Analysis of dynamic characteristic for misalignment-spline gear shaft based on whole transfer matrix method

2.1. Dynamic model of SGF

A great deal of special coupling structures makes it more difficult to analyze the dynamic characteristics of SGF with traditional method, therefore, WTMMR is employed to analyze kinetic characteristics of SGF on this account in this paper. The complex rotor system is divided into several substructures according as its own structure and crucial parts with WTMM, afterwards, adding the virtual units (both length and quality are 0) in the appropriate position in order to have the equal number and consistent structure of all these substructures. For the sake of fitting modular analysis, the same state vector (passing from left to right as a whole) should include the section state parameters of the same number of all substructures. The number of frequency equations is only related to the number of substructures, independent of the number of supported and coupled connections.

This is a hypothesis that the inner and outer casing of supporting system is not taken into account, therefore, the simplified model of SGF consisting of five shafts (I, II, III, IV, V) is shown as Fig. 2. From this picture, shaft I and shaft II are coupled by the bevel gear pair, similarly, shaft IV and shaft V are coupled by the bevel gear pair, too. In addition, the central transmission shaft is made up of shaft II and shaft III and shaft IV with involute spline coupling. The power is introduced from the propeller at shaft I, then driving generator at shaft I after the transmission of energy through shaft II and shaft III and shaft IV.

As shown in Fig. 3, SGF is divided into 2 units (1 and 2) with the model division principle of WTMM in each substructure, besides, the substructure of less than two units is filled with virtual units in order to have the equal number and consistent structure of all these substructures. Then, the partitioned units are marked into 9 segments (shaft units and coupling units) in this paper.

2.2. Calculation of whole transfer matrix for SGF

Considering the lateral and torsional vibrations of SGF, the state vector of each substructure contains ten elements (5 generalized forces and 5 generalized displacements) in this paper. The overall state vector in section $i$ can be expressed as:

In formula:

where: $Z_i^I$, $Z_i^{II}$, $Z_i^{III}$, $Z_i^{IV}$, $Z_i^V$ is the state vector of substructure (I, II, III, IV, V) in section $i$ respectively; $Q_x^j$ is the component of shear force at $X$ axis; $M_y^j$ is the component of bending moment at $Y$ axis; $Q_y^j$ is the component of shear force at $Y$ axis; $M_x^j$ is the component of bending moment at $X$ axis; $T^j$ is the axial torque; ${\theta }_y^j$ is the component of rotation at $Y$ axis; $X^j$ is the component of deflection at $X$ axis; ${\theta }_x^j$ is the component of deflection at $Y$ axis; ${\phi }^j$ is the axial torsional angle; $i=$1, 2,…, 9; $j=$ I, II,…, V.

Supposing that there is a coupling in segment $n$ ($1\le n\le 9$, $n\in z$), the whole transfer matrix of SGF can be represented as:

The following principles need to be followed when the whole transfer matrix $T_i$ is constructed owing to the difference of unit’s number for each substructure and location for coupling unit while the states of each substructure are transmitted simultaneously with the overall state vector $Z$.

(1) If the state of each axis is transmitted independently in a certain period, the transfer matrix of each single axis is written to the corresponding position in the whole transfer matrix according to the method of sitting in the right seat, in addition, the coupling between the each axis is 0.

(2) If the state of an axis isn’t transmitted, the corresponding positions are represented by the unit matrix $E$ (the same order as the single axis transfer matrix) to generate the whole transfer matrix.

(3) If the coupling unit of the whole transfer matrix isn’t 0 due to the presence of coupling units between two axes, the whole transfer matrix is determined by the coordination relation of specific force balance and deformation.

Based on the above construction principle of whole transfer matrix $T_i$, the whole transfer matrix of each segment can be determined as Eq. (4):

where: $E_1$, $E_2$, $E_3$, $E_4$ are 5 order, 10 order, 15 order, 20 order unit matrix respectively; $T_I^{\mathrm{①}}$, $T_{II}^{\mathrm{①}}$, $T_{III}^{\mathrm{①}}$, $T_{IV}^{\mathrm{①}}$, $T_V^{\mathrm{①}}$ are respectively the transfer matrix of shaft I, shaft II, shaft III, shaft IV, shaft V at units $\mathrm{①}$; $T_I^{\mathrm{①}}$ and $T_V^{\mathrm{①}}$ are respectively transfer matrix of shaft I and shaft V at units $②$; $T_{11}^{C_m}$, $T_{12}^{C_m}$, $T_{21}^{C_m}$, $T_{22}^{C_m}$ are elements of coupled transfer matrix $\left[\begin{array}{ll}{T}_{11}^{C_m}& T_{12}^{C_m}\\ T_{21}^{C_m}& T_{22}^{C_m}\end{array}\right]$ for coupling unit $C_m$ respectively ($m=$1, 2, 3, 4).

