The variable axial transmission system composed of universal joint transmission shafts and a gear pair has been applied in many engineering fields. In the design of a drive system, the dynamics of the gear pair have been studied in detail. However few have 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 (carbased 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.
In the 1970s, to meet customer demand, the crossover was modified from cars in Europe and the United States. It stemmed from SUVs, gradually developed into an arbitrary combination of car, SUV, MPV, and pickup. It set the comfort, fashion, and appearance standards for such cars; it had the control of an SUV and the capacity of an MPV internally. Crossovers are divided into SAV, CDV, vans, and other models. In recent years the production and sales of crossovers have developed rapidly in China. The technology has also been greatly improved through absorption and digestion. However the coupling technology between the transmission shafts on the chassis and gears in the main reducer has been a hot research topic for the industry. There remains the effect on car NVH due to their coupling. Qualitative research into the influence of transmission shaft angle and intermediate support stiffness on meshing vibration of gears remains sparse.
A CDV car produced in China suffers from excessive internal noise (60 db). The performance of an NVH does not meet the level required of a passenger vehicle. The main reason for this is shock and vibration from its transmission system [
This work focuses on the drive shaft—rear axle main reducer and gear [
The key factors affecting vibration and noise in the main reducer are the driving gear and driven gear’s meshing movement [
The “transmission shaft—driving gear and driven gear” system is analyzed dynamically. The complex mechanical system is simplified by the lumped mass method. The torsional vibration mathematical analysis model for this elastic mechanical system is established by Lagrange’s equation: because the system components are both numerous and complex in structure, the form of movement is not only translational but also rotational. It is necessary to calculate only after simplification. In the torsional vibration model, all components connected 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 bodies, after equivalent treatment, are connected by a series of equivalent bodies with elasticity and no moment of inertia; thus an equivalent analysis model of the vehicle’s variable axial transmission system is established. The structure of the vehicle transmission shaft and the main reducer investigated here is shown in Figure
Universal transmission device structure investigated.
The torsional vibration mathematical model of the transmission shaft, driving gear, and driven gear system is established by considering the transmission system torsional vibrations: a simplified model is shown in Figure
Torsional vibration analysis model of the transmission shaft, driving gear, and driven gear system.
The transmission shaft comprises three crossshaft universal joints. Suppose that there is a model with a node in each universal joint (nodes 1, 2, and 3). Torque transferred from the engine by the gearbox acts on node 1. The two section transmission shafts are simplified as torsional springs and mass points. The intermediate support is simplified as a torsional spring of stiffness
Equation (
When the power transmission shaft is stable, the rotation angle between transmission shaft axes [
The relationship between the different transmission shaft torques is as follows:
Uniting (
To obtain the numerical solution, (
Set the output variable of system:
Select the state space variables:
Equation (
At the same time, the state space expression for the torsional mathematical analysis model output variable is obtained:
The parameters of the mathematical model are substituted into the state space expression. The torsional vibration response of the system can then be obtained by RungeKutta method.
The transmission shaft’s main parameters for the car researched here are listed in Table
Main parameters of the transmission shaft.
Parameters  Shear modulus 
Length of the shaft 
Shaft tube section outside diameter 
Shaft tube section inside diameter 

First shaft  79.4  524  63.5  59.9 
Second shaft  79.4  875  63.5  59.9 
Main parameters of gears.
Name and code  Driven gear  Driving gear 

The number of teeth ( 
41  8 
Surface cone angle (DELTA 
76° 55′ 40′′ ≈ 76.93°  18° 14′ 10′′ ≈ 18.24° 
Reference cone angle (DELTA 
75° 55′ 50′′ ≈ 75.93°  12° 56′ ≈ 12.93° 
Root angle (DELTA 
70° 16′ 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 ( 
170.601 mm  
Equivalent radius ( 
73.18 mm  20.5 mm 
Mean spiral angle (BETA)  26° 46′ 25′′ ≈ 26.77°  49° 59′ 48′′ ≈ 49.9967° 
Cutter diameter (BETA 
152.4 mm  
The gear width (BF)  25 mm  33.3 mm 
Base circle diameter (Db) 
Db = 
The torsional stiffness of the transmission shaft is
The transmission shaft of the car is modelled as a thinwalled circular crosssection:
The intermediate support and transmission shaft total equivalent torsional stiffness values are as follows [
There are many calculation methods available for the linear meshing stiffness of gears: as far as accuracy is concerned, a single gear tooth’s linear meshing stiffness is calculated by the ISO method. Results that have little difference to the actual meshing stiffness may be obtained by this method. The hypoid gears of Gleason are adopted in the main reducer of the vehicle with a gear meshing stiffness with a linear value [
The value of the gear’s equivalent meshing damping is
The average torsional stiffness of the gear is as follows:
The equivalent polar moment of inertia of the first transmission shaft at node 1 of the universal joint is
The 3d model of transmission shaft and main reducer is imported into UG software. Values of
Values of
Moment of inertia 




About the 
0.000538185  0.617267494  0.002713877 
About the 
0.000129689  0.001702246  0.000535828 
About the 
0.0005382  0.617267479  0.002981971 
According to (
Using the same method, the crossshaft and shaft fork’s moments of inertia can be calculated: these act from the second universal joint and their values are listed in Table
Values of
Moment of inertia 




About the 
0.342704517  3.720747326  0.342704517 
About the 
0.000541772  0.012791622  0.000541772 
About the 
0.342442366  3.731261792  0.342442366 
The parameters for (
Main parameters of transmission system.
Name  Code  Parameters  Name  Code  Parameters 

The series equivalent stiffness of the first transmission shaft’s tube and its intermediate support 


The series equivalent stiffness of the second transmission shaft’s tube and its intermediate support 




The damping coefficient of the first transmission shaft 

0.002  The damping coefficient of the second transmission shaft 

0.002 


The equivalent polar moment of inertia of the first transmission shaft at node 1 of the universal joint 

0.309234412 kg·m^{2}  The equivalent polar moment of inertia of the first transmission shaft at node 2 of the universal joint 

0.309234412 kg·m^{2} 


The equivalent polar moment of inertia of the second transmission shaft at node 2 of the universal joint 

1.866514293 kg·m^{2}  The equivalent polar moment of inertia of the second transmission shaft at node 3 of the universal joint 

2.324507441 kg·m^{2} 


The driving gear and driven gear’s average meshing stiffness in reverse 

2.48 × 10³ N·m/rad  The driving gear and driven gear’s average meshing damping 

47.8 (N·s/m) 


The equivalent polar moment of inertia of the main reducer driving gear 

1.236925720 kg·m^{2}  The equivalent polar moment of inertia of the main reducer driven gear 

0.027998012 kg·m^{2} 
Here, considering fluctuations in main reducer input torque and load torque, the input torque and load torque are set as follows [
Having established the mathematical relationship between the angle of the transmission shafts and the output variables of the state space using a MATLAB simulation, the system response curves for angular velocity and angular displacement are realised as shown in Figures
Response curve: driving gear angular velocity.
Response curve: driven gear angular displacement.
In Figure
The changes in the driven gear’s angular displacement are shown in Figure
From an analysis of Figure
Transmission error: angular displacement—time curve.
In Figure
Angular acceleration plots for the driven gear for different stiffness values.
The main factor causing vibration is meshing of the gears. The vibration of the main reducer in the rear bridge directly affects NVH performance. Therefore, vibration of the main 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 between transmission shafts are changed in a fullscale car test. Through changes in the aforementioned factors, NVH performance status inside the car and vibration of the main reducer would verify the validity of the study in actual operating conditions.
The test systems comprised a portable vibration tester developed by Wuhan University of Technology for the automobile main reducer and a portable LMS.Test.Lab NVH noise tester from LMS Co., (Belgium).
The portable vibration testing system has four acquisition channels. One channel acquisition records the transmission shaft speed signal; the other two channels record vibration acceleration signals in the vertical and horizontal directions from the main reducer. The recorded signals are displayed as plots of speed and the main reducer vibration acceleration on the computer screen in realtime. The LMS.Test.Lab NVH noise tester can test noise inside the car in realtime and reflect the overall noise performance of the automobile.
The experiment does not change the box, car clutch, front and rear suspensions, or any other factors: the only changes were made to the installation angle between transmission shafts and the intermediate support stiffness. The automotive engine was run over its full working range of up to 6000 rpm. The experimental installation is shown in Figure
Experimental installation.
When the angle between transmission shafts is 7°, the intermediate support stiffness is approximately 700 N/mm, and the experimental result is shown in Figure
Experimental results.
When the angle between transmission shafts is 5°, the intermediate support stiffness is approximately 500 N/mm, and the experimental result is shown in Figure
Experimental results.
To further verify the effects of a change in installation angle and intermediate support stiffness on the rear axle, the experiment also tests the NVH (noise) inside the vehicle. The results are shown in Figure
Experimental results: noise test inside vehicle. (
Comparing Figures
The research shows that the angle between transmission shafts has a great influence on vibration and shock in the rear axle main reducer. The numerical matching of the relationships between angle, intermediate support stiffness, and gears can be found by numerical solution. The angle between transmission shafts and the intermediate support stiffness can then be quantified. According to the CDV car studied in this research, the installation angle between transmission shafts should be designed to be 5° as far as possible and 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 main reducer can be decreased. There is no other problem caused after longterm test. The method applied in this research has reference value for vibration and noise reduction in variable axial transmission systems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
It was partially supported by the WUT Innovation Fund 145204003.