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Abstract

Most of the time, mass eccentricity, and misalignment exist at the same time with aviation spline coupling working. Therefore, in this paper, the function of dynamic meshing force between multi-teeth and a non-linear dynamic model of involute spline coupling system in aero-engine with mass eccentricity and misalignment were presented. And then, the non-linear dynamic meshing force of spline coupling in aero-engine on different misalignment and mass eccentricity was investigated. The result shows that when the mass eccentricity and the misalignment are both small, the aviation involute spline coupling can run steady. And with the increase of mass eccentricity or misalignment, the dynamic load coefficient of the aviation involute spline coupling gradually increase. At the same time, as the mass eccentricity or misalignment increases, some teeth suffer more load, some teeth suffer less load, and some teeth are out of engagement so that they do not suffer any load. The running state of spline coupling becomes more and more unstable.

Keywords

Involute spline coupling mass eccentricity misalignment non-linear dynamic model

Introduction

Involute spline coupling of aero-engine are suffering periodic fluctuation loads while taking-off, cruising, and landing. Such loads will induce alternating contact stress on the working surface of spline couplings, and cause a vibration with small amplitude between contact surfaces. Due to this vibration caused by the internal and external excitations cannot be eliminated and avoided, it leads to fretting wear of spline couplings. The damage of fretting wear is invisible, which can seriously weaken the stability, safety, and reliability of the spline coupling in aero-engine. Therefore, the estimation of fretting wear of involute spline coupling in aero-engine remains a hot topic in the field of engineering.^1–3 And from the calculation formula of fretting wear, it can be seen that the wear coefficient, relative slide distance, and contact stress are all the basic parameters for the calculation of fretting wear.^4–6 The relative slide distance is a combination of the slide distance between the friction surfaces of two spline couplings and their vibration displacement under fluctuating loads.^5,6 However, the relative slide distance and the contact stress of involute spline coupling must be calculated based on the dynamic force between spline teeth. Thus, the dynamic load is the key to the design and optimization of spline coupling, as well as an important basis for accurately estimating the fretting wear of spline coupling.

In recent years, the studies on the dynamic meshing force of the spline coupling are mainly in two aspects: the vibration characteristics of spline couplings and the contact mechanics model of splines. For the study of the vibration characteristics of involute spline couplings, a piecewise linear dynamic model of a spline joint subjected to both backlash and circumferential tooth position errors was proposed by Kahraman. The equation of motion with a piecewise linear displacement function was obtained in the dimensionless form. For cases with a large number of spline teeth, a generalized approximation that reduces the piecewise linear displacement function into a piecewise non-linear one was proposed, and its accuracy was demonstrated using a case of linearly varying tooth position errors.⁷ Nataraj and Kappaganthu⁸ studied the nonlinear dynamics of two rigid rotors that are connected by splines, and the coupling had the Coulomb friction. The study indicated that the system had three unstable fixed points and a limit cycle when the system ran at a speed above the critical value, and the response near the limit cycle showed signs of chaos. Park⁹ performed all-round numerical calculations and experimental studies on loose-fit spline coupling and the rotor-bearing system. The results showed that the stability of the rotor system could be improved by lubricating the spline. At the same time, misalignment also affects the dynamic behavior of the rotor system.^10,11 Zhao et al.¹² established a dynamic model of spline considering the misalignment, and calculated and tested the dynamic characteristics of the spline-rotor system. The study showed that misalignment could cause complex double-frequency vibration in the shaft system, and loose-fit sleeves/spline coupling are subject to self-oscillation.^13,14 Tuckmantel and Cavalca proposed the comparison between two approaches for modeling the forces and moments generated by a metallic disc coupling under angular misalignment. The first is a well-established model based on the linear bending flexure of the disc packs, assuming the misalignment efforts are the sum of the first four harmonic components. In the second approach, a structural analysis of the coupling was conducted through finite element method, where the cyclic nature of coupling efforts was captured by the application of consecutive shaft spin angles.¹⁵ For the spline contact mechanics model, Sum et al.¹⁶ pointed out that the local mesh can be refined by MPCs. This is an effective way to realize the finite element analysis of splines under asymmetric loads. Liu et al.¹⁷ calculated the contact characteristics of the connection of spline couplings in aero-engine by the finite elements and verified it with tests. Peng and Li¹⁸ established a bending and squeezing deformation model for the spline with single-tooth meshing, and derived the formulas for bending, shearing, and squeezing of single-tooth meshing. Silvers et al.¹⁹ proposed a sequential expansion model and a statistical analysis model for predicting spline meshing. Barrot et al.^20,21 derived the torque between spline teeth, and they also studied the stress and axial load distribution of the spline coupling by the finite element method and established equations for the torsional stiffness as well as the sectional moment of inertia. Medina and Olver²² built meshes by the finite elements to perform boundary element integration. The stress of each node on spline was obtained. Tjernberg²³ obtained an accurate equation of stress concentration factor, and conducted the fatigue testing combined with finite element analysis. The results showed that spline shafts can withstand higher stresses if they are heated and quenched, and the average axial load distribution can reduce the stress concentration at the root of teeth. Zhu and Yao²⁴ derived the circumferential force and connection stiffness of the involute splines without clearance. Although the above researchers have done a lot of research on the vibration characteristics and contact mechanics models of spline couplings, and even some studies have also considered the impact of misalignment,^25–27 but nearly none of them emphasized on the dynamic meshing force of spline couplings in aero-engine. In fact, during the operation of the involute spline coupling of aero-engine, there are gaps, misalignment and eccentricity between teeth due to the manufacturing errors. Even though, at the beginning, the spline coupling was dynamically balanced before installation, and there was no mass eccentricity. However, in the operation process, due to the uneven backlash on the tooth side, only part of the teeth participated in the meshing. As the number of load cycles increased, the wear amount at different positions of the spline tooth surface was different, so the mass eccentricity gradually occurred, and the mass eccentricity became larger and larger. These three issues may exist in the operation individually or simultaneously, which seriously affected the system’s stability during operation. The dynamic parameters of the involute spline coupling of aero-engine and its dynamic meshing force will be significantly affected by these three factors. However, the above literature did not consider the coupling effect of these factors to analyze the dynamic meshing force of the involute spline coupling of aero-engine. In addition, in these researches, the calculation of the meshing force is based on the meshing with single tooth and meshing line with a fixed position. For the multi-tooth meshing of involute splines, each single tooth has a meshing line at the corresponding position. The position of meshing line is a function of the rotational position and the number of the teeth pair due to the existence of rotation. For the single-tooth meshing of involute splines, the position of the meshing line is time-varying. The teeth with different numbers of pairs have meshing lines in different directions, and it is resulting in different components of teeth meshing forces on the coordinate axis. Besides, there are different clearances and meshing deformations, as well as different meshing forces because of vibration.

