# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 1 Introduction
According to Rayleigh model, unbalance responses are expressed as a function of position on the rotor-shaft, rotor unbalance and bearing coefficients [33]. Using the Rayleigh beam model and Timoshenko model, mathematical modeling of continuous multi-stepped rotor-bearing systems was presented and the partial differential equations were discretized to obtain a close-form solution [34; 35]. According to [PERSON] theory, [PERSON], [PERSON] and [PERSON] presented methods for multi-stepped beams with attachments [36], multi-stepped rotors [37] and multi-span multi-stepped shafts resting on multiple bearings [38] and rotors with multiple concentrated masses [39]. Using Timoshenko beam theory combined with FEM, [PERSON] [40] studied dynamic responses of bidirectional functionally graded rotor shaft. It is found that the obtained dynamic responses are influenced by temperature variation, power-law index and internal viscous and hysteretic damping. [PERSON] [41] modelled the rotor-shaft as a spinning Timoshenko beam and derived analytical expressions for critical solutions to analyze damping gyroscopic effects on the stability of parametrically excited continuous rotor systems. [PERSON] [42] obtained equations of motion for a Timoshenko shaft with flexible disks which were solved numerically to obtain the time domain response. Although [PERSON] [30; 33] proposed an analytical solution of a continuous rotor-baring system supported with two bearings and a disc, the solution cannot be applied to rotors with multi bearing and muti disc. Moreover, with regard to forces of the bearings and disc, the continuous model of a rotor-bearing system is a fourth-order non homogeneous partial differential equation set with homogeneous boundary condition. There is no universal method to solve them so far. When using integral transform method to solve the equation, it is difficult to achieve the inverse transformation. While for the separation of variablesmethod, it can only be used to solve the homogeneous partial differential equation set with homogeneous boundary condition. In this paper, the continuous rotor dynamic analysis method (CRDAM) is proposed for multi-disc and multi-span rotor represented by the homogeneous and elastic Rayleigh beam. Forces of the bearings and discs are considered as point forces. free-ended boundary condition is used in modelling the rotor. A continuous model, which is a fourth-order non-homogeneous partial differential equation set with homogeneous boundary condition, is developed. An analytical solution is proposed to express the unbalance response as a function of position, rotor unbalance and the stiffness and damping coefficients of bearings. Numerical simulations are conducted to validate the proposed method. The accuracy of the proposed method is higher than the Ricatti method and the calculating speed is faster than FEM. It can be applied to the identification of rotor unbalance and bearing coefficients for multi-disc and multi-span rotors. The remainder of this paper is organized as follows. Section 2 describes the continuous model of the multi-disc and multi-span rotors and its analytical solution. Section 3 is the validation of the proposed method by numerical simulations of rotors supported with rolling bearings and oil-journal bearings. Section 4 describes the main conclusions.