A fixture for securing a cylindrical or shell thin-walled component includes a support having one or more securing devices for securing the component to the support. The fixture also includes an inflatable pressure element which, in use, is positioned within the secured component and is adapted to press outwardly, when inflated, against a liner inserted between the pressure element and the inner thin wall of the component. The liner has one or more elastic members, which bias it against the outward expansion of the inflated pressure element. The inflated pressure element and the liner damp vibrations in the component during machining of the outer side of the thin wall.