Dynamic modeling of double helical gears

Abstract

There are few publications on the dynamic model of double helical gears considering axial vibration. In the present dynamic model, it is thought that the axial force of the left and right ends generated by tooth meshing can offset each other. Because of the existence of manufacture and installation, the axial force of the left and right ends cannot completely offset which will cause the axial vibration. In this paper, a dynamic model of double helical gears is proposed. Firstly, the calculated model of axial displacement caused by the errors of manufacture and installation under low speed is proposed. Secondly, the dynamic model of double helical gears considering axial vibration and backlash is built. On this basis, the dynamic equations of this system are established and solved. Finally, a pair of double helical gears is taken as an example. The axial displacement is calculated out and compared with that from the experiment. The dynamic behaviors of double helical gears are studied.

Keywords

Double helical gears axial displacement axial vibration backlash dynamic model

1. Introduction

The reduction of vibration and noise is now paid more and more attention by researchers. Double helical gears are widely used in high-speed marine ship transmission because of their high loading capability, steady transmission and small axial force compared with helical gears (Amendola, 2006). The noise of marine ships threatens their own safety and affects the comfort of seamen and passengers. High-frequency dynamic force is created by the gear meshing which generates structure-borne noise and it is a major component of marine ship noise. Therefore, reduction of vibration and noise of gear meshing becomes an important research direction in gear design of marine ships.

Vibration and noise are generated in the dynamic state. Numerous analytical models of gear dynamic including spur, helical, bevel, face gears, etc. can be found in the literature (Amarnath et al., 2012; Huang et al., 2011; Stringer et al., 2011). However, the literatures on double helical gears is quite limited. Jauregui and Gonzalez (1999) proposed a single-degree dynamic model. Ajmi and Velex (2001) proposed a twelve-degree dynamic model. Prashant and Ahmet (2013) built a dynamic model of a double-helical planetary gear set. Wu and Zhu (2001) established the double helical gear’s theoretical analyzing vibration model by studying and analyzing vibration patterns of spur and helical gears. Bu et al. (2012) developed a generalized dynamic model for herringbone planetary gear train. The models thought that the axial force of the left and right ends generated by tooth meshing could offset each other, so the axial vibration is neglected. Because of the existence of manufacture and installation, the axial force of the left and right ends cannot completely offset which causes the axial vibration. Therefore, the model considering the axial vibration is necessary to be established.

The axial displacement caused by the errors of manufacture and installation under low speed will turn into the vibration excitation under high speed. Therefore, for double helical gears, the axial displacement cannot be ignored in the course of vibration calculation. Besides, it is essential that the accurate calculation of inner excitations including stiffness excitation, error excitation and impact excitation is completed for the establishment of a dynamic model.

Computerized simulation of meshing and contact (tooth contact analysis (TCA)) was developed for spiral bevel and hypoid gear drives (Litvin and Gutman, 1981; Litvin et al., 1995). Then, it is applied to other gears. TCA technology can accurately simulate the generation of tooth surface and the meshing process with different errors including installation errors and manufacture errors. Also, TCA offers the accurate geometrical data of tooth surface for loaded tooth contact analysis (LTCA).

Loaded tooth contact analysis is an important method for numerical simulation of the gear meshing process. It is a bridge between the geometry design and the mechanical analysis of the gear. Many contributions are available in helical gears, spiral bevel gears, hypoid gears and worm gears (Li, 2002; Litvin et al., 2007).

Combining TCA and LTCA, the meshing characteristics of double helical gears under the actual working conditions can be obtained. Based on the above results, the inner excitations can be accurately calculated.

Specific objectives of this study are as follows:

Based on TCA and LTCA, the calculated model of axial displacement caused by the errors of manufacture and installation under low speed is proposed. The error excitation is calculated.

Based on the results of TCA and LTCA, stiffness excitation and gear-into impact excitation are calculated.

The dynamic model of double helical gears considering axial vibration and backlash is built up and the dynamic equation of system is solved. The internal dynamic excitations produced by stiffness excitation, error excitation and gear-into impact excitation are considered in this model.

It should be pointed out that TCA and LTCA of double helical gears can be found in the references (Wang and Fang, 2009; Wang et al., 2010, 2012a).

