gear web and rim thickness optimization to improve...

Gear web and rim thickness optimization to improve vibration and fatigue reliability of a rotary UAV gearbox

E.E. Taskinoglu1, V.Y. Öztürk1 , Y. Paça1

1 TAI – Turkish Aerospace Industries, Inc.,Integrated Helicopter Systems ODTU Teknokent, A-Blok, 06531 Ankara, Turkey e-mail: [email protected]

Abstract Helicopter structures suffer from excessive vibration originated from many sources including dominant main and tail rotor excitations. Some of these vibration contributors can be identified very easily because of their unique frequency characteristics, such as blade passing frequencies. Additionally, transmission gearboxes are also key sources that show themselves at several gear mesh frequencies. In a gearbox, the transmission error is the phenomena that turn meshing gears into a powerful vibration source at several helicopter zones. Transmission error is the difference in translated rotation between gears due to gears deformation in contact and manufacturing errors. Since, there is a direct correlation between the transmission error and gear induced vibration, reducing the peak to peak transmission error in a gear mesh cycle is a crucial step in gearbox design. In general, the transmission error reduction is done by modifying the tooth profiles in micro scale by using transmission error analytical models. These models are not suitable to reflect the effect of gear web topology and rim thicknesses. So FEA is mandatory to take into consideration the effect of web topology and rim thicknesses on gear pair global stiffness which in turn plays a role in peak to peak transmission error calculation. While reducing the transmission error, other concerns such as tooth root stress and gearbox modal characteristics should also be controlled in order to have a feasible design solution. So design of a gearbox becomes a multi-objective optimization problem with several manufacturing and sizing constraints. In the scope of this study, a spur gear is optimized by using web and rim thicknesses as design variables while aiming to reduce vibration, tooth root and contact stresses. In each iteration, objective function is obtained by performing non-linear static analyses followed by modal analysis in which the mesh stiffnesses obtained in non-linear static analyses is employed to realistically represent the gearbox global stiffness. Static analyses are performed by rotating the gear pairs in very small increments and solving for transmission error, root and contact stresses at each increment in order to obtain peak to peak transmission error, maximum tooth root and contact stresses in a gear mesh cycle. Due to FEA procedure employed, static analyses are performed by uncoupling the gear pairs while in modal analysis a FEM that represents all the gears and suporting structures as well as with gears mesh stiffnesses is used. CONMIN open source optimization code is used with an in-house code developed to automate the incremental static and modal analyses and to link the optimization code with the structural analyses. MSC MARC® and MSC Nastran® are used as the finite element solvers. Transmission error, eingenvalue, root and contact stress sensitivies to web and rim thicknesses are investigated. Current limitations on FEA procedure employed in this study are highlighted as well as with advantages and disadvantages observed.

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1 Introduction

The requirements that intensively have arisen during recent years for lightweight and compact size design of gears, such as for those gears used in drive-line systems of rotorcrafts, have put the emphasis on the necessity to understand and properly evaluate the gear output parameters such as tooth root stresses, contact stresses, static transmission error (STE) etc. Most of the time, optimization of gear design parameters needed in order to achieve the conflicting requirements of having lightweight gears with high fatigue lives. Noise has also become an important issue in gear design; therefore when combined with its effect on fatigue, vibration characteristics (natural frequencies, mode shapes) can also be critical depending on the application area and the rotating speeds of the gears under consideration. In this study, the main gearbox of a rotary-wing unmanned air vehicle is optimized via CONMIN, based on the results obtained from FE analyses. Parameters under consideration for this study were selected as the natural frequencies of the gears, tooth bending stresses, contact stresses and static transmission error values. The frequency ranges to be avoided are selected as the frequencies at which the main rotor is rotating, its multiples and the mesh frequency of gears respectively, in order to avoid the danger of a natural frequency clash of the gear blanks which will possibly result in damaging of the transmission system. The tooth root stresses and contact stresses are also desired to be kept in certain limits in order to achieve the fatigue requirements of the gears. The last parameter in the optimization routine is the static transmission error (STE), which is the main source of the gear noise and vibration [1].

1.1 Rotary UAV Gearbox Design

Some of the important design parameters of the gears under consideration are given below in Table 1. Due to the confidential properties of the gears, the details of the design cannot be disclosed here; however it is worth to mention that the gears have addendum modifications applied on them, therefore the operating centre distances are not the same as the theoretical ones. Output speed and power of the gearbox is 540 rpm and 54 hp respectively.

