Numerical Analysis and Demonstration: Transmission Shaft Influence on Meshing Vibration in Driving and Driven Gears

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Hindawi Publishing Corporation Shock and Vibration Volume 2015, Article ID 365084, 10 pages http://dx.doi.org/10.1155/2015/365084

Research Article Numerical Analysis and Demonstration: Transmission Shaft Influence on Meshing Vibration in Driving and Driven Gears

Xu Jinli, Su Xingyi, and Peng Bo

Wuhan University of Technology, Wuhan 430070, China

Correspondence should be addressed to Su Xingyi; [email protected]

Received 23 November 2014; Accepted 3 April 2015

Academic Editor: Wen Long Li

Copyright © 2015 Xu Jinli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The variable axial transmission system composed of universal joint transmission shafts and a gear pair has been applied inmany engineering fields. In the design of a drive system, the dynamics of the gear pair have been studied in detail. However fewhave paid attention to the effect on the system modal characteristics of the gear pair, arising from the universal joint transmission shafts. This work establishes a torsional vibration mathematical model of the transmission shaft, driving gear, and driven gear based on lumped masses and the main reducer system assembly of a CDV (car-based delivery vehicle) car. The model is solved by the state space method. The influence of the angle between transmission shafts and intermediate support stiffness on the vibration and noise of the main reducer is obtained and verified experimentally. A reference for the main reducer and transmission shaft design and the allied parameter matching are provided.

1. Introduction described as a “universal joint—transmission shaft—gear meshing system” (known as a variable axial transmission In the 1970s, to meet customer demand, the crossover was system). This kind of flexible transmission system can meet modified from cars in Europe and the United States. It the requirements of torque output floating3 [ , 4]. There is no stemmed from SUVs, gradually developed into an arbitrary accurate quantitative analysis on angle between the transmis- combination of car, SUV, MPV, and pick-up. It set the com- sion shafts according to the CDV car. In the situation of free- fort, fashion, and appearance standards for such cars; it had dom (car suspension), the initial installation angle changes the control of an SUV and the capacity of an MPV internally. ∘ ∘ from 0 to 10 . Without considering influence of car weight Crossovers are divided into SAV, CDV, vans, and other andloadontheinitialinstallationangle,anglebetweenthe models. In recent years the production and sales of crossovers have developed rapidly in China. The technology has also transmission shafts changes during driving, while few studies been greatly improved through absorption and digestion. have been involved in quantitative analysis on intermediate ∼ However the coupling technology between the transmission support stiffness. The stiffness value (600 N/mm 700 N/mm) shaftsonthechassisandgearsinthemainreducerhas is determined by analogy and experience analysis, especially been a hot research topic for the industry. There remains themutualcouplingeffectofinstallationanglebetween the effect on car NVH due to their coupling. Qualitative transmission shafts and intermediate support stiffness on research into the influence of transmission shaft angle and vibration of driving and driven gear is less involved currently. intermediate support stiffness on meshing vibration of gears This work focuses on the drive shaft—rear axle main remains sparse. reducer and gear [5, 6] system of the vehicle, creates a ACDVcarproducedinChinasuffersfromexcessive torsional vibration model, and solves its equations by the internal noise (60 db). The performance of an NVH does lumped mass method. Influences of transmission shaft angle not meet the level required of a passenger vehicle. The main and intermediate support stiffness on the meshing vibration reason for this is shock and vibration from its transmission of both driving and driven gears are studied [7, 8], and system [1, 2]. The transmission system of that car can be the installation angle and intermediate support stiffness are 2 Shock and Vibration

The output Universal The first Intermediate shaft of joint transmission support The second transmission shaft transmission shaft Main reducer

𝜔1 𝜔2 1 2 𝛼1 𝜃2 𝜔3 𝜃1 M2 M M1 3 𝜃3 3 𝛼2 𝜔4 4

M4 𝜃4

Figure 1: Universal transmission device structure investigated.