In this work, in order to get the more accurate dynamic load, the model and equation of meshing force between the involute spline coupling was obtained considering the coupling effect of mass eccentricity, misalignment, and clearance, and then, the non-linear meshing force of spline coupling on different misalignment, mass eccentricity, and clearance was studied. The most important is that the model, equation and results of meshing force were all based on the multi-tooth meshing theory. It provides a good foundation to the predict on the fretting wear and fretting fatigue of spline coupling, and it is very important to design the involute spline couplings in aero-engine of high reliability, high accuracy, and high performance, as well as provides a good idea and method for dynamic load analysis of other parts with mass eccentricity and misalignment.

Calculation of dynamic meshing force of the system

Calculation of the displacement on meshing line

In Figure 1, a ray OK was drawn through the intersection point A of reference circle and external spline tooth profile, taking the center of circle O of the external spline shaft as the starting point. And then a segment AN tangent to base circle of external spline at point N was drawn, taking point A as the starting point.

Figure 1.

Schematic diagram of the rotational angle of a single tooth on the external spline.

So, it can be obtained from the property of the involute that $∠ AON = α_{0}$ . Here, $α_{0}$ is the pressure angle on reference circle; segment AN is perpendicular to the tooth profile (although the tooth shape has been simplified, some attributes are still calculated as the involute). Therefore, the rotational angle of a tooth on the external spline at one time is defined as:

θ_{1_{i}} = \frac{2 π}{z} i + ω t + θ_{0}

(1)

where, $z$ is the total number of teeth; $i$ is the tooth number; $ω$ is the angle velocity; $t$ is the time; $θ_{0}$ is the half angle of tooth thickness of the reference circle, $θ_{0} = π / 2 z$ ; the angle $φ_{1_{i}}$ between the working teeth profile and the x direction is defined as equation (14):

φ_{1_{i}} = θ_{1_{i}} - α_{0}

(2)

As shown in Figure 2, the intersection point of the working teeth profile and the reference circle is point A at the initial moment. Due to the transverse vibration displacement, the working teeth profile of the external spline moved from the black oblique line to the red oblique line. Accordingly, point A moved to $A_{2}$ . The direction of the direction of the meshing line was along the direction of JA and $K A_{1}$ . So, the working teeth profiles at these two positions were also parallel to each other. Meanwhile, the segment $LJ$ perpendicular to the line $A A_{3}$ and the line segment $LK$ perpendicular to the line $A_{1} A_{2}$ . It can be got that:

∠ L A_{2} A_{3} = φ_{1_{i}}

(3)

∠ KL A_{2} = ∠ L A_{2} A_{3} = φ_{1_{i}}

(4)

∠ A_{3} AL = ∠ L A_{2} A_{3} = ∠ KL A_{2} = φ_{1_{i}}

(5)

Figure 2.

The displacement of single tooth of external spline along the meshing line ( $x_{1} > 0, y_{1} > 0$ ).

If the external spline in Figure 2 is considered moved by a distance of $x_{1}$ along the x-axis firstly and then a distance of $y_{1}$ along the y-axis. As can be seen from the Figure 9, the relative displacement on the meshing line is the length of the segment $A A_{3}$ or $A_{1} A_{2}$ , and it equals to:

l_{A A_{3}} = l_{JA} - {l_{JA}}_{3}

(6)

and then:

l_{K A_{2}} = l_{L A_{2}} \sin φ_{1_{i}} = x_{1} \sin φ_{1_{i}}

(7)

l_{JA} = l_{LA} \cos φ_{1_{i}} = y_{1} \cos φ_{1_{i}}

(8)

Therefore, the relative displacement distance on the meshing line on the external and the internal splines are shown as follows, respectively:

Δ n_{1} = - x_{1} \sin φ_{1_{i}} + y_{1} \cos φ_{1_{i}}

(9)

Δ n_{2} = - x_{2} \sin φ_{2_{i}} + y_{2} \cos φ_{2_{i}}

(10)

Due to the analysis processes for the external and the internal spline moving along the x-axis and the y-axis in other quadrants are similar as above, that is to say $φ_{2_{i}} = φ_{1_{i}}$ , and the angle between the working tooth profile and x direction in the formula of next sections of this work is all expressed by $φ_{1_{i}}$ Therefore, the total relative displacement on meshing lines between the internal and external spline is shown as follows:

\begin{matrix} Δ n = Δ n_{2} - Δ n_{1} + r_{b} (θ_{2} - θ_{1}) \\ = (x_{1} - x_{2}) \sin φ_{1_{i}} - (y_{1} - y_{2}) \cos φ_{1_{i}} + r_{b} (θ_{2} - θ_{1}) \end{matrix}

(11)

Dynamic parameters

As for spline coupling, it is characterized as a multi-tooth meshing, and the number of tooth pair in-mesh is varying with the vibration, as well as the mesh deformation of each pair of teeth is also different. Therefore, the comprehensive meshing stiffness suitable for gear dynamics and pure torsion models of involute splines is no longer applicable. Here, in order to calculate the meshing force, the single-tooth meshing stiffness is calculated firstly, and then the single-tooth meshing force is obtained based on the single-tooth meshing stiffness. Finally, the total meshing force of the involute spline coupling is obtained by summing the single-tooth meshing force.