GEAR1 GEAR2 GEAR3

Number of teeth 19 20 86

Module [mm] 4 4 4

Pressure angle [°] 25 25 25

Tooth thickness 40 40 40

Operating center distance 80 216

Table 1: Gear Design Parameters

When assessing the tooth bending stresses, AGMA standard formulation for tooth bending stress, which applies certain factors depending on the operating conditions, is utilized; however the input stress value to this formulation is the Von Mises stress value obtained via FEA, instead of the Lewis formula-based stress value [2]. Desired stress range can be adjusted depending on the required margin of safety (MS) value. MS values are obtained using the allowable stress values based on the gear material, life expectation from the gears and reliability levels desired.

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Similar to the tooth bending stress assessment, contact stresses are also evaluated using AGMA standard formulation for pitting resistance. In a similar manner, input stress value to the formulation is the one obtained from the FE analysis, instead of the Hertzian-based contact stress value of the mentioned formulation. As stated earlier, STE is the main source of the gear vibration and noise. Transmission error can be defined as the difference between the theoretical and actual relative angular rotations between a pinion and a gear. As the difference between the maximum and minimum STE gets larger, rotational vibration amplitudes grow to larger values, especially when the gears are rotating at the first few harmonics of the mesh frequencies. Since STE is known to be more dominantly dependent on the load applied and the gear profile properties rather than the web and rim thicknesses, a major improvement is not expected by modifying the web and rim thicknesses of the gears and the target STE values are entered to the optimization routine keeping this fact under consideration. When calculating the STE from the FE analysis, gear pairs are assumed to be perfect, that is no tooth geometry errors and mounting errors are present on the gearbox and STE is purely due to the elastic deformation of the teeth in contact.

1.2 Finite Element Model

Same finite element model geometry is used both for stress calculations, modal analysis and STE calculations (Figure 1). The models are prepared in pre-processors Patran® and Mentat® of MSC Software®. For modal analysis, SOL 103 solver of MSC Nastran® is used, whereas for stress analysis and STE calculations, MSC MARC® is used in order to capture the contact phenomena.

Figure 1: Overview of the model used in modal analysis

In the modal analysis model, shafts are also represented as bar elements and connected to the gear blanks via rigid multi-point constraint elements (MPCs). Contact between the gears is simulated via CBUSH elements representing the contact stiffnesses in the direction of the pressure angle. Those contact stiffness values are calculated from the stress analyses results and they are inversely proportional with STE values. Gear blanks are modeled with 2D CQUAD and CTRIA elements. The ends of the input and output shafts are fixed, being the boundary conditions for the modal analysis. Stress and STE analyses are carried out separately for both gear meshes using Marc (Figure 2).

DYNAMICS OF ROTATING MACHINERY 1363

Figure 2: Overview of the model used in modal analysis

The FE mesh around the possible contact areas is kept as fine as possible in order to capture the contact phenomena accurately. The analyses in Marc are run for the whole mesh cycles of the gear meshes with 0.5º increments of rotation of the pinions and the respective rotation of the mating gears through the gear ratio. The reason for implementing such an approach is the fact that the contact characteristics change from position to position during a mesh cycle. In order to obtain the most critical stress states and the peak-to-peak STE values, it is necessary to analyze the whole mesh cycle. A procedure file is prepared in MSC Mentat® in order to automate the running of the consequent analysis which also allows for the changing of the parameters in the optimization stage of this study. Analyses are carried out in a quasi-static manner, i.e. during the analyses, torque is applied to the center of the pinion and the center of the mating gear is fixed both in translational and rotational directions. Resulting rotation of the pinion gives the STE in rotational units, which is convertible to the metric units. Stress distribution data is also available to observe as the outcome of the analyses (Figure 3).

Figure 3: Sample contact and bending stress distributions

2 Optimization

2.1 Method and Tools

In this study CONMIN open source gradient based optimizer which uses Constrained Function Minimization algorithm, is utilized. CONMIN is an optimizer developed by G.N.Vanderplaats [6]. Algorithm uses method of feasible directions by moving from a feasible location to an improved location. Problems that have linear or nonlinear constraints can be dealt with this algorithm. The problems that are solved with CONMIN are generally in the following form.


Minimize :

Subject to constraints:

0

The problem that are functions of multivariable, , limited by lower and upper band side constraints, and , subjected to constraint can be minimized or maximized using this algorithm.

Details of the method are not the scope of this study. In [7], detailed mathematical investigation of the method is given.

2.2 Procedure The objective of the problem under consideration is a function of transmission error , Tooth root stress , and contact stress , . These parameters are normalized in a way that similar order of

magnitude is guaranteed. Objective function that is to be minimized is given in following formulation;

, , , , , 1,2 (1)

By levering the weight coefficients in the objective function one can emphasize any of the design parameters more or less with respect to remaining function parameters. The design goal of this study is to keep modal frequencies of the gear combination away from a specific frequency band while reducing the stress and transmission error values indicated in above equation. Critical frequencies of the gear system are 90 Hz and 774 Hz, which are the blade passing frequency lying in modal frequency range of the gear system and the mesh frequency, respectively. Minimum 5% safety margin is aimed for the design. This criterion can be summarized as follows;

0.95x774 Hz < Gear System Modal Frequencies < 1.05x774Hz 0.95x90 Hz< Gear System Modal Frequencies < 1.05x90 Hz

It is aimed that the distance between all frequencies of the gear combination and critical frequencies indicated above is maximized. In this study MSC Nastran® and MSC MARC® are used for the FE based calculations. Results obtained from these tools are sent to the optimizer to evaluate the objective function and the related constraints. The general flowchart of the program structure is given in Figure 4.


Figure 4: Optimization Flow Chart

In this design optimization, the gears thicknesses are employed as design variables to achieve the necessary objective formed by tooth root stresses, contact stresses and peak to peak transmission errors as well as with modal frequency of the gear system as design constraints. Design variables are;

= thickness of gear rims

, = web thickness of 86 tooth gear



In each iteration, these design variables are sent to MSC Mentat® through a procedure file (.proc file) which constructs the new MSC Mentat® database file with new thickness definitions and employs MSC


MARC® for nonlinear static analysis. Non Linear Static analyses are performed by rotating the gear pairs in very small increments and solving for transmission error, root and contacts stresses at each increment in a gear mesh cycle. The analyses in MSC MARC® are run for the whole mesh cycles of the gear meshes with 0.5º increments of rotation of the pinions and the respective rotation of the mating gears. Due to FEA procedure employed, static analyses are performed by uncoupling the gear pairs. For each iteration, the peak-to-peak transmission error is calculated by collecting the analyses results during the whole mesh cycle and combining the minimum and maximum among the transmission error results, while the maximum contact and the maximum tooth root stresses are extracted from the stress results obtained for a whole mesh cycle. As a second analysis step in an iteration, the obtained average values of the transmission errors are converted to contact stiffnesses that replace the stiffnesses of the bush elements in MSC Nastran® input file which are used to reflect the effect of contact stiffness change in modal analysis due to gear thicknesses alteration. When all the thickness and stiffness parameters are updated for combined model, normal modes analysis is performed using MSC Nastran®. Calculated modal frequencies are extracted and sent to CONMIN as constraint parameters.

3 Results

Although there is approximately 3% of weight increase, tooth root stresses are reduced by 6.2% and 4.12% while the tooth contact stresses are reduced by 7.4% and 4.47%. In addition, peak to peak transmission errors are reduced by 2.38% and 2.53%. The main achievement in this optimization study is the change in modal frequency. The initial design of the gear system has a modal frequency that coincides with the blade passing frequency of 90 Hz. Thus, the initial design is an infeasible design. At the end of the optimization study, both frequency constrains are satisfied .Indeed, the closest modal frequency to the related blade frequency is approximately 26% apart from the blade frequency, 90 Hz. The design variables and results history of the optimized structure are given in following figures;

