In this paper, a continuous rotor system is modelled by considering some critical factors, like the gyroscopic and rotary inertia effects of disc and shaft cross-sections, large shaft deformation, and restriction to shaft axial motion at the bearings. The bearings are replaced by springs along horizontal and vertical directions. Governing partial differential equations (PDE’s) for the vibrations of the disc along the horizontal and vertical directions are derived by employing the Hamiltonian principle. The governing equation is then transformed into a set of ordinary differential equations (ODEs) using method of modal projection. The large deformation and restriction to axial motion of the shaft yields nonlinearities in the system governing equations. Only the linear system is analysed in this first part. The parameters in the dimensionless form of the governing equations are functions of some independent variables which are associated to the material and geometrical properties of the rotor system. Understanding the effect of the system properties on the response characteristics will lead to an appropriate design. Exact analytical solutions for the motion along horizontal and vertical directions are obtained. The effect of the independent parameters on the system dynamics are analysed wherein the variation of the dependent parameters are also monitored. The system dynamics are explored using time-displacement responses, phase-plane plots, frequency response diagrams, phase angle plots and Campbell diagrams.