And the single-tooth meshing stiffness can be expressed as in Xue et al.:¹⁵

k_{m} = \frac{1}{Δ_{T}}

(13)

where, $Δ_{T}$ is the total flexibility of a pair of teeth, which can be expressed as follows:

Δ_{T} = Δ_{b 1} + Δ_{b 2} + Δ_{s 1} + Δ_{s 2}

(14)

where, $Δ_{bi}$ is the bending flexibility of the spline; $Δ_{si}$ is the shear flexibility of the spline; $i$ = 1 and 2 mean external spline and internal spline, respectively. The calculation process can be found in the literature.¹⁵ And the torsional stiffness of external spline kT1 and the torsional stiffness of internal spline kT2 are calculated by:

k_{T_{i}} = \frac{π G d_{i}^{4} min}{32 l_{i}}

(15)

$d_{i}$ is the equivalent diameter of shafts, mm; $l_{i}$ is the equivalent length of shafts, mm; G is modulus of rigidity. As well as the meshing damping of single teeth and the torsional damping of each spline are calculated as follows:

c_{m} = 2 ζ_{m} \sqrt{k_{m} \frac{{r_{b}}^{2} J_{1} J_{2}}{J_{1} + J_{2}}}

(16)

{\begin{matrix} c_{T 1} = 2 ζ_{T} \sqrt{k_{T 1} \frac{J_{M} J_{1}}{J_{M} + J_{1}}} \\ c_{T 2} = 2 ζ_{T} \sqrt{k_{T 2} \frac{J_{L} J_{2}}{J_{L} + J_{2}}} \end{matrix}

(17)

In formula (16) and (17), $ζ_{m}$ is mesh damping ratio, which is 0.1;²⁸ $ζ_{T}$ is torsional damping ratio of material, which is 0.005.²⁸

Nonlinear dynamic meshing force

Due to the clearance, the meshing force of a single tooth is a piece-wise linear function. Whether positive or negative of its value is mainly related to the clearance. If the relative displacement on the meshing line in the positive direction is greater than the clearance of the working teeth profile, the working teeth profile is regarded as being meshed and the meshing force is negative; if the relative displacement on the meshing line in the negative direction is greater than the clearance of the non-working teeth profile, the non-working teeth profile is regarded as being meshed, and the meshing force is positive; and if the relative displacements on the meshing line both in the positive and negative direction are less than the clearance of the working teeth profile and the non-working teeth profile, it is considered that there is no meshing force. Therefore, the meshing force of single tooth is (the value of the meshing force on the external spline and the internal spline is the same, but the direction is opposite):

F_{n_{i}} = {\begin{matrix} k_{m} (Δ n - c'_{0}) + c_{m} & \overset{\cdot}{Δ} n Δ n > c'_{0} \\ 0 & - c_{0} \leq Δ n \leq c'_{0} \\ k_{m} (Δ n + c_{0}) + c_{m} & \overset{\cdot}{Δ} n Δ n < - c_{0} \end{matrix}

(18)

where, $k_{m}, c_{m}$ are the meshing stiffness and meshing damping, respectively; $c_{0}$ is the clearance of the working teeth profile; $c'_{0}$ is the clearance of the non-working teeth profile.

The formula of the total meshing force and meshing torque are the summation of the meshing force components and meshing torque components along x and y direction, respectively:

{\begin{matrix} F_{mx} = - \sum_{j = 1}^{z} F_{nj} \sin φ_{1_{i}} \\ F_{my} = \sum_{j = 1}^{z} F_{nj} \cos φ_{1_{i}} \\ T_{m} = r_{b} \sum_{j = 1}^{z} F_{n_{i}} \end{matrix}

(19)

where, $F_{mx}$ is the component of the total meshing force along the x-axis; $F_{my}$ is the component of the total meshing force along the y-axis; $T_{m}$ is the total meshing torque. The whole process is shown in Figure 3.

Figure 3.

A flow chart for the calculation of the dynamic meshing force of the system.

Dynamic models and equations

Dynamic models

In this work, the lumped mass method was used to simplify the system, and the dynamic model of the aviation involute spline coupling system was established by comprehensively considering the effect of mass eccentricity, misalignment, and clearance (as shown in Figures 4 and 5). In Figure 4, four ellipses were used to represent the components, which are prime mover, external spline, internal spline, and load respectively. In Figure 4, x₁ axis of the coordinate system where the center of the external spline was located coincides with the x₂ axis of the coordinate system where the center of the internal spline was located. The center (x₂, y₂) of the internal and spline moved along the x-axis for a distance $l_{x}$ , which was taken as the initial parallel misalignment amount in the x-direction. The situation of the y-direction was the same as the x-misalignment amount, and there is no further showing here. The meanings of parameters in Figures 4 and 5 are shown in Table 1.

Figure 4.

The dynamic model of involute spline coupling in aero-engine.

Figure 5.

Mass eccentricity and misalignment diagram.

Table 1.

Value of dynamic parameters in Figures 4 and 5.

