NUMERICAL AND EXPERIMENTAL MODAL ANALYSIS OF GEAR WHEEL

NUMERICAL AND EXPERIMENTAL MODAL ANALYSIS OF GEAR WHEEL ANDREA CIZIKOVA 1, KATARINA MONKOVA 1, JAROMIR MARKOVIC 2, JAKUB CIENCALA 3, 1 Techical Uiversity of Kosice Faculty of Maufacturig Techologies Presov, Slovak Republic 2 Slovak Metrological Society, Baska Bystrica, Slovakia 3 Techical uiversity of Ostrava Faculty of Egieerig, Ostrava, Czech Republic DOI : 10.17973/MMSJ.2016_11_201654 e-mail: kataria.mokova@tuke.sk The paper deals with the theoretical ad practical aspects that are ecessary for idetifyig of modal parameters of the gear wheel as a compoet of the plaetary gear mechaism, which are basic for the dyamic aalysis. The paper presets thematically itegrated part of the results of umerical ad experimetal modal aalysis coducted o a particular gear wheel. The paper aimed to create experimetal ad computatioal model ad subsequet verificatio of selected theories i defiig the modal parameters, comparig results obtaied umerical ad experimetal modal aalysis ad assessmet of dyamic characteristics of the compoet with respect to its operatig coditios. Numerical aalysis was carried out i a computer CAD/CAM program PTC Creo, which works o the Fiite Elemet Method. The results obtaied by experimetal modal aalysis were evaluated by measurig system PULSE.Experimetal measuremets have show that the values of atural frequecies alog with the atural shapes are adequate ad that o atural frequecy is ot the same or close to the gear frequecy, so the compoet ca be used i the real practice. KEYWORDS modal aalysis, vibratios, modal parameters, excitatio, frequecy respose fuctio 1 INTRODUCTION I the past two decades, modal aalysis has become a major techology i the quest for determiig, improvig ad optimizig dyamic characteristics of egieerig structures. To appreciate its sigificace i the moder egieerig area ad its potetial for future sciece ad techology, it is appropriate to capture some of the backgroud facts which will help to uderlie this uique techology. Curret modal aalysis methods for idetifyig structures are usually based either o very simple ad flexible techiques, or o very sophisticated computatioal oes. Nowadays, the modal aalysis is cosidered to be a moder study of the dyamics, which allows to determie the modal properties of the examied object. The modal properties are used to obtai the dyamic descriptio of the mechaical system. I techical practice, most of the problems associated with excessive oise or mechaical vibratio, caused by the characteristics of the mechaical system [Cacko 2014]. These properties are called "modal". It is obvious that thaks to the well-kow modal parameter it is possible to prove ad to predict the resultig properties of the mechaical system. The modal characteristics may be calculated ad also evaluated uder the test modal aalysis[baro 2016, Hloch 2008]. To describe the modal parameters ad behaviour of mechaical systems, there are used possibility of degradatio of complex oscillatory plot ito sub happeig, each of which is characterized by atural frequecies ad mode shapes its ow. [Milacek 1992] The resultig parameter of modal aalysis belogs to the atural frequecy of the system, custom shapes vibratio ad modal dampig of the system. Modal aalysis is used to solve a umber of techical problems such as the specificatio of modal frequecy system which, i accordace with excitatio frequecies ca lead to resoace; the critical speed, etc., whe verify the reliability of simplified mathematical models assembled i the so-called geometric coordiates, comparig the results of experimetal measuremets, the modificatio of mechaical systems coectig additioal elemets to such retue their out of bad harmful effects. [Krolczyk 2014]. 