Most of the information presented in this section comes from a great book - “Spring design and manufacture” by Tubal Cain ISBN 978-085242-925-9. This Tubal Cain is an English chap who has written loads of good technical books not the other good (American?) Tubal Cain who has lots of interesting machining videos on youtube. It seems to me that anyone who is called Tubal Cain is a good bloke!
So far I've only found these equations mentioned on this internet site =>
Springs. But I'll put them here as well for the sake of keeping the flow going.
I've presented the equations in a slightly different format from the site linked above – they are more in keeping with the book I referenced above. Below you can see a list of properties for the two springs in the OM61X piston vacuum pump.
Stiffness is defined as the force required to compress or extend the spring by a unit amount. I'm using the metric system and because the springs are small the values are coming out as Newtons per millimetre (N/mm). This means that to compress the large spring by 1mm you need to apply a force of 13.48 N. Plotting applied load (or force) against the deflection of the spring you end up with a straight line relationship. This straight line relationship is often referred to as a linear relationship. If the graph was a curve it would be non-linear. The spring will “go non-linear” when it reaches the extremes of its travel; in compression the number of independent coils (n) reduces as the coils begin to touch; in tension the effective diameter of the coil spring (D) reduces or necks.
To check the calculation given above I devised a pretty gash way of measuring the force and deflection of each spring. In an ideal world I'd use a decent way of measuring spring deflection with say a depth gauge or a LVDT but the equipment I've got wouldn't allow it. In an ideal world I'd have used a calibrated force gauge to measure the force – something like that is beyond my play budget at the moment. All I had was an old fashioned set of bathroom scales. There is no way that these measurements that I have made are reliable in any commercial sense (let alone to a decent research standard) but they do show the general method that can be applied and copied.
The measurements I made, however, do provide a good indication of what is likely and because I'm comparing these measurements with an established theory they show that they are kind of believable! The measurements are only presented as an indication of what's likely to be going on. They are made to a unique guy in a shed standard that isn't recognised by ISO. Feel free to take these results with a pinch of salt or not...
To make the measurements I used my hydraulic press to compress the spring. If I had been able to connect a low pressure hydraulic gauge I could have worked out the applied force at the end of the piston but I went for a quicker option – using the bathroom scales! Old fashioned scales are not known for being particularly accurate but I checked them with some known weights and they were coming up with consistent repeatable values. (Please note digital bathroom scales are of no use for this method as they tend to reset themselves at low values) This was a quick and dirty solution that didn't involve too much messing about.
If I could have got my depth gauge to fit then I would have used that to measure the deflection but again no such luck. Quick and dirty solution number two was to mark the deflection on a back of an envelope.
This method doesn't seem very accurate – I know – but take a look at the results. I made three sets of measurements compressing each of the springs in my piston vacuum pump. These results are presented as force (kg X 9.81 m/s2) versus deflection.
Stiffness is the slope of these lines so you can see I measured a stiffness of about 11.2 N/mm for the big spring (compare with calculation of 13.48 N/mm) and I measured about 10.2 N/mm for the small spring (compare with calculation of 12.24 N/mm). Not bad eh? Especially when you consider this has been achieved with a bit of paper a ruler and a some dodgy bathroom scales. I was planning to build my own spring force / deflection measuring rig as commercially these are seriously expensive – I'm not so sure I'll bother now...
Next thing I wanted to check was how the two springs behaved together one inside the other just as they are fitted in the vacuum pump. Theoretically the stiffness should be added together because they are in parallel. This is like saying that they are in operation side by side rather than in series which would be one spring on top of the other. Making a measurement as before came up with these force verses deflection lines.
As you can see the measured stiffness (22.1 N/mm) is about the same as the combined calculated value (13.48 + 12.24 = 25.72 N/mm) and the combined two independently measured values (11.2 + 10.2 = 21.4 N/mm). Again not too bad. It is all believable.
This has all been very clever (I can hear some of you saying) but what's the point? Well with all of this data I can work out how much the force the return spring is pushing the cam follower onto the roller coaster cam on the timing device.
I could of course have just measured the deflection of the cam follower being pushed into an assembled vacuum pump – but where's the educational fun in that? To be honest I did that anyway – so here's the result – but don't forget I also had to check the validity of using those dodgy bathroom scales...
From the graph above you can see that the return springs apply more force to the cam follower than the opposing force provided by the vacuum “suck” (this has been calculated above to be about 317N in the most likely situation to 373N in the most perfect case possible on planet earth where an OM61X is likely to be operating). When fully released the minimum force needed to move the piston is about 400N; and when pushed to the end of the stroke you need about 620N. But this is just the static case – what happens when the engine is running?