# A New Mathematical Modeling Method for Four-Stage Helicopter Main Gearbox and Dynamic Response Optimization

Authors: Yuan Chen; Rupeng Zhu; Guanghu Jin; Yeping Xiong; Jie Gao; Meijun Liao
Journal: Complexity
 (2019)

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## Abstract

A new mathematical modeling method, namely, the finite element method and the lumped mass method (LMM-FEM) mixed modeling, is applied to establish the overall multinode dynamic model of a four-stage helicopter main gearbox. The design of structural parameters of the shaft is the critical link in the four-stage gearbox; it affects the response of multiple input and output branches; however, only the meshing pairs were frequently shown in the dynamic model in previous research. Therefore, each shaft is also treated as a single node and the shaft parameters are coupled into the dynamic equations in this method, which is more accurate for the transmission chain. The differential equations of the system are solved by the Fourier series method, and the dynamic response of each meshing element is calculated. The sensitivity analysis method and parameter optimization method are applied to obtain the key shaft parameters corresponding to each meshing element. The results show that the magnitude of dynamic response in converging meshing pair and tail output pair is higher than that of other meshing pairs, and the wall thickness has great sensitivity to a rotor shaft. In addition, the sensitivity analysis method can be used to select the corresponding shaft node efficiently and choose parameters appropriately for reducing the system response.


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## Body

## 1. Introduction

The four-stage helicopter main gearbox has three input branches; they converge on the central gear and go one way to the rotor and the other way to the tail chain. The transmission chain includes many parts, which make its structure very complicated. The meshing pairs include internal excitations such as time-varying meshing stiffness and transmission error; in addition, external excitations like torques of the engine, rotor, and tail load also have an important effect on the system. The dynamic response has a profound influence on the fatigue life of the system; therefore, it is necessary to carry out the dynamic analysis for the four-stage helicopter gearbox.Regarding gearbox modeling, Kubur et al. [1] established the dynamic model of a multishaft helical gear reduction system formed by flexible shafts and studied the influence of some of the key system parameters under forced vibrations. Raclot and Velex [2] simulated the contributions of shape deviations and mounting errors to the dynamic behaviour of multistage geared systems. Choy et al. [3] presented a multistage multimesh gear transmission system; the individual modal component responses and the overall system dynamics of the gearbox were predicted. Dzitkowski and Dymarek [4] obtained the multistage gear mechanical characteristics by properly selecting the dynamical properties of the system based on using the active synthesis method. Chen et al. [5] established the vibration model of a four-stage main transmission system in a helicopter through the lumped mass method, and the influence of the torsional stiffness of shafts on the first five orders of the system’s natural frequency was studied. However, in this dynamic model, the shaft was not regarded as a node and all the differential equations are related to the meshing pairs.In the area of gear response analysis, Parker et al. [6] analyzed the dynamic response of a helicopter planetary gear system under different ranges of operating speeds and torques based on a finite element method and focused on the gear contact conditions. Velex and Flamand [7] calculated dynamic tooth loads and response on a planetary gear set by the Ritz method and compared the results with those given by direct integrations for highly reduced computation times. Chaari et al. [8] compared the dynamic response of healthy planetary gears with cracked planetary gears in both the time and frequency domains applied with the Wigner-Ville distribution method. Walha et al. [9] investigated dynamics of a two-stage gear system involving backlash and time-dependent mesh stiffness, and the decomposition of a nonlinear system into some linear systems was solved by the Newmark iterative algorithm. Zhou et al. [10] developed a coupled lateral-torsional nonlinear dynamic model with 16 degrees of freedom (16-DOF) of gear-rotor-bearing transmission system considering the nonlinear features, and the mean load excitation had a complicated influence on the coupled system; they concluded that the torsional vibration was the dominant response in the geared system. Chen et al. [11] detected the key shafts of the four-stage helicopter gearbox and analyzed their sensitivity to each branch of the system through the lumped mass method.Gearbox parameter optimization plays an important role in the helicopter design. Chen and Shao [12] demonstrated the effectiveness of the proposed mesh stiffness model under the influences of the tooth profile modification, not only applied in low contact ratio, but also in high contact ratio. Yang et al. [13] presented the gearbox parameter optimization method by the artificial bee colony algorithm when diagnosing the gear faults and verified the theory through a two-stage parallel shaft gearbox. Bozca [14–16] studied gearbox geometric parameter optimization to reduce rattle noise in an automotive transmission based on a torsional vibration model, and the module, number of teeth, axial clearance, and backlash could be improved through this method. However, system dynamic optimizations were mostly through gear parameters or profile modification; the shaft parameters were not regarded as variables in these researches.About the sensitivity analysis to system parameters, Lin and Parker [17] and Guo and Parker [18] investigated the natural frequency and vibration mode sensitivities to system parameters. Chen et al. [19] studied the response sensitivity to system parameters like gear mesh stiffness, damping, diameter ratio, and gear mass unbalance in a coupled gear system. However, sensitivity analysis based on multinode dynamic modeling has not been captured by these researches.In summary, the research of most scholars focuses on the planetary gear chain of the helicopter main gearbox. The dynamic equations also seldom reflect the characteristics of the shaft parameters as independent nodes; thus, it is meaningful to improve the dynamic modeling and propose the optimization method.