Supposing the solution for anisotropic support rotor systems in the form of $x=X{e}^{\omega t}$, $\theta =\Theta e^{\omega t}$, $\phi =\mathrm{\Phi }{e}^{\omega t}$, the whole transfer matrix of supporting rotor system can be expressed as Eq. (5) when the motion of the rotor in the vertical and horizontal directions are considered and the damping of the rotor system is not taken into account:

In formula:

where: $Z_{xx}=m{\omega }^2-k_{xx}$; $Z_{yy}=m{\omega }^2-k_{yy}$; $\gamma =6E_i{J}_p/k_s{G}_i{A}_i{l}_i^2$; $k_s=$2/3; $m$ is the lumped weight of the rotor system; $J_p$ and $J_d$ are polar moment of inertia and radial inertia of rotation respectively; $k_{xx}$, $k_{yy}$, $k_{yx}$ and $k_{xy}$ are the anisotropic bracing stiffness respectively; $\mathrm{\Omega }$ and $\omega$ are spin speed and revolution speed respectively.

Fig. 2The simplified model of SGF

The whole transfer matrix of unsupported shaft can be obtained by making the anisotropic bracing stiffness ($k_{xx}$, $k_{yy}$, $k_{yx}$ and $k_{xy}$) being 0 in Eq. (5). As shown in Fig. 2, the state is transmitted by the coupling unit ($C_1$ and $C_4$) of orthogonal bevel gear between shaft I and shaft II, shaft IV and shaft V, moreover, the shaft I and shaft IV are coupled to the shaft II and shaft V with meshing force respectively. The meshing force varies with the displacement produced by vibration generated during gear transmission. For orthogonal bevel gear, the following assumptions are made as: (1) the main part of orthogonal bevel gear is rigid, (2) the tooth of a bevel gear is flexible, (3) the meshing force is concentrated force and doesn’t change along the direction of the mesh line, (4) the acting point of meshing force is at the midpoint of contact line, (5) the meshing stiffness $K_s$ is stiffness in meshing direction of orthogonal bevel gear, (6) the positive direction of the line displacement and force is the positive direction of the coordinate axis, (7) the direction of angular displacement and torque is determined by the right-hand rule. For the coupling unit $C_1$, the state vectors at the right end and the left end can be assumed to be Eq. (6):

where: $h=R$, $L$.

According to the relation between right state vector and left state vector of orthogonal bevel gear $A$ and $B$, the transfer matrix of coupling unit can be obtained as Eq. (6) through literature [13]:

Fig. 3Unit division model of SGF with WTMM

3.1. Calculation of meshing force

As Fig. 4, two coordinate systems are established in order to analyse expediently meshing force of misaligned spline coupling:

1. $\left\{S:O-XYZ\right\}$: the coordinate system of internal spline coupling;

2. $\left\{S^f:O^f-X^f{Y}^f{Z}^f\right\}$: the coordinate system of the offset external spline relative to internal spline.

In Fig. 4, the dotted line is the position of the $i$th external spline teeth when the external spline coupling isn’t offset. Points A and B A(B) is the point of addendum circle and involute for the $i$th internal (external) spline teeth) are located on involute line of the $i$th internal spline teeth. $R_i^a\left(R_i^b\right)$ are the distance between point A (point B) and axis O of internal spline, namely, the addendum radius of the $i$th internal spline teeth (external spline teeth). When the axis of internal spline and external spline aren’t offset, $l_i^{ab}$ (the length of meshing line for the $i$th spline teeth) can be obtained by principle of involute generation:

where: $r_b$ is the radius of base circle of internal spline.

Fig. 4Meshing model of misaligned involute spline coupling

It is assumed that the axis of external spline moves distance $u\left(v\right)$ in the positive direction of $X\left(Y\right)$ axis relative to the axis of internal spline. Owing to the smaller distance $u$ and $v$, the change of the radian for meshing involute isn’t considered caused by offsets of the axis for external spline. The actual meshing situation is shown as thick lines in Fig. 4 when the external spline rotates a certain angle. The intersection of addendum circle and involute moves from point A to point C in the $i$th internal spline teeth, and $R_i^c$ is the distance between point C and axis O of internal spline. The length $l_i^{ab}$ of meshing line for the $i$th spline teeth can be obtained by principle of involute generation:

where: $R_i^c=\sqrt{u^2+v^2}{C}_{{\theta }_i}+\sqrt{{\left(R_i^b\right)}^2-(u^2+v^2)S_{{\theta }_i}^2}$ ($C_{{\theta }_i}=\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_i$, $S_{{\theta }_i}=\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_i$, the rest as so on), ${\theta }_i=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(u/v\right)-{\phi }_i$, ${\phi }_i$ is spline position angle between the forward direction of $X$ axis and the radius $R_i^c$ of addendum circle for the $i$th external spline teeth.

In Fig. 5, the meshing arc length $l_i^{ac}$ in the $i$th spline teeth is subdivided into $q$ parts with equal arc length $∆l$, consequently, the arc length $l_{ij}^{ap}$ between the $j$th point $P_{ij}$ and point A, the radius $R_{ij}$ of the $j$th point $P_{ij}$ in the $i$th spline teeth relative to the axis of inner spline, the pressure angle ${\alpha }_{ij}$ of the $j$th point $P_{ij}$ in the $i$th spline teeth can be represented as:

where: $\mathrm{\Delta }l=l_i^{ac}/n$, $j=$1,…, $q$.