Figure 5: Design Variable History


Figure 6: Total Mass vs Iteration

Figure 7: Tooth Root Stress vs Iteration

Figure 8: Tooth Contact Stress vs Iteration


Figure 9: Peak to Peak Transmisson Error vs Iteration

4 Conclusions

The main objective of this study is to build an automatic procedure to calculate transmission error, tooth root stresses, contact stresses and modal frequencies of a gear system. By doing so, it will be possible to perform optimization studies that ask for FEA to evaluate objective function and design constraints. This study is performed in order to control and check the stability and the reliability of the procedure in a real design problem. In this study, a spur gear system is optimized by using web and rim thicknesses as design variables while aiming to reduce vibration, tooth root and contact stresses. In each iteration, objective function is obtained by performing non-linear static analyses followed by modal analysis in which the mesh stiffnesses obtained in non-linear static analyses is employed to realistically represent the gearbox global stiffness. The vibration and noise of a gear system is mainly originated from the transmission error. In practice, the transmission error reduction is done by changing the gear tooth root profile. Indeed, the optimization results show that the changes in gear thicknesses are not so effective to reduce the transmission error, only 2.53% change in transmission error accompanied with a 3% weight increase. However, the modal frequencies of the gear system should also be an important design parameter to consider for an acceptable gearbox vibration level. The modal frequencies should be apart from main excitation frequencies. In this study, vibration reduction is done, aiming to shift the modal frequencies apart from the blade passing and mesh frequencies, by using two inter-related mechanisms; global stiffness change due to gear thickness change which results in alteration of the transmission error. Change in gear thicknesses results in transmission error alteration thus contact stiffness change. So, indirectly, the global stiffness of the gear system is modified due to gear contact stiffness alteration. Evaluation of the optimization results reveals that the effect of changing the contact stiffness has little effect on modal frequency shift. So the main contribution comes from global stiffness variation due to change in overall gear stiffnesses. Tooth root stress plays an important role in design of a gear system with acceptable fatigue life. So the reduction in tooth root stresses should be evaluated while performing trade-off studies concerning other design criterias, such as vibration and noise. This is the reason why the tooth root stresses are used in the objective function formulation. Due to the non-linear nature of the S-N data, the 6.2% change in tooth root stress will result in a fatigue life increase significantly higher than 6.2%. So, the current achievement in tooth root stress is very valuable in terms of fatigue life considerations. In gear design, contact stress is a key parameter to avoid scoring, pitting and wear. So, reducing the contact stresses will be helpful to prevent these failure mechanisms. Unfortunately, there is no reliable analytical or numeric prediction method for these failures. So keeping the contact stress values at


permissible minimum is the best design strategy. In the current optimization study, the contact stresses are reduced by maximum 7.4%. One of the most significant disadvantages of utilizing a gradient -base optimization algorithm is the high probability to stack at a local optimum. Indeed, for a gradient-based algorithm, the probability of a randomly chosen starting point to give the global maximum is 1% [10]. So the optimization procedure needs to be repeated several times by changing the initial design and the optimization parameters to avoid it in some extent. When the run-time for a gradient base optimization is evaluated, it is seen that repeating the procedure several times becomes acceptable when compared with other optimization algorithms such as Genetic algorithm. For the current study, the evaluation of the objective function and the constraints take significant time due to necessity to perform non linear static analyses for a whole gear mesh cycle within small increments of angular rotation. So this limitation results in lack of repetition of the optimization procedure from different starting points, initial designs. An alternative and time effective way to calculate the necessary output parameters need to be investigated but when we think of the necessity to introduce the change in gear stiffnesses due to gears thickness alteration, the FE analyses seem to be the only alternative for the purpose of this study. Another possible way to decrease the run-time can be employment of a response surface algorithm for objective function and constraints evaluation, but the suitability of response surface algorithms for non-linear problems is questionable. As a further study, topological optimization of the gears is planned to be performed, as well as with sizing optimization. By this way, better optimization results with less weight increase (less than 3%) can be possible. Also the design variables for sizing optimization can be extended to cover gear design parameters additional to gear thicknesses by employing re-meshing algorithms or other specific tools (in-house programs etc.) in the optimization procedure.


References

[1] H.N. Özgüven and D.R. Houser, Dynamic Analysis of High Speed Gears Using Loaded Static Transmission Error, Journal of Sound and Vibration, 125, 71-83 (1988).

[2] E.J. Shigley and C.R. Mischke, Mechanical Engineering Design ,McGraw Hill, 2006. [3] A. Kahraman ve R. Singh, Non-linear Dynamics of a Spur Gear Pair , Journal of Sound and

Vibration, 142, 49-75, (1990). [4] J.D.Smith , Gear Noise , Marcel Dekker publ.(1999). [5] V.K. Tamminana, A Kahraman and S. Vijayakar, A Study of the Relationship Between the Dynamic

Factors and the Dynamic Transmission Error of Spur Gear Pairs, Journal of Mechanical Design, 129, 75-84, (2007).

[6] G.N. Vanderplaats, CONMIN — A Fortran program for constrained function minimization , Technical Memorandum TM X-62282, NASA Ames Research Center, Moffett Field, California(1973).

[7] G. Zoutendijk, Methods of Feasible Directions , Elsevier Publishing Co. ,Amsterdam(1960). [8] R.T. Haftka, Z. Gurdal, Elements of Structural Optimization , Kluwer Academic Publishers, (1990) [9] A.D. Belegundu, T.R. Chandrupatla , Optimization Concepts and Applications in Engineering ,

Prentice-Hall Inc.(1999). [10] P. Charbonneau , An Introduction to Genetic Algorithms for Numerical Optimization , NCAR

Technical Note (March 2002). [11] S. Rajeev , C.S. Krishnamoorty, Discrete Optimization of Structures using Genetic Algorithms,

Journal of Structural Engineering ASCE 118(5): 1233}50, (1992).


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