2. The calculated model of axial displacement for double helical gears

The axial displacement caused by the errors of manufacture and installation under low speed will turn into the excitation of vibration under high speed. Therefore, for double helical gears, the axial displacement excitation should not be ignored in the course of dynamic calculation. Considering the errors of manufacture and installation, the axial displacement under low speed can be obtained by TCA and LTCA.

When the errors of manufacture and installation exist, unbalance loading between the right-side gear pair and the left-side gear pair unavoidably appears. The axial displacement of floating pinion achieves the equilibrium of transfer torque of two ends. Because the pitch radius of two end pinions is nearly equal, the transfer torque equilibrium can be translated into the equilibrium of two ends axis forces, that is,

\sum_{l = 1}^{n} p_{l I} \cos γ_{l I} + \sum_{l = 1}^{n} p_{l Π} \cos γ_{l Π} = \sum_{l = 1}^{n} p_{l III} \cos γ_{l III} + \sum_{l = 1}^{n} p_{l IV} \cos γ_{l IV}

(1)

Where, it is assumed that left-end and right-end have two teeth (I, II, ІІІ and ΙV) simultaneously meshing in a meshing position, respectively. The numbers of meshing pairs of teeth in a position are calculated by the TCA and then are confirmed by the LTCA. Normal force of contact point pair can be calculated by LTCA. Schematic diagram of axial force balance is shown in Figure 1. Detailed calculation procedure can be found in the references (Wang et al., 2012a).

Equation (1) is placed in the calculation of LTCA. When the two ends’ normal forces in contact position are not equal, a small axial displacement ΔZ is artificially given. TCA and LTCA are recalculated until Equation (1) is satisfied. In this way, the final axial displacement can be obtained. The calculation flowchart of axial displacement for double helical gears is shown in Figure 2.

The synthetic meshing error of gear pair includes frequency error of gear (short period error) and frequency error of shaft (long period error). Among them, the long period error also includes N short period errors.

Tooth contact analysis and LTCA are calculated in a meshing period (i.e., short period). Therefore, the axial displacement is also obtained in a short period. In order to obtain the axial displacement of long period, TCA and LTCA are calculated in N short periods. By soft MATLAB, the values of axial displacement are integrated and expanded into periodic function.

3. The calculation of inner excitations for double helical gears

The vibration and noise of gear system originate in the dynamic excitations which include inner excitations and outer excitations. In the paper, only inner excitations including stiffness excitation, error excitation and impact excitation are considered.

Based on the results of TCA and LTCA, inner excitations can be accurately calculated. The flowchart for calculation of internal excitations is shown in Figure 3.

3.1. The calculation of time-variant stiffness excitation and impact excitation

By LTCA, contact force and contact deformation in instantaneous meshing position can be obtained. Then, time-variant stiffness of gear pair can be calculated. Time-variant stiffness of gear pair is composed of meshing stiffness of single-pair teeth engaged. Therefore, by inverse transform, meshing stiffness of single-pair teeth can be calculated. Detailed calculations of stiffness excitation and gear-into impact excitation can be found in the literature (Wang et al., 2012b).

3.2. The calculation of excitation error

Numerous literatures only consider the profile errors and pitch errors. They are synthesized system equivalent errors according to the gear machining accuracy grade and finally used harmonic function to simulate. The axial displacement caused by the errors of manufacture and installation under low speed will turn into the excitation of vibration under high speed. Therefore, for double helical gears, excitation error should include axial displacement errors under low speed. In section 2, the calculation model of axial displacement has been introduced.

4. The dynamic model of double helical gears considering the axial vibration and backlash

The character of double helical gears is that the floating installs of pinion achieves the transfer torque equilibrium (i.e., the axial constraint does not exist in the two ends of pinion). According to meshing characteristics of load sharing for double helical gears, the linear time-variant dynamic model of the double helical gears is established by the method of concentrated parameter. The model of dynamic analysis for double helical gears is shown in Figure 4 (where, the torsional stiffness, the torsional damping, the bending stiffness and the bending damping of axis are not shown). Without considering the friction force of contact tooth surface, the dynamic model of double helical gears is a twelve-degree vibrating system. The twelve degrees are represented as

{δ} = {y_{q 1} z_{q 1} θ_{q 1} y_{p 1} z_{p 1} θ_{p 1} y_{q 2} z_{q 2} θ_{q 2} y_{p 2} z_{p 2} θ_{p 2}}^{T}

(2)