The input J J shaft e1 K K e2 1 0 󳰀 1 2 Je2

M 𝜃 1 1 M 𝜃 M 𝜃 2 2 K2 6 6 M3 𝜃3 Je3 K3 34 M4 𝜃4

M5 𝜃5

Figure 2: Torsional vibration analysis model of the transmission shaft, driving gear, and driven gear system. quantified. Vibration and shock from the variable axial the form of movement is not only translational but also transmission system are reduced, thereby improving the rotational. It is necessary to calculate only after simplification. vehicle’s performance. This study provides useful reference In the torsional vibration model, all components connected for the design of automobile chassis. with the rotating shaft are treated as absolutely rigid bodies with a moment of inertia. The shaft section moment of inertia equivalent is transferred to the two ends of the shaft. The rigid 2. The ‘‘Transmission Shaft—Driving Gear bodies, after equivalent treatment, are connected by a series and Driven Gear’’ Mathematical Model of equivalent bodies with elasticity and no moment of inertia; thus an equivalent analysis model of the vehicle’s variable The key factors affecting vibration and noise in the main axial transmission system is established. The structure of the reducer are the driving gear and driven gear’s meshing move- vehicle transmission shaft and the main reducer investigated ment [9, 10]. It is also one of the factors influencing vehicle hereisshowninFigure1. NVH. As an elastic mechanical system, both the driving gear The torsional vibration mathematical model of the trans- anddrivengearwillhavevibrationresponsestodynamic mission shaft, driving gear, and driven gear system is excitation during transmission time. The dynamic excitation established by considering the transmission system torsional ofthedrivinganddrivengearsmainlyincludestransmission vibrations: a simplified model is shown in Figure 2. shaft input excitation and road load excitation11 [ ]. As a result, The transmission shaft comprises three cross-shaft uni- when researching the problem of vibration and noise caused versal joints. Suppose that there is a model with a node in by the meshing of the driving gear and driven gear, it is each universal joint (nodes 1, 2, and 3). Torque transferred important to determine the main characteristics of both of fromtheenginebythegearboxactsonnode1.Thetwo their excitations. section transmission shafts are simplified as torsional springs and mass points. The intermediate support is simplified 2.1. Establishing a Torsional Vibration Model: “Transmission as a torsional spring of stiffness 𝐾0.Theoutputendof Shaft—Driving Gear and Driven Gear”. The “transmission the second transmission shaft is connected to the driving shaft—driving gear and driven gear” system is analyzed gear; because the driving gear shaft stiffness is large, the dynamically. The complex mechanical system is simplified influence of the driving gear shaft on torsional vibration of bythelumpedmassmethod.Thetorsionalvibrationmath- the system is ignored. Load torque is simplified and applied ematical analysis model for this elastic mechanical system to the main reducer’s driven gear. According to Lagrange’s is established by Lagrange’s equation: because the system equation and Newton’s second law, the corner is selected components are both numerous and complex in structure, as the origin: the generalised coordinates are expressed as Shock and Vibration 3

𝑞=(𝜃1,𝜃2,𝜃3,𝜃4,𝜃5,𝜃6). The system’s torsional vibration The relationship between the different transmission shaft model is torquesisasfollows: ̈ ̇ ̇ 𝐽𝑒1𝜃1 −𝐾1 (𝜃2 −𝜃1)−𝐶1 (𝜃2 − 𝜃1)=𝑀1, 𝑀𝑖𝜔𝑖 =𝑀𝑙𝜔𝑙. (5) ̈ ̇ ̇ 𝐽𝑒2𝜃2 +𝐾1 (𝜃2 −𝜃1)+𝐶1 (𝜃2 − 𝜃1)=−𝑀2, Uniting (1) to (5) allows the torsional vibration analysis 󸀠 ̈ ̇ ̇ modeltobenumericallysolved. 𝐽𝑒2𝜃3 −𝐾2 (𝜃4 −𝜃3)−𝐶2 (𝜃4 − 𝜃3)=𝑀3, (1) ̈ ̇ ̇ 𝐽𝑒3𝜃4 +𝐾2 (𝜃4 −𝜃3)+𝐶2 (𝜃4 − 𝜃3)=−𝑀4, 2.2. Numerical Solution of the System Torsional Vibration Model. To obtain the numerical solution, (2) is solved by a 𝐽 𝜃̈ +𝑇 =𝑀, driving 5 𝑛 5 combination of state space method and Runge-Kutta method. The state space method is analytical and is a comprehensive 𝐽 𝜃̈ −𝑇 =−𝑀, driven 6 𝑛 6 method for the dynamic characteristics of the system in modern control theory. The analysis process sets state space 𝐽 where 𝑒1 is the equivalent polar moment of inertia of the first variables which can identify and characterise the system. 𝐾 transmission shaft at node 1 of the universal joint; 1 is the Then the dynamic characteristics of the system are identified series equivalent stiffness of the first transmission shaft tube by the first-order differential equations composed of the state 𝐶 and its intermediate support; 1 is the damping coefficient space variables. The transfer function between system input 𝐽 of the first transmission shaft; 𝑒2 is the equivalent polar and output can be obtained by the state space equation. If moment of inertia of the first transmission shaft at node the input is known, then the value of the state variables 2 of the universal joint; is the equivalent polar moment of can determine the changing characteristics of the output inertia of the second transmission shaft at node 2 of the completely. 𝐾 universal joint; 2 is the series equivalent stiffness of the Set the output variable of system: second transmission shaft tube and its intermediate support; 𝐶2 is the damping coefficient of the second transmission 𝑌1 =𝑍2, shaft; 𝐽𝑒3 is the equivalent polar moment of inertia of the second transmission shaft at node 3 of the universal joint; 𝑌2 =𝑍4, 𝐽 is the equivalent polar moment of inertia of the main driving 𝑌3 =𝑍6, reducer driving gear; 𝐽 is the equivalent polar moment (6) driven 𝑌 =𝑍, of inertia of the main reducer driven gear; 𝑇𝑛 is the dynamic 4 8 𝑀 𝑀 meshing force of the driving gear and driven gear; 1, 6 are 𝑌5 =𝑍10, theengineoutputtorqueandtheequivalenttorqueloading 𝑌 =𝑍 . onthedrivengear. 6 12 Equation (1) is transferred to the torsional vibration equation of the system in matrix form: Select the state space variables:

{𝐽𝑒}⋅{𝑞}̈ + {𝐶𝑒}{𝑞}̇ + {𝐾𝑒}{𝑞}={𝑀 (𝑡)} , (2) 𝑍1 =𝜃1, {𝐽 } ̇ where 𝑒 is the equivalent polar inertia matrix of the system; 𝑍2 = 𝜃1, {𝐶 } {𝐾 } 𝑒 is the equivalent damping matrix; 𝑒 is the equivalent 𝑍 =𝜃, stiffness matrix; and {𝑀(𝑡)} isthetorquevectormatrixofthe 3 2 ̇ system. The dynamic meshing force between the driving gear 𝑍4 = 𝜃2, and the driven gear is as follows [12, 13]: 𝑍5 =𝜃3, ̇ ̇ 𝑇𝑛 =𝐾𝑚 ⋅(𝜃5 −𝜃6)+𝐶𝑚 ⋅(𝜃5 − 𝜃6), (3) ̇ 𝑍6 = 𝜃3, where 𝐾𝑚, 𝐶𝑚 aretheaveragemeshingstiffnessofreverseand 𝑍7 =𝜃4, (7) average meshing damping between gears. ̇ When the power transmission shaft is stable, the rotation 𝑍8 = 𝜃4, angle between transmission shaft axes [14] (Figures 1 and 2) 𝑍 =𝜃, is 9 5 𝑍 = 𝜃̇ , tan 𝜃2 = tan 𝜃3 ⋅ cos 𝛼1, 10 5 𝜃 = 𝜃 ⋅ 𝛼 , tan 4 tan 3 cos 2 𝑍11 =𝜃6, cos 𝛼1 𝜔 = 𝜔 , (4) ̇ 3 2 2 2 𝑍12 = 𝜃6. 1−sin 𝛼1 ⋅ cos 𝜃2 𝛼 𝜔 = cos 2 𝜔 . 4 1+ 2𝜃 ⋅( 2𝛼 −1) 3 Equation (2) is transformed into the state space expres- sin 3 cos 2 sion [15]: 4 Shock and Vibration

𝜃̇ ̇ 1 𝑍1 [ ] [ ] [𝜃̈ ] [ 𝑍̇ ] [ 1] [ 2 ] [ ̇ ] [ ̇ ] [𝜃2] [ 𝑍 ] [ ] [ 3 ] [𝜃̈ ] [ ̇ ] [ 2] [ 𝑍4 ] [ ] [ ] [𝜃̇ ] [ 𝑍̇ ] [ 3] [ 5 ] [ ] [ ] [𝜃̈ ] [ 𝑍̇ ] [ 3] = [ 6 ] [ ̇ ] [ ̇ ] [𝜃4] [ 𝑍7 ] [ ] [ ] [𝜃̈ ] [ 𝑍̇ ] [ 4] [ 8 ] [ ̇ ] [ ̇ ] [𝜃 ] [ 𝑍9 ] [ 5] [ ] [ ̈ ] [𝑍̇ ] [𝜃5] [ 10] [ ] [ ] [ ̇ ] 𝑍̇ [𝜃6] [ 11] ̈ 𝑍̇ [𝜃6] [ 12] 010000000000 [ ] [−𝑘1 −𝑐1 𝑘1 𝑐1 ] [ 00000000] [ 𝐽 𝐽 𝐽 𝐽 ] 𝑍 [ 𝑒1 𝑒1 𝑒1 𝑒1 ] 1 [ 000100000000] [ ] [ ] [ 𝑍 ] [ 𝑘 𝑐 −𝑘 −𝑐 ] [ 2 ] [ 1 1 1 1 00000000] [ 𝑍 ] [ ] [ 3 ] [ 𝐽𝑒2 𝐽𝑒2 𝐽𝑒2 𝐽𝑒2 ] [ ] [ 000001000000] [ 𝑍4 ] [ ] [ ] [ ] [ 𝑍 ] [ −𝑘2 −𝑐2 𝑘2 𝑐2 ] [ 5 ] [ 0000 0000] [ ] [ 𝐽󸀠 𝐽󸀠 𝐽󸀠 𝐽󸀠 ] [ 𝑍 ] = [ 𝑒2 𝑒2 𝑒2 𝑒2 ] ⋅ [ 6 ] [ 000000010000] [ 𝑍 ] [ ] [ 7 ] [ 𝑘 𝑐 −𝑘 −𝑐 ] [ ] [ 2 2 2 2 ] [ 𝑍8 ] (8) [ 0000 0000] [ ] [ 𝐽𝑒3 𝐽𝑒3 𝐽𝑒3 𝐽𝑒3 ] [ 𝑍 ] [ ] [ 9 ] [ 000000000100] [ ] [ ] [𝑍10] [ −𝑘𝑚 −𝑐𝑚 𝑘𝑚 𝑐𝑚 ] [ ] [ 00000000 ] [𝑍 ] [ 𝐽 𝐽 𝐽 𝐽 ] 11 [ driving driving driving driving ] [ 000000000001] [𝑍12] [ ] [ 𝑘 𝑐 −𝑘 −𝑐 ] 00000000 𝑚 𝑚 𝑚 𝑚 𝐽 𝐽 𝐽 𝐽 [ driven driven driven driven ] 0000 0 0 [ ] [ 1 ] [ 000 0 0] [𝐽 ] [ 𝑒1 ] [ 0000 0 0] [ ] [ 1 ] [ 0− 00 0 0] [ 𝐽 ] [ 𝑒2 ] 𝑀 [ 0000 0 0] 1 [ ] [ ] [ 1 ] [𝑀 ] [ 00 00 0] [ 2] [ 󸀠 ] [𝑀 ] [ 𝐽𝑒2 ] [ 3] + [ ] ⋅ [ ] . [ 0000 0 0] [𝑀 ] [ ] [ 4] [ 1 ] [𝑀 ] [ 000− 00] 5 [ 𝐽𝑒3 ] [ ] [𝑀6] [ 0000 0 0] [ ] [ 1 ] [ 0000 0 ] [ 𝐽 ] [ driving ] [ ] [ 0000 0 0] [ 1 ] 0000 0− 𝐽 [ driven ] Shock and Vibration 5

At the same time, the state space expression for the a single gear tooth’s linear meshing stiffness is calculated by torsional mathematical analysis model output variable is the ISO method. Results that have little difference to the obtained: actual meshing stiffness may be obtained by this method. The hypoid gears of Gleason are adopted in the main reducer of 𝑌1 𝑍2 [ ] [ ] the vehicle with a gear meshing stiffness with a linear value [𝑌 ] [ 𝑍 ] [ 2] [ 4 ] [17] [ ] [ ] [𝑌 ] [ 𝑍 ] [ 3] [ 6 ] 𝑘 = 1.84 × 106 / . [ ] = [ ] 𝑚𝑏 N m (13) [𝑌 ] [ 𝑍 ] [ 4] [ 8 ] [ ] [ ] The value of the gear’s equivalent meshing damping is [𝑌5] [𝑍10]

[𝑌6] [𝑍12] 1 1 𝐶𝑚 =2𝜉√𝑘𝑚𝑏 ( + ) N ⋅ s/m. (14) 𝑚1 𝑚2 𝑍1 [ ] [ 𝑍 ] 𝑚 𝑚 𝜉 [ 2 ] 1, 2 are the driven and driving gear’s qualities and is the [ ] [ 𝑍 ] damping ratio, with a value of 0.1 here. [ 3 ] [ ] (9) The average torsional stiffness of the gear is as follows: 010000000000 [ 𝑍 ] [ 4 ] [ ] [ ] 𝐾 (𝑟2 +𝑟2 ) [000100000000] [ 𝑍5 ] 𝑚𝑏 𝑏1 𝑏2 3 [ ] [ ] 𝐾𝑚 = =2.34×10 N ⋅ m/rad, (15) [ ] [ ] 2𝑟2 𝑟2 [000001000000] [ 𝑍6 ] 𝑏1 𝑏2 = [ ] ⋅ [ ] . [000000010000] [ 𝑍 ] [ ] [ 7 ] where 𝑟𝑏1 is the pitch circle radius of the driven gear and 𝑟𝑏2 [ ] [ ] [000000000100] [ 𝑍 ] is the pitch circle radius of the driving gear. [ ] [ 8 ] [ ] The equivalent polar moment of inertia of the first 000000000001 [ 𝑍 ] [ ] [ 9 ] transmission shaft at node 1 of the universal joint is [ ] [𝑍 ] [ 10] 𝐽 = 0.5 ⋅ (𝐽 +𝐽 )+𝐽 . [ ] 𝑒1 𝑠𝑧𝑒1 𝑐𝑑𝑒1 𝑧𝑐𝑒1 (16) [𝑍11] 𝐽𝑠𝑧𝑒1 is the first universal joint cross-shaft’s equivalent polar [𝑍12] moment of inertia about the first transmission axis; 𝐽𝑐𝑑𝑒1 The parameters of the mathematical model are substi- is the first transmission shaft’s equivalent polar moment of tuted into the state space expression. The torsional vibration inertia about its geometric centre; 𝐽𝑧𝑐𝑒1 is the first joint shaft response of the system can then be obtained by Runge-Kutta fork’sequivalentpolarmomentofinertiaaboutthefirst method. transmission axis. The 3d model of transmission shaft and main reducer is 𝐽 𝐽 𝐽 3. Determining Parameters in the Model imported into UG software. Values of 𝑠𝑧𝑒1, 𝑐𝑑𝑒1,and 𝑧𝑐𝑒1 are calculatedbyUG:thevaluesarelistedinTable3. 2 According to (16) 𝐽𝑒1 = 0.309234412 kg⋅m .Asthesame, 3.1. Determining Coefficients in the Model. The transmission 2 shaft’s main parameters for the car researched here are listed 𝐽𝑒2 =𝐽𝑒1 = 0.309234412 kg⋅m . in Table 1. The main parameters of both driving and driven Using the same method, the cross-shaft and shaft fork’s gears are listed in Table 2. moments of inertia can be calculated: these act from the The torsional stiffness of the transmission shaft is second universal joint and their values are listed in Table 4. 󸀠 2 󸀠 2 𝐽𝑒2 = 1.8665142935 kg⋅m , 𝐽𝑒3 = 2.324507441 kg⋅m , 󸀠 𝐺𝐼𝑃 𝐾 = N ⋅ m/rad. (10) 𝐽 = 1.236925720 ⋅ 2 𝐽 = 𝑖 𝐿 driving kg m ,anddriven 2 0.027998012 kg⋅m . 𝐺 7.94×1010 ⋅ −2 𝐿 is the shear modulus, N m ;and is the length The parameters for (1) canbefoundinTable5. of the shaft. The transmission shaft of the car is modelled as a thin- 3.2. Determination of Car Main Reducer External Excitation. walled circular cross-section: Here, considering fluctuations in main reducer input torque 3 −6 4 𝐼𝑝 ≈2𝜋𝑑 (𝐷−𝑑) = 4.86 × 10 m . (11) andloadtorque,theinputtorqueandloadtorquearesetas follows [18]: The intermediate support and transmission shaft total equivalenttorsionalstiffnessvaluesareasfollows[16]: 𝑀1 (𝑡) =𝑀1 +𝜎cos 𝜔𝑘𝑡, (17) 1 1 1 𝑀 (𝑡) =𝑀 +𝜎 𝜔 𝑡, = + . 6 6 cos 𝑘 󸀠 (12) 𝐾𝑖 𝐾𝑂 𝐾𝑖 where 𝑀1 is the input torque from the engine; 𝑀6 is the load There are many calculation methods available for the lin- torque; 𝜎 is the coefficient of torque fluctuation (its value is ear meshing stiffness of gears: as far as accuracy is concerned, 0.1); and 𝜔𝑘 is the angular velocity of the transmission axis. 6 Shock and Vibration

Table 1: Main parameters of the transmission shaft.

Shaft tube section Shaft tube section Shear modulus Length of the shaft Parameters outside diameter inside diameter 𝐺 (GPa) 𝐿 (mm) 𝐷 (mm) 𝑑 (mm) First shaft 79.4 524 63.5 59.9 Second shaft 79.4 875 63.5 59.9

Table 2: Main parameters of gears.

Name and code Driven gear Driving gear The number of teeth (𝑍)418 ∘ 󸀠 󸀠󸀠 ∘ ∘ 󸀠 󸀠󸀠 ∘ Surface cone angle (DELTA-𝐴)7655 40 ≈ 76.93 18 14 10 ≈ 18.24 ∘ 󸀠 󸀠󸀠 ∘ ∘ 󸀠 ∘ Reference cone angle (DELTA-𝐵)7555 50 ≈ 75.93 12 56 ≈ 12.93 ∘ 󸀠 󸀠󸀠 ∘ ∘ 󸀠 󸀠󸀠 ∘ Root angle (DELTA-𝐹)7016 47 ≈ 70.28 12 2 35 ≈ 12.04 Offset distance 30 mm (lower) ∘ 󸀠 ∘ The average pressure angle (ALPHA) 21 15 ≈ 21.25 Pitch circle diameter (𝐷) 170.601 mm Equivalent radius (𝑅) 73.18 mm 20.5 mm ∘ 󸀠 󸀠󸀠 ∘ ∘ 󸀠 󸀠󸀠 ∘ Mean spiral angle (BETA) 26 46 25 ≈ 26.77 49 59 48 ≈ 49.9967 Cutter diameter (BETA-𝑅)152.4mm The gear width (BF) 25 mm 33.3 mm Base circle diameter (Db) Db = 𝐷∗COS(ALPHA)

Table 3: Values of 𝐽𝑠𝑧𝑒1, 𝐽𝑐𝑑𝑒1,and𝐽𝑧𝑐𝑒1.

