Formula 1: Fast Cars Fast Simulation

Transcription

1 KISSsoft AG Frauwis Hombrechtikon SWITZERLAND Phone: Fax: web: info@kisssoft.ch Formula 1: Fast Cars - Fast Simulation Formula 1: Fast Cars Fast Simulation One of this season again successful F1 Racing Teams, bets for Layout, Optimization and theoretical Verification of its gearboxes using the KISSsys Integrated Simulation Tool. With it, it is possible to evaluate the Service Life of its Driveline components such as Gears, Shafts and Bearings, in an inexpensive and understandably way and that, in a very short time. The parameter variations allowing for the sensivity considerations concerning service life, lead to smaller, lighter but still sufficiently dimensioned gearboxes. The next victory is already around the corner... The goals to reach with the help of the simulation model are: Attain similar service life for all gears and bearings Estimate service life for different gearbox configurations Quickly examine influencing factors such as lubrication, production tolerances, etc. Tune the gearbox for specific track sectors Reduce the number of Test Runs. 1 Formula 1 Gearboxes Formula 1 Gearboxes must have between a minimum of four and a maximum of seven forward speeds; Continuously Variable Transmission (CVT) Gearboxes are not allowed. According to FIA (Fédération Internationale de l Automobile) rules, a reverse speed is also mandatory. Speed shifting is semiautomatic, i. e. the pilot uses the two toggle switches on the steering wheel and there is no need for a clutch. The shift fork is electro-hydraulically driven since it will be shifted around 2600 times during the race. The shifting takes place unsynchronized, sequentially and shifting times amount to less than two hundredths of a second. 1 of 11

2 A typical gearbox has three stages. Just after the engine, comes the shift stage with the fixed wheels on the driving shaft and the loose wheels on the driven shaft, see the gearbox diagram in Fig. 5. These, are mounted on the hollow shaft with needle bearings. After the shift stage, comes a spiral bevel gear stage. Follows another spur wheel stage with a reduction of about 1:3 in which the driven spur wheel simultaneously serves as carrier for the intermediate differential spur wheels. Typically, the spur wheel differential consists in seven planet pairs as displayed in Fig. 1 (only one planet pair is shown). Transmission housings are, e. g., made out of titanium alloys or also of carbon fiber composites. Figure.1: Structure of an F1 Gearbox. Simplified Display inspired in the Calculation Model. Blue: Gears; Yellow: Bearings; Gray: Shafts; Red: Couplings The selection of high-resistant materials, close production tolerances together with the inherent and precise material- and manufacturing specifications, comprehensive quality controls, high engineering charges and high test costs, leads towards gearbox costs of about per piece. Nowadays, F1 motors reach over rpm, have maximal torques of over 350 Nm and powers around 850 HP. Time-dependent data, such as shift position, real-time motor rpms and the correspondent torque, are available for each race or training session. A frequency distribution of torques, rpms and shift positions is created based upon these real time measures. It is thus possible to show the gearbox loading during a complete racetrack. Fig. 2 shows a typical load spectrum fort he 5 th gear. It shows the frequency against torque and number of revs; there are equivalent spectra for the other speed gears. 2 of 11

3 Frequency [%] Nº. of Revs [rpm] Torque [Nm] Figure 2: Load Spectrum for the 5 th Gear. Frequency Distribution as a Function of Torque and Number of Revs. Disregarding gearbox dynamic effects (driveline vibration, external impacts, tooth flank disengagement), this data are sufficient to make a calculated estimate of the gearbox components service life. 2 Service Life Calculation At present, the basis for the conventional calculation of service life estimation, is the linear cumulative damage hypothesis which, apparently independently, was simultaneously developed by Palmgren (in 1924), [1] and Miner (in 1945, only) [2] for the calculation of rolling bearings. This procedure is in general applicable but it has precision limitations. The primordial idea is the assumption that the damage to a machine component, subject to a cyclic loading under a constant amplitude, increases linearly with the number of applied loading cycles and reaches a damage value of S=1 in case of fracture. The model for the damage mechanism goes under the assumption that, the fluctuation of the forces imposed upon the component, originates small cracks or increases existing ones. The formation of cracks is quantified by the damage S, which increases linearly with the number of cycles until the component fails, i. e., until S=1. The basis for the damage calculation is the Wöhler Line, established by Gassner and Wöhler. The component s Wöhler Line applies the maximal sustainable stress amplitude on a number of cycles. It is determined based upon the material s Wöhler Line in which the differences between component and sample sizes, surface rugosity, etc., will be taken into consideration. On the other hand, the material s Wöhler Line is determined by measurements made on standardized samples. See [3]. 3 of 11

