Introduction This webpage includes notes on methods of calculating radial deflections on
shafts transmitting torques and subject to lateral loads and moments which tend to bend the shaft.
The notes are provided to enable basic outline calculations to be carried out. For detailed design work
it is recommended that quality reference documents are used.
Examples of using this principle are found on web page Shear Bending moment diagrams. Shafts of same diameter
It is often sufficient to calculate the deflections based on the shaft being one diameter. This clearly results in a conservative value if the diameter used is the minimum diameter. Shafts of with many diametersFor more complicated shafts with more than one diameter various methods are available including the Area Moments method ref. Beam Theory and Energy methods (Castigliano's method) . ref. .Energy Methods However it is often easier to use numerical integration to obtain shaft deflections which are reasonably accurate. This basis for this method is the differential equation below .The simple basis of this method is that the change in slope between two points Δθ = Δx.(M/EI) .... Δx being the
distance betwen the points. The resulting The method is based on one provided in the book "Machine Design: Theory and Practice.. by A.D Deutshman, W.E.Michels, C.E.Wilson, (McMillan Publishing Co Inc)" A version of the spreadsheet below has be created with the excelcalc.com repository which enables convenient access to the method of calculating deflections of stepped shafts ref ExcelCalcs.com calculation Shaft Deflections The first step is to evaluate all forces and reactions in the shaft and identify points along the shaft at each force and reaction and change of section. It is also more accurate if long sections of constant diameter with no loads are divided into smaller intervals with associated points. This method is illustrated using a shaft example below (vertical and horizontal planes ).
Shaft Details Material Steel The following step by step approach is used to complete this procedure 1) Construct a table with the columns as shown. Calculate the bearing forces.
Calculate the valuesof EI for each section. Calculate the bearing centre distance. Note: The slopes calculated have all been relative to the slope at point 1. They are relative not absolute values. The resulting calculated deflections are also relative to the tangent of the shaft at point 1 - they are not absolute values. The total deflection between the bearings is zero and thus the integration constant is the proportional error away from the correct value (error /m) . This is used to correct all of the intermediate values as follows 13) Multiply the total integration constant (K) by the x values between points. Add this relative
integration constant value in column 12 in the same row as the deflection increment. 14)The corrected deflection at each point is the product of the relative integration constant and the calculated deflection increment in the preceding row. Enter this in the point row in column 13. To this add the corrected deflection of the previous point to make up a running total. The slopes calculated are also relative to the tangent at point 1. To arrive at the absolute
slope /angle at any point simply add the angle correction value. θ
To calculate the shaft angle,in the vertical planes, at the bearing centres Points 1 and 8. Simply add θ Deflections in Horizontal plane The same method is applicable for obtaining the deflections in the horizontal plane
To calculate the shaft angle,in thehorizontal planes, at the bearing centres Points 1 and 8.
Simply add θ Calculating shaft deflectionThe numerical methods above enable calculation of the deflection of the shaft in the horizontal and vertical planes. To obtain the total resultant deflection and the angle of the resultant deflection . The following calculations are completed δ Example..based on shaft above.. |

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