I have an older model Mansfield with a traditional ball cock float valve. When I flush the toilet, the water fills the tank until the float valve closes. As soon as the ball cock rises up, there is a loud vibration for about 10 seconds. What is causing this? How can I fix it?

And your vibration is caused by the washer that is not seated properly. Turn the water off to the toilet. Remove the three screws from the top of the valve and pull up. Pry the washer out and turn it over. Reseat it firmly. Place a pan over the valve and turn the water on to flush out the supply. Reassemble and test. Good luck, Tom

I am having the same issue but no ballcock float and I do not see but one screw on the valve and that simply holds the handle on. Please explain. Do I need to replace the valve shutoff?

Try this How to Repair a Moaning Toilet - ballcock valve replacement - YouTube