2 CONDITIONS OF THE EXPERIMENTS The aim of experimet was to idetify the modal parameters of the gear wheel as the compoet of plaetary gear. Material of gear wheel was carbo steel C45. The PULSE aalyser, model 2827-002 with its additioal modules, was used for data processig. PULSE aalyser cosists of a measurig module type 3109 ad commuicatio module type 7533. The measurig module is used to create the geometry model, ad costructig the measurig poits. The commuicatio module is used to export ad import data, which is ecessary for example at the base averagig FRF (Frequecy Respose Fuctio) fuctio. As the vibratio sesor was used piezoelectric accelerometer type 4374. Whe choosig a acceleratio sesor, it is ecessary to take ito accout its weight durig measuremet that is ot a case of distortio of measuremet data. For applicatios of the sesor of acceleratio it is ecessary to emphasize o its orietatio. Correct orietatio of the acceleratio sesor is based o compliace of its coordiate system with coordiate measuremet systems structure. Wakig ivestigated of wheel was carried out usig a modal hammer Brüel & Kjær, type 8203, which had a plastic tip. To amplify the sesor sigal, the type of amplifier 2627 A was used, which could also be used as a simple coverter i default mode with a sesitivity of 1 mv / pc. I the experimet, two amplifiers of the same type were used, oe for amplifyig the sesor sigal ad the secod oe for the respose to the alarm sigal amplificatio i modal hammer. The measurig set is show i Figure 1. Figure 1. The measurig set 3 METHODS USED AT THE EXPERIMENTS Cotemporary desig of complex mechaical, aeroautical or civil structures requires them to become icreasigly lighter, more flexible ad yet strog. These striget demads ofte made them more susceptible to uwated vibratios. Where the vibratio of a structure is of cocer, the challege lies o 1232

better uderstadig its dyamic properties usig aalytical, umerical or experimetal meas, or a combiatio of them. The process of determiig the iheret dyamic characteristics of a system i forms of atural frequecies, dampig factors ad mode shapes, ad usig them to formulate a mathematical model for its dyamic behaviour is called modal aalysis. The formulated mathematical model is referred to as the modal model of the system ad the iformatio for the characteristics is kow as its modal data. The most used method to evaluate the umerical calculatios is Fiite Elemet Method - FEM. The umerical method used to simulate the stresses, deformatios, atural frequecies, heat flow, electromagetics, fluid flow, ad so o. created the physical model. The priciple of this umerical method is based o discretizatio of cotiuum to some umber of fiite elemets, ad these are beig ivestigated parameters set out i the idividual grid poits. The research method used at experimetal method was the plaed experimet. The most appropriate method to the processig of dyamic sigals is a "Fourier Trasform" (FFT - Fourier Trasform). Fourier trasform fuctios by covertig the time depedece of the measured values i the frequecy domai. It also compesates the primary fuctio of harmoic sequece features that have differet frequecies ad phases. Compesatio of the process is the eed of makig the product of simple waves approximately the shape of primary fuctios. Fourier trasformatio is defied as the sum of the substitutio x t of harmoic fuctios [Malotova 2016]. of ay fuctio Expressio of Fourier trasformatio ca be writte i the followig form [Bilosova 2011]: x a 2 t 2t (1) T T 0 t a cos b si, where 2 1 a ad b are the Fourier coefficiets of fuctios. Applyig the Fourier trasformatio i digital form requires the creatio of appropriate algorithms for processig discrete data, which is quite difficult. Curretly optimized algorithm called "Fast Fourier trasform". Experimetal modal aalysis to determie modal parameters examied i the housig with a frequecy respose, respectively frequecy trasfer fuctio. FRF - Frequecy Respose Fuctio has bee implicated i respose to the excitatio system. It is essetial to describe the relatioship of liear mechaical systems. I geeral, the frequecy trasfer fuctio ca be defied as a measuremet of the time course of the dyamic mechaical excitatio system f t ad the correspodig time course of respose to the mechaical system xt i the frequecy domai. The mai reaso for the frequecy trasfer fuctio lies i its simple