An approximate, closed form solution of sealing gap induced lateral forces for imperfect sealing land geometries

It is a well known fact that even small deviations from the perfect cylindrical geometry of either the bore or the sealing land of spool valves may cause substantial lateral forces and, in turn, a sticking of the valve. The reason is the pressure distribution in the sealing gap which looses its axisymmetry in such cases. This effect is counteracted by a number of circumferential grooves which even the pressure distribution. That valve sticking effect can be studied by solving the Reynolds equation for such gap geometries and computing the resulting radial force. From a design viewpoint it is valuable to have a compact understanding of the role of design parameters; in the case of non-uniform sealing gaps, of the role of the imperfection parameters. Gaining compact understanding of cause effect relations with numerical solutions is difficult if more than two parameters are involved since a graphical representation by diagrams is impossible. In this paper an approximate but closed form solution is derived for a class of imperfect sealing gap geometries. They provide that immediate insight by a compact formula for the resulting lateral force. The study is based of a Ritz method for a direct solution of the variational formulation of the Reynolds equation and asymptotic expansions concerning the role of the imperfection parameters.