## 2. LMM-FEM Mixed Modeling

The dynamic modeling process of a typical four-stage helicopter main gearbox is shown in Figure1 [20]. Figure 1(a) is the system dynamic model, there are seven meshing elements (A, B, C, D, E, and F) in the system, and the elements F and G are internal and external meshing pairs in the planetary system, which contains one sun gear, six planet gears, and one carrier. θ is the rotational degree of freedom (DOF) of each node. Figure 1(b) is a comparison of the LMM modeling method and the LMM-FEM mixed modeling method. In this new method, the gearbox is regarded as finite critical nodes, which include a shaft and a meshing pair; each DOF corresponds to each node of the system. The system’s generalized DOF coordinate vector X is
(1)X=θ1j,θ2j,θ3j,θ4j,θ5j,θ6j,θ7j,θ8j,θ9,θ10,θ11,θ12,θ13,θ14θ15,θ16,θ17,θ18,θ19,θ20,θ21,θ22,θ23,θ24T.Figure 1
Dynamic modeling for the multinode system.
(a)
Dynamic model
(b)
Modeling comparison


### 2.1. Dynamic Modeling of the Meshing Pairs

For the seven meshing pairs (A, B, C, D, E, F, and G) highlighted in Figure1, the time-varying meshing stiffness can be expressed in the Fourier series with meshing frequency ω [21]:
(2)kt=km+kasinωt+β,kspit=km,spi+ka,spisinωt+βspi,krpit=km,rpi+ka,rpisinωt+βrpi,where kt, kspit, and krpit are time-varying meshing stiffness of each gear pair; km and ka are the average and maximum variable meshing stiffness; β is the initial phase of meshing stiffness; and ω is the fundamental meshing frequency.The transmission errors are shown in the same way:(3)et=em+easinωt+φ,espit=Aspisinωt+φspi+EpisinωpHt+φpi+α+EssinωsHt+φs−2πi−1N+α,erpit=Arpisinωt+φrpi+EpisinωpHt+φpi−α+ErsinωrHt+φr−2πi−1N−α,where et, espit, and erpit are time-varying transmission errors of each gear pair; em and ea are static and dynamic transmission error amplitude; φ is the initial phase of transmission error; ωpH, ωsH, and ωrH are rotational frequency of planet gear, sun gear, and carrier; α is the pressure angle; and N is the number of planet gear.Tooth deflection along the meshing line is defined as the dynamic response and presented as follows:(4)Xt=θdrd−θprp−et,where θd and θp are rotational DOF of drive gear and driven gear and rd and rp are the radii of the base circle in the drive gear and driven gear.The dynamic forces of each gear pairFt are defined as follows:
(5)Ft=Fpt+Fdt,Fpt=ktXt,Fdt=ctẊt,where ct is the meshing damping and Ẋt is the relative velocity along the meshing line.


### 2.2. Differential Equation of Multinode System

According to the dynamic modeling above, the differential equation of the multinode dynamic model can be deduced through Newton’s law:Node 1: shaft node(6)J1θ¨1j+G1πD1432l11−d4D4θ1j=TEjNodes 2–3: meshing pair A(7)J2θ¨2j+F2_3pjt+F2_3djtr1+G1πD1432l11−d4D4θ2j−θ1j=0,J3θ¨3j−F2_3pjt+F2_3djtr1+G4πD4432l41−d44D44θ3j−θ4j=0Node 4: shaft node(8)J4θ¨4j−G4πD4432l41−d44D44θ3j−θ4j+G4πD4432l41−d44D44θ4j−θ5j=0Nodes 5–6: meshing pair B(9)J5θ¨5j+F5_6pjt+F5_6djtr5+G4πD4432l41−d44D44θ5j−θ4j=0,J6θ¨6j−F5_6pjt+F5_6djtr6+G7πD7432l71−d74D74θ6j−θ7j=0Node 7: shaft node(10)J7θ¨7j−G7πD7432l71−d74D74θ6j−θ7j+G7πD7432l71−d74D74θ7j−θ8j=0Nodes 8–10: meshing pairs C and D(11)J8θ¨8j+F8_9pjt+F8_9djtr8+G7πD7432l71−d74D74θ8j−θ7j=0,J9θ¨9−∑j=13F8_9pjt+F8_9djtr9+G15πD15432l151−d154D154θ9−θ15+F9_10pt+F9_10dtr9=0,J10θ¨10−F9_10pt+F9_10dtr10+G11πD11432l111−d114D114θ10−θ11=0Node 11: shaft node(12)J11θ¨11+G11πD11432l111−d114D114θ11−θ12−G11πD11432l111−d114D114θ10−θ11=0Node 12: meshing pair E(13)J12θ¨12+F12_13pt+F12_13dtr12+G11πD11432l111−d114D114θ12−θ11=0,J13θ¨13−F12_13pt+F12_13dtr13+G14πD14432l141−d144D144θ13−θ14=0Node 14: shaft node(14)J14θ¨14+G14πD14432l141−d144D144θ14=TtNode 15: shaft node(15)J15θ¨15+G15πD15432l151−d154D154θ15−θ16−G15πD15432l151−d154D154θ9−θ15=0Nodes 16–23: meshing pairs F and G(16)J16θ¨16+∑i=16Fspipt+Fspidtr16+G15πD15432l151−d154D154θ16−θ15=0,Jpiθ¨pi−Fspipt+Fspidtrpi+Fcpipt+Fcpidtrpi=0,J23θ¨23−∑i=16Fcpipt+Fcpidtrpi+G24πD24432l241−d244D244θ23−θ24=0Node 24: shaft node(17)J24θ¨24+∑i=16Fcpipt+Fcpidtrpi+G24πD24432l241−d244D244θ24−θ23=−Trwhere TEj is denoted as the torque of engine j (j=1,2,3); D and d are the outer and inner diameters of the hollow shaft; G is the shear elastic modulus of the shaft; Tt is the output torque of the tail chain; and Tr is the output torque of the rotor shaft.The equations for each DOF could be written as the following matrix-vector form:(18)MX¨+CẊ+KX=F.The excitationF could be expanded to the Fourier series with the fundamental frequency as well; kth order excitation is
(19)Fk=A1ksinωkt+A2kcosωkt.The excitation causes the system to generate a response [22]:
(20)Δxk=B1ksinωkt+B2kcosωkt,where B1k and B2k could be solved by the following equation:
(21)−ωk2M+K¯−ωkCωkC−ωk2M+K¯B1kB2k=A1kA2k.The dynamic response is the linear superposition of the results corresponded by each order:(22)Δxt=∑k=15B1ksinωkt+B2kcosωkt.


## 2.1. Dynamic Modeling of the Meshing Pairs

For the seven meshing pairs (A, B, C, D, E, F, and G) highlighted in Figure1, the time-varying meshing stiffness can be expressed in the Fourier series with meshing frequency ω [21]:
(2)kt=km+kasinωt+β,kspit=km,spi+ka,spisinωt+βspi,krpit=km,rpi+ka,rpisinωt+βrpi,where kt, kspit, and krpit are time-varying meshing stiffness of each gear pair; km and ka are the average and maximum variable meshing stiffness; β is the initial phase of meshing stiffness; and ω is the fundamental meshing frequency.The transmission errors are shown in the same way:(3)et=em+easinωt+φ,espit=Aspisinωt+φspi+EpisinωpHt+φpi+α+EssinωsHt+φs−2πi−1N+α,erpit=Arpisinωt+φrpi+EpisinωpHt+φpi−α+ErsinωrHt+φr−2πi−1N−α,where et, espit, and erpit are time-varying transmission errors of each gear pair; em and ea are static and dynamic transmission error amplitude; φ is the initial phase of transmission error; ωpH, ωsH, and ωrH are rotational frequency of planet gear, sun gear, and carrier; α is the pressure angle; and N is the number of planet gear.Tooth deflection along the meshing line is defined as the dynamic response and presented as follows:(4)Xt=θdrd−θprp−et,where θd and θp are rotational DOF of drive gear and driven gear and rd and rp are the radii of the base circle in the drive gear and driven gear.The dynamic forces of each gear pairFt are defined as follows:
(5)Ft=Fpt+Fdt,Fpt=ktXt,Fdt=ctẊt,where ct is the meshing damping and Ẋt is the relative velocity along the meshing line.


