Compound Pendulum

After writing the article on the simple pendulum calculation, I had fully intended to follow it up with an article on the Compound pendulum, but as it turned out my efforts to do the calculation for this proved to be based on the wrong equations so after struggling for some time to find the right solution I finally appealed to the many clock builders on my mailing list for help.

It didn't take long to get 4 responses with suggestions as to how to do it. 

The first was from Roger Bunce who sent a page from his copy of  'Workshop Calculations, Tables, and Formulae by F. J. Camm'. 

You use the formula to calculate the equivalent pendulum length of a simple pendulum from the lengths a and b in the compound pendulum. The resulting length can then, of course, be used in the simple pendulum formula to calculate the period.

It should be noted that the length b in the diagram should be to the centre of the top weight.

Next was a link from Rus Thomas to an Excel file on Paul Rogers website for his 'compound pendulum period calculator'. This is an excellent resource offered by Paul and allows you to calculate both simple and compound pendulums. I have used this further down to illustrate how it can be used for your own calculations.

Next came a suggestion from Robert Miller who suggested searching on the works of Henry Kater, which I did but must admit I struggled to understand how to use the data provided.

Finally from Guy Winslow who provided a link to a calculator called  Eureqa. I watched a couple of the videos and couldn't figure out how I would actually be able to use it. 

So, in the end, I have settled on using the Excel file from Paul Rogers.

Excel Calculator

The reason for wanting to do this calculation was to use a much shorter 'seconds' pendulum in a mantle clock is a clock that could sit on a shelf without having the 1-meter long pendulum protruding through it. The double ended pendulum allows this because when you put a second weight above the pivot point it slows the clock down so you can shorten the length to main weight quite considerably.

If you are interested in doing this you can download a copy of the Excel file here. I have modified this slightly so that only the cells that you need to interact with are visible.

The image above shows my modified file with the inputs in the pale orange and the calculated values in green. 

My first calculation assumes that both are going to be equal which simplifies the initial setup, but this is not necessarily how it will finish up as you can always make the bottom weight larger than the top which is quite normal. Not sure that it would work so well in practice if you make the top larger than the bottom.

It should be noted that the weights are assumed to be cylinders so you need for a start to input the diameter and length of the weights you are going to use. If your weight is not a cylinder in this orientation then you need to adjust the values of diameter and length to the proportions of your weight and adjust until you get it approximately right. 

Now input the values for H1 and H2 and you should see immediately a value for the Period in seconds. 

I have a value of 2 seconds for the period (well almost)  which is 1 second to swing in one direction and then 1 second to swing back again. I have achieved this by inputting the value for H2 which is the maximum length I can have for the clock that I am designing, and then keep on adjusting  H1 until the value is reached.

At this stage, you could also introduce some changes to the sizes of the weights to arrive at a solution that produces a better aesthetic to the pendulum.

A couple of things to note here are that the values for H1 and H2 are assumed to be to the centre of mass, in reality, should consider the weight of the rod as well, but it is a quite small difference so it has been ignored here.

The other thing is there may be other factors that will affect the movement of the pendulum over time, and in the way, the clock is constructed, so adjustment should always be allowed for in positioning the weights on the pendulum when it is finally built and running.

Generally speaking, the following rules apply:-

To increase the period:

Reduce H2

Increase H1

To decrease the Period:

Increase H2

Decrease H1

Keep H1 smaller than H2

Keep M1 smaller or equal to M2

Goal Seek

Too enable you to more quickly adjust the values for length and or weight, Excel has a function called Goal Seek that can be used to quickly zero in on a value that you are seeking. For instance, I require a value of 2 seconds for the period but it would take a lot of time to manually increment the value of H1 to achieve this so Goal Seek can be used to speed up this process.

To get there go to the Data tab and click, then go to 'What-if Analysis' and then click on 'Goal Seek', that will bring up the screen shown below.

To fill out the Goal seek box first click in the 'Set cell' box and then click in the box containing the value for the Period. 

Next, enter the value you want in seconds in the 'To value' box.

Now click in the 'By changing cell' and then click in the box containing the value for H1.

Finally, click OK and it will work out the value for H1.

The result is shown above, the period is now very close to 2 seconds and the H1 dimension has been reduced to 196.6784399 to achieve it. You can do the same thing for H2 or the weights.  

It seems to work better if you fix H2 and recalculate for H1 than the other way around.

The photograph above shows the test rig for the woodenclocks gravity escapement fitted with the double ended used to test the calculated results from the Excel file.