Considering the basal-deformation and shearing-deformation, the deformation caused by the transmission load can be divided into the following three categories based on Weber-Eenergy-Method as Fig. 6: (1) bending deformation ${\delta }_b$, shear deformation ${\delta }_a$, axial compression deformation ${\delta }_a$ at spline tooth; (2) deformation ${\delta }_e$ at the round corners and deformation ${\delta }_m$ at basal body; (3) local contact deformation ${\delta }_c$ caused by contact stress. The deformation caused by contact stress in the $j$th part is equal to the deformation caused by forces when $q$ is big enough.

Fig. 5The model of equal division for meshing arc length

Fig. 6The deformation of spline teeth

a) Bending deformation

b) Shear deformation

c) Axial-compression deformation

d) Deformation at round corners and basal-body

e) Local contact deformation

The total deformation $\delta$ of the $j$th part in the $i$th spline teeth can be represented as:

The deformation of the internal spline teeth and external spline teeth is different under the influence of load when the spline coupling transmits torque. There’s a hypothesis that ${\delta }_{ij}^A$ and ${\delta }_{ij}^B$ are the comprehensive deformation of the $j$th part for the $i$th internal spline teeth and the $i$th external spline teeth respectively, and ${\delta }_{ij}^C$ is the contact deformation of the $j$th part for the $i$th spline teeth. Therefore, the comprehensive meshing stiffness of the $j$th part for the $i$th spline teeth can be expressed as:

The whole comprehensive meshing stiffness of the $i$th spline teeth can be expressed as:

Transfer torque $T$ can be expressed with the meshing force $W_{ij}^T$ caused by torsion:

where: $W_{ij}^T=\varphi (R_{ij}-R_f)K_{ij}$, and $\varphi$ is twist angle for each key deformation.

Therefore, the meshing force caused by torsion of the $j$th part for the $i$th spline teeth can be expressed in the form of Eq. (15):

The Eq. (16) is the status parameter of right side and left side for internal spline $C$ and internal $D$:

where: $h=R$, $L$.

The internal spline $C$ and the external spline $D$ are respectively the driving shaft and the driven shaft, and the displacement of the $j$th part for the $i$th internal spline teeth in the meshing direction can be synthesized by the components in the direction of the $X$ axis and $Y$ axis:

where: ${\psi }_{ij}={\phi }_i+{\gamma }_i-{\alpha }_{ij}-{\theta }_{ij}-\pi /2$, ${\phi }_i$ is spline position angle of the $i$th external spline teeth, ${\gamma }_i$ is the included angle between $R_{ij}$ and the midline of the $i$th external spline teeth, ${\alpha }_{ij}$ is the pressure angle of the $j$th part for the $i$th internal spline teeth, ${\theta }_{ij}$ is the unfolding angle of ${\alpha }_{ij}$.

The meshing force caused by vibration displacement of the $j$th part for the $i$th spline teeth can be expressed in the form of Eq. (18):

Consequently, the comprehensive meshing force of the $j$th part for the $i$th spline teeth can be calculated as:

The meshing force $F_x$ at $X$ axis and the meshing force $F_y$ at $Y$ axis can be calculated by the comprehensive meshing stiffness $K_{ij}$ and ${\tau }_{ij}$ in the direction of meshing and the angle of meshing force ${\mathrm{\Psi }}_{ij}$ (the angle between normal at the $j$th part for the $i$th spline teeth and the positive direction of $X$ axis):

3.2. Establishment of coupled transfer matrix

There is a hypothesis that the form of solution for status parameter can be represented as: $x=X{e}^{\omega t}$, $\theta =\Theta e^{\omega t}$, $\phi =\mathrm{\Phi }{e}^{\omega t}$. The centrifugal force of weight for spline coupling will appear as a result of vortex motion when the spline coupling runs at high speed. The moment of inertia for spline coupling can be obtained by the relative centroid theorem. The force analysis of the spline coupling is shown in Fig. 7, and the kinematics equation of internal spline $C$ can be established according to the D’Alembert principle:

where: $m_C$ is the weight of internal spline $C$ (drive shaft), $I_d^C$ and $I_p^C$ are diameter moment of inertia and polar moment of inertia respectively, $\mathrm{\Omega }$ is the spin velocity of spline coupling.

Fig. 7Equilibrium analysis of force and moment for spline coupling

a)

b)

c)

d)

e)

Similarly, the kinematics equation of external spline $D$ can be established as Eq. (22):

where: $m_D$ is the weight of internal spline $D$ (drive shaft), $I_d^D$ and $I_p^D$ are diameter moment of inertia and polar moment of inertia respectively.

The displacement equation of left and right sides for spline coupling agrees with the Eq. (23) according to the continuity condition of displacement:

According to the relationship about the state vectors of left and right sides, the coupled transfer matrix of spline coupling can be represented as:

where: $T_{21}=T_{22}=T_{41}=T_{42}=$0: