(2) Modelling and Analysis of Automatic Transmission

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MODELLING AND ANALYSIS OF A MODERN AUTOMATIC TRANSMISSION GEARBOX Y Song, A Tylee-Birdsall, E Roeloffzen Romax Technology Ltd, Rutherford House, Highfield Science Park, Nottingham NG7 2PZ, UK

SUMMARY A new approach for 3D modelling and analysis of a modern automatic transmission system is presented in this paper. The gear pairs, planetary gear system, bearings, shafts and clutches are represented as analysis objects and the planetary carriers and housing are represented as stiffness quality finite element components. They are all combined together into one whole hyperstatic system in RomaxDesigner. The complex planetary gear carriers and gearbox housing are modelled by means of reduced stiffness matrices created in a finite element analysis package Nastran. A 6-speed automatic gearbox was modelled in RomaxDesigner based on above method and the full operating duty cycle of the transmission has been input and used for the analysis. This method allows to dramatically reduce the size of the model and can thus be carried out to a time scale that enables quick concept designs as well as responsive problem solving in existing gearboxes. 1: CURRENT ISSUES OF MODELLING TRANSMISSION SYSTEM An automatic transmission gearbox is a very complex mechanical system which includes planetary gear train, shafts, bearings, clutches, shifting mechanism and gearbox housing [1,2]. All these components are connecting with each other through gear meshing, bearing mounting and other connections. The different gears are selected by switching on or off a combination of different clutches on the planetary gear trains. The housing is normally mounted on the engine or chassis or both. All those components are interacting with each other and hence when changing certain parameters in any component will inevitably affect other components. In the process of transmission design, it is important to optimise it considering the changes in these parameters but at the same time to maintain or not to compromise the performance of the transmission’s internal components. It is difficult to achieve


those targets using conventional transmission analysis methods in which the gears and the housing are analysed separately and the influence of housing flexibility on gear mesh misalignment is obtained by making a series of approximations. These approximations, e.g. linear bearings, rigid housings, result in a considerable reduction in accuracy. An alternative would be to use a full finite element (FE) model of the whole transmission system. However, even the most sophisticated FE model, which requires great expertise and involves immense time and solving cost, there is no guarantee that the results can be relied upon [3,4]. For a full model to be realistic, it requires extensive model correlation for all components. At present it is a requirement to correlate individual transmission component in isolation with experimented modal analysis techniques. This has several disadvantages, for example, a prototype is required for each component, and when the correlated components are connected, assumptions are made about their boundary conditions. This can result in a model which is not analogous to reality and would require a large amount of ‘model correlation’ to yield acceptable results. To simulate the loaded FE contact analysis of a set of gears using surface-tosurface contact elements in current FE technology will require a great deal of effort. In order to achieve an acceptable accuracy, the elements used in defining the tooth surface should be small enough to capture the gear meshing characteristics. Thus it requires a considerable size model and a great deal of computer resource to solve the nonlinear tooth contact analysis. Imagine half a dozen of gear pairs in simultaneous meshing in a modern automatic transmission, the modelling effort and the solution expense are really a big challenge, not to mention the complex gearbox housing, different types of nonlinear bearings and clutch components. In this paper, a new approach to 3D modelling and analysis of a modern automatic gearbox using specialised transmission software RomaxDesigner, a commercial software developed by Romax Technology Ltd, is presented. RomaxDesigner allows modelling and analysis of a complete automatic transmission in which gears, planetary gear system, bearings and shafts are modelled as analysis objects with correlation to validated tests. RomaxDesigner can calculate all the gear meshing points, forces, load distribution and take into account all the boundary connections. The planetary carriers and housing are meshed in a commercial finite element package Visual/Nastran. The housing finite element model is condensed to a reduced stiffness matrix and coupled with the internal transmission through the bearing nodes. The transmission system model built by this approach is very compact compared to conventional finite element models. Since the gears, shafts,


bearings and clutches are all defined as objects in RomaxDesigner, it is much easier and faster to obtain these 3D components by just keying in the design parameters and editing their attributes. The modelling time for those components and assemblies are significantly reduced and any possible mistakes, which may inevitably happen in a conventional modelling process, can be avoided. The above modelling method is illustrated by a case study. A 6-speed modern automatic transmission is modelled in RomaxDesigner (See Figure 1) and a full duty cycle in a series of different gear selections is analysed. Figure 2 shows a graphical representation of the gearbox system deflection under load. The case study was modelled to determine the bearing and gear life in a modern automatic gearbox.

Figure 1:

Figure 2:

A modern 6-speed automatic transmission

Shaft deflections of the automatic transmission


2: MODELLING THE SHAFTS The gear shafts are modelled as beam elements; the outer diameter and bore of the shaft and its position are the only input parameters required for the modelling. The loading is applied to the shaft by specifying its position, direction and magnitude. The gears and bearings and other components are mounted on the shaft in the shaft assembly window. The connection and interaction of the shaft with other shafts is realised through the meshing of the mounted gears and through interconnecting bearings. The positioning of the shaft in a gearbox is done by editing its properties. The shaft can be rotated in RomaxDesigner relative to a global co-ordinate system and can be repositioned in a later stage if necessary. Figure 3 shows the input shaft modelled in RomaxDesigner.

Figure 3:

Input shaft modelled in RomaxDesigner

3: MODELLING THE BEARINGS In order to achieve a high accuracy in the prediction of shaft deflection, the classic “simply supported beam analysis” is not used here. Instead, a non-linear stiffness of the bearing is calculated based on standard methods [6] which take into account the applied load, internal geometry and internal clearance as well as the reaction of the internal geometry to the applied load with respect to the actual contact angle of the elements. The 6x6 stiffness matrix for each bearing is obtained by linearizing the nonlinear behaviour of the bearing close to the operation condition [3,7]. Figure 4 shows an example of bearings modelled in RomaxDesigner.


Figure 4:

3D bearings model and its mounting on the shaft

4: MODELLING HELICAL GEAR SETS Spur and helical gears are treated as objects in RomaxDesigner. The modelling of these gears as part of a gear train is done in two stages: stage 1 concept design and stage 2 detailed design. Stage 1: In the concept design stage, only a few essential inputs are required such as tooth number, module or diametral pitch, centre distance, pressure angle and face width in which the centre distance is determined by the position of the gears. The gears are either defined as gear pair or a gear train. The meshing of gears in a gear train can be specified interactively. Stage 2: In the second stage, the concept design is converted into a detailed design based on a standard rack or a user-modified rack. The modifications of gear tooth profiles like tooth thickness, tooth tip and edge chamfers and root fillet radius are easily manipulated in a editing window. In the case study, the required modifications are input in the detailed gear design window from which the 3D gear was modelled. Figure 5 shows a helical gear and a helical gear train created in RomaxDesigner. The gears are defined with stiffness under contact and a backlash that is derived from the assembled and deflected conditions. The behaviour of the gear mesh is nonlinear.


Figure 5:

Helical gear and helical gear train modelled in RomaxDesigner

5: MODELLING THE PLANETARY GEARS AND CARRIER A simple planetary gear train normally consists of one sun gear, one ring gear, and some arbitrary number of planets. The gear ratio changes are controlled by activating a set of clutches. Depending on the clutch selection, RomaxDesigner automatically calculates the planetary gear train’s ratios. The modelling of a planetary gear train is similar to that of a helical gear train except that the meshing among the planets, sun gear and ring gear has to be set more carefully. Figure 6 shows a planetary gear train in the automatic transmission model from the case study.

Figure 6:

A planetary gear train modelled in RomaxDesigner


The carrier is an irregular 3D component (see Figure 7(c)). In the case study, an advanced method, i.e. a reduced stiffness matrix method, was used to represent the carrier. Reduced stiffness matrices allow a large and complex geometry to be represented by a relatively small matrix. The planet carrier interfaces with the rest of system through the planet pins and the planet carrier support bearings. In the finite element model and RomaxDesigner model, each support bearing is represented by a single node at its centre at the bearing position and this node is connected to the carrier surface nodes at the bearing positions by rigid elements. In static or steady state analysis, the interaction between carrier, bearing nodes, and planet pins is accurately represented by the reduced stiffness matrix[5]. The carrier FE mesh is imported into RomaxDesigner model and then the bearing nodes and planetary gear pins are connected to it. The connected FE mesh is exported to a standard FE package (Nastran) to extract the carrier’s stiffness matrix. The stiffness matrix is then again imported into the RomaxDesigner model to represent the carrier component.

(a) Planetary gears

Figure 7:

(b) Planetary gears and Carrier FE mesh

(c) Planetary gears and Carrier 3D model

Illustration of the incorporation of a planetary carrier finite element component

6: MODELLING THE CLUTCH The clutch model is simplified to two parts, one inner part and one outer part (See Figure 8). For a planetary system, the inner part is connected to, for example, the ring gear while the outer part is connected with the housing. A rigid connection is used to simulate the connection between the outer part and the housing.


Outer part Inner part

Figure 8:

Typical simplified representation of clutch in RomaxDesigner


Figure 9:

Illustration of combining gearbox housing FE component and internal transmission components

The gearbox housing influences the deflection of the mounted bearings and therefore affects the meshing of the gear trains. In order to accurately predict the gear transmission error, the influence of the housing must be considered. However, the gearbox housing has a very complex geometry and cannot be modelled as a simple component in RomaxDesigner. The inclusion of the influence of the housing on system deflection is achieved by condensing the housing finite element model into a reduced stiffness matrix and incorporating it into the current system model. The reduced stiffness matrix represents the interaction between the bearings and the housing. In the case study, the finite


element gearbox model is created in Visual/Nastran. After the model is imported into the RomaxDesign model, the bearing nodes and clutch nodes are connected to the housing FE mesh. The connected housing model is then exported to Visual/Nastran. After applying the boundary conditions, which simulate the mountings of the transmission gearbox to the engine and chassis, the reduced stiffness matrix is extracted in Visual/Nastran and imported into the RomaxDesigner model to represent the housing (See Figure 9). By incorporating the housing stiffness matrix, a fully coupled non-linear algorithm to analyse the shaft/bearing/gear/housing hyperstatic system is created. Since only the reduced stiffness matrix is needed, the gearbox housing finite element mesh can be deleted after the stiffness matrix is imported into the system. This significantly reduces the model size for a complex gearbox. 8: DUTY CYCLE ANALYSIS When the gearbox model is completed, before running the duty cycle, a powerflow can be run first to check the various torsional connections between gears, shafts, bearings, clutches, carriers and housing under different operation conditions. Once the powerflow is checked, the model can be submitted for duty cycle analysis. In the case study, the driving input power is 104.7KW and the input torque is 250NM. Six forward gears driving, 1 reverse gear driving and 6 forward coasting and 1 reverse coasting gear load cases are analysed in this duty cycle. The calculating time for the 14 load cases only take about 28 minutes on a Pentium 4 PC with 3GHz CPU and 2G RAM. The bearing damage, bearing life, shaft stress, shaft fatigue life, gear mesh misalignment, gear life, gear contact stresses and bending stresses were calculated. Table 1 shows the bearing life summary for the duty cycle analysis. One potential bearing failure is identified and the customer has been advised to consider a different bearing. The bearing contact fatigue life and rolling contact damage are calculated based on both the ISO standard and the Romax adjusted life method. In the ISO standard, it is assumed that the bearing is purely radial loaded, with no clearance, no change in the contact angle and no misalignment. The RomaxDesigner adjusted life method, however, considers those conditions and adjusts the bearing life accordingly using a load zone factor. The accumulated total bearing life damage under a duty cycle for all gear selections or a specific bearing’s damage under individual load case can be output (See Figure 10). The gear bending and contact stresses are also calculated and output as part of a standard duty cycle report (See Figure 11). The gear meshing misalignment values are important factors in transmission


designs. Figure 12 shows one example of the calculated mesh misalignments for one gear pair under both gears of driving and coasting conditions. Table 1 Summary of bearing life of calculation

Total accumulated duty cycle damage for different bearings

A bearing’s duty cycle damage under individual load case

Figure 10: Bearing life damage chart


Figure 11: Gear bending and contact stresses in all gears 35

Misalignment (um)

30 25 20 15 10 5 0 1st gear- 1st gearcoast drive

2nd gearcoast

2nd geardrive

3rd gearcoast

3rd geardrive

4th gear- 4th gear- 5th gear- 5th gear- 6th gear- 6th gear- Reverse Reverse coast drive coast drive coast drive geargearcoast drive

Operating Gear Conditions

Figure 12: Mesh misalignment of a gear pair under different operating conditions

9: CONCLUSIONS A new approach for the modelling and analysis of a modern automatic transmission system with transmission software RomaxDesigner has been presented. The modelling and analysis of the major components of such a transmission system like the shafts, gears, bearings, clutches, planetary gear, planetary carriers and gearbox housing were covered and illustrated by a case study of a 6-speed automatic gearbox. The housing’s interaction with the engine and chassis through the transmission mountings are also included. The


planetary carriers and gearbox housing are incorporated into the internal transmission system by means of a reduced stiffness matrix which is extracted using a commercial finite element package. This new approach makes the model very compact and significantly reduces the modelling and analysis time of a complex automatic transmission system. The model can predict the shaft, gear and bearing life much faster than a conventional finite element analysis model without compromising the accuracy. By taking the advantage of modelling the gearbox system as a whole, the model can predict the gear mesh misalignment accurately, allowing transmission designers to efficiently modify the gear tooth micro-geometry to compensate for the effect of misalignment prior to the prototyping. REFERENCES 1. Design Practices: Passenger Car Automatic Transmissions, third edition, AE-18, Society of Automotive Engineers, Inc., USA, 1994 2. Design Manual for Enclosed Epicyclic Metric Module Gear Drives, ANSI/AGMA 6123-A88, October 1988 3. HARRIS, J, JAMES, B M AND WOOLLEY, A M, -Predicting the Effects of Housing Flexibility and Bearing Stiffness on Gear Misalignment and Transmission Noise using a Fully Coupled Non-Linear Hyperstatic Analysis, Proceedings, Institution of Mechanical Engineers, C577/005/2000, May 2000. 4. POON, S, Y, -A New Approach to Transmission Design, Romax Internal Document, 2004 5. Visual/Nastran 2002 User Manual, MSC Software Corporation, Santa Ana, USA, 2002 6. ALLAN, R K, -Rolling Bearings, Sir Issac Pitman & Sons, Ltd, London, 1954 7. RomaxDesigner User Manual, Romax Technology Ltd, Nottingham, UK, 2003

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