In Figure 4, the relation between the vibration displacement of center point q₁, p₁, q₂ and p₂ and generalized displacement of driving gear and driven gear are represented as

{\bar{y}}_{ij} = y_{ij} + θ_{ij} R_{ij}

(3)

{\bar{z}}_{ij} = z_{ij} - {\bar{y}}_{ij} \tan β

(4)

The normal meshing stiffness of double helical gears can be calculated from section 3.1. The normal damping is calculated through a kind of usual method, that is,

c = 2 ζ_{v} \sqrt{K \cdot m}

(5)

The existence of backlash can generate impact between engaged teeth, which influences the stability of gear transmission. In the paper, only the gear backlash along meshing line is considered. The clearance function f can be expressed as

f (x) = | \begin{matrix} x - b \begin{matrix} x > b \end{matrix} \\ 0 \begin{matrix} x < = | b | \end{matrix} \\ x + b \begin{matrix} x < - b \end{matrix} \end{matrix}

(6)

Considering the interaction between left and right end gear pair, the dynamic meshing forces along the tangential direction are represented as

\begin{matrix} F_{y i j} = k_{y j} f ({\bar{y}}_{q j} - {\bar{y}}_{p j} - e_{y j}) + c_{y j} ({\dot{\bar{y}}}_{q j} - {\dot{\bar{y}}}_{p j} - {\dot{\bar{e}}}_{y j}) \\ + {(- 1)}^{s + 1} [k_{i x} (θ_{i 1} - θ_{i 2}) + c_{i x} ({\dot{θ}}_{i 1} - {\dot{θ}}_{i 2})] \\ = \cos β [k_{j} f ({\bar{y}}_{q j} - {\bar{y}}_{p j} - e_{y j}) + c_{j} ({\dot{\bar{y}}}_{q j} - {\dot{\bar{y}}}_{p j} - {\dot{\bar{e}}}_{y j})] \\ + {(- 1)}^{s + 1} [k_{i x} (θ_{i 1} - θ_{i 2}) + c_{i x} ({\dot{θ}}_{i 1} - {\dot{θ}}_{i 2})] \end{matrix}

(7)

The dynamic meshing force along the axial direction is represented as

\begin{matrix} F_{z i j} = k_{z i j} ({\bar{z}}_{q j} - {\bar{z}}_{p j} - e_{z j}) + c_{z i j} ({\dot{\bar{z}}}_{q j} - {\dot{\bar{z}}}_{p j} - {\dot{\bar{e}}}_{z j}) \\ +^{(- 1) j} k_{z i j} z_{i e} +^{(- 1) j} c_{z i j} {\dot{z}}_{i e} \\ = \sin β [k_{i j} ({\bar{z}}_{q j} - {\bar{z}}_{p j} - e_{z j}) + c_{i j} ({\dot{\bar{z}}}_{q j} - {\dot{\bar{z}}}_{p j} - {\dot{\bar{e}}}_{z j}) \\ +^{(- 1) j} k_{i j} z_{i e} +^{(- 1) j} c_{i j} {\dot{z}}_{i e}] \\ = \sin β {k_{i j} [z_{q j} - z_{p j} - ({\bar{y}}_{q j} - {\bar{y}}_{p j}) \tan β - e_{z j}] \\ + c_{i j} [{\dot{z}}_{q j} - {\dot{z}}_{p j} - ({\bar{y}}_{q j} - {\bar{y}}_{p j}) \tan β - {\dot{e}}_{z j})] \\ +^{(- 1) j} k_{i j} z_{i e} +^{(- 1) j} c_{i j} {\dot{z}}_{i e}} \end{matrix}

(8)

Where, i = q,p; if i = q, j = 1 and i = p, j = 2, then s = 1, otherwise, s = 2.

According to Figure 3, the dynamic equations of this system are established by Newton’s second law (see Appendix 1).

The relative displacement between meshing points of two teeth surfaces along the meshing line is represented as

λ_{j} = {\bar{y}}_{qj} - {\bar{y}}_{pj} - e_{yj}

(9)

The phase separation of rotation angle between two ends gear pair is represented as

γ_{i} = θ_{i 1} - θ_{i 2}

(10)

We define λ_j and γ_i as new free degrees respectively, t = τw_n as a time dimension and b_c as a displacement dimension, the identical dimensionless equations are deduced by the method of eliminating relative displacement and identical dimensionless.