Sign	Value	Sign	Value
k _px1	5 × 10⁶ (N/m)	k _px2	5 × 10⁶ (N/m)
k _py1	5 × 10⁶ (N/m)	k _py2	5 × 10⁶ (N/m)
c _px1	5 (Ns/m)	c _px2	5 (Ns/m)
c _py1	5 (Ns/m)	c _py2	5 (Ns/m)
ρ ₁	2/4/5/6 × 10⁻⁴ (m)	ρ ₂	0
T _d	30.66 (Nm)	T _L	28 (Nm)
l _x	2/3/4/5 × 10⁻⁴ (m)	c_j	2.5/5/7.95/12.5 × 10⁻⁵ (m)
l _y	0	c′_j	2.5/5/7.95/12.5 × 10⁻⁵ (m)

While $x_{1}, y_{1}, θ_{1}$ and $x_{2}, y_{2}, θ_{2}$ are the vibration displacements along the x-axis and y-axis, and the torsional displacement around the axis of the external spline and the internal spline respectively; $θ_{M}, θ_{L}$ are the torsional displacements of prime mover and loads, $θ_{M}$ and $θ_{1}$ , $θ_{1}$ and $θ_{2}$ , and $θ_{2}$ and $θ_{L}$ are all different from each other due to torsional deformation of the axis and spline teeth; $ρ_{1}$ and $ρ_{2}$ are the mass eccentricity of the external spline and the internal spline respectively; $m_{1}$ and $m_{2}$ are the concentrated mass of the external and the internal splines, respectively; $J_{M}, J_{L}$ are the moment of inertia of the external and the internal splines; $k_{T 1}$ is the torsional rigidity of the drive shaft between the prime mover and the external spline, and $k_{T 2}$ is the torsional rigidity of the drive shaft between the internal spline and the loads; $k_{p 1}, k_{p 2}$ are the supporting stiffness on the external and the internal splines (the components of them along x and y axial are $k_{px 1}, k_{px 2}, k_{py 1}$ , and $k_{py 2}$ ); $c_{T 1}$ is the torsional damping of the drive shaft between the prime mover and the external spline, and $c_{T 2}$ is the torsional damping of the drive shaft between the internal spline and the loads; $c_{p 1}, c_{p 2}$ are the supporting damping of the external and the internal splines (the components of them along x and y axial are $c_{px 1}, c_{px 2}, c_{py 1}$ , and $c_{py 2}$ ); $F_{mx}, F_{my}$ are the components of the meshing force of the spline coupling along the x-axis and y-axis; $l_{x}$ is the offset distance in the x-axis direction, $l_{y}$ is the offset distance along y axial; $T_{d}$ is the driving torque, and $T_{L}$ is the load torque:

{p} = [x_{1} y_{1} x'_{2} y_{2} θ_{m} θ_{L} θ_{1} θ_{2}]'

(20)

and $x'_{2} = x_{2} + l_{x}$ , according to Newton’s second law, the dynamic equation of the involute spline coupling of aero-engine shown in Figures 10 and 11 can be expressed as follows:

[M] {\overset{\cdot\cdot}{p}} + [C] {\overset{\cdot}{p}} + [K] {p} + [C_{0}] = [F]

(21)

in which: $[M]$ is quality matrix; $[C]$ is damping matrix; $[K]$ is stiffness matrix; $[C_{0}]$ is clearance matrix; $[F]$ is generalized force matrix. And:

[M] = [\begin{matrix} m_{11} & 0 & 0 & 0 & 0 & 0 & m_{17} & 0 \\ 0 & m_{22} & 0 & 0 & 0 & 0 & m_{27} & 0 \\ 0 & 0 & m_{33} & 0 & 0 & 0 & 0 & m_{38} \\ 0 & 0 & 0 & m_{44} & 0 & 0 & 0 & m_{48} \\ 0 & 0 & 0 & 0 & m_{55} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & m_{66} & 0 & 0 \\ m_{71} & m_{72} & 0 & 0 & 0 & 0 & m_{77} & 0 \\ 0 & 0 & m_{83} & m_{84} & 0 & 0 & 0 & m_{88} \end{matrix}]

(22)

[C] = [\begin{array}{l} c_{11} & c_{12} & c_{13} & c_{14} & 0 & 0 & c_{17} & c_{18} \\ c_{21} & c_{22} & c_{23} & c_{24} & 0 & 0 & c_{27} & c_{28} \\ c_{31} & c_{32} & c_{33} & c_{34} & 0 & 0 & c_{37} & c_{38} \\ c_{41} & c_{42} & c_{43} & c_{44} & 0 & 0 & c_{47} & c_{48} \\ 0 & 0 & 0 & 0 & c_{55} & 0 & c_{57} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{66} & 0 & c_{68} \\ c_{71} & c_{72} & c_{73} & c_{74} & c_{75} & 0 & c_{77} & c_{78} \\ c_{81} & c_{82} & c_{83} & c_{84} & 0 & c_{86} & c_{87} & c_{88} \end{array}]

(23)

[K] = [\begin{matrix} k_{11} & k_{12} & k_{13} & k_{14} & 0 & 0 & k_{17} & k_{18} \\ k_{21} & k_{22} & k_{23} & k_{24} & 0 & 0 & k_{27} & k_{28} \\ k_{31} & k_{32} & k_{33} & k_{34} & 0 & 0 & k_{37} & k_{38} \\ k_{41} & k_{42} & k_{43} & k_{44} & 0 & 0 & k_{47} & k_{48} \\ 0 & 0 & 0 & 0 & k_{55} & 0 & k_{57} & 0 \\ 0 & 0 & 0 & 0 & 0 & k_{66} & 0 & k_{68} \\ k_{71} & k_{72} & k_{73} & k_{74} & k_{75} & 0 & k_{77} & k_{78} \\ k_{81} & k_{82} & k_{83} & k_{84} & 0 & k_{86} & k_{87} & k_{88} \end{matrix}]