Moment of inertia 𝐽𝑠𝑧𝑒1 𝐽𝑐𝑑𝑒1 𝐽𝑧𝑐𝑒1 20000 About the 𝑥-axis 0.000538185 0.617267494 0.002713877 About the 𝑦-axis 0.000129689 0.001702246 0.000535828 About the 𝑧-axis 0.0005382 0.617267479 0.002981971 17000

󸀠 󸀠 󸀠 Table 4: Values of 𝐽 , 𝐽 ,and𝐽 . 𝑠𝑧𝑒2 𝑐𝑑𝑒2 𝑧𝑐𝑒2 14000 󸀠 󸀠 󸀠 Moment of inertia 𝐽𝑠𝑧𝑒2 𝐽𝑐𝑑𝑒2 𝐽𝑧𝑐𝑒2 About the 𝑥-axis 0.342704517 3.720747326 0.342704517 𝑦 11000

About the -axis 0.000541772 0.012791622 0.000541772 (deg) velocity angular gear Driving About the 𝑧-axis 0.342442366 3.731261792 0.342442366

8000 0 12345678910 Considering when the CDV car is running at an engine speed Time (s) of 3500 rpm, 𝑀1 = 151.20 N⋅m, the vibration and noise were Angle 12 deg Angle 6 deg large. The normal mass of the car is 1500 kg, and the wheel 10 5 𝑀 = 750 ⋅ Angle deg Angle deg effective radius is 290 mm, giving 6 N m; the speed Angle 8 deg Angle 3 deg of transmission shaft is 2500 rpm. Figure 3: Response curve: driving gear angular velocity. 4. Response Analysis of the System’s Torsional Vibration In Figure 3,theinstallationangle𝛼 is taking values ∘ ∘ ∘ ∘ ∘ ∘ ∘ 4.1. Effects of the Angle between the Axes on Vibration of 0 ,3,5,6,8,10,and12 in turn. The transmission in the Main Reducer. Having established the mathematical system consists of multiparts. During engine start-up phase, relationship between the angle of the transmission shafts and coupling effect between multiparts causes the change of the output variables of the state space using a MATLAB angular velocity fluctuations at different installation angles, simulation, the system response curves for angular velocity while angular velocity fluctuations are stable in a certain area and angular displacement are realised as shown in Figures 3 and periodic vibrating by a certain time. When installation ∘ and 4. angle is 12 , the unstable time of fluctuation is longest and Shock and Vibration 7

Table 5: Main parameters of transmission system.

Name Code Parameters Name Code Parameters The series equivalent The series equivalent stiffness of the first 7.36 × 105 ×𝐾 stiffness of the second 4.41 × 105 ×𝐾 𝐾 𝑂 ⋅ 𝐾 𝑂 ⋅ transmission shaft’s tube 1 7.36 × 105 +𝐾 N m/rad transmission shaft’s tube 2 4.41 × 105 +𝐾 N m/rad and its intermediate 𝑂 and its intermediate 𝑂 support support The damping coefficient of The damping coefficient of 𝐶1 0.002 the second transmission 𝐶2 0.002 the first transmission shaft shaft The equivalent polar The equivalent polar moment of inertia of the 2 moment of inertia of the 2 𝐽𝑒1 0.309234412 kg⋅m 𝐽𝑒2 0.309234412 kg⋅m first transmission shaft at first transmission shaft at node 1 of the universal joint node 2 of the universal joint The equivalent polar The equivalent polar moment of inertia of the moment of inertia of the 󸀠 2 2 second transmission shaft 𝐽𝑒2 1.866514293 kg⋅m second transmission shaft 𝐽𝑒3 2.324507441 kg⋅m at node 2 of the universal at node 3 of the universal joint joint The driving gear and driven The driving gear and driven 3 gear’s average meshing 𝐾𝑚 2.48 × 10 N⋅m/rad gear’s average meshing 𝐶𝑚 47.8 (N⋅s/m) stiffness in reverse damping The equivalent polar The equivalent polar 𝐽 ⋅ 2 𝐽 ⋅ 2 moment of inertia of the driving 1.236925720 kg m moment of inertia of the driven 0.027998012 kg m main reducer driving gear main reducer driven gear

0.8 1.5 0.7 0.6 1 0.5 0.4 0.3 0.5 0.2

Angular displacement (deg) displacement Angular 0.1 0 0.0 0∘ 1∘ 2∘ 3∘ 4∘ 5∘ 6∘ 7∘ 8∘ 9∘ 10∘

Angle between transmission shafts (deg) −0.5 Transmission error (deg) error Transmission Figure 4: Response curve: driven gear angular displacement.

−1

5∘∼6∘ stabilize fluctuation is largest. While angle is ,driving −1.5 gear can quickly enter smooth transmission and fluctuation 012345678 of angular velocity is minimal. Time (s) The changes in the driven gear’s angular displacement are 3 shown in Figure 4. With an analysis of response curve and Angle deg Angle 10 deg 5 12 variationcurve,itisseenthattheanglebetweentransmission Angle deg Angle deg Angle 8 deg shafts has an influence on vibration and noise in both gears. With the change in angle between the transmission shafts, the ∘ Figure 5: Transmission error: angular displacement—time curve. driven gear’s amplitude of vibration increases at first from 0 ∘ to 3 and then decreases to a series of small fluctuations of ∘ ∘ stable amplitude at installation angles of 3 to 6 . When the angle increases, the torsional vibration intensifies. the transmission error fluctuates cyclically. Different angles From an analysis of Figure 5, when the transmission between transmission shafts lead to different transmission shaft stiffness is the same as that of its intermediate support, accuracies. When the angle between the transmission shafts 8 Shock and Vibration