## 2.2. Differential Equation of Multinode System

According to the dynamic modeling above, the differential equation of the multinode dynamic model can be deduced through Newton’s law:Node 1: shaft node(6)J1θ¨1j+G1πD1432l11−d4D4θ1j=TEjNodes 2–3: meshing pair A(7)J2θ¨2j+F2_3pjt+F2_3djtr1+G1πD1432l11−d4D4θ2j−θ1j=0,J3θ¨3j−F2_3pjt+F2_3djtr1+G4πD4432l41−d44D44θ3j−θ4j=0Node 4: shaft node(8)J4θ¨4j−G4πD4432l41−d44D44θ3j−θ4j+G4πD4432l41−d44D44θ4j−θ5j=0Nodes 5–6: meshing pair B(9)J5θ¨5j+F5_6pjt+F5_6djtr5+G4πD4432l41−d44D44θ5j−θ4j=0,J6θ¨6j−F5_6pjt+F5_6djtr6+G7πD7432l71−d74D74θ6j−θ7j=0Node 7: shaft node(10)J7θ¨7j−G7πD7432l71−d74D74θ6j−θ7j+G7πD7432l71−d74D74θ7j−θ8j=0Nodes 8–10: meshing pairs C and D(11)J8θ¨8j+F8_9pjt+F8_9djtr8+G7πD7432l71−d74D74θ8j−θ7j=0,J9θ¨9−∑j=13F8_9pjt+F8_9djtr9+G15πD15432l151−d154D154θ9−θ15+F9_10pt+F9_10dtr9=0,J10θ¨10−F9_10pt+F9_10dtr10+G11πD11432l111−d114D114θ10−θ11=0Node 11: shaft node(12)J11θ¨11+G11πD11432l111−d114D114θ11−θ12−G11πD11432l111−d114D114θ10−θ11=0Node 12: meshing pair E(13)J12θ¨12+F12_13pt+F12_13dtr12+G11πD11432l111−d114D114θ12−θ11=0,J13θ¨13−F12_13pt+F12_13dtr13+G14πD14432l141−d144D144θ13−θ14=0Node 14: shaft node(14)J14θ¨14+G14πD14432l141−d144D144θ14=TtNode 15: shaft node(15)J15θ¨15+G15πD15432l151−d154D154θ15−θ16−G15πD15432l151−d154D154θ9−θ15=0Nodes 16–23: meshing pairs F and G(16)J16θ¨16+∑i=16Fspipt+Fspidtr16+G15πD15432l151−d154D154θ16−θ15=0,Jpiθ¨pi−Fspipt+Fspidtrpi+Fcpipt+Fcpidtrpi=0,J23θ¨23−∑i=16Fcpipt+Fcpidtrpi+G24πD24432l241−d244D244θ23−θ24=0Node 24: shaft node(17)J24θ¨24+∑i=16Fcpipt+Fcpidtrpi+G24πD24432l241−d244D244θ24−θ23=−Trwhere TEj is denoted as the torque of engine j (j=1,2,3); D and d are the outer and inner diameters of the hollow shaft; G is the shear elastic modulus of the shaft; Tt is the output torque of the tail chain; and Tr is the output torque of the rotor shaft.The equations for each DOF could be written as the following matrix-vector form:(18)MX¨+CẊ+KX=F.The excitationF could be expanded to the Fourier series with the fundamental frequency as well; kth order excitation is
(19)Fk=A1ksinωkt+A2kcosωkt.The excitation causes the system to generate a response [22]:
(20)Δxk=B1ksinωkt+B2kcosωkt,where B1k and B2k could be solved by the following equation:
(21)−ωk2M+K¯−ωkCωkC−ωk2M+K¯B1kB2k=A1kA2k.The dynamic response is the linear superposition of the results corresponded by each order:(22)Δxt=∑k=15B1ksinωkt+B2kcosωkt.


## 3. Numerical Calculation

### 3.1. System Parameter and Response Calculation

The parameters of nodes are shown in Table1. In addition, the system is powered by three engines, the maximum output power of each engine is 1500 kW, and engine speed is 10,000 rpm. The output power of the rotor is 4000 kW, and the rotor speed is 300 rpm. The transmission mechanical efficiency is 95%.Table 1
Four-stage helicopter main gearbox parameters.
Element
Node
Tooth number
Module
Face width (m)
Transmission error (μm)
Initial phase of transmission error
Initial phase of meshing stiffness
ShaftD/d (m)
Shaft length (m)
Shaft element
1(i)
N/A
N/A
N/A
N/A
N/A
N/A
0.04/0.03
0.8
Meshing element A
2(i)
28
5
0.04
12;
π/3;
0; 0; 0
N/A
N/A
18;
π/9;
14
4π/3
3(i)
86
5
0.04
12;
π/3;
0; 0; 0
N/A
N/A
18;
π/9;
14
4π/3
Shaft element
4(i)
N/A
N/A
N/A
N/A
N/A
N/A
0.06/0.05
1.6
Meshing element B
5(i)
38
4.5
0.35
22;
−π/9;
π/3;
N/A
N/A
17;
−4π/3;
π/9;
15
0
π/3
6(i)
88
4.5
0.35
22;
−π/9;
π/3;
N/A
N/A
17;
−4π/3;
π/9;
15
0
−π/3
Shaft element
7(i)
N/A
N/A
N/A
N/A
N/A
N/A
0.06/0.04
2.2
Meshing elements C and D
8(i)
65
3
0.42
8;
−π;
0; 0; 0
N/A
N/A
12;
0;
10
−π/6
9
144
3
0.42
8;
−π;
0; 0; 0
N/A
N/A
12;
0;
10
−π/6
10
70
3
0.42
10
−π/6
0
N/A
N/A
Shaft element
11
N/A
N/A
N/A
N/A
N/A
N/A
0.04/0.03
1.4
Meshing element E
12
40
3.5
0.47
12
π/9
0
N/A
N/A
13
70
3.5
0.47
12
π/9
0
N/A
N/A
Shaft element
14
N/A
N/A
N/A
N/A
N/A
N/A
0.04/0.03
1.1
Shaft element
15
N/A
N/A
N/A
N/A
N/A
N/A
0.14/0.11
0.8
Meshing elements F and G
16
68
5
0.4
13; 8;
−4π/3; π/9;
π; 0;
N/A
N/A
6; 8;
−π/7; −π/9;
4π/7;
12; 5
π/3;-π/9
0; −π/3; 0
17
37
5
0.4
13; 8
−4π/3; −π
π; −π/3
N/A
N/A
18
37
5
0.4
8; 9
π/9; 0
0; 4π/3
N/A
N/A
19
37
5
0.4
6; 7
−π/7; 4π/3
4π/7; −π
N/A
N/A
20
37
5
0.4
8; 6
−π/9; −π/9
0; −4π/3
N/A
N/A
21
37
5
0.4
12; 6
π/3; π/9
−π/3; 0
N/A
N/A
22
37
5
0.4
5; 8
−π/9; −5π/3
0;π
N/A
N/A
23
142
5
0.4
−π; 0;
−π/3;
N/A
N/A
8; 9; 7
4π/3; −π/9;
4π/3;
6; 6; 8
π/9; −5π/3
−π; 4π/3;
0;π
Shaft element
24
N/A
N/A
N/A
N/A
N/A
N/A
0.12/0.09
2.0The calculation results can be seen from Figure2, and the dynamic response exhibits periodic vibrations under multifrequency excitation caused by time-varying meshing stiffness and transmission errors; thus, the response amplitudes are different. The gear rotational frequency reduces with the transmission chain following the output branch; in other words, the deceleration effect is obvious.Figure 2
Dynamic response regarding tooth deflection in each gear pair.
(a)
Meshing pair A
(b)
Meshing pair B
(c)
Meshing pair C
(d)
Meshing pair D
(e)
Meshing pair E
(f)
Meshing pair F
(g)
Meshing pair GBy comparison, it indicates that the dynamic response of meshing pair E has the maximum amplitude due to the heavy tail output load. In addition, the magnitude of the response in meshing pair C, especially pair C3, has a relatively higher response as well, as a result of the three-input branches converging at the central gear. Regarding the planetary gear train, the response of planetary gear 1 is greater than that of the other planetary gears due to its largest transmission and manufacturing error.