(24)

[C_{0}] = [λ_{cx} - λ_{cy} - λ_{cx} λ_{cy} 0 0 - r_{b} λ_{cn} r_{b} λ_{cn}]

(25)

\begin{matrix} [F] = [m_{1} ρ_{1} {\overset{\cdot}{θ}}_{1}^{2} \cos θ_{2} m_{2} ρ_{1} {\overset{\cdot}{θ}}_{1}^{2} \sin θ_{1} - m_{1} g m_{2} ρ_{2} {\overset{\cdot}{θ}}_{2}^{2} \cos θ_{2} \\ m_{2} ρ_{2} {\overset{\cdot}{θ}}_{2}^{2} \sin θ_{2} - m_{2} g T_{d} - T_{L} 0 0] \end{matrix}

(26)

in which, the nonzero elements in matrix $[M]$ , $[C]$ , and $[K]$ are shown in the Appendix 1. And $λ_{1} ~ λ_{5}$ are the meshing force coefficients; $λ_{cn}$ is the sum of clearance of spline; $λ_{cx}$ is component along x axis of the sum of clearance, $λ_{cx} = λ_{cn} \sin φ_{j}$ ; is component along y axis of the sum of clearance, $λ_{cx} = λ_{cn} \cos φ_{j}$ . All of these parameters are expressed as follows:

λ_{1} = \sum_{j = 1}^{z} λ_{1 j}, λ_{1 j} = {\binom{\sin φ_{j} Δ n_{j} (t) > c'_{j} & Δ n_{j} (t) < - c_{j}}{0 - c_{j} \leq Δ n_{j} (t) \leq c'_{j}}

(27)

λ_{2} = \sum_{j = 1}^{z} λ_{2_{j}}, λ_{2_{j}} = {\begin{matrix} \cos φ_{j} Δ n_{j} (t) > c'_{j} & Δ n_{j} (t) < - c_{j} \\ 0 - c_{j} \leq Δ n_{j} (t) \leq c'_{j} \end{matrix}

(28)

λ_{3} = \sum_{j = 1}^{z} λ_{3_{j}}, λ_{3_{j}} = {\begin{matrix} \sin φ_{j} \cos φ_{j} & Δ n_{j} (t) > c'_{j} & Δ n_{j} (t) < - c_{j} \\ 0 & - c_{j} \leq Δ n_{j} (t) \leq c'_{j} \end{matrix}

(29)

λ_{4} = \sum_{j = 1}^{z} λ_{4_{j}}, λ_{4_{j}} = {\begin{matrix} \sin^{2} φ_{j} & Δ n_{j} (t) > c'_{j} & Δ n_{j} (t) < - c_{j} \\ 0 & - c_{j} \leq Δ n_{j} (t) \leq c'_{j} \end{matrix}

(30)

λ_{5} = \sum_{j = 1}^{z} λ_{5_{j}}, λ_{5_{j}} = {\begin{matrix} \cos^{2} φ_{j} & Δ n_{j} (t) > c'_{j} & Δ n_{j} (t) < - c_{j} \\ 0 & - c_{j} \leq Δ n_{j} (t) \leq c'_{j} \end{matrix}

(31)

λ_{cn} = \sum_{j = 1}^{z} λ_{c_{jn}}, λ_{c_{jn}} = {\begin{matrix} - k_{m} c'_{j} & Δ n_{j} (t) > c'_{j} \\ 0 & - c_{j} \leq Δ n_{j} (t) \leq c'_{j} \\ k_{m} c_{j} & Δ n_{j} (t) < - c_{j} \end{matrix}

(32)

The angular velocity ${\overset{\cdot}{θ}}_{M}$ of the prime mover was assumed to be a constant value. In order to eliminate the rigid body displacement of the equation (20), new degrees of freedom were introduced as follows:

{\begin{matrix} Δ_{1} = r_{b} (θ_{1} - θ_{M}) \\ Δ_{2} = r_{b} (θ_{2} - θ_{1}) \\ Δ_{3} = r_{b} (θ_{2} - θ_{L}) \end{matrix}

(33)

The dimensionless reference parameter $ω, l$ was introduced to non-dimensionalize the above equation (20). And then the equation (20) is as follows after the eliminating the rigid body displacement and non-dimensionalize:

[\bar{M}] {\bar{\overset{\cdot\cdot}{p}}} + [\bar{K}] {\bar{p}} + [\bar{C}] {\bar{\overset{\cdot}{p}}} + [{\bar{C}}_{0}] = [\bar{F}]

(34)

in which:

[\bar{M}] = [\begin{matrix} {\bar{m}}_{11} & 0 & 0 & 0 & 0 & 0 & {\bar{m}}_{17} & 0 \\ 0 & {\bar{m}}_{22} & 0 & 0 & 0 & 0 & {\bar{m}}_{27} & 0 \\ 0 & 0 & {\bar{m}}_{33} & 0 & 0 & 0 & 0 & {\bar{m}}_{38} \\ 0 & 0 & 0 & {\bar{m}}_{44} & 0 & 0 & 0 & {\bar{m}}_{48} \\ 0 & 0 & 0 & 0 & {\bar{m}}_{55} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\bar{m}}_{66} & 0 & 0 \\ {\bar{m}}_{71} & {\bar{m}}_{72} & 0 & 0 & 0 & {\bar{m}}_{77} & 0 \\ 0 & 0 & {\bar{m}}_{83} & {\bar{m}}_{84} & 0 & 0 & 0 & {\bar{m}}_{88} \end{matrix}]

(35)