The identical dimensionless of equation (7) is represented as

f_{yij} = \cos β [ϕ_{ij} f (ψ_{j}) + 2 ζ_{ij} {\overset{\cdot}{ψ}}_{j}] + ϕ_{ix} χ_{i} + 2 ζ_{ix} {\overset{\cdot}{χ}}_{i}

(11)

The identical dimensionless of equation (8) is represented as

f_{zij} = \sin β {ϕ_{ij} [Z_{qj} - Z_{pj} - ψ_{j} \tan β - 2 E_{zj}] + 2 ξ_{ij} [{\overset{\cdot}{Z}}_{qj} - {\overset{\cdot}{Z}}_{pj} - {\overset{\cdot}{ψ}}_{j} \tan β - 2 {\overset{\cdot}{E}}_{zj}] + (- 1)^{j} ϕ_{ij} Z_{ie} + (- 1)^{j} 2 ξ_{ij} {\overset{\cdot}{Z}}_{ie}}

(12)

The identical dimensionless equations are shown in Appendix 2.

5. The solution and analysis of dynamic model of double helical gears

The solution of dynamic of double helical gears is considered as an example. The parameters of double helical gears are given in Table 1. Usually, backlash of ordinary gear is between 0.07 mm and 0.15 mm. Where, the backlash of two ends gear pairs are all selected 0.1 mm.

Table 1.

Parameters of double helical gears.

Parameters	Pinion	Gear
Tooth number	31	102
Normal modulus /(mm)	4.5
Normal pressure angle /(^°)	20
Helical angle /(^°)	28.34
Face width /(mm)	90×2
Helical direction	Right left	Left right
Run-out groove width /(mm)	70
Rotation speed of pinion /(r/min)	2881
Output torque /(N·m)	2000

5.1. The measurement of axial displacement

The main technical parameters are described in Table 2. The errors of manufacture are measured with a gear tester. The partial measurement results of teeth are shown in Figure 5 (where, only measurement of left gear pair is given, and that of right is similar). The horizontal axis represents all teeth which include 31 teeth in the example. The axial displacements are measured by an eddy current sensor under low rotation speed. The test gear box is shown in Figure 6 and Figure 7. The measured data are integrated by soft MATLAB, the axial displacement from experiment is shown in Figure 8 (a). For the example listed in Table 1, it can be found that a long period includes 3162 short meshing periods. Therefore, by 3162 calculations of meshing period, the axial displacement from calculation is shown in Figure 8 (b). Table 3 shows the axial displacement amplitude of experiment and calculation. In the example, the installation errors are not considered. In the practical calculation, they are considered in TCA.

Table 2.

Operation parameters.

Rotation speed of motor/(r/min)	Rotation speed of gear /(r/min)	Rotation speed of pinion /(r/min)
1420	1.943	6.39

Table 3.

Amplitude of axial displacement from experiment and calculation.

Torque (N·m)	Amplitude of measurement/um	Amplitude of calculation/um
2000	24.3	23.98

From Figure 8 and Table 3, we can find that the calculated values of axial displacement are in good agreement with the measured values, which shows the correctness of the proposed method.

5.2. The solution and analysis of dynamic model of double helical gears

Based on the results of TCA and LTCA, the meshing synthetic stiffness, meshing stiffness of single-pair teeth, the axial displacement under low rotation speed and the gear-into impact excitation are calculated. The discrete values are integrated and expanded into periodic function. Corresponding curves are shown in Figure 9. The curve of manufacture errors is also shown in Figure 9, where, the manufacture precision grades of double helical gears is 5.

The identical dimensionless equations are solved by the method of the fourth-order Runge–Kutta algorithm with variable step lengths. Taking the left end gear pair as an example, the vibratory responses of acceleration in time–domain spectrum are shown in Figure 10, respectively. From Figure 10, it is found in the example, the axial vibration cannot be neglected. The vibratory response of acceleration along y-direction in frequency spectrum is shown in Figure 11.

The literature (Wang et al., 2016) proposed a method of modified optimization of double helical gears based on the minimum transmission errors. The verification test of vibration and noise is completed. The paper uses part of the test results to compare with the proposed theory. The detection point of vibration and noise of double helical gears is shown in Figure 12 (Wang et al., 2016). The collected frequency spectrum of acceleration along y-direction (measuring points 2 in Figure 12) is shown in Figure 13.