4 Is there a load spectrum, i. e., a data set with a defined number of load cycles with the required frequency in a certain number of cycles, it will be assumed that the damages originated in the individual cycles can be added up. Presenting the damage S=S(n,σ a ) as a function oft he number of cycles and the stress amplitude, it results from the linear damage increase, for a constant stress amplitude: S( n, σa) = n S(1, σa) Should σ amax be the maximal sustainable stress amplitude for N number of cycles, it follows from the requirement S(N, σ amax )=1 and (1): 1 = S(N, σ α ) = N S(1, σα max) S(1, σα max) max = 1 N With it, results for a tension σ a0 after n 0 cycles, a partial damage of S(n 0, σ a0 ): n0 S( n0, σ a) = n0 S(1, σα 0) = N Should S i be the partial damage of the i th load scenario, N i the maximal sustainable number of cycles for the corresponding stress and n i the actual number of cycles of the last scenario considered, the cumulative damage S will be: S = Si = i i n N i i The Wöhler Line was subject to several modifications; see Fig. 3 according to [4]. According to Miner Original (line a), the stresses smaller than the permanent sustainable values will not be considered as contributing to the damage. This methodology normally leads to an optimistic calculation because the stresses also lay bellow the fatigue limit contributing to the cumulative damage. In the diagram, the Wöhler Line has an inflexion point, and goes downwards, at a certain number of cycles (mostly at 1x10 6 cycles). In the elementary Miner Rule according to Corten-Dolan (line c), the damage contribution lying under the fatigue limit will be considered, while the fatigue limit line keeps the inclination without inflexion. This methodology normally tends to conservative results. In the Miner Rule according to Haibach (line b), the fatigue limit line introduces an inflexion with about half the inclination (2k-1) at the break point at N 00. With that, the stresses under the fatigue limit will also be considered and the calculated service live lay between the Miner s original and the elementary. For this reason, this Miner Rule modification is frequently be used as a compromise. 4 of 11

5 Figure 3: Wöhler Line Modifications, a) Miner Original, b) Haibach, c) Miner Elementary (Corten / Dolan) The damage model described above considers exclusively the damages caused by stress fluctuations. Other effects, such as, corrosion or thermal influences, will not be considered here. This could be of considerable importance for machines running relatively slow but in the case of F1 Racing these effects could be neglected though. 3 Calculation Model Obviously, the primary objective in car racing is a lightweight gearbox with a satisfactory service life. With this in mind, arose the need to create a calculation model allowing for the fast calculation of the service life of the whole driveline (gears, rolling bearings, shafts and shafthub connections) taking into consideration all relevant ancillary conditions (e. g., materials, lubrication, temperature, road surface irregularities, etc.). This model should clarify whether the influence of a certain weight-reducing measure in the service life is or is not sustainable. As any calculating engineer knows, such a calculated service life estimate has plenty of unknown errors, which could only be reduced by comparing calculation results against bench test runs and field experience. Factors influencing the calculated service life must be adjusted until the calculated service life coincides with measured values. For that, the quality of the loading assumption, the considered load- and safety coefficients and, naturally, the calculation methodology must also be maintained during a long period of time. It is easier to make qualitative prognoses, e. g., to determine by which factor the service life changes when an influential factor is changed (Relative Miner Rule, e. g. [5]. Correspondingly, it is also good to make a prognosis about a service life comparison of two components, which are in a similar environment under similar load regimes. Thus, at present, it is not yet the intention of eliminating the gearbox final testing on a test bench or in trial runs, but that of trying different gearbox combinations without having to carry out the very expensive service life preliminary tests. In the future, out of time and cost arguments, this first stage in certifying a car racing gearbox should be mainly done by using calculation methodologies. 5 of 11

