A method of component mode synthesis is presented for the analysis of multishaft rotor-bearings systems. The motion of each component of the system is described as the superposition of constraint modes associated with boundary coordinates and constrained precessional modes associated with internal coordinates. The constrained precessional modes for each component are truncated and the reduced component equations are assembled to yield a set of system equations. The nonsymmetric nature of the general problem requires the utilization of biorthogonality relations between right and left vector sets in order to decouple the component precessional modes. The method is developed for damped whirl speed/stability analysis and comparative results are presented for various levels of mode truncation for two example systems.

TECHNICAL NOTES 795 ratios. (As seen in Fig. 2a, the effect of the structural damping is relatively small for this range of the mass ratio.) The considerable decrease of the flutter frequency with increasing Mach number is also seen in Fig. 2b. In Figs. 3a and 3b, the corresponding flutter modes are plotted. The remarkable feature of these flutter modes is that the phase difference ( /7 a ) between h motion and a motion rapidly decreases with increasing Mach number and the mass ratio. This implies an interesting fact as to the mechanism of the flutter. The large phase difference (4> h a ) between the h motion and a. motion at M=0.6 indicates that the flutter at this Mach number is essentially a classical-type (bending-torsion) flutter for which the phase difference between the degrees of freedom is playing the dominant role. 10 ' 11 On the other hand, the phase difference ( hta ) at M= 0.90-0.95 tends to approach zero as the mass ratio increases. (This tendency is further confirmed by calculating the more extreme case of ^ = 300 (M=0.95), for which becomes less than 1 deg.) Since the zero phase difference between the h motion and the a motion indicates the existence of the pivotal point at x pv =a -(h/b)/ct, u the flutter mode, for instance, at M-0.95 and \L -60 is essentially a pitching oscillation with the pivotal point at x pv = -3.75 (if we neglect the phase difference of 5.6 deg), which is very close to the first natural mode shown in Fig. 2a. From these facts, we can derive an interesting conclusion that the mechanism of the single-degree-of-freedom flutter is dominating the flutter of this system at the bottom of the transonic dip. It is also evident from the mechanism of single-degree-of-freedom flutter as discussed in the previous section that the large time lag between the aerodynamic pressures and the airfoil motion in the transonic region, which is caused by the compressibility effect, is the main cause of the transonic dip phenomenon.