I’m surprised that you are limiting your deliberations to such a narrow temperature band. You haven’t said where in the world you will be running this vehicle so let’s guess at somewhere in the temperate zones of the northern hemisphere. It is easy to imagine that, after running at high speed for a long time on a rough track on a hot day, the oil temperature could easily be as high as 70 degrees centigrade. Conversely, on a particularly cold day in winter (but not so cold that no work can be done) you might need to consider the vehicle having to operate with the oil temperature as low as -20 degrees centigrade. Now take these values as a ratio of absolute temperatures and you have 343/253 = 1.35. So that’s a 35% difference in gas volume for the same pressure at the two extremes in temperature. This may not be insignificant and, as you suspected, warrants further investigation.
Disregard the accumulators for a minute and imagine that the hydraulic cylinders in your suspension were all connected together and piped to a hand pump with a pressure gauge on its outlet. With the vehicle stationary and on level track you could gradually pump up the suspension and create a plot of pressure against ride height. This will be a function of the weight of the vehicle, the number of wheels, the size of your cylinders and the geometry of your mechanisms. The fact that there are no accumulators in circuit won’t change the pressures – it just means if you ran like this you would have a very hard ride and probably generate pressures high enough to damage something.
As the ride height varies so will the pressure needed to support the vehicle at that height – this is because you have a fancy geometry involved in your mechanism. It’s for you to work out the trigonometry, but, for the purposes of an example, let’s image that each cylinder had a bore of 50 mm and a stroke of 100 mm and that the pressure varied linearly from 80 bar (lowest ride height) to 130 bar (highest ride height). It’s also up to you to work out how the stroke of the cylinder varies with ride height but let’s represent the distance from the back of the piston to the rear of the cylinder (inside) by the letter X. So when the cylinder is fully compressed (lowest ride height) X=0 and when the cylinder is fully extended (highest ride height) X=100. The expression relating pressure (P) to piston position (X) will be P=80 + X/2. These will be gauge pressures; strictly speaking you should work in absolute pressures but that’s a complication which isn't worth the effort.
Now let’s think about those accumulators you are going to install to act as the springs (not the dampers). You wouldn't want the accumulator bladder to be fully expanded before the suspension cylinder was fully extended so we can assume you would still want ~10% of the accumulator volume filled with oil. We might also assume that you wanted the cylinders to be at 50% stroke at 20 deg C. So let’s just pick an accumulator and see how it works out.
A typical “10 Litre” accumulator has a gas volume of 9.4 Litres. You want there to be 10% of oil still in the accumulator when the cylinder is fully expanded so the gas volume will be 8.46 Litres (8640 cc) at this point. The example cylinder has a total swept volume of 196 cc so when at 50% stroke the gas volume will be 8542 cc. When at 50 mm cylinder stroke the pressure would be 105 bar (just using my example geometry and numbers). This lets you define the accumulator pre-charge pressure at 20 deg C. Your 8542 cc of gas at 105 bar will reduce its pressure by the ratio 8542/9400 and will be just 95.4 bar. (If we bothered to do the calculations according to absolute pressures the number would be 95.3 bar and even that’s not accurate because we haven’t included compressibility factors which is why it’s safe to ignore the “use absolute pressures” stricture).
Then the temperature changes...imagine it’s now 40 deg C. If the oil circuit were an enclosed volume the gas pressure would rise. But the volume isn't closed so the cylinder rod will extend a little. The complication is that your suspension geometry means any change of stroke causes there to be a change of pressure – and this will affect the gas volume. So here’s how you get round it:
In a spreadsheet enter your equation for pressure in terms of cylinder stroke (P=80+X/2) and arrange this so that you can input a value of X and get out a value for P. This will be the pressure needed to sustain the vehicle at that particular height. Call this “P(geometric)”
In a nearby cell enter the expression that allows you to calculate the gas pressure for any value of X and at any new temperature T (Kelvin). It goes like this (Boyle’s law):
P x V = N x T
[Where P is in bar, V is the gas volume in cc, N is a constant (for that size accumulator and that particular pre-charge pressure) and T is the temperature in Kelvin.]
We know that at 20 deg C (293 Kelvin) the gas volume is 8542 cc when the pressure is 105 bar. Plug in all these numbers and you get N = 3061 (can’t be bothered to work out the units here). Now the gas volume was 8542 cc when the cylinder was at 50 mm stroke, so we could say that the gas volume was:
V = 8444 + 196X/100 , i.e. when X = 50, V = 8542.
Now we can enter a formula on the spreadsheet to give us the gas pressure for different values of X (and T) for our current selection of accumulator size and pre-charge pressure:
P = (3061 x T)/(8444 + 196X/100)
This will be the gas pressure at any particular cylinder stroke at any gas temperature. When you plug in T=293 and X=50 you will get P=105. Call this “P(gas)”
Then, in another cell, put in “P(gas)/P(geometric)”. This is the ratio of calculated pressures and the answer will should be 1.000 because both pressures are 105 bar (actually it won't quite be 1.0 because the constant 3061 was rounded down a little). You can then correct 3061 if you want but I wouldn't bother – this is, after all, only an example.
Now put in a new value for the temperature (40 deg C = 313 Kelvin). The spreadsheet will calculate a new P(gas) value. Now – and this is the crafty bit - click on the cell with the ratio of calculated pressures and use the spreadsheet’s “goal seek” function to make the ratio equal to 1.000 by changing the value of X. You will then get a new solution where the gas pressure dictates a particular cylinder displacement which uses exactly the same pressure to support the vehicle. In the simple example I've used here the cylinder displacement changes to 63 mm. Remember that this is the change of cylinder stroke – you still need to calculate the new ride height based on that particular cylinder position.
No-one can tell you whether the change in cylinder stroke between 20 deg C and 40 deg C is significant because it depends on so many factors and also on your opinion on what change is acceptable. You might want to investigate a much wider temperature range though. If you are a little more sophisticated with the spreadsheet you can use it as a design tool to investigate the effect of different accumulator sizes, cylinder sizes, pre-charge pressures etc.
Hope this method gives you the steer you were looking for to get a feel for how your design will perform.
DOL
