# Deflection curve of a Compound Spur Gear Shaft

1. Mar 24, 2013

### pepster

Hi All!

I would really appreciate your help on something that has been bothering me for the past week.

I am uncertain on how to derive the deflection curve equation for a certain shaft. Loads are acting on both ends of the shaft, with the reaction forces being provided by two separate bearings that are located towards the middle of the shaft.

Is it correct to assume that by finding the Bending moment equations for each section of the shaft and then double integrating I will find deflection for each section of the shaft? I have attached a picture of my most recent attempt so far.

I am fairly certain that this is the right approach, however I am not sure as to whether or not I need to somehow combine all these moment equations into one equation so that I have a smooth deflection curve as opposed to a disjointed one.

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Last edited: Mar 24, 2013
2. Mar 26, 2013

### timthereaper

If you're looking for an analytical method, try using Castigliano's theorems. The second theorem is the best for determining deflections across an entire beam. It states that the partial derivative of the strain energy in the beam w.r.t. the load yields the deflections at the location of the load. The strain energy is easier to compute than you might think (or I thought when I first learned about this).

If you're interested in the deflection of the beam at a point other than where the load is applied, the trick is to place a fake force at that location. Include that fake force term in the equation until after solving the integral, then just set it equal to zero. If you do that at enough locations on your beam, you can then plot a nice curve.