solve for the 6 unknowns. The 6 boundary conditions are as follows:
1. The deflection under the 1st bearing is equal to 0
2. The deflection under the 2nd bearing is equal to 0
3. The continuity equations, where the slope of the shaft is equal (Under the gear # 3)
4. The continuity equations, where the deflection under the gear # 3 must be equal.
5. The continuity equation at the location of the 2nd bearing is
Using the above boundary conditions the 6 equations are solved for the 6 constants.
The deflections under the gear # 3 and gear # 4 in the y –axis are calculated by substituting these values in equations Eqn # 2 or Eqn # 4 ( for gear # 3) and Eqn # 6 ( for gear # 4). In these equations
----- is the youngs modulus for carbon steel.
NOTE: The above analysis has been done only for the y-axis, a similar analysis has to be done in the z-axis too and the deflections under the gear # 3 and gear # 4 must be calculated for the z-axis too.
Once the deflections under the gear # 3 ( and ) and under the gear # 4 (and ) are calculated the resultant deflections are to be calculated as follows:
Step # 2 : Calculation of the first critical speed
By using the Rayleigh method the first critical speed is calculated by using the equation which is given below.
is the resultant force acting at the position 3 under the gear # 3
is the resultant force acting at the position under the gear # 4
Check whether the first critical speed is greater then 1000 rpm.
If it is not greater than 1000 rpm then you have to increase the diameter of the shaft and repeat the calculations.
If it is less than 1000 rpm then, select a design speed for the shaft, which should be equal to
For the bearing selection we have to go to Eqn (11-9) which is as follows:
is the rated load
is the design load ( in our case RA and RB )
= 106 rev
is the design life = 8000 hrs to 14000 hrs
is the design speed = (0.6 to 0.8)
is the reliability = 90 %
for deep groove ball bearings = 3
Once you calculate the rated load for each bearing, the load ratings have to be calculated by
Taking this value of appropriate bearings have to be selected from the Table (11-3).
NOTE: Do convert the units of to KN before you go to the Table (11-3).
After the bearing selection, do finalize the diameters of the shaft.
Step # 1: A material for the key has to be assumed from the list of materials given to you in the text book. One can choose any CD material or any HR material, except that it should satisfy only one condition.
Step # 2: From the table (8-15), a square key has to be chosen depending upon the range of the diameters of the shaft. Interpolate the data for shaft sizes greater than . Select the width and the height which are equal in a square key. Lets assume that it is "a".
Step # 3: Calculate the length for the key by solving Shear and Beraing stress equations for the key.
Bearing Stress: (Solve for l)
Shear Stress: (Solve for l)
The final value of the length (l) of the key will be the larger of the two values calculated for each location. Generally, key sizes must fall within the following range by length.
All the best and have a nice vacation. But do not forget to submit me the report on your analysis, in the class immediately after the vacation.