[\bar{K}] = [\begin{matrix} {\bar{k}}_{11} & {\bar{k}}_{12} & {\bar{k}}_{13} & {\bar{k}}_{14} & 0 & 0 & {\bar{k}}_{17} & {\bar{k}}_{18} \\ {\bar{k}}_{21} & {\bar{k}}_{22} & {\bar{k}}_{23} & {\bar{k}}_{24} & 0 & 0 & {\bar{k}}_{27} & {\bar{k}}_{28} \\ {\bar{k}}_{31} & {\bar{k}}_{32} & {\bar{k}}_{33} & {\bar{k}}_{34} & 0 & 0 & {\bar{k}}_{37} & {\bar{k}}_{38} \\ {\bar{k}}_{41} & {\bar{k}}_{42} & {\bar{k}}_{43} & {\bar{k}}_{44} & 0 & 0 & {\bar{k}}_{47} & {\bar{k}}_{48} \\ 0 & 0 & 0 & 0 & {\bar{k}}_{55} & 0 & {\bar{k}}_{57} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\bar{k}}_{66} & 0 & {\bar{k}}_{68} \\ {\bar{k}}_{71} & {\bar{k}}_{72} & {\bar{k}}_{73} & {\bar{k}}_{74} & {\bar{k}}_{75} & 0 & {\bar{k}}_{77} & {\bar{k}}_{78} \\ {\bar{k}}_{81} & {\bar{k}}_{82} & {\bar{k}}_{83} & {\bar{k}}_{84} & 0 & {\bar{k}}_{86} & {\bar{k}}_{87} & {\bar{k}}_{88} \end{matrix}]

(36)

[\bar{C}] = [\begin{matrix} {\bar{c}}_{11} & {\bar{c}}_{12} & {\bar{c}}_{13} & {\bar{c}}_{14} & 0 & 0 & {\bar{c}}_{17} & {\bar{c}}_{18} \\ {\bar{c}}_{21} & {\bar{c}}_{22} & {\bar{c}}_{23} & {\bar{c}}_{24} & 0 & 0 & {\bar{c}}_{27} & {\bar{c}}_{28} \\ {\bar{c}}_{31} & {\bar{c}}_{32} & {\bar{c}}_{33} & {\bar{c}}_{34} & 0 & 0 & {\bar{c}}_{37} & {\bar{c}}_{38} \\ {\bar{c}}_{41} & {\bar{c}}_{42} & {\bar{c}}_{43} & {\bar{c}}_{44} & 0 & 0 & {\bar{c}}_{47} & {\bar{c}}_{48} \\ 0 & 0 & 0 & 0 & {\bar{c}}_{55} & 0 & {\bar{c}}_{57} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\bar{c}}_{66} & 0 & {\bar{c}}_{68} \\ {\bar{c}}_{71} & {\bar{c}}_{72} & {\bar{c}}_{73} & {\bar{c}}_{74} & {\bar{c}}_{75} & 0 & {\bar{c}}_{77} & {\bar{c}}_{78} \\ {\bar{c}}_{81} & {\bar{c}}_{82} {\bar{c}}_{83} & {\bar{c}}_{84} & 0 & {\bar{c}}_{86} & {\bar{c}}_{87} & {\bar{c}}_{88} \end{matrix}]

(37)

{\bar{p}} = [\bar{x_{1}} \bar{y_{1}} \bar{x'_{2}} \bar{y_{2}} \bar{Δ_{1}} \bar{Δ_{2}} \bar{Δ_{3}}]

(38)

[{\bar{C}}_{0}] = [\bar{λ_{cx 1}} - \bar{λ_{cy 1}} - \bar{λ_{cx 2}} \bar{λ_{cy 2}} - \bar{λ_{cn 1}} \bar{λ_{cn 2}} \bar{λ_{cn 3}}]

(39)

[\bar{F}] = [{\bar{f}}_{1} {\bar{f}}_{2} {\bar{f}}_{3} {\bar{f}}_{4} {\bar{f}}_{5} 0 {\bar{f}}_{7}]

(40)

where the nonzero elements in matrix $[\bar{M}]$ , $[\bar{C}]$ , $[\bar{K}]$ , $[\bar{C_{0}}]$ , and $[\bar{F}]$ are shown in the Appendix 1.

Analysis of dynamic load of spline coupling under multi-factors action

The geometric parameters of an aviation involute spline are as follows: the modulus m is 2.5 mm; the number of teeth is 22; the pressure angle is 30°; the meshing stiffness of single spline tooth is km = 1.5972 N/m; $d_{1} = d_{2} = 30$ mm; $l_{1} = 81$ mm; $l_{2} = 82$ mm; n = 6000 r/min. And the rest parameter values are shown in Table 1.

The results of nonlinear dynamic meshing forces of the system under the action of multiple factors were discussed in section 4.

