MBS Gear-tooth Stiffness Model

Transcription

1 MBS Gear-tooth Stiffness Model Implementation of a new coupling model for fast and accurate simulation of gear pairs using stiffness characteristic arrays Faysal Andary M.Sc. 1), Dipl.-Ing. Daniel Piel 2), Dipl.-Ing Matthias Wegerhoff 1) 1) Institut für Maschinenelemente (IME), RWTH Aachen, Prof. Dr.-Ing. G. Jacobs 2) Werkzeugmaschinenlabor (WZL), RWTH Aachen, Prof. Dr.-Ing. C. Brecher Abstract The simulation of gear trains is important to understand the behavior of systems and their components in many industrial applications, especially in regards to acoustics, fatigue, and wear. However, there is a visible mismatch in the degree of detail between the existing gear simulation models. On one hand, most MBS models use rigid tooth-bodies with constant stiffness that is independent of the contact point and the load case. This results in simulation inaccuracies, especially regarding proper load application and stiffness distribution which leads to imprecise acoustic and fatigue estimations. On the other hand, most of the current elastic tooth stiffness models use in-depth contact and impact calculations which result in a slow and computationally-intensive simulation process that cannot be applied to large multibody dynamic systems. MBS Gear-tooth stiffness is an FVA project that aims to help bridge the gap between the two aforementioned approaches. It is implemented as a Simpack user-routine, which provides a flexible and powerful modeling environment with advanced processing tools. It contains five different calculation approaches Constant stiffness, Stiffness Array (STIRAK import), Fourier Function, Gear Data Input, and Constant Stiffness with DIN 3990 models that define the stiffness distribution along the tooth flank, and can be accessed in the parameter window through a selection menu:. The Stiffness Array model is the main focus of this paper. It uses pre-calculated stiffness characteristics from the FVA program FE-Stirnradkette (STIRAK) developed by the WZL. These arrays are calculated based on the material and geometry of the teeth and gear bodies and can be imported to Simpack as an external file. This model represents a more realistic behavior of gears than the standard MBS models without delving into contact and impact anal-

2 ysis, and therefore can be used in large multibody systems without being computationally intensive. Hence, MBS Gear-tooth stiffness provides the first step in bridging the gap between the degrees of complexity in the current models. Future research is also planned to further expand the model by incorporating elastic tooth bodies and different representations of load distribution along the flank. State of the Art: SIMPACK The field of multi-body simulation (MBS) is one of the fastest developing fields in mechanics, especially nowadays with the advancement of computation power. SIMPACK is a leading Multi-Body Simulation software used for the dynamic analysis of mechanical and mechatronic systems. These systems consist of various components that are interconnected by means of markers and force-elements. SIMPACK offers a large array of multi-body components within its intrinsic libraries. These libraries can be also expanded with a user-routines interface which allows users to define their own programmable components to incorporate into the multi-body environment. SIMPACK libraries include force elements to simulate the behavior of gear pairs, the most important of which is Force Element 225: Gear Pair: This force element is able to accurately position the forces acting on the gear teeth and it supports force application on elastic bodies as of SIMPACK 9.8. However, the stiffness distribution on the teeth is calculated using the ISO 6336 standard [1], which does not take into account the load dependency of the stiffness distribution. STIRAK The software FE-Stirnradkette (STIRAK) was developed at the Institute for Machine Tools and Production Engineering (WZL). It allows advanced simulation of the meshing process of gear pairs using detailed flank geometry in which the system is represented using a Finite- Element model. This software also enables the considerations of the various effects at the contact zone in regards to different determining factors, e.g. flank modifications, different operational points, shaft elasticity, joint geometry or material parameters. STIRAK is under continuous development and validation through past and on-going FVA projects. STIRAK uses FE-based tooth contact analysis to determine and optimize the operating behavior of gear pairs. The gear flank geometry and tooth root geometry is described by a manufacturing process. By solving the FE-Model the load depending stiffness behavior is saved in influence coefficients. This enables a quick calculation of different tooth flank topographies and various loads. With the calculus of variation it is possible to optimize the load carrying and running

3 Output Input behavior of gears [4, 5, 6]. The input file contains information related to gear layouts, gear data, tool geometry, flank topography, gear body geometry, and the geometries of the shaft and bearing. The output of the calculation contains the contact geometries, excitation characteristics, load capacity, and flank corrections [2]. This process is summarized in Figure 1 However, this FE-based application is only suited so far for the analysis of quasi-stationary contact behavior and it is not able to perform transient analysis of entire dynamic systems because of the detailed and focused approach used on the gear mesh calculations. For this reason, it proved necessary to develop an interface that would allow the results from STIRAK to be used in a 6-DOF multi-body environment such as the one SIMPACK has to offer. y E 1 Layout Gear Data C Tool Geometry Flankentopografie Topography Gear Body Shaft / Bearing b E 2 x symmetric/ asymmetric symmetric/ asymmetric D max d f FE-Stirnradkette Ease-off Transmission Error Course of Stiffness Hertzian Pressure Variation- Calculation Contact Pattern Fourier-Spectrum Excitation Table Tooth Root Stresses f Hα Contact Geometries Excitation Characteristics Load Capacity Corrections Figure 1: Input and output of the FE-based tooth contact analysis Objective The aim of this research project is to implement the results of the detailed calculation scheme performed by STIRAK into a multi-body environment capable of simulating large dynamic systems. This results in a wider range of applications for the STIRAK data, as well as a more accurate simulation of the behavior of gear pairs in large multi-body systems f Hβ Force Element Model Description The MBS-Gear force element was developed at the Institute for Machine Elements and Machine Design (IME) of RWTH Aachen in cooperation with the Institute for Machine Tools and