### 3.2. Sensitivity Analysis of Responses Influenced by Shaft Wall Thickness

The sensitivity analysis method is widely used to study the key variables under many uncertainties, which can provide a parametric reference for the design of the four-stage helicopter gearbox. In this paper, the ratio of the inner and outer diameters of the shaft is regarded as a variable, the sensitivity coefficient of each pair is calculated, and the key shafts affecting the response characteristics are investigated.The ratio of the inner and outer diameters of the shaft is defined asα; the sensitivity coefficient of the response xt to α is defined as follows:
(23)Sαx=xα′′−xα/xαα′−α/α×100%,where x and x′ are the response amplitudes corresponding to the ratio of α and α′.By changing the diameter ratioα, the sensitivity coefficient of each meshing pair is shown in Figure 3. According to Figure 3(a), the sensitivity coefficients of node 1i to meshing pairs (A, B, and C) are greater than 50%, and the coefficient is gradually increased with the transmission chain. Therefore, the diameter ratio of node 1 is the most sensitive to meshing pair C. In addition, node 1 is also about 28% sensitive to the response of the planetary gear system, and the other meshing pairs are insensitive to this node.Figure 3
Sensitivity coefficient of each meshing pair influenced byα.
(a)
Node 1
(b)
Node 4
(c)
Node 7
(d)
Node 11
(e)
Node 14
(f)
Node 15
(g)
Node 24According to Figure3(b), the sensitivity coefficient of node 4 and meshing pair A is the highest, which directly affects the dynamic response of the first-stage deceleration, and the sensitivity of the subsequent stages is gradually reduced. The sensitivity of node 7 in Figure 3(c) has the similar influence law as that of node 1, and the sensitivity coefficients are quite close as well. In Figure 3(d), node 11 has the highest sensitivity (14.5% and 38%) with its adjacent meshing pair D and meshing pair E. In Figure 3(e), node 14 is highly sensitive to the meshing pairs of the tail chain, which includes meshing pairs D and E. In Figure 3(f), the wall thickness of node 15, the input shaft of the sun gear, directly affects the response of the planetary gear train; thus, it is the most critical node of meshing pair F. Node 24 in Figure 3(g), the rotor shaft, has the sensitivity of 17% to meshing pair G, and the coefficient is much larger than that of other nodes.From these figures, the sensitivity correspondence and shaft selection method regarding wall thickness could be generalized in Table2, which could be a reference for reducing the vibration shock.Table 2
Sensitivity correspondence of wall thickness.
Meshing pair
Most sensitive node
A, B, C
Node 1 and node 7
D
Node 14
E
Node 11
F
Node 15
G
Node 24


