ANALELE UNIVERSITĂłII EFTIMIE MURGU REŞIłA ANUL XVII, NR. 2, 2010, ISSN 1453-7397 Petru Florin Minda, Ana Maria Budai Application o Mathematica Sotware or Estimate the Fatigue Lie Time Duration o Mechanical System The paper present how we can use Mathematica to solve the equations types usually used to determinate the maximum stress cycles that can be support by a mechanical system until he will be out o use. To illustrate the type o equations used in specialized literature to estimate atigue lie time duration was chosen a speciic case o mechanical structure applied to atigue. It is about lever button o runner blade mechanism o Kaplan turbine, that in unction support a very intensive alternative strain. Keywords: atigue, equation, algorithm, mechanism 1. Introduction Lately, saety and reliability have become bywords to the mechanical industry at a time when energetically economy has dictated the use o the most sophisticated design procedures to optimize the use o material without increasing the probability o component ailure. With the need to provide an economical design through mass reduction, many o the older design procedures have been set aside in lieu o the more recent and comprehensive design procedures. Taking in consideration the extensive development o computer sotware package the time and costs o saety designing are greatly reduced. The sotware packages not only calculate the low o loads through components and the stresses at stress concentrations, but also are able to encompass large volumes and many channels o real time loading histories, and combine the two or atigue lie evaluations o complete mechanical systems [1]. The great disadvantage o that kind o sotware package is the costs and the technical limitation imposed by the huge data base necessary to be created or a most precisely results. Usually, the method used or estimate atigue lie duration presume determination o real stress in key point o a structure (the point with the 203
greatest risk or crack) and than to determinate the maximum stress cycles that can be supported by the structure. The irst part o the problem is resolved using the statically linear stress analyse (inite elements method). Currently, to ind number o cycles until ailure, is used Wıhler diagram o the analysed system material. To rise such kind o diagram supposes o lot o time and supplementary costs. Like an alternative, or solving the second part o the problem, a ew researchers developed analytical equations that, even still disputed, are already applied. 2. Initial consideration The two general equation models about us taking beore are the Morrow and Smith-Watson-Topper (SWT) models [1]. The Morrow equation (1) modiied the elastic term o the strain lie equation or introducing the local mean stress into the strain lie equation: ε σ σm b = ( 2N ) ( ) c + 2N 2 E ε (1) where: ε - total strain range; σ - true racture strength (value o σa at one reversal); σm - the mean stress; E modulus o elasticity; ε - atigue ductility coeicient; N - number o cycles to ailure; thereore reversals to ailure; b - atigue strength exponent; c - atigue ductility exponent. 2N is equal to the number o The SWT model assumes that by atigue lie or any condition o local mean stress depends on the product o( σ + σ ε : m a) a ε 2 2b ( ) ( ) ( ) ( ) b + σ c a+ σm = σ 2N + σ ε 2N (2) 2 Both the Morrow and SWT approaches are in current use by the atigue community and no consensus exists as which one is superior to other. Overall, the belie o the atigue community is that the SWT approach provides good results and it is a good choice or general use. However, a review o current literature suggests that the Morrow result provides better results or compressive mean stresses while SWT gives non-conservative results. 204
To obtain more approach results rom real exploitation conditions must calculate the number o cycle or every representative regime that occur in time on service. To select these regimes is necessary to have a good loads history usually obtained rom monitoring process that always exists in modern industry. Ater we have all the number o cycles the last problem is to apply a cumulative damage methodology or calculating the atigue lie. On these paper we will ocus on the method used or solved such equation like the one propose by Morrow or SWT. To resolve equation (1) will be used specialized sotware capable to obtain real answers rom equations with a high degree o diiculty. It is about sotware named Mathematica a ully integrated sotware environment or technical and scientiic computing. As we already said without such o sotware, actually, the problem can not be solved Mathematica is one o the world's most respected sotware systems, and an essential tool or leaders in science and technology across the globe. Legendary or its sophisticated capabilities, yet easy enough to be used by children, Mathematica has considered as the most powerul general computation system ever created-- and a complete computational environment or millions o people. Whether they have tasks that involve numbers, ormulas, unctions, graphics, data, documents, or interaces, Mathematica gives automatic access to by ar the largest collection o algorithms ever assembled[5]. 2. Results and comments The study system, used to exempliy how the sotware solve such equations is represented by lever s button rom runner blade mechanism o a Kaplan turbines. Material o laver is 30CrNiMo8 and his mechanical properties are in accordance with current standards. The work regime taking in considerations or this analyze are: o H = 25m and ϕ = 10 With H is noted the head waterall and φ represent the angular positions o runner blades. The equation choused, in accordance with speciic exploitation conditions o the turbine, to typiy the sotware algorithm are the Morrow relation (1). The entry dates necessary to apply relation (1) are: - characteristically dates or regime one necessary to calculate the number o stress cycles N [2] : - σ = 262, MPa ; a 3 - σ = 229,6MPa. - true racture strength [4]: σ = 1230MPa. - atigue ductility coeicient [4]: ε = 0, 7. - total strain range [4]: ε = 0, 0025. m 205
- modulus o elasticity: E= 205000MPa. The entry dates, except modulus o elasticity E, are calculate using measured dates in situ, respectively common ormula proposed by specialized literature. Having all the necessary entry dates it is obtained an equation having the ollowing orm: 0,07 0,7 0,00464 N + 0,4309 0,00125= ( ) ( ) 0 N As seen, such o equation is practically impossible to solve without a specialized sotware. Here intervene the Mathematica sotware. Built into Mathematica is the world's largest collection o both numerical and symbolic equation solving capabilities with many original algorithms, all automatically accessed through a small number o exceptionally powerul unctions. Mathematica's symbolic architecture allows both equations and their solutions to be conveniently given in symbolic orm, and immediately integrated into computations and visualizations.[3] Solve exact solutions to equations and systems NSolve general numerical solutions to equations and systems FindRoot numerically ind local roots o equations DSolve exact solutions to dierential equations NDSolve numerical solutions to dierential equations NSolve gives as a general way to ind numerical approximations to the solutions o polynomial equations. Finding numerical solutions to more general equations, however, can be much more diicult, FindRoot gives as a way to search or a numerical root o a unction or a numerical solution to an arbitrary equation, or set o equations. The Mathematica sequence or inding the root o 0,07 0,7 ( ) + 0,4309 ( ) 0,00125= 0 0,00464 N is in igure 1. N 206
is: Figure 1. The result o compilation with Mathematica sotware, or our speciied system 8 N = 1,38 10 cycles Finally must be speciied that the result obtained, in that orm, it is know useul. That kind o equation can be applied only when we have mechanical system operating to constant operating loads. It is known, by practice, that such systems do not exist. For a real system is necessary to known the history o exploitation conditions, the made many such determination and then apply a cumulative damage methodology to ind the inal result. 4. Conclusions The paper presents an alternative method that easier can be used to determinate de maximum number o stress cycles that can be support by a mechanical system, in known operating conditions. The Mathematica sotware provides an inexpensive method whit minimal time allocated, to solve a very important problem o modern design concept. Only the design engineer is able to decide which method wants to apply. The most part o engineers that already used the analytic method to solve real problem, considers that this alternative way is precisely enough to be used in designing mechanical systems. 207
Reerences [1] ***** Automotive Steel Design Manual, Desktop Engineering Int l Inc., CARS ASDM, 2009. [2] ***** Lie time duration calculcus or lever o runner blade mechanism o CHE Portile de ier I turbine, CCHAPT, Technical Report no. U-09-400-289, November, 2009. [3] Minda A., Stoica D., Tomescu M., Solving nonlinear equations with Mathematica, Analele Universitatii Etimie Murgu ReşiŃa, Anul XV, Nr.1, 2008, [4] Oliviu R., s.a., Oboseala metalelor, Vol I si II, Ed. Tehnica, Bucuresti, 1992. [5] *****http://reerence.wolram.com/mathematica/guide/equationsol ving.html Addresses: Petru Florin Minda, Etimie Murgu University o ReşiŃa, PiaŃa Traian Vuia, nr. 1-4, 320085, ReşiŃa, lorin.minda@just.ro Univ. Assist. Eng. Ana Maria Pittner, Etimie Murgu University o ReşiŃa, PiaŃa Traian Vuia, nr. 1-4, 320085, ReşiŃa, am.pittner@uem.ro 208