4 Production Engineering (WZL). The force element was built into a SIMPACK user library using the user-routines interface. The force element also comes with two additional marker models that serve as the meshing points on each of the gear wheels. The force element is defined between these markers in the multi-body system rather than the default body reference markers. The incorporation of these markers adds several advantages to the model, such as the ability to quickly and effortlessly generate a gear pairing. Since the gear parameters are entered in the from-marker, the force element can be used with any geometry, and on any axis. A representation of this model is shown in Figure 2 Figure 2: Representation of the force element between two gear wheels The force element window contains a selection menu parameter that allows the users to choose their desired calculation approach that describes the stiffness distribution along the tooth flank. The user-force currently supports five different modules: Constant stiffness Stiffness Array (STIRAK import) Fourier Function Gear Data Input Constant Stiffness with DIN 3990 The constant stiffness models assume a constant stiffness distribution throughout the tooth flank, the Fourier function model represents the stiffness using a Fourier function, and the gear data input for direct input of geometric data. The stiffness array model is the one that uses the characteristic arrays from STIRAK, and the main topic of this manuscript.

5 Tooth Contact Analysis In order to calculate the load dependent stiffness characteristics, STIRAK uses the mathematical spring model at the contact area to generate a system of equations based on the number of considered contact positions through an iterative process. The procedure starts with an initial contact distance as well as a set of influence coefficients. This combination is used to determine the initial load distribution, which in turn is used to calculate the distance for the first iteration. After that, the array of current distances of contact Δ i for each step is used to calculate the load distribution at the contact points, which is then used to the distance for the following calculation stage Δ i 1. This process is described in Figure 3. Spring Model for Contact Area Stress F tot Influence Coefficients Pinion Distance of Contact α i,pinion Δ i { α i,gear α i,j Gear Tooth Width b Δ i F t, 1 2 n tot System of Equations n 1 b 1,1 2,1,1 1 F1 1 F 2 1 F n 0 f Figure 3: Principle of calculation of load distribution and rigid body motion with the mathematical spring model [2] Stiffness array generation 1 1, n 2, n n, n A x In order to export the stiffness array from STIRAK into an appropriate format that can be read by the user-force in SIMPACK, an additional application was developed at WZL in order to automate the generation process. The application bears the name D2S_Kennfeld.exe, and was developed as a part of the FVA project DRESP2SIMPACK [3]. It combines the relevant computational core of STIRAK together with a utility for the preparation and conversion of the resulting array. The output is a three dimensional stiffness array that describes the mesh stiffness as well as the unloaded transmission error. tot n: Number of Positions in the considered Joint Rolling Motion α res = α pinion + α gear Load Distribution F i, Rigid-Body-Motion f tot f tot Stress F tot F 1 F 2 F 3 F n

6 STIRAK STIRAK STIRAK The application expects two input files: Eingabedaten.fsk: contains the input data for the specific gear stage, including gear and tooth geometry (for STIRAK). D2S_Kennfeld.fsk: contains the configuration options for the application, such as the type of geometric variation and load dependencies. The calculation process is described in Figure 4. Two possible calculation approaches exist based on the choice of variation in the configuration file. In case the flank line deviation method is selected, the application will first calculate the initial tooth geometry based on the input parameters. After that, an FE-mesh is generated, and solved for the corresponding load distribution. Then the iteration procedure described in Figure 3 commences until a set of conversion solutions is achieved. This results in a detailed simulation of flank modifications, as well as a record of the stiffness distribution as a function of each load case. The results are then written into the stiffness array and exported to the output file. In case where the variation of center distance method is selected, the distance from the center is modified during the iteration procedure, and a new FE-mesh is generated for every step. This also results in a stiffness array that s exported an output file. Input data files Eingabedaten.fsk D2S_Kennfeld.fsk Stiffness Array D2S_Kennfeld.exe Caculus of Variation Variation of Flank Line Deviation Variation of Center Distance Calculation of Geometry i th Center of Distance FE - Mesh Generator Calculation of Geometry FE-Solver FE - Mesh Generator i th Value of f hβ FE-Solver Calculation of Geometry Strength Calculation Unloaded Transmission Error Mesh Stiffness Strength Calculation i th Stiffness Array i th Stiffness Array Editing Stiffness Arrays Figure 4: Calculation procedure of the application D2S_Kennfeld.exe