### 3.3. Sensitivity Analysis of Responses Influenced by Shaft Length

The sensitivity coefficient of the responsext to the shaft length is defined as follows [23]:
(24)Slx=xl′′−xl/xll′−l/l×100%,where x and x′ are the response amplitudes corresponding to lengths l and l′.By changing the only variable parameter shaft lengthl, the sensitivity coefficient of each meshing pair is calculated in the same way, as is shown in Figure 4. From Figures 4(a)–4(c), it can be seen that the three meshing pairs of node 1, node 4, and node 7 have the highest sensitivity to three meshing pairs in the input branch, but the sensitivity coefficient of node 4 is relatively low. In Figures 4(d) and 4(e), node 11 and node 14 are similar in the influence law; they both have about 10% sensitivity coefficient to meshing pairs D and E. In Figure 4(f), the coefficient of node 15 to meshing pair F in the planetary gear system is 44%; with the coefficient depicted in Figure 4(f), it could be concluded that the wall thickness and shaft length of node 15 are critical parameters to internal meshing pair in the planetary gear train. In Figure 4(g), although node 24 is the most sensitive to the external meshing pair of the planetary gear system, the sensitivity coefficient is only 0.12%. In addition, the highest response sensitivity correspondence regarding shaft length is generalized in Table 3.Figure 4
Sensitivity coefficient of each meshing pair influenced by shaft length.
(a)
Node 1
(b)
Node 4
(c)
Node 7
(d)
Node 11
(e)
Node 14
(f)
Node 15
(g)
Node 24Table 3
Sensitivity correspondence of shaft length.
Meshing pair
Most sensitive node
A, B, C
Node 1
D, E
Node 11
F
Node 15
G
None


### 3.4. Structural Parameter Optimization Based on Response Sensitivity Analysis

By the sensitivity analysis of the meshing pairs based on the wall thickness and the shaft length, the most sensitive parameter has been investigated according to Tables2 and 3. From Figure 2, it could be noted that the maximum response amplitude of the four-stage system is located at meshing pair E, according to the sensitivity analysis; the corresponding node is node 11. The second largest response amplitude is situated at meshing pair C3, and the corresponding node is node 1.Therefore, a parameter optimization method is provided by changing the structural parameters of the shaft node without considering other errors and phase changes. It is initially set that the design range of the shaft diameter ratioα of node 1 is 0.6 to 0.9, and the range of the shaft length l is 0.6 to 1.4 m. After going through the parameters in the range, the selected diameter ratio and shaft length are shown in Table 4 as an improved group. Similarly, α11 is limited to 0.75~0.9, and l11 is 1.2~1.8 m; the original group and improved group are selected in the same way. Calculate the dynamic response of the original group and the improved group separately, as shown in Figure 5. From the figure and table, it can be concluded that the response of meshing pair C3 is reduced by 34.7% and that of meshing pair E is decreased by 36.1%, and the improvement effect is quite obvious.Table 4
Parameter optimization of the hollow shaft.
Parameter
Original group
Improved group
Optimization effect
α1
0.75
0.6
N/A
l1
0.8
0.6
N/A
α11
0.75
0.9
N/A
l11
1.4
1.8
N/A
XC3
0.0262
0.0171
−34.7%
XE
0.0343
0.0219
−36.1%Figure 5
Dynamic response optimization comparison regarding tooth deflection.