Dynamic load of spline coupling under different mass eccentric

When $l_{x}$ = 2 × 10⁻⁴ m, $l_{y}$ = 0, $ρ_{2} = 0$ , the results of total nonlinear dynamic meshing force in the x-axis direction, frequency spectrum corresponding to the meshing force and dynamic load coefficient of the aviation involute spline coupling under different eccentric mass of external spline are shown in Figures 6 to 13, in which, Figure 6(a) and (b) show the component of total meshing force in the x-axis direction and its frequency spectrum when the mass eccentricity is $ρ_{1} = 2 \times 10^{- 4} m$ . It can be seen that the total meshing force in the x-axis direction fluctuates in a range from 275 N to 600 N, and there is a larger force amplitude when the frequency value is 102.5 Hz, and it approximately equals to the working frequency of the spline which is 100 Hz (6000/60). Figure 6(c) shows the dynamic load coefficient changing trend of the spline coupling. It shows that the dynamic load coefficient is steady and the value of it is around 0.9571. Figure 7 shows the meshing force on each tooth of the external spline at 80.03 s when the mass eccentricity is $ρ_{1} = 2 \times 10^{- 4}$ m. Due to the existing of the initial misalignment and mass eccentricity, the force on each tooth of the external spline are not uniform. From Figure 7(b), it can be seen that the sum of the force of each tooth is not zero, and the sum is roughly the same as the result in Figure 6(c). Figures 7 and 8 are the dynamic load situation of spline coupling when the mass eccentricity is $ρ_{1} = 4 \times 10^{- 4}$ m. In Figure 8(a) and (b), the peak values of the total meshing force in the x-axis direction change compared with Figures 6(a) and 7(b), it changes from 600.5 N to 778.5 N, and the amplitude corresponding the frequency 102.5 Hz increase to 254 N. At the same time, from Figure 8(c), it can be seen that the fluctuation range of dynamic load coefficient increases than that when the mass eccentricity is $ρ_{1} = 4 \times 10^{- 4}$ m, it is 0.904–1.017. From Figure 9, it can be seen that with the mass eccentricity increasing from $ρ_{1} = 2 \times 10^{- 4}$ m to $ρ_{1} = 4 \times 10^{- 4}$ m, the dynamic force along the normal direction and x axial of each tooth is getting more non-uniform, even some teeth are not meshed.

Figure 6.

Dynamic load of involute spline coupling when $ρ = 2 \times 10^{- 4}$ m: (a) total nonlinear dynamic meshing force in the x-axis direction, (b) frequency spectrum corresponding to the meshing force, and (c) dynamic load coefficient.

Figure 7.

Dynamic load of each tooth when $ρ = 2 \times 10^{- 4}$ m and t = 80.03 s: (a) normal force on each tooth of the external spline and (b) dynamic meshing force in the x-axis direction of each tooth.

Figure 8.

Dynamic load of involute spline coupling when $ρ = 4 \times 10^{- 4}$ m: (a) total nonlinear dynamic meshing force in the x-axis direction, (b) frequency spectrum corresponding to the meshing force, and (c) dynamic load coefficient.

Figure 9.

Dynamic load of each tooth when $ρ = 4 \times 10^{- 4}$ m and t = 80.03 s: (a) normal force on the each tooth of external spline and (b) dynamic meshing force in the x-axis direction of each tooth.

Figure 10.

Dynamic load of involute spline coupling when $ρ_{1} = 5 \times 10^{- 4}$ m: (a) total nonlinear dynamic meshing force in the x-axis direction, (b) frequency spectrum corresponding to the meshing force, and (c) dynamic load coefficient.

Figure 11.

Dynamic load of each tooth when $ρ_{1} = 5 \times 10^{- 4}$ m and t = 80.03 s: (a) normal force on each tooth of the external spline and (b) dynamic meshing force in the x-axis direction of each tooth.

Figure 12.

Dynamic load of involute spline coupling when $ρ_{1} = 6 \times 10^{- 4}$ m: (a) total nonlinear dynamic meshing force in the x-axis direction, (b) frequency spectrum corresponding to the meshing force, and (c) dynamic load coefficient.

Figure 13.

Dynamic load of each tooth when $ρ_{1} = 6 \times 10^{- 4}$ m and t = 80.03 s: (a) normal force on each tooth of the external spline and (b) dynamic meshing force in the x-axis direction of each tooth.

Figure 10 shows the dynamic load situation of the spline coupling when the mass eccentricity is $ρ_{1} = 5 \times 10^{- 4}$ m. In Figure 10(c), the total meshing force in the x-axis direction has become unstable, the fluctuation range of the force has increased and there are some negative values. It means the vibration becomes seriously. And in Figure 10(b), it can be seen that in addition to the frequency component corresponding to the largest amplitude of meshing force, the other four components with very small amplitude can be seen. They are 300.3 Hz, 439.5 Hz, 498 Hz, and 542 Hz, respectively. There is no obvious relationship among these four frequency components. The fluctuation of dynamic load coefficient in Figure 10(c) also increases, with the fluctuation range of 0.785–1.135, also indicating that the system running is gradually becoming unstable. At the same time, compared with Figure 9(a) and (b), when the eccentricity is $ρ_{1} = 5 \times 10^{- 4}$ m, the dynamic load of each tooth in Figure 10 does not change much, and the number of teeth un-loading increases.

When the mass eccentricity increases to $ρ_{1} = 6 \times 10^{- 4}$ m, it can be seen from Figures 11(a) and 12(b) that the peak value of total meshing force in the x-axis direction increases from 920 N when the mass eccentricity is $ρ_{1} = 5 \times 10^{- 4}$ m to 1300 N and when the mass eccentricity is $ρ_{1} = 6 \times 10^{- 4}$ m, and the changing trend is more chaotic, while the value of negative meshing force increases than before. Also, in Figure 12(c), though the range of dynamic load coefficient is almost the same as in Figure 10(c), the dynamic load coefficient becomes more unstable than before. Meanwhile, from Figure 13 it can be seen that the number of teeth un-meshing becomes larger and larger, which seriously affects the distribution and the sharing of dynamic load on the spline, and easily causes the wear and fatigue failure of the spline teeth.

