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What you are describing is definitely not wind buffeting. If your hood is shaking as you state it is I would suggest you check the rims for roundness, tires and suspension
You've either got some.hyperbole going on, or a mechanical issue. Could be bad wheels and tires or a hood not adjusted properly. Maybe a combo of the two.
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I've put 400 miles on it so far driving over 50MPH most of the time, usually averaging 60MPH. I also ran it a bit over 90 for a minute on a very long overpass once. Until the top came off today, the car was smooth on smooth roads, no indication of out of balance tires or out of round wheels at any speed. The car does feel unstable over bumps at any speed as-is, but I never saw any actual flex in the front end before, and I never felt like the car was shaking apart. I was under the impression removing the Targa top did not affect torsional rigidity at all. I stand corrected. Maybe the shocks are so far gone, and/or the springs are too, and that all came together with "a little extra flex" to really compound the issue. Top is back on and will stay on for now. Thank you.
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Please note... the top off the car should handle and behave just fine...I have seen pics of people on this forum auto-crossing with a stock C4 with targa top off (so their helmet will fit!). The "tortional rigidity" with the top off I am referring to would likely be a small amount of deflection allowed that can perhaps cause "cowl shake" or some slight additional flex but not dangerous or significant from a handling perspective... The only time I've seen problems here with significant "cowl shake" or issues such as yours there was something with a wheel/tire or suspension that was the primary root cause. With the top on it was either much less noticeable or not allowed to resonate. Some cars (years?) may be more prone to this too...but I have nothing but anecdotal information from this forum on that.
You said that the car feels "unsettled" over bumps and perhaps has suspension or wheel related issues, I would start there and ignore the cowl shake for now. Get it handling properly first then if the open top issues are still present and it's bothering you address those. good luck
My previous 89 coupe had terrible cowl shake with the targa off. Thought that was normal. Current 89 roadster has additional factory bracing in multiple places and no cowl shake at all.
New bushings... which ones? The control arm bushi gs are a real.pain in the ***, and so many people don't do them. They could have just been sway bar bushings and called it a day, like most people. I did them on my 92 at roughly 130k and had a mix of "barely a bushing" and "looks basocally new". I painstakingly did a prothane kit woth poly bushings and they were a huge improvement. Kick the tires and look for movement, or lift the car and do it the right way.
Shocks. Definitely possible. You can do a real quick test by shoving each corner of the suspension down.
Alignment. Get one done. I'd probably do this after ruling out the bushings, because you'll need an alignment after doing them, but definitely get one, it could easily cause this issue.