A modelling technique for linear time invariant non-gyroscopic rigid multibody systems with tree structure is stated. The equations of motion are derived analytically on the basis of the Roberson-Wittenburg method. The interconnection structure between the bodies within the multibody is established by a directed system graph and the dynamics are based on a variational analysis. The resulting differential equations are described with generalized Lagrangian coordinates. Several types of forces are formulated explicitly and mathematical expressions for each of them are derived. Formulae for the classical tire model and the tire model of Crolla are incorporated in the equations of motion. As a result, each system matrix in the differential equations is written as a combination of simple and easily constructed matrices which are functions of the geometry and the mechanical parameters of the mechanism.