12 reducer and the NVH noise performance can reflect meshing vibration of the gears directly. To verify the validity of the study, intermediate support stiffness and installation angle 10 between transmission shafts are changed in a full-scale car test. Through changes in the aforementioned factors, NVH ) 2 performance status inside the car and vibration of the main 8 reducer would verify the validity of the study in actual operating conditions. The test systems comprised a portable vibration tester 6 developed by Wuhan University of Technology for the automobile main reducer and a portable LMS.Test.Lab NVH noise 4 tester from LMS Co., (Belgium). The portable vibration testing system has four acquisition Angular acceleration (deg/s acceleration Angular channels. One channel acquisition records the transmission 2 shaft speed signal; the other two channels record vibration acceleration signals in the vertical and horizontal directions fromthemainreducer.Therecordedsignalsaredisplayedas 0 plots of speed and the main reducer vibration acceleration 012345678910 onthecomputerscreeninreal-time.TheLMS.Test.LabNVH Time (s) noise tester can test noise inside the car in real-time and reflect the overall noise performance of the automobile. Support stiffness Support stiffness 1500 N/mm 800 N/mm The experiment does not change the box, car clutch, front Support stiffness Support stiffness andrearsuspensions,oranyotherfactors:theonlychanges 1000 N/mm 500 N/mm were made to the installation angle between transmission shafts and the intermediate support stiffness. The automotive Figure 6: Angular acceleration plots for the driven gear for different stiffness values. engine was run over its full working range of up to 6000 rpm. The experimental installation is shown in Figure 7 and vibration is tested by point-to-point measurement, which ∘ ∘ avoidstheinfluenceofothervibrations.Thenoisetester(LMS changes from 3 to 5 , the transmission error fluctuation is ∘ Co., Belgium) is used to verify the data of which the project small. While the angle is 5 , the transmission error is min- always recorded two sets to allow later comparison. imised; therefore it affords a better transfer of engine speed ∘ When the angle between transmission shafts is 7 ,the through the universal joint. As a result, rear axle vibration and intermediate support stiffness is approximately 700 N/mm, noise caused by speed fluctuation can be reduced. Based on and the experimental result is shown in Figure 8. the above analysis, the angle between transmission shafts not ∘ When the angle between transmission shafts is 5 ,the only influences the transmission characteristics of the shaft, intermediate support stiffness is approximately 500 N/mm, but also influences angular speed fluctuation and torsional and the experimental result is shown in Figure 9. vibration of the gear system. To improve transmission system ∘ To further verify the effects of a change in installation comfort, the design angle should be 5 . angle and intermediate support stiffness on the rear axle, the experiment also tests the NVH (noise) inside the vehicle. The 4.2. The Response Analysis of Intermediate Support Stiffness results are shown in Figure 10. on Main Reducer Vibration. In Figure 6,theinputtorque Comparing Figures 8 and 9, when the installation angle ∘ andloadtorquehavecosine-momentum.Withtheinterme- is 5 and the intermediate support stiffness is 500 N/mm, diate support’s stiffness changing, the angular acceleration the effect on the rear axle vibration is obviously smaller. of the driven gear will undergo cyclical fluctuations. In the Seen from Figure 10, when the transmission shaft angle simulation, the intermediate support is variable and is set to and intermediate support stiffness are changed, the NVH 500 N/mm, 800 N/mm, 1000 N/mm, and 1500 N/mm in turn, performance inside the car improved. Through experimental other settings remaining unchanged. Through comparative verification, data simulation, and analysis the data coincide analysis of the four plots in Figure 6, the driven gear’s with experimental verification thereof. angular acceleration fluctuations are different. When the intermediate support stiffness is 500 N/mm, the angular 6. Conclusion acceleration fluctuation is relatively stable. The effect on both gears’ meshing vibration is smaller. The research shows that the angle between transmission shafts has a great influence on vibration and shock in the 5. Experimental Verification rear axle main reducer. The numerical matching of the rela- tionships between angle, intermediate support stiffness, and The main factor causing vibration is meshing of the gears. gears can be found by numerical solution. The angle between The vibration of the main reducer in the rear bridge directly transmission shafts and the intermediate support stiffness affects NVH performance. Therefore, vibration of the main can then be quantified. According to the CDV car studied Shock and Vibration 9

The vertical direction

The axial direction

Photoelectric sensor

Figure 7: Experimental installation.