## 3.1. System Parameter and Response Calculation

The parameters of nodes are shown in Table1. In addition, the system is powered by three engines, the maximum output power of each engine is 1500 kW, and engine speed is 10,000 rpm. The output power of the rotor is 4000 kW, and the rotor speed is 300 rpm. The transmission mechanical efficiency is 95%.Table 1
Four-stage helicopter main gearbox parameters.
Element
Node
Tooth number
Module
Face width (m)
Transmission error (μm)
Initial phase of transmission error
Initial phase of meshing stiffness
ShaftD/d (m)
Shaft length (m)
Shaft element
1(i)
N/A
N/A
N/A
N/A
N/A
N/A
0.04/0.03
0.8
Meshing element A
2(i)
28
5
0.04
12;
π/3;
0; 0; 0
N/A
N/A
18;
π/9;
14
4π/3
3(i)
86
5
0.04
12;
π/3;
0; 0; 0
N/A
N/A
18;
π/9;
14
4π/3
Shaft element
4(i)
N/A
N/A
N/A
N/A
N/A
N/A
0.06/0.05
1.6
Meshing element B
5(i)
38
4.5
0.35
22;
−π/9;
π/3;
N/A
N/A
17;
−4π/3;
π/9;
15
0
π/3
6(i)
88
4.5
0.35
22;
−π/9;
π/3;
N/A
N/A
17;
−4π/3;
π/9;
15
0
−π/3
Shaft element
7(i)
N/A
N/A
N/A
N/A
N/A
N/A
0.06/0.04
2.2
Meshing elements C and D
8(i)
65
3
0.42
8;
−π;
0; 0; 0
N/A
N/A
12;
0;
10
−π/6
9
144
3
0.42
8;
−π;
0; 0; 0
N/A
N/A
12;
0;
10
−π/6
10
70
3
0.42
10
−π/6
0
N/A
N/A
Shaft element
11
N/A
N/A
N/A
N/A
N/A
N/A
0.04/0.03
1.4
Meshing element E
12
40
3.5
0.47
12
π/9
0
N/A
N/A
13
70
3.5
0.47
12
π/9
0
N/A
N/A
Shaft element
14
N/A
N/A
N/A
N/A
N/A
N/A
0.04/0.03
1.1
Shaft element
15
N/A
N/A
N/A
N/A
N/A
N/A
0.14/0.11
0.8
Meshing elements F and G
16
68
5
0.4
13; 8;
−4π/3; π/9;
π; 0;
N/A
N/A
6; 8;
−π/7; −π/9;
4π/7;
12; 5
π/3;-π/9
0; −π/3; 0
17
37
5
0.4
13; 8
−4π/3; −π
π; −π/3
N/A
N/A
18
37
5
0.4
8; 9
π/9; 0
0; 4π/3
N/A
N/A
19
37
5
0.4
6; 7
−π/7; 4π/3
4π/7; −π
N/A
N/A
20
37
5
0.4
8; 6
−π/9; −π/9
0; −4π/3
N/A
N/A
21
37
5
0.4
12; 6
π/3; π/9
−π/3; 0
N/A
N/A
22
37
5
0.4
5; 8
−π/9; −5π/3
0;π
N/A
N/A
23
142
5
0.4
−π; 0;
−π/3;
N/A
N/A
8; 9; 7
4π/3; −π/9;
4π/3;
6; 6; 8
π/9; −5π/3
−π; 4π/3;
0;π
Shaft element
24
N/A
N/A
N/A
N/A
N/A
N/A
0.12/0.09
2.0The calculation results can be seen from Figure2, and the dynamic response exhibits periodic vibrations under multifrequency excitation caused by time-varying meshing stiffness and transmission errors; thus, the response amplitudes are different. The gear rotational frequency reduces with the transmission chain following the output branch; in other words, the deceleration effect is obvious.Figure 2
Dynamic response regarding tooth deflection in each gear pair.
(a)
Meshing pair A
(b)
Meshing pair B
(c)
Meshing pair C
(d)
Meshing pair D
(e)
Meshing pair E
(f)
Meshing pair F
(g)
Meshing pair GBy comparison, it indicates that the dynamic response of meshing pair E has the maximum amplitude due to the heavy tail output load. In addition, the magnitude of the response in meshing pair C, especially pair C3, has a relatively higher response as well, as a result of the three-input branches converging at the central gear. Regarding the planetary gear train, the response of planetary gear 1 is greater than that of the other planetary gears due to its largest transmission and manufacturing error.


## 3.2. Sensitivity Analysis of Responses Influenced by Shaft Wall Thickness

The sensitivity analysis method is widely used to study the key variables under many uncertainties, which can provide a parametric reference for the design of the four-stage helicopter gearbox. In this paper, the ratio of the inner and outer diameters of the shaft is regarded as a variable, the sensitivity coefficient of each pair is calculated, and the key shafts affecting the response characteristics are investigated.The ratio of the inner and outer diameters of the shaft is defined asα; the sensitivity coefficient of the response xt to α is defined as follows:
(23)Sαx=xα′′−xα/xαα′−α/α×100%,where x and x′ are the response amplitudes corresponding to the ratio of α and α′.By changing the diameter ratioα, the sensitivity coefficient of each meshing pair is shown in Figure 3. According to Figure 3(a), the sensitivity coefficients of node 1i to meshing pairs (A, B, and C) are greater than 50%, and the coefficient is gradually increased with the transmission chain. Therefore, the diameter ratio of node 1 is the most sensitive to meshing pair C. In addition, node 1 is also about 28% sensitive to the response of the planetary gear system, and the other meshing pairs are insensitive to this node.Figure 3
Sensitivity coefficient of each meshing pair influenced byα.
(a)
Node 1
(b)
Node 4
(c)
Node 7
(d)
Node 11
(e)
Node 14
(f)
Node 15
(g)
Node 24According to Figure3(b), the sensitivity coefficient of node 4 and meshing pair A is the highest, which directly affects the dynamic response of the first-stage deceleration, and the sensitivity of the subsequent stages is gradually reduced. The sensitivity of node 7 in Figure 3(c) has the similar influence law as that of node 1, and the sensitivity coefficients are quite close as well. In Figure 3(d), node 11 has the highest sensitivity (14.5% and 38%) with its adjacent meshing pair D and meshing pair E. In Figure 3(e), node 14 is highly sensitive to the meshing pairs of the tail chain, which includes meshing pairs D and E. In Figure 3(f), the wall thickness of node 15, the input shaft of the sun gear, directly affects the response of the planetary gear train; thus, it is the most critical node of meshing pair F. Node 24 in Figure 3(g), the rotor shaft, has the sensitivity of 17% to meshing pair G, and the coefficient is much larger than that of other nodes.From these figures, the sensitivity correspondence and shaft selection method regarding wall thickness could be generalized in Table2, which could be a reference for reducing the vibration shock.Table 2
Sensitivity correspondence of wall thickness.
Meshing pair
Most sensitive node
A, B, C
Node 1 and node 7
D
Node 14
E
Node 11
F
Node 15
G
Node 24


