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Charger SRT Hellcat
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Some observations: I've had a an 2006 SRT Charger, 2017 Hellcat Charger, and 2020 Charger Hellcat which all had 20 inch tires. I don't remember what tires were on the 2006, but the 2017 and 2020 both had Pirellis. They all shook at higher speeds. I work at a dealer and used to balance the tires on the 2006 almost weekly (probably wasn't doing it right) and never got it to not shake until I finally replace the tires with Potenzas. They were pretty darn smooth. Then I sold it. I had Discount Tire road force balance the Pirellis on the 2017. It got good, at times, but never SMOOTH. I just lived with it. I had the tires on the 2020 road forced balance by Discount. after moving tires around repeatedly. I finally got to where the steering wheel didn't shake at all......usually. If I drove the car the day before, it was almost perfect. If it sat a few days or more, it shook for a time, as others have mentioned. Here's an interesting phenomenon: there is a long street that leads to the freeway where I typically do 70 mph or more (in Mexico). It has a long sweeper to the left. If I was going fast, it would often began to shake about halfway through this sweeper. At slower speeds it didn't. I'll assume the tire began to deform to the side causing this. I'll be getting my new 2021 in a few weeks and fully expect it to shake. It's a shame that they can't make a Charger with 20s that doesn't shake. Interestingly, my 2018 Pacifica with 20 inch tires is smooth. Always! Also, my 2008 GT500, 2010 GT500, 2012 GT500, and 2013 GT500 NEVER shook at all.
 

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SRT DEMON ONESEAT
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@Diel11 This is normal in some respects. The tires are flat spotting. Some do it more than others. If you took off and drove around town and never exceeded 35 MPH you likely would not notice it. But if you hit the freeway it will be obvious. I reccomend finding an initial speed that will avoid the shake. Speed up or slow down a bit. I have had Drag Radials flat spot after 20 minutes in the staging lanes. What most don't understand is I was in the middle of 12 hot laps. The car only shook on that ONE pass. Why? It's because those 2 flat spots just "happened" to be in Sync when I staged to make that pass. That is why your car may shake more on one day than others and the "happy speed" on the freeway may vary a bit.
 

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There can arise flat spots from the car sitting too long. To avoid this the factory inflates the tires to IIRC 45psi.

Once the car has been sold the tire pressures should be 32psi but check your car's door tag for proper inflation pressure.

With a number of cars with high performance Pirelli, Michelin, Continental and Bridgestone tires under some conditions the tires will over night or two have some signs of flat spotting.

This is normal. I found the problem is most likely when it gets hot during the day but then cools off quite a bit at night. If it stays hot all night long the tires don't cool down enough to develop a flat spot and if the day is not hot enough the tires do not get hot enough to deform slightly once the car is parked and even if it cools down.

I was about a mile or so from the freeway. Often by the time I got the freeway the tires had smoothed out. It was the rare event the tires took any freeway time to lose all the flat spotting but it happened. I can't recall the circumstances but it must have been a perfect combination of hot enough day time temperature and the tires getting plenty hot then after the car put away for the night the night time temperature dropping pretty fast and dropping quite a bit.

Really as long as the tires smooth out after a few minutes/miles (worst case) driving the situation is normal but harmless. I would be very reluctant to run the tires over the recommended door jam tire pressure. 50psi as someone suggested I think is above the max inflation pressure on the sidewall. This should not be exceeded and if you do want to experiment with tire pressures you should allow for the fact the tires will heat up and the tire pressure will go up by a few psi. Even with 32psi I have seen the tire pressures go to 36psi after some running in hot weather.

I would also be reluctant to replace the high performance tires with all season tires. While the all season tires probably won't flat spot they are in my experience lousy tires to fit on a high performance car. I would rather put up with just a bit of mild flat spotting that goes away in nearly no time than to live with lousy tires all the time.
The PROBLEM with your premise is that the stock Pirellis suck like a mosquito riding a vacuum cleaner into a black hole. Almost ANY tire is better than ANY Pirelli made in the Hellcat size.

I have continued to say that the Pirellis seem like a giant F YOU by Pirelli for Dodge wrangling some stupid-cheap price from them for their tires, or maybe Pirelli has some embarrassing pictures of Dodge execs driving Chevies or Fords?

I dunno, but somehow the same maker that makes tire in size for fast, portly SUV's from Europe seems to have reserved the tires from the "It's okay, Luigi, we'll just sell them to tugboats to put along their gunwhales" bin for America.
 
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