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ok..... Keep it simple.
What are the advantages and disadvantages of struts(dampeners/shocks) point upwards or "inverted" pointing downward?
An inverted strut is often used in a mcpherson strut front suspension as it is stronger and more rigid than a conventional damper. A mcpherson strut suspension design inevitably applies some level of bending force into the strut as the car is loaded up under brakes and heavy cornering. The damper itself however really is best at handling loads in compression and tension. What this can mean is that some small amount of flex can be expected in a conventional mcpherson strut damper which affects accuracy. An inverted damper supports any bending load with the outside of the damper body which supported by guide bushes in the outer tube.
Downsides are that the inverted damper is more expensive to produce.
Two questions, and my thoughts - which could be wrong...
When you say "strut", do you mean as used in the McPherson and Chapman "strut", or do you mean 'coil-overs', where the shock absorber (spring) is fixed around the damper? They may appear similar but there are critical differences.
Which do you consider the 'normal' orientation and which do you consider 'inverted'?
With the classic 'strut, I know of no instance where the shaft is NOT to the top and the main body to the bottom. This is because of the design making it very difficult to invert it. The lower end (body) has to have the stub axle housing, lower wishbone/track rod, steering arm, etc, fastened to it and making them, or attachment points, part of the body is easy. It is also easy to have the top plate designed to bolt to the shaft end and incorporate the chassis and springs plates, as well as pivots and any NVH insulation deemed needed.
With the coil-over, or even just the damper, I would suspect it is primarily 'shaft up' because that way the dampening fluid will just sit at the bottom of the damper body, where the piston can move through it. There may aslo be some consideration for the shaft seals having to seal against 'splash' fluid, rather than being immersed.
When gas pressurised damper designs were introduced, the need to consider the fluid was largely taken out of the picture, as it was now kept in a pressurised reservoir. This new design also meant the damper was equally effective in 'normal', shaft up, orientation as it was in inverted, shaft down.
This is where I think it starts getting interesting - what affect does this have on the sprung/unsprung relationship? I suspect some smart cookies realised the ratio was improved by inverting them - especially if additional cooling fins, fluid reservoirs, etc, were used.
The sprung/unsprung question may also affect the position of the springs, adjusters, etc with coil-overs as their mass will have to be consided, as will larger diameter shafts.
Lastly, don't overlook the packaging - they may not fit in the preferred orientation - and, often, 'fashion' may dictate as 'monkey see, monkey do'.
Inverted shocks, used in McPherson suspension, handles alot more sideways forces, especially in rally/rallycross where the are facing very high demands in track conditions and race incidents.
As a performance aspect, I don't think there is any specific gain, if their setup identical other than inverted (upside down).
Jumping in here to add to an already good discussion with a few more points -
McPherson struts exist in the OEM world because they are easier to package in meany instances (take up less space) and therefore leave more room for people, fuel, luggage etc. They also tend to be economical to produce at scale relative to something like a double wishbone. Without these requirements from the road car world, I'm not sure it would exist as it's a pretty poor solution for a motorsport application in general.
In a McPherson strut design, the damper shaft ends up taking side loads, both in cornering and braking. These side loads have two main negatives
The compliance just means you get a lot of flex in the system from the damper shaft flexing. It's always less structurally efficient (just the total weight dividing by the stiffness) the use a McPherson strut relative to a proper wishbone.
The friction is partly from the relatively small bushings that the damper shaft runs on and also partly from the damper shaft flexing. The result of these is the damper starts to lock up, sometimes you see the term "stiction" used for this. This is always a bad thing, it makes the suspension behave as if it were stiffer and will reduce grip.
Inverting the damper just allows you to use the thicker part of the damper (that would otherwise be at the bottom of the housing doing nothing) as the sliding surface. This reduces compliance a lot, because the sideload in the damper is a bending moment on the damper shaft. Increasing the diameter even a small amount decreases the bending deflection of a shaft massively.
Another advantage is you can now use much larger bushings or bearings as the sliding face which helps everything run with less friction (less stiction).
An additional downside to what has been discussed already is it can make bump rubber tuning more time-consuming. With an inverted damper, the bump rubbers will now be inside the strut unit and the damper will need to be removed from the strut to change bump rubbers or packers.