## 3.3. Sensitivity Analysis of Responses Influenced by Shaft Length

The sensitivity coefficient of the responsext to the shaft length is defined as follows [23]:
(24)Slx=xl′′−xl/xll′−l/l×100%,where x and x′ are the response amplitudes corresponding to lengths l and l′.By changing the only variable parameter shaft lengthl, the sensitivity coefficient of each meshing pair is calculated in the same way, as is shown in Figure 4. From Figures 4(a)–4(c), it can be seen that the three meshing pairs of node 1, node 4, and node 7 have the highest sensitivity to three meshing pairs in the input branch, but the sensitivity coefficient of node 4 is relatively low. In Figures 4(d) and 4(e), node 11 and node 14 are similar in the influence law; they both have about 10% sensitivity coefficient to meshing pairs D and E. In Figure 4(f), the coefficient of node 15 to meshing pair F in the planetary gear system is 44%; with the coefficient depicted in Figure 4(f), it could be concluded that the wall thickness and shaft length of node 15 are critical parameters to internal meshing pair in the planetary gear train. In Figure 4(g), although node 24 is the most sensitive to the external meshing pair of the planetary gear system, the sensitivity coefficient is only 0.12%. In addition, the highest response sensitivity correspondence regarding shaft length is generalized in Table 3.Figure 4
Sensitivity coefficient of each meshing pair influenced by shaft length.
(a)
Node 1
(b)
Node 4
(c)
Node 7
(d)
Node 11
(e)
Node 14
(f)
Node 15
(g)
Node 24Table 3
Sensitivity correspondence of shaft length.
Meshing pair
Most sensitive node
A, B, C
Node 1
D, E
Node 11
F
Node 15
G
None


## 3.4. Structural Parameter Optimization Based on Response Sensitivity Analysis

By the sensitivity analysis of the meshing pairs based on the wall thickness and the shaft length, the most sensitive parameter has been investigated according to Tables2 and 3. From Figure 2, it could be noted that the maximum response amplitude of the four-stage system is located at meshing pair E, according to the sensitivity analysis; the corresponding node is node 11. The second largest response amplitude is situated at meshing pair C3, and the corresponding node is node 1.Therefore, a parameter optimization method is provided by changing the structural parameters of the shaft node without considering other errors and phase changes. It is initially set that the design range of the shaft diameter ratioα of node 1 is 0.6 to 0.9, and the range of the shaft length l is 0.6 to 1.4 m. After going through the parameters in the range, the selected diameter ratio and shaft length are shown in Table 4 as an improved group. Similarly, α11 is limited to 0.75~0.9, and l11 is 1.2~1.8 m; the original group and improved group are selected in the same way. Calculate the dynamic response of the original group and the improved group separately, as shown in Figure 5. From the figure and table, it can be concluded that the response of meshing pair C3 is reduced by 34.7% and that of meshing pair E is decreased by 36.1%, and the improvement effect is quite obvious.Table 4
Parameter optimization of the hollow shaft.
Parameter
Original group
Improved group
Optimization effect
α1
0.75
0.6
N/A
l1
0.8
0.6
N/A
α11
0.75
0.9
N/A
l11
1.4
1.8
N/A
XC3
0.0262
0.0171
−34.7%
XE
0.0343
0.0219
−36.1%Figure 5
Dynamic response optimization comparison regarding tooth deflection.


## 4. Conclusion

Based on the new dynamic modeling method and the response characteristics analysis, the overall multinode dynamic model of a four-stage helicopter main gearbox is established. By solving coupling differential equations, the sensitive laws of wall thickness and length of the hollow shaft are obtained. Therefore, the structural parameter optimization method is proposed to improve the original parameter group. The results enable us to draw the following conclusions:(1)
The magnitude of dynamic response in meshing pairs E and C3 is higher than that of other meshing pairs due to the load torque and converging influence
(2)
Each meshing pair has the most sensitive node according to the sensitivity analysis; parameter sensitivity correspondence tables are proposed for the main gearbox design
(3)
Regarding the rotor shaft (node 24), the wall thickness has great sensitivity to response; nevertheless, the influence of shaft length is negligible
(4)
In the design of the four-stage helicopter main gearbox, the shaft corresponding to node 1 should be relatively short and thick, and the shaft corresponding to node 11 should be comparatively long and thin, which can effectively reduce the response of the system
(5)
The sensitivity analysis method can be applied to quickly select the corresponding shaft node and its parameters for reducing the response, which has a good auxiliary effect on system modification. The improvement effect is obvious according to the group comparison



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*Source: 5274712-2019-02-26.xml*