7 Using the results in an MBS model After generating the file containing the stiffness arrays, the next action is to use these result in an MBS model. The first step in preparing the model is to create the boundary markers on the bodies of the components where the gear stage takes place. The actual geometry of the body primitives in SIMPACK has no effect on the calculation, so the markers can be placed directly on shafts without spending time on the creation of geometrically accurate gear wheels. There are two user markers of designated types -19 and -18 accompanying the force element that are used for the from and to bodies respectively. The from and to markers are determined according to the direction of torque propagation [3]. After the markers are placed on their respective bodies, the gear dimensions can be entered into the parameter fields of the markers. These parameters include the body reference markers and joints of the gear wheels, as well as their pitch diameters, the helix angle, the pressure angle, and the gear ratio. The type of the gear stage should also be specified which can be spur, planetary, or bevel gear pairs. If all the parameters are entered correctly, then the markers should be located at the expected meshing gear meshing point as soon as the changes are applied. When the markers are fully established, the force element can be created. The from and to markers are input in their respective fields in the parameter window. The stiffness array file can be imported into its corresponding parameter field (parameter 7) using the global search path. Other parameters should also be supplied, such as the values of damping, gear backlash, and phase angle. The user can also choose an appropriate extrapolation method for stiffness values outside the defined arrays. After the gear stage is completely defined through the markers and force element, time simulation can be started and the output of the system can be visualized in SIMPACK Post-Processor. Along with the various body kinematics and joint force outputs provided by SIMPACK, several output channels have also been programmed into the MBS-Gear force element. These outputs include but are not limited to: elastic compression in normal direction, force in normal direction, relative rotational speed, and the equivalent meshing tooth stiffness at every time step. Through SIMPACK Post-Processor, these output channels can then be exported as text or graphical data.

8 Since the force element applies loads on assumed rigid tooth bodies, the simulation time is relatively fast compared to force elements that consider node deformation of elastic bodies in real time. This is important for obtaining both quick and accurate results on complex multi-body systems before investing in large amounts of time and computational power. Figure 5: A summary of the process to import the stiffness array into an MBS model in SIM- PACK Sample output In order to illustrate the effect of the stiffness array calculation approach, an example model of one gear stage has been built and simulated in SIMPACK using two different calculation approaches to describe the stiffness. One model used a stiffness array generated from STIRAK using D2S_Kennfeld.exe, while the second model uses Fourier coefficients to describe the

9 stiffness distributions. The lower and higher regions in the curves describe the points of single and double tooth engagements, respectively. Figure 6: Stiffness distribution at the contact point as a function of the rotation angle using the stiffness array from STIRAK (above) as compared to the Fourier stiffness approach The shape of the stiffness as a function of angle for two load cases is shown in Figure 6. The center plot is used as reference, and it shows an example of the matching stiffness curves under a constant load. The same system is then subjected to an increasing load. This resulting stiffness curves are shown in the bottom-most plot. The stiffness array model responds to the change in the load and the average tooth stiffness increases in the second part of the plot, whereas the stiffness of the Fourier model remains constant. The response of the stiffness array model to load variation offers several advantages over traditional gear models, such as: More precise calculations of acoustic behavior and fatigue estimation than constant and linear stiffness models Shorter simulation time and lower memory requirements than the in-depth models for realtime elastic body simulation or impact analysis, especially for larger multi-body systems. This is due to the use of a pre-calculated stiffness arrays.

10 Summary and Outlook The advanced calculation approach implemented by the mathematical spring model in STIRAK has been successfully implemented into the multi-body simulation software SIMPACK by means of a user-defined force element along with associated marker elements. An interfacing application has also been developed in order to facilitate the transfer of the output stiffness data into a single file that can be easily imported into the force element. This results in a more accurate representation of the tooth stiffness in large multi-body systems which leads to the possibility of more precise acoustic and fatigue estimations. Further development of the MBS-Gear user force is planned in order to expand it with more advanced features, such as support for the force application on elastic tooth bodies, along with improvements to the simulation speed. Acknowledgement The authors gratefully acknowledge financial support by the German Research Association (FVA) [FVA No. 603 I & II] for the achievement of the project results. Special thanks belong to the supporting FVA working committee AG-MKS. References [1] SIMPACK, "Documentation for Simpack 9.7," München, [2] Brecher, C. and Weck, M.: "Benutzeranleitung zum Programm FE-Stirnradkette v4.0," FVA Nr. 377, [3] Jacobs, G. Schelenz, R. and Augustino, R.: "DRESP2SIMPACK: Portierung von Berechnungsroutinen aus DYLA und DRESP nach SIMPACK," FVA Nr. 603, [4] Neupert, B.: "Berechnung der Zahnkräfte, Pressungen und Spannungen von Stirn und Kegelradgetrieben. Dissertation," RWTH Aachen, [5] Bong, B.: "Erweiterte Verfahren zur Berechnung von Stirnradgetrieben auf der Basis numerischer Simulationen. Dissertation," RWTH Aachen, [6] Cao, J.: "Anforderungs und fertigungsgerechte Auslegung von Stirnradverzahnungen durch Zahnkontaktanalyse mit Hilfe der FEM. Disseretation," RWTH Aachen, 2002.