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.  

    2. 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

     

     

    wpe4.jpg (19676 bytes)

     

     

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

      2.  

      wpe1.jpg (3833 bytes)

      ………………….. eqn. 1

      ………………….. eqn. 2

       

    2. For A ≤ x ≤ A+B

      3.  

       

      wpe2.jpg (4904 bytes)

      ………………….. eqn. 3

       

      ………………….. eqn. 4

       

       

    3. For A+B ≤ x ≤ A+B+C

     

    wpe3.jpg (5311 bytes)

     

     

     

    ………………….. 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

    Mathcad, Maple, Mathematica or Matlab

     

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

    4.  

    where:

     

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

    5. where:

     

  5. 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.