A THE TRANSFER MATRIX — COMPONENT MODE SYNTHESIS FOR ROTORDYNAMIC ANALYSIS —

The transfer matrix-component mode synthesis has been developed for the analysis of critical speed, response to imbalance and rotordynamic optimal design of multispool rotor system. This method adopted the advantages of the transfer matrix method for the train structure and the component •mode synthesis for reducing degrees of freedom. In this method, the whole system is divided into several subsystems at the boundary coordinates. The constrained vibration modes and the static deflection curves of the constrained rotor subsystems are analysed by the improved transfer matrix method. The whole system is connected together by the component mode synthesis in accordance with the coordinate transformation. Numerical examples show that this method is superior to the traditional transfer matrix method and the component mode synthesis by FEM. This method has been successfully used for the rotordynamic analysis and optimal design of the compressors and the gas turbine aeroengines. key words: transfer matrixcomponent mode synthesis, rotordynamics, critical speed, response to imbalance, optimal design.


MODE SYNTHESIS
The TMCMS is based on the concept of the superposition of two kinds of mode shapes. One of them is constrained vibration mode shape and the other is the static deflection curve. The constrained vibration mode shape is the free vibration of the constrained undamped subsystem when all of the boundary coordinates are fixed. The static deflection curve is obtained when the boundary coordinate is released in turn.
The equation of motion for free vibration of a rotor system is as follows, M2 icaCat-l-K Z±C2= 0 (1) where the sign " -" in front of the second term refers to the forward whirl and the sign "+" the backward whirl.
For the use of the component mode synthesis, the whole system is divided into several subsystems at some special points where the subsystems are coupled or the characteristics is complicated such as bearing and squeeze film damper etc. as shown in The constrained vibration mode shape p and eigenvalue C2, can be obtained by solving these equations.
Let yi=1(i=1,-k ) in turn for each time, equation (7) will become nonhomogeneous, solving corresponding equations for non-rotating condition, the static deflection curves 6 can be obtained. The

CRITICAL SPEEDS
From the determinant of the coefficients of equation (10), the eigensolutions can be obtained by a iterative procedure. A u +B, B. =0 (11)

13b, A bb +B it
Then, the vibration mode shapes can be obtained by substibuting the eigenvalue into equation (10). FIG. 3 shows the scheme of a dual rotor system with one intershaft bearing and the calculation model. The data of this rotor system are listed in Table 1: Some of the calculation results are listed in

RESPONSE TO IMBALANCE
For the calculation of response to imbalance, the equation of motion (1) should be changed into nonhomogeneous form: where F is the imbalance force and can be expressed as When the eccentricity e has been given the response to imbalance can be calculated by the equation (16). The steady response to imbalance in physical coordinate will be obtained by the modal transformation matrix (8). FIG. 4 shows the steady response to imbalance of a gas turbine rotor by TMCMS.
operating ranges FIG. 4 Response to imbalance of a gas turbine

OPTIMAL DESIGN
The most important things in the optimal design of the rotordynamics are to keep the critical speeds far away from the operating ranges and to minimize the response to imbalance at any operating ranges.
The most effective way to these ends is the optimal design of the stiffness of the support and the damping coefficient of the damper. The rotor system has many supports and maybe also many dampers.
So that, the sensitivity analysis of the stiffnesses of the supports to the critical speeds and the damping coefficients of the dampers to the responses to ac, 6 aC, The sensitivity to the physical coordinate will be : where in, n is the number of inner and boundary coordinate respectively.
The stiffness coefficients of the supports and the damping' coefficients of the dampers will be changed by a optimal procedure in accordance with the results of the sensitivity analysis. The optimal goal is to get the critical speeds far away form the operating range and to obtain the minimum vibration response during passing through the critical speeds and in the normal operating range.
The comparison of the response to imbalance before and after the optimal design is shown in FIG. 4, where curve 1 represents the original response, curve 2 the peak response remove away from the operating ragion by stiffness optimal design and curve 3 the response after stiffness and damping optimal design.

CONCLUSIONS
The transfer matrix-component mode synthesis presented in this paper is a suitable method for calculating the critical speed and. response to imbalance as well as for the optimal design of the complex multi -spool rotor system. The static deflection curve and the constrained vibration mode is analysed by the improved transfer' matrix method.
The component mode synthesis is employed for analyzing the dynamic characteristics of the whole rotor system and the sensitivity of the parameters of the supports and the dampers to the critical speeds and responses to imbalance for the optimization of rotordynamics.