70 300

) 250 2 200 60 150 100

Vibrating (m/s Vibrating 50 50 0 0 1000 2000 3000 4000 5000 6000 Engine speed (r/min) Pa (db) Pa 40 Vertical vibration—speed curve (speed up) Vertical vibration—speed curve (speed down)

Figure 8: Experimental results. 30

) 300 20 2 250 1200 1600 2000 2400 2800 3200 3600 200 (rpm) 150 A 100 B

Vibrating (m/s Vibrating 50 55 db 0 0 1000 2000 3000 4000 5000 6000 Figure 10: Experimental results: noise test inside vehicle. (𝐴:noise ∘ Engine speed (r/min) effect; the angle of transmission shafts is7 and the intermediate support stiffness is approximately 700 N/mm; 𝐵:noiseeffect;the Vertical vibration—speed curve (speed up) ∘ angle of transmission shafts is 5 and the intermediate support Vertical vibration—speed curve (speed down) stiffness is approximately 500 N/mm). Figure 9: Experimental results.

Acknowledgment in this research, the installation angle between transmission ∘ It was partially supported by the WUT Innovation Fund shafts should be designed to be 5 as far as possible and 145204003. the intermediate support stiffness should be 500 N/mm as predicted by theory and verified experimentally. As a result, the transmission shaft’s effect on vibration and noise from the References main reducer can be decreased. There is no other problem [1] B. Gao, H. Chen, Y. Ma, and K. Sanada, “Design of nonlinear caused after long-term test. The method applied inthis shaft torque observer for trucks with automated manual trans- research has reference value for vibration and noise reduction mission,” Mechatronics, vol. 21, no. 6, pp. 1034–1042, 2011. in variable axial transmission systems. [2] T. P. Turkstra and S. E. Semercigil, “Vibration control for a flexible transmission shaft with an axially sliding support,” Conflict of Interests Journal of Sound and Vibration,vol.206,no.4,pp.605–610,1997. [3] C. L. Yu, L. Zhang, J. G. Wu, and Z. J. Xie, “Modeling and The authors declare that there is no conflict of interests simulation of a mini-bus with dynamics of multi-body systems,” regarding the publication of this paper. Journal of Chongqing University,vol.26,pp.35–38,2003. 10 Shock and Vibration

[4] M. Gnanakumarr, S. Theodossiades, H. Rahnejat, and M. Men- day, “Impact-induced vibration in vehicular driveline systems: theoretical and experimental investigations,” Proceedings of the InstitutionofMechanicalEngineersPartK:JournalofMulti-Body Dynamics,vol.219,pp.1–12,2005. [5] J. L. Xu, “Analysis of contact stress of hypoid gear of micro-car real rear axle,”in Proceedings of the 2nd International Conference on Electronic & Mechanical Engineering and Information Tech- nology (EMEIT '12),vol.9,pp.949–954,2012. [6] K. Liu, Research of the Soft Body Dynamics of the Gears in Main Reducer of Car Drive Axle, Wuhan University of Technology, Wuhan, China, 2012. [7] J. Baumann, D. D. Torkzadeh, A. Ramstein, U. Kiencke, and T. Schlegl, “Model-based predictive anti-jerk control,” Control Engineering Practice, vol. 14, no. 3, pp. 259–266, 2006. [8] L. Webersinke, L. Augenstein, and U. Kiencke, “Adaptive linear quadratic control for high dynamical and comfortable behavior of a heavy truck,” SAE Technical Paper 0534, 2008. [9] A. Artoni, M. Gabiccini, and M. Kolivand, “Ease-off based compensation of tooth surface deviations for spiral bevel and hypoid gears: only the pinion needs corrections,” Mechanism and Machine Theory, vol. 61, pp. 84–101, 2013. [10]D.Park,M.Kolivand,andA.Kahraman,“Predictionofsurface wear of hypoid gears using a semi-analytical contact model,” Mechanism and Machine Theory,vol.52,pp.180–194,2012. [11] Q. Wang and Y.-D. Zhang, “Coupled analysis based dynamic response of two-stage helical gear transmission system,” Journal of Vibration and Shock,vol.31,no.10,pp.87–91,2012. [ 1 2 ] W. W. Z h e n g , Mechanical Principle,HigherEducationPress, Beijing, China, 1997. [13] Z. Y. Zhu, Theoretical Mechanics (I), Peking University Press, Beijing, China, 1984. [14] Q. Li and J. J. Zhang, “Analysis and arrangement of mine car driveline system,” Mechanical Research & Application,vol.4,pp. 56–61, 2010. [15] L. H. Wang, Y. Y. Huang, and Y. F. Li, “Study on nonlinear vibration characteristics of spiral bevel transmission system,” China Mechanical Engineering, vol. 18, no. 3, pp. 260–264, 2007. [16] L. Liu, Finite Element Analysis of Car Propeller Shaft Supporting Bracket, vol. 1, Drive Systems Technology, Shanghai, China, 2007. [17]O.D.Mohammed,M.Rantatalo,andJ.-O.Aidanpa¨a,¨ “Improv- ing mesh stiffness calculation of cracked gears for the purpose of vibration-based fault analysis,” Engineering Failure Analysis, vol. 34, pp. 235–251, 2013. [18] M. Pang, X.-J. Zhou, and C.-L. Yang, “Simulation of nonlinear vibrationofautomobilemainreducer,”Journal of Zhejiang University (Engineering Science),vol.43,no.3,pp.559–564, 2009. International Journal of

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