6 The gearbox calculation model was developed with the KISSsoft / KISSsys software package: See Fig. 4. Whereas KISSsoft can calculate and optimize individual machine elements, with KISSsys it is possible to model and calculate a complete driveline in an easy and fast way. KISSsys defines also global parameters (which for gearboxes are torque, number of revolutions, external forces and others such as temperature or lubrication). The individual system parts will then be dimensioned and optimized with the KISSsoft s integrated calculation modules. From the global parameters, KISSsys administrates all these data and calculates local loads and revs passing them to the appropriate module for the calculation of an individual component. The proven and tested KISSsoft programs will then calculate the individual components. In the course of this, the user will get in his system different views: A two-dimensional functional chart, displaying the kinematics and the torque flows. A tree-type display of the systems logical structure Tabular views for handling the system parameters and displaying the detailed calculations data. Three-dimensional display of the geometry, for optical control and collision proofing. A programming language integrated in KISSsys, allows the definition of the relationships between individual dimensions, in which can also be used functions such as, e. g., known table calculation functions. Figure 4: KISSsys Calculation Model: Gearbox as Tree-structure- and graphical Display, In- and Output Tables, 3D graphical Display. 6 of 11

7 The Formula 1 gearbox was modeled for all gears, shafts and bearings. Based upon the calculation model, the resultant kinematics shows the complete gearbox properties, especially the activation / deactivation of the shifted / not shifted pairings, the change of sense for the reverse gear as well as the power distribution in the spur gear differential. Next to the gear data, the input includes also the following shaft- and bearing data: Chosen shift Nominal number of revs and torque Load spectrum Load / revs distribution in the differential Data on lubrication temperature Load coefficient and theoretical safeties Calculation methodologies for gears, bearings and shafts. The user can decide whether to import a load spectrum from a file, or to define it manually. The load spectrum will then be saved in a table and can be displayed as a graphical display. With this imported (or manually defined) spectrum, a damage calculation for the gears and shaft bearings will now be carried out. Now, in a first step, the damage calculation activates the relevant shift stage, calculates the current number of revs and torque at the input and, with these settings, carries out the kinematics calculation anew. Fig. 5 shows the gearbox diagram resultant from the kinematics calculation. The gearbox schematics shown in Fig. 5, results from the kinematics calculation. Figure 5: Gearbox Schematics with active Power Path (in Red), Gears (in Black) and Bearings (in Blue). 7 of 11

8 With the loads on the individual components resulting from this, (the non-shifted stages are not subject to load) only a partial damage will be determined taking into consideration load coefficients and theoretical safeties. This partial damage will now be cumulated for all of the stages. To calculate an equivalent load for this application is meaningless because the Wöhler Line slope for the different components varies and the gearbox kinematics change with time (with the speed shifting). The calculation of the root- and flank service life, as well as the differential static proof, takes place according to DIN 3990, ISO 6336 or AGMA For the bearings, this can be done either following the Standard L 10 calculation, the extended service life calculation according to the FAG Catalogue or to ISO 281. The strength of the materials employed, as well as the lubricant properties, will be taken from a KISSsoft database. 4 Adjusting between Test and Calculation It is clear that the calculation model has certain limitations: The calculation does not consider load sequence. Also not considered are the external impacts (going over pavement unevenness. These load peaks as far as they do not cause any fracture- can result in either an increase or a reduction of the service life. High tensile stresses may cause internal compressive stresses (in tooth root) which could even reduce damage due to subsequent tensile stresses. Likewise, an inversion of the power flow, i. e., when the wheels decelerate the engine, is also difficult to take into consideration. In these cases, flexural fluctuations on individual teeth occur, but with a very low number of cycles. That is why we worked with an estimated flexural fluctuation factor of K Wb =0,95. Also because of these limitations, the calculated service life will not match the measured values, neither could this be expected. However, the real gearbox service life data come from bench test runs (although these test cannot reproduce the real racing conditions) and from them, the calculated service life estimates can be adjusted. Should it be possible to create a certain factor separate for the individual gears and bearings- with which to over- or underestimate the service life calculation as compared with the test results, it would be reasonable to suggest considering this factor for future service life calculation. Now, this can be achieved by, instead of making the cumulative damage S=1, give it a value S V corrected by the relationship between measured and calculated service life, see Fig. 6. This procedure, known since around 1970 as Relative Miner Rule, leads to a higher accuracy in the calculation of the service life. However, the cost of determining such corrective factors for the drive line components are high and can only be supported by financially strong and well organized companies. 8 of 11

