# Continuous Rotor Dynamics of Multi-Disc and Multi-Span
## 1 Introduction  
Rotor Dynamics, which has been concerned all the time, is an old research field. It is the mechanical basis for the design, manufacture, safe operation and fault diagnosis of rotating machinery. There are forward and inverse problems of rotor dynamics. The forward problem is to study dynamic responses of rotors with known rotor parameters and loads, while for the inverse problem, it is identification of rotor parameters and loads from dynamic responses. This paper is focused on the forward problem which is continuous model and analytical solution of unbalance response of multi-disc and multi-span rotor. The continuous rotor dynamic analysis method proposed can be applied to studying the inverse problems which are rotor unbalance identification algorithm and bearing coefficients identification algorithm, respectively. In order to determine the unbalance response of the system, the Ricatti method [1; 2; 3; 4; 5; 6] and the Finite Elements method (FEM) [7; 8; 9; 10; 11; 12; 13] are proposed, which are based on discrete model. Although these methods can be used to predict the unbalance vibration for a complicated rotor system, it remains hard to be applied to the identification of bearingdynamic parameters, estimation of rotor unbalance and balancing of rotor systems. It is because their model are discrete and the unbalance response can not be expressed as a function of position, rotor unbalance and stiffness and damping coefficients of bearings. Hence, it's difficult to identify rotor unbalance and bearing dynamic parameters by solving the equations which are built using the unbalance responses at some positions on rotor-shaft as known conditions. For the continuous beam model of rotor systems, in the literature there exist limited numbers of studies that concentrate on the analytical modeling of rotor systems. Euler-Bernoulli beam [14], Rayleigh beam [14; 15; 16; 17; 18] and Timoshenko beam [19] are three kinds of continuous beam models for unbalanced rotor-bearing system. Based on Euler-Bernoulli model, vibration analysis of beams with multiple steps and different shapes of cross section were conducted [20]. [PERSON] [21] carried out free vibration analysis of the Euler-Bernoulli shaft with two rigid bearings and a flexible disk to study the influence of the disk's location and aspect ratio on natural frequencies and critical speeds. [PERSON] [22] presented a continuous rotor model for rubbing applications based on Euler-Bernoulli continuous model considering transverse shear, rotatory inertia, and gyroscopic moments and Wavelet Techniques was used to study the response for detection and characterization of rubbing phenomenon. [PERSON] provided a method to obtain the analytical imbalance response of the Jeffcott rotor under constant acceleration [23]. [PERSON] [24] studied an axially-loaded Euler-Bernoulli shaft, clamped at one end with an intermediate spring support. That work placed the equations of motion into state-space form to find critical speeds. The same approach was applied to shafts with rectangular cross-sections for examining the stability of pretwisted simply-supported rotating beams [25] and various end conditions that are different in orthogonal directions relative to the rotating shaft [26]. [PERSON] and [PERSON] [27] modeled an Euler-Bernoulli shaft with a single bladed disk using the Ritz method for both shaft and blades. [PERSON] and [PERSON] [28] incorporated flexible disks into the Ritz series analysis for modeling hard disk drives. More specific applications using this method were performed by [PERSON] et al. [29] for a rotating, cracked Euler-Bernoulli shaft. [PERSON] [30] proposed an analytical solution of unbalance responses for single-span and single-disc rotor-bearing systems and an identification method for rotor unbalance was given. To investigate rotor dynamics, such as the vibration problem, the influence of the number of blade and rotating speed on the natural frequencies, Euler-Bernoulli theory were employed along with FEM [31; 32].