From Figure 11 and Figure 13, we can find: (1) the peak frequency of acceleration along y-axis almost is near 2945 Hz which is 10 times the meshing frequency; and (2) the values calculated and measured are not exactly equal. The reason is that the vibration of the box is not considered in the calculated value.

6. Conclusions

Based on TCA and LTCA, the calculated model of axial displacement caused by the errors of manufacture and installation under low speed is proposed. The meshing synthetic stiffness, meshing stiffness of single-pair teeth, the error excitation and the gear-into impact excitation are calculated, respectively.

The dynamic model of double helical gears considering axial vibration and backlash is built up and the dynamic equations of system is established and solved. The internal dynamic excitations produced by stiffness excitation, error excitation and gear-into impact excitation are considered in this model.

A pair of double helical gears is taken as an example. The total cumulative pitch error of left and right tooth surface for pinion and gear are measured. The axial displacements are measured by an eddy current sensor under low rotation speed. The calculations of axial displacement are obtained according to the measured errors, which are in good agreement with the measurements. Taking the left end gear pair as an example, the dynamic behaviors of double helical gears are studied and analyzed.

Footnotes

Acknowledgements

The authors wish to acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 51475210), Taishan Scholar Project Special Funds (2016–2020) during the course of this investigation. The authors would also like to thank the editor and anonymous reviewers for their suggestions for improving the paper.

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 received financial support of the National Natural Science Foundation of China (Grant No.51475210), Taishan Scholar Project Special Funds (2016–2020) during the course of this investigation.

Nomenclature

P_l(l = 1, 2, … , n)

normal force of contact point pair (l-l’)

γ_l

acute angle between normal force and axis, which can be calculated by tooth contact analysis

j = 1, 2

subscript

i = q, p

subscript

y_ij

translational vibration displacement of two ends helical gear center point along the y-direction (Figure 3)

z_ij

translational vibration displacement of two ends helical gear center point along the z-direction (Figure 3)

θ_ij

angular vibration displacement of two ends helical gear center point (Figure 3)

R_ij

base circle radius

helical angle

meshing normal stiffness

ζ_v

relative meshing damping coefficient of double helical gear pair, we select ζ_v = 0.070

equivalent mass of double helical gear pair

average mesh stiffness of double helical gear pair

half backlash

k_j

torsional stiffness of two ends gear pair

c_j

torsional damping of two ends gear pair

k_yj

meshing stiffness along the tangential direction

c_yj

meshing damping along the tangential direction

k_zj

meshing stiffness along the axial direction

c_zj

meshing damping along the axial direction

e_yj

meshing error along the tangential direction

e_zj

meshing error along the axial direction

k_qx

torsional stiffness along X-direction of pinion shaft (Figure 3)

c_qx

torsional damping along X-direction of pinion shaft

k_px

torsional stiffness along X-direction of gear shaft

c_px

torsional damping along X-direction of gear shaft

k_qy

torsional stiffness along the Y-direction of pinion shaft (Figure 3)

c_qy

torsional damping along the Y-direction of pinion shaft

k_py

torsional stiffness along the Y-direction of gear shaft

c_py

torsional damping along the Y-direction of gear shaft

k_qz

tensile (compression) stiffness along the Z-direction of pinion shaft (Figure 3)

c_qz

tensile (compression) damping along the Z-direction of pinion shaft

k_pz

tensile (compression) stiffness along the Z-direction of gear shaft

c_pz

tensile (compression) damping along the Z-direction of gear shaft

z_ie

axial displacement under low speed calculated by the loaded tooth contact analysis

subscript

α_θ, α_z, α_y

relative vibration acceleration along the contact line, axial direction and transverse direction, respectively

f_θ, f_z, f_y

amplitude of relative vibration acceleration along the contact line, axial direction and transverse direction, respectively

torque acting on the gear

f, ϕ, ξ, ψ, Y, X,

Z, E, χ

corresponding identical dimensionless value

γ_i

phase separation of rotation angle between two ends gear pair

least common multiple of tooth number of driving gear and driven gear

Δz₁

axial displacement in a short period

Δz

axial displacement in a long period

k_d

meshing stiffness of single-pair teeth

impact force

manufacture error

rotation time of pinion

Appendix 1

Appendix 2

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