usage, i which it is possible to describe the respose of the actual mechaical system [Trebua 2012, Petru 2013]. Frequecy trasfer fuctio is defied by the formula [Frakovsky 2011]: X m H. (2) F N From the equatio (2) expresses the frequecy trasfer fuctio as the ratio of output ad iput to the system ad also represets the property of represetig the compliace of the aalysed mechaical dyamic system [Hosedl 2014] system. The values of the modal parameters, which are acquired by computig methods are compared with measured values acquired by the experimetal aalysis. I the techical practice these values coicide oly occasioally. For the theoretical modal aalysis, will draw up a motio equatio ad the resultig values of the modal parameters is obtaied by applyig the "modal trasformatio". Basis modal trasformatio cosists i the replacemet of the system with each other of the commitmet of homogeeous differetial equatios by a system of idepedet, isolated to resolvig, homogeeous differetial equatios. Complex calculatios of the system of physical equatios is i may cases ecessary to udergo a eeded simplificatio. The theoretical modal aalysis becomes idispesable i the case of the absece of the actual mechaical system [Stojadiovic 2014, Pada 2014]. To implemet the theoretical modal aalysis ca be summarised i the followig steps, amog which iclude [Bilosova 2011]: 1. compilatio of the motio equatio - physical model, 2. aalysis of the free oscillatio - modal model, 3. aalysis of forced vibratio for harmoic excitatio - respose model. I a experimetal modal aalysis will determie their ow frequecy, their ow shapes vibratio ad modal dampig of the mechaical system usig the experimetally determied set of data frequecy trasfer fuctio. These fuctios are ofte referred to as "frequecy characteristics". However, oly the properly doe experimetal modal aalysis determies what the most accurate real modal properties of a mechaical system. [Ugureau 2011, Krehel 2016] The opposite procedure is i the implemetatio of the experimetal modal aalysis, which icludes the followig steps [Metelski 2015]: 1. measurig appropriate sets of frequecy respose fuctios - respose model, 2. the aalysis of measured data - modal model, 3. the implemetatio of additioal calculatios - physical model. I the cotext of experimetal measuremets was made modal aalysis of the gear wheel by settig the aalyser i the measurig system of the PULSE. I the measurig system, it was defied the apparatus such as the modal hammer ad accelerometer with amplifiers with their basic parameters. The ext step was the creatio of the geometry i the software MTC Hammer. O the model, 81 measurig poits were highlighted. (Figure 2) 4 EXPERIMENTAL AND NUMERICAL MODAL ANALYSIS Modal aalysis ca be performed i two differet ways either i the theoretical plae as a calculatio or o a practical level by performig experimetal measuremets o the real physical Figure 2. Geometry models created i the measurig system PULSE After geometry creatio, the ext step was the locatio of the sesor, its orietatio ad the defiitio of excitatio place. It was also importat to defie the referece poit that was 1233

determied i excitatio poit with the umber 10, show i Figure 3. aalysis) simulates ucostraied body, the same coditios were setup at umerical aalysis. Cosequetly, it was ecessary to defie the frequecy rage i which the aalysis was carried out. Figure 3. Locatio of the referece poit o the gear wheel Whe settig the parameters of the FFT aalysis, it was ecessary to defie the umber of discrete time data, which are used for the calculatio of the spectrum. The set value for the umber of discrete time data was 6 400 of spectral lies. Discrete time data are ecessary whe calculatig the spectrum. Ivestigated the frequecy has bee set i the iterval 0-25,6 khz with a frequecy resolutio of 4 Hz. At the studied gear wheel, drew measurig poits usig the paper rulers. The distace of the measurig poits was determied by the software whe creatig the geometry of the model. A importat part of the experimet was savig the referece