I just finished restoring my 67 convertible. It runs very smooth at low speeds but when I get above 50 mph it starts shaking pretty badly. I rebuilt the front suspension but didn't do much to the back other than rebuilding the drum brakes.

I am thinking I installed the drive shaft 180 degrees off. If this was the case, would I feel the vibration at lower speeds also?

 

Wheel alignment, tire balance, tire out of round, bent wheel.

 

or bent axle

 

you can try the driveshaft, that's quickly done ... but driveshaft vibrations usually only come in at 70-80mph (at least with my ratio and tyres its about 3200rpm on driveshaft)

if you feel it in the steering it's the front. if it's in the back then check at what speed exactly you get it in 4th gear and try the same in 3rd. if it re-occurs at the same speed, but different gear then it's behind the gearbox (driveshaft, diff, axles, wheels)

maybe it's just as simple as someone balanced the wheels on-car you turned it and now it's changed

anyway, check for loose parts before you drive that again, make sure wheels are mounted correctly

 

when a tire sits a long time it will develop a flat spot that will work out as tire heats up over time. You can hear this type of noise at 20+ mph.

 

If you alignment is off or steering components loose then should feel vibration through steering wheel.

Sometimes wheel weights fall off and cause vibration.

I would address issues already mentioned.

Rotate drive shaft even though this is not my suspect here.

Just for kicks I would check tightness of every bolt/nut I could find.

I would get wheels balanced and verify alignment is proper.
If your alignment shop does not use shims I would find another shop.

Double check front wheel bearing preload.

 

Welcome to the 50 mph shake and shudder club.This seems to be a common problem in 67-68 Mustang convertibles.I have a 68 with rebuilt front suspension.New balanced drive shaft.New motor mounts.New transmission.New tires balanced many times.The problem is not going away.I had another 68 convertible and it did the same thing.The car being a convertible is hitting a certain resonance at that speed.If you you look at it another way.A shoe box with a top is very strong.Pull on both sides.Very little give.Remove the top and pull on the top and you can see how easy the bottom flexes.Convertible are not as strong as a hardtop or fastback.There is not much you can do except stay out of that speed range.

 

This is referred to as cowl shake.The car being a convertible has too little unsprung weight,

 

I also had the car aligned and brakes redone.New wheel bearings.Still shakes at 50 mph then smooths out.It will drive you crazy but it ain't going away.

 

Tire(s) out of round. Have them shaved and rebalanced.