The dynamic model of the five shaft geared rotor system is shown in Fig。 2。 kiL , ciL and kiR, ciR are the stiffness parameters and damping parameters of the left TPJBs and right TPJBs on the five shafts respectively, which are dependent on the loading charac- teristic。 Fi is the meshing force。 Unlike the traditional one-to-one gear pair's transmission, the geared system in Fig。 2 is consisted of a bull-gear (M) and four pinion gears, and the bull-gear meshes with the other four gears at the same time, which cause a much stronger coupling effect。 This compact structure can offer several advantages such as smaller floor space, lower energy consump- tion, and higher power density。 However, it also means that much more critical speeds must disperse in a certain interval of speed, which brings much more difficulties in critical speed designing。

3。6-DOFs dynamic modeling method of geared rotor system

The finite element method is used to set up the dynamic  model of the geared rotor system shown in Fig。 3, and the assumptions

Fig。 4。  Rotating rigid discs   model。

and simplifications are introduced as follows: (1) All the shafts are linear elastic。 (2) The bearing houses are rigid。 (3) A gear pair is represented  by linear equivalent  stiffness  along  its  pressure line。

(4) A shaft section with a larger radius  can  be  simplified as rigid disc。

The beam element shown in Fig。 3 is chosen to describe shafts, which has 12 degrees of freedom (DOFs) and is defined as

uij ¼ h xi      yi      zi      θxi      θyi      θzi      xj      yj      zj      θxj      θyj      θzj i

The dynamic equation of the beam element is described as

Fig。 5。  The dynamic model of gear mesh  effect。

section is simplified to the combination of the rectangular section

Mbu€ ij þðCb þ ΩGbÞu_ ij þ Kbuij  ¼ Fb ð2Þ

where Ω is the rotating speed around axes-x; Fb is the force vector of the beam element; Mb, Cb , Gb and Kb are the mass matrix, damping  matrix,  gyroscopic  matrix  and  stiffness  matrix  respec-

where  Fmesh    is  the  mesh  force  acting  on  the  tooth;  δBr     is    the

tively, where Cb   is defined as Rayleigh damping and described    as

Cb  ¼ α U Mb þβ U Kb ð3Þ

where α and β are the scaling   factors。

The impellers, wheels and gears are regarded as rigid rotating discs (with gyroscopic effect), and simplified to be lumped mass and moment of inertia acting on the corresponding node, which is shown in Fig。 4。 The rigid disc has 6 degrees of freedom, which is defined as

corresponding bending deformation of rectangular section; δBt is the corresponding bending deformation of trapezoid section; δs is the corresponding shear deformation; and δG is the corresponding elasticity deformation caused by the root tilting。 Assume i and j are

the nodes of the gear pair, where i is on the driving pinion, which is shown in Fig。 5。 ψ ij is defined as the angle between the tangent plane of the gear contact surfaces and the axis-z and described as

( ϕij þαij driving gear rotate counterclockwise