Machine Design - II (ME 308 Lab)
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:
.. eqn. 1
.. eqn. 2
.. eqn. 3
.. eqn. 4
.. 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
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).
The equations can be solved easily using any of the available math software like
Mathcad, Maple, Mathematica or Matlab
Now, to solve for the deflection under the gear in the y-direction, substitute for C1 and C2 in equation 2 to obtain:
To find the slope of the shaft at the 1st bearing, substitute for C1 in equation 1 to obtain:
To find the slope of the shaft at the 2nd bearing, substitute for C3 in equation 3 to obtain:
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.