elemet, that was the particular purpose for which the modal test shall be coducted. The gear wheel was studied positioed o a soft foam that was show o Figure 1. The way of the excitatio was elected by meas of modal hammer. The choice of the method of the excitatio was due to the small size of the wheel, the shape ad stiffess. After settig all the ecessary data, it was possible to realize the measuremet of the modal data. The measuremet of the modal data cosisted of the sesor of the acceleratio attached to the referece poit usig the beeswax, see Figure 3. After coectio of sesor with referece poit, the excitatio of gear wheel by modal hammer followed. Excitatio, the gear wheel o each of the measuremet poits took place o the stroke of modal hammer. Experimetal modal aalysis i the idividual poits of measuremet carried out i two directios - radial ad axial directio. A modal hammer to each of the measurig poits struck five times i oe directio, because of the averagig FF (Fast Fourier Trasformatio) fuctio. This process is the repeated the experimet i makig the secod directio stroke of hammer. To view the various modes of vibratios was used module REFLEX PULSE system. Parallel with the experimetal modal aalysis, the mathematical model of the gear wheel was prepared. The calculatio was carried out by the method of fiite elemets i the software PTC Creo. It is preseted i Figure 4. Computig part of the modal test cosists first i the creatio of a model of the wheel ad selectig the material - steel. After you create the model, defie the material properties of the wheel. Amog the basic material properties iclude the specific desity of the material, whose value is 7,82708 10-6 kgmm -3, the value of the Youg's modulus is 2,1.10 8 kpa, Poisso's value is 0,27. The value of the Youg's module ad Poisso's umbers correspod to the tabular value for steel. I the ext, steel as the material was assiged to compoet. After this step, it was ecessary to create a fiite elemets grid that has bee defied with desity of 5 mm. I relatio to the fact that the soft foam (o which the gear was positioed durig experimetal Figure 4. Virtual model for umerical aalysis created i PTC Creo 5 RESULTS AND DISCUSSION The samplig frequecy was setup at 6400 Hz with the accuracy 2 Hz. Fial FRF is preseted i the Figure 5. The atural frequecies of the gear were defied by meas of the RFP (Ratioal Fractio Polyomial) method o the base of the measured data. They are listed i the Table 1, where the values of frequecies achieved by meas of FEM method i software PTC Creo are also preseted. Figure 5. Fial FRF fuctio Table 1. Acquired values of the atural frequecies Natural frequecies Numerical method Experimetal method f1 [Hz] f2 [Hz] f3 [Hz] f4 [Hz] 4207,6 8 109,0 12535,3 13640,4 4 289,3 8 212,2 12 751,4 13 861,3 Moreover, the first four mode shapes, that correspod to the atural first four frequecies, were evaluated withi the both methods of modal aalysis. They are show i the Table 2. 1234

Table 2. The first four atural shapes of gear wheel obtaied by umerical ad experimetal modal aalysis From the Table 1 ad ew measurig ad computig resources, without which it would be impossible to obtai, ad the to process the measured data. Today there are may scietific papers, studies ad publicatios o the use of modal aalysis ad also thaks to the possibility of purchasig the ecessary software with devices ad udoubtedly good facilities of laboratories. Alog with the developmet of moder computer techology, experimetal modal aalysis has become the mai tool for solvig complex structural vibratio problems. For a existig egieerig structure, it provides vital iformatio o its dyamic behaviour, thus permittig itelliget solutios to vibratio problems the structure may be experiecig. I the article two approaches to the modal aalysis were preseted focused o the umerical ad experimetal methods. Obtaied data of modal aalysis will be the base for the dyamic aalysis ad for the ext experimets that authors are goig to perform. Table 2 it is clear that the data is comparable, so the boudary