The dynamic equation of the rotating rigid discs is described as

where αij is the pressure angle of the gear pair and φij is the angle between the centerline of the gear pair and the axis-y (Fig。 5)。

Mdu€ þ ΩGdu_ ¼ Fd ð5Þ

The  helix  angle βij

is defined as positive when with a    right-

where Ω is the rotating speed around axes-x; Fd is the force vector of the rotating rigid discs; Md and Gd are the mass matrix and gyroscopic matrix respectively and described as

hand helix driving gear, whereas negative when with a   right-hand

helix driving gear。

The dynamic equations of the nodes i  and j  connected by  the

spring-damper element for gear mesh without considering the damping effect can be described as

where Fm  is the force vector of the meshing; uij  is the displace-

where m is the mass of the disc; Jp and Jd are the polar moment of inertia and the diametral moment of inertia respectively。

The effective stiffness of the gear mesh refers to the mesh effect

The   mesh  stiffness   matrix  of   the  gear  pair   Kij    in    global

coordinate system with 12th order can be described as

0 k1;1 …  k1;12 1

of the involved tooth in the whole mesh area and depends on the

tooth elastic deformation and the gear contact ratio。 The equiva- lent mesh stiffness can be obtained by a popularly used formula (known as Ishikawa Formula) stated in Eq。 (7), where the  tooth

aij ¼½ — sin βij sin  ψ ij   cos βij cos ψ ij   cos βij       sgnri   cos βij       sgnri   sin ψ ij   sin βij       sgnri   cos ψ ij   sin     βij

sin βij       — sin ψ ij   cos βij       — cos ψ ij   cos βij       sgnrj   cos βij       sgnrj   sin ψ ij   sin βij       sgnrj   cos ψ ij   sin βij ]

Fig。 6。  Five-pad TPJB。 (a) Geometry parameters and (b) force diagram。

The symbol sgn is defined as

( 1 driving gear rotate counterclockwise

sgn— 1    driving gear rotate clockwise

where B is the bearing length; D is the bearing bore diameter; η is the dynamic viscosity; and ω is the angular velocity of the shaft。 The static operating behavior of the bearing can be explained by

the static equilibrium line γ ¼ γðεÞ and the dimensionless static load capacity Sy0  in the y-direction and Sz0  in the z-direction    on

As  early as  1980s,  many  scholars  have  carried  out extensive

researches in the dynamic model of TPJB, and many simplified models of TPJB are provided [17,18]。  In order to simplify   calculat-

the  eccentricity ε。摘要——本文研究了齿轮平行转子系统五个轴的可变临界速度和共振频率，其变化是受到因不同外部负载而改变的含油轴承刚度的影响。首先，一个现代的离心式压缩机系统作为本次研究的对象。通过使用旋转梁单元的6个自由度和啮合螺旋齿轮的网状线性刚度来建立它的一个有限元模型。含油轴承的刚度和阻尼都是基于传统的油膜流公式计算的。然后，基于以上的模型，计算在不同的负载下整个系统的三个输出轴的Campbell图。最终，通过比较获得的临界速度和负载范围来揭露可能导致转子系统共振的临近运行速度。