9 Load values measured during the race Assumed, idealized load values Load Spectrum determined from measured values Load Spectrum determined from assumed load values Calculated Service Life, N A.calc with S=1 Calculated Service Life, N V.calc with S=1 Test run measured Service Life N V.exp S V =N V.exp / N V.calc N A.calc.corrected =N A.calc * S V Figure 6: Relative Miner-Calculation s operational Flowchart concerning Service Life. 5 Summary In car racing, the objective behind the gearbox layout is to attain a predetermined service life (in the range of few hours) with a very minimum of weight. An overview of the service life of all components is extremely useful because, on the one hand, it immediately detects possible weak points and, on the other hand, the over-dimensioned components (thus, too heavy) are quickly identified. For instance, with a small modification of the tooth face width of the individual shift stages it is possible to even out the service life of the different gears without any problems. The toothing optimization costs, as well as the number of variants to consider, increase very quickly. Using a calculation model such as the one described above, a lot of time can be saved since various modifications can be simultaneously considered. A bonus benefit is the possibility of calculating a gearbox for different racing tracks (load spectra) with minimal additional costs. It is thus possible to quickly quantify the influence of lubrication, gear material or profile modification on the service life; with the increasing experience, it will be more and more possible to reduce although not completely eliminatecostly testing. 6 Bibliography [1] Palmgren, A., Die Lebensdauer von Kugellagern, VDI-Z 58, 1924 [2] Miner, M.A., Cumulative damage in fatigue, Journal of Applied Mechanics, Vol [3] Deutsches Institut für Normung, DIN 50100, Werkstoffprüfung, Dauerschwingversuch, Februar 1978 [4] Haibach, E., Betriebsfestigkeit, Verfahren und Daten zur Bauteilbeurteilung, Springer, 2002 [5] Gudehus H., Zenner H., Leitfaden für eine Betriebsfestigkeitsrechnung, Verlage Stahleisen, of 11

10 What is KISSsoft / KISSsys? KISSsoft for Analysis of Machine Elements KISSsoft is the leading software for Design, Optimization and Analysis of power transmissions and machine elements such as spur, helical, bevel, planetary, worm, face, rack / pinion gears, gear trains shafts and axles, roller and, journal bearings shaft-hub connections: splines, interference fit, keys, pins high strength bolts, hardness conversion, springs, belts, chains, KISSsoft enables you to calculate the strength or the Service Life of such elements based on standards such as ISO, DIN, AGMA, and BS or on standardized analysis methodology as provided by VDI or FKM. All calculations can be automatically documented using the reporting function. KISSsys for Systems Modelling KISSsys: Within KISSsys, a complete system of machine elements is compiled. KISSsys calculates the power flow and administers links between the various variables. For strength calculations of single machine elements, KISSsys uses the proven KISSsoft program. The calculated results appear tabulated or graphed in KISSsys, whereby the strength and service life of all elements in the construction can be viewed at all times. Kinematic calculation: Connect bevel, helical, worm and face gears Add shafts, bearings and shaft-hub connections to the model Include chain and belt drives Activate / de-activate couplings Add external loads and coefficient of efficiency 3D modelling: Based on the tree structure and gear / shaft geometry, a 3D geometrical model is generated for use in collision checks or for CAD export. The kinematics model defines the arrangement of the components in space. Special features: Calculation of load spectra for all elements included in the model (bearings, shafts, gears, splines ) Use of gearbox variants in the same KISSsys model Automatically perform sensitivity analysis, Model epicyclical or planetary gears like Wolfrom or Ravigneaux Automatically generate analysis documentation for a complete gearbox, Use scripting language for automation of routine tasks. Why use KISSsoft / KISSsys? Draft, optimize and verify your design with a single and easy to use tool. Improve the efficiency and safety of your design. Share data with manufacturers, customers or engineering consultants. Automatically get full documentation for each analysis (*.rtf format). Five languages available: English, German, French, Italian, Spanish. Programmed by fully trained, experienced mechanical engineers. Initially developed by a Swiss gearbox manufacturer: application-oriented tools for the engineer, it goes on being our focus. How to get started with KISSsoft / KISSsys Request more information, demo or test version from KISSsoft. Get started using the tutorials: Supported by trained mechanical engineers. Documentation, manuals and tutorials available. 10 of 11

11 KISSsoft AG Frauwis Hombrechtikon SWITZERLAND Tel: Fax: info@kisssoft.ch KISSsoft USA 3719 North Spring Grove Road Johnsburg (IL) USA Tel: Fax: dan.kondritz@kisssoft.com Sales, customer support, engineering Hanspeter.Dinner@KISSsoft.ch Stefan.Beermann@KISSsoft.ch Dan.Kondritz@KISSsoft.com (USA) Software development Ulrich.Kissling@KISSsoft.ch Markus.Raabe@KISSsoft.ch Please send me O Information on software modules and prices O Test version of software O Information on training seminars O Please specify O Please contact me Name. Company Full postal address Phone.. 11 of 11