coditios ad other settigs were defied correctly. The differeces i frequecies betwee the values achieved by umerical ad experimetal aalysis were probably caused by the fact that real body was ot totally ucostraied (it was positioed o the soft foam) ad also by uequal umbers of fiite elemets. While at the evaluatio of experimetal aalysis, the grid of fiite elemets was row; it meas that the elemets were several times greater i compariso with umerical modal aalysis i software PTC Creo. The coformity of the achieved results also presets the Figure 6, where the first eleve atural frequecies obtaied by both method are compared. Figure 6. Compariso of the depedecies of atural frequecies measured at idividual atural modes Natural frequecies were also compared with the teeth frequecies of the gear fg. There are three various umbers of teeth at the gear, so the teeth frequecies fg 2, fg 3, fg 4were computed o the base of rotatioal frequecy of shaft fs ad umber of teeth of the gear 2, 3, 4: It is clear that the highest calculated teeth frequecy is 1708 Hz ad it is sufficietly away from lowest measured atural frequecy 4 207,6 Hz. 6 CONCLUSION Modal aalysis is curretly oe of the fields of sciece that is developig rapidly. This is primarily due to the availability of (3) (4) (5) ACKNOWLEDGMENT The paper was published thaks to the grats VEGA 1/0614/15; KEGA 087TUKE-4/2015; KEGA 013TUKE-4/2014 with direct support of Miistry of Educatio of the Slovak Republic. REFERENCES [Baro 2016] Baro, P. et al. Proposal of the kowledge applicatio eviromet of calculatig operatioal parameters for covetioal machiig techology, Key Egieerig Materials, Volume 669: (2016) pp. 95-102 [Bilosova 2011] Bilosova, A. Experimetal modal aalysis. VSB TU Ostrava, 2011. (i Czech) [Cacko 2014] Cacko, P., Kreicky, T., Dobrasky, J. Impact of a excessive wear of bearig o the mechatroic devices. I: Applied Mechaics ad Materials. Vol. 460 (2014), pp. 99-106. ISSN 1660-9336 [Frakovsky 2011] Frakovsky, P. et al. Use of modal aalysis i the diagosis of the vibratio of machiery, 2011 (i Slovak) [Hloch 2008] Hloch, S. et al. Experimetal study of surface topography created by abrasive waterjet cuttig, Strojarstvo. Volume 49(4): (2008) pp. 303-309 [Hosedl 2014] Hosedl S. Theory based maagemet ad software support of property drive desigig of techical products, MM Sciece Joural, Vol. 10, (2014) pp. 496 502, DOI: 10.17973/MMSJ.2014_10_201413 [Krehel 2016] Krehel, R., et al. Diagostic aalysis of cuttig tools usig a temperature sesor. Key Egieerig Materials, Volume 669: (2016) pp. 382-390 [Krolczyk 2014] Krolczyk, G. M. et al. Ifluece of techological cuttig parameters o surface texture of austeitic stailess steel, Applied Mechaics ad Materials, Volume 693: (2014) pp. 430-435 [Malotova 2016] Malotova, S., Cep, R., Cepova, L. et al. Roughess Evaluatio of the Machied Surface at Iterrupted Cuttig Process I: Maufacturig Techology, Volume 16, Issue 1, (2016) pp. 168-173 [Metelski 2015] Metelski, A. et al. Taguchi desig of experimet versus dyamic programmig approach i the optimizatio of turig process, Applied Mechaics ad Materials, ISSN 1662-7482, Vol. 808, (2015) pp 66-71 [Milacek 1992] Milacek, S. Modal aalysis of mechaical vibratios.cvut Praha, Czech Republic, (1992) ISBN 80-0100- 872X [Pada 2014] Pada, A. et al. Progressive techology diagostics ad factors affectig machiability, Applied Mechaics ad Materials, Volume 616: (2013) pp. 183-190 1235

[Petru 2013] Petru, J. et al. Ifluece of cuttig parameters o heat-affected zoe I: Techical Gazette Vol. 20/ 2, pp. 225-230 [Stojadiovic 2014] Stojadiovic, S. M., Majstorovic, V. D. Developig egieerig otology for domai coordiate metrology, I. FME Trasactios, vol. 42(3), (2014) pp. 249-255 [Trebua 2012] Trebua, F., Simcak, F., Huady, R. Vibratio ad modal aalysis of mechaical systems, Kosice, (2012), ISBN 978-80-442-1206-4 (i Slovak) [Ugureau 2016] Ugureau, M. et al. Iovatio ad techology trasfer for busiess developmet, Procedia Egieerig, Volume 149: (2016) pp. 495-500 CONTACTS: Eg. MSc. Adrea Cizikova, PhD. Assoc. Prof. