Machine Design - II (ME 308 Lab)

Rigidity Analysis

1. According to the rigidity criteria specified in the problem:
2.

1. The shaft at the gear shall not deflect more than 0.001"
2.

3. The shaft slope through the bearings shall not exceed 1°

and

The following analysis explains the successive integration (double integration) method. It shows how to find both the deflections and slopes in the vertical direction (y-axis).

You are required to complete the required calculations and to find the slopes and deflections in the horizontal direction (z-axis) by yourself.

In the double integration method, the following equations form the basis for the analysis:

 Load Equation Shear Equation Moment Equation Slope Equation Deflection Equation

3. Finding the moment, slope and deflection equations:
1. For 0 ≤ x ≤ A
2.

………………….. eqn. 1

………………….. eqn. 2

3. For A ≤ x ≤ A+B
4.

………………….. eqn. 3

………………….. eqn. 4

5. For A+B ≤ x ≤ A+B+C

………………….. eqn. 5

………………….. eqn. 6

 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: The deflection under the 1st bearing is equal to 0 y(x=0) = 0 ……………. B.C. # 1 The deflection under the 2nd bearing is equal to 0 y(x=12-) = 0 ……………. B.C. # 2 y(x=12+) = 0 ……………. B.C. # 3 The continuity equations, where the slope of the shaft is equal to 0 (under the gear) y(x=6-) = y(x=6+) ……………. B.C. # 4 ……………. B.C. # 5   The continuity equation at the location of the 2nd bearing is   ……………. B.C. # 6     The 6 boundary conditions above are applied to the 6 equations (equations 1 through 6 ) and the resultant equations are solved simultaneously to find the 6 unknowns (C1 through C6).

Note:

The equations can be solved easily using any of the available math software like

4. Now, to solve for the deflection under the gear in the y-direction, substitute for C1 and C2 in equation 2 to obtain:
5.

where:

6. To find the slope of the shaft at the 1st bearing, substitute for C1 in equation 1 to obtain:
7. where:

8. To find the slope of the shaft at the 2nd bearing, substitute for C3 in equation 3 to obtain:

where:

REPEAT the same procedure to find the deflection of the shaft and slopes in the horizontal (z-direction) and then finally check against the rigidity criteria given.