Misalignment

When the mass eccentricity is $ρ_{1} = 2 \times 10^{- 4}$ m, $ρ_{2} = 0$ , $l_{y}$ = 0, the results of total nonlinear dynamic meshing forces in the x-axis direction, frequency spectrum corresponding it, and dynamic load coefficient of the aviation involute spline coupling under different misalignment $l_{x}$ are shown in Figures 14 to 19. Figures 14 and 15 show the load situation of involute spline coupling when $l_{x} = 3 \times 10^{- 4}$ m. From Figure 14(a) and (b), it can be seen that the maximum value of the total meshing force of the external spline teeth in the x-axis direction changed greatly compared with the value when $l_{x} = 2 \times 10^{- 4}$ m, and the maximum value increased to 811.6 N, but the corresponding frequency spectrum has no obvious change. Figure 14(c) shows that the dynamic load coefficient also increases too with the misalignment increasing, and the fluctuation range of it is 0.87–1.04. At the same time, it can be concluded from Figures 6 and 15 that with the increase of misalignment, the load of each spline tooth increases, and the number of meshing teeth also changes. Compared with Figure 7, the teeth with less load are no longer loaded, but the teeth with more load are loading heavier. Similarly, after analysis of Figures 16 to 19, it can be found that the total meshing force of the external spline in the x-axial direction, the corresponding frequency spectrum, and the dynamic load coefficient all become more unstable when $l_{x} = 4 \times 10^{- 4}$ m. The number of the teeth loaded of the spline has a further reduction, and the difference of the load between each tooth is increased too.

Figure 14.

Dynamic load of involute spline coupling when $l_{x} = 3 \times 10^{- 4}$ m: (a) total nonlinear dynamic meshing force in the x-axis direction, (b) frequency spectrum corresponding to the meshing force, and (c) dynamic load coefficient.

Figure 15.

Dynamic load of each tooth when $l_{x} = 3 \times 10^{- 4}$ m and t = 80.03 s: (a) normal force on each tooth of the external spline and (b) dynamic meshing force in the x-axis direction of each tooth.

Figure 16.

Dynamic load of involute spline coupling when $l_{x} = 4 \times 10^{- 4}$ m: (a) total nonlinear dynamic meshing force in the x-axis direction, (b) frequency spectrum corresponding to the meshing force, and (c) dynamic load coefficient.

Figure 17.

Dynamic load of each tooth when $l_{x} = 4 \times 10^{- 4}$ m and t = 80.03 s: (a) normal force on each tooth of the external spline and (b) dynamic meshing force in the x-axis direction of each tooth.

Figure 18.

Dynamic load of involute spline coupling when $l_{x} = 5 \times 10^{- 4}$ m: (a) total nonlinear dynamic meshing force in the x-axis direction, (b) frequency spectrum corresponding to the meshing force, and (c) dynamic load coefficient.

Figure 19.

Dynamic load of each tooth when $l_{x} = 5 \times 10^{- 4}$ m and t = 80.03 s: (a) normal force on each tooth of the external spline and (b) dynamic meshing force in the x-axis direction of each tooth.

Conclusion

Therefore, it can be concluded by investigating the non-linear dynamic load of involute spline coupling in aero-engine with mass eccentricity and misalignment that:

when the mass eccentricity and misalignment were all small, the total meshing force in the x-axial direction and dynamic load coefficient of the aviation involute spline coupling are relatively stable. That is to say, the aviation involute spline coupling can run steady under this situation.

the frequency component corresponding to the meshing force peek value approximately meets the double frequency relationship, and the lowest frequency approximately equals to the double rotation frequency, but it is not a very accurate mathematical relationship. It because that the spline belongs to the multitooth meshing, and the separation and engagement of teeth do not obey the strict law while the spline running, which has a certain impact on the frequency of the meshing force. However, with the increase of mass eccentricity or misalignment, all of the maximum value of the total meshing force along x direction, and the fluctuation range of the dynamic load coefficient of the aviation involute spline coupling gradually increase.

from Figures 7 to 13, it can be concluded that the load sharing between each tooth are very uneven with the increase of eccentricity of the spline coupling. It can be seen that some teeth were suffering more load, but some teeth suffering less load, and even some teeth were not suffering any load at all. And with the increase of eccentricity, misalignment, or clearance, the teeth number of out of engagement on the spline coupling were more and more, which poses a serious threat to the strength of spline coupling. As well as it made the spline coupling becomes more and more unstable. When the eccentricity, misalignment, or clearance is large enough, some teeth were suffering too much load, the spline coupling leads to spline failure. Therefore, the auxiliary centering structure should be designed while designing the spline coupling, so that the spline coupling can operate under the condition of absolute alignment or only small misalignment; at the same time, in the processing and manufacturing, the machining error and assembly error should be strictly controlled, so that the mass eccentricity and tooth clearance can be well controlled, and finally the load distribution of the spline coupling will be uniform, and the number of teeth involved in the meshing will be as many as possible.

and also, the function of non-linear dynamic load and non-linear dynamic model of involute spline coupling in aero-engine presented here provides a good method to the analysis on multi-teeth meshing structures with mass eccentricity and misalignment, and gives a good and important reference to the designing the involute spline with high accuracy, high reliability, and high strength.

Footnotes

Appendix I

Parameters and the corresponding formulas before and after the eliminating the rigid body displacement and non-dimensionalize.

Where:

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge financial support from the National Natural Science Foundation (Grant No. 52005312), the National Natural Science Foundation of Shaanxi in China (Grant No. 2019JQ-457), and Special Scientific Research Plan Project of Shaanxi Provincial Department of Education in China (Grant No. 19JK0147).

ORCID iD

Xiangzhen Xue

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Nonlinear dynamic load analysis of aviation spline coupling with mass eccentricity and misalignment

Abstract

Keywords

Introduction

Calculation of dynamic meshing force of the system

Calculation of the displacement on meshing line

Dynamic parameters

Nonlinear dynamic meshing force

Dynamic models and equations

Dynamic models

Analysis of dynamic load of spline coupling under multi-factors action

Dynamic load of spline coupling under different mass eccentric

Misalignment

Conclusion

Footnotes

Appendix I

Declaration of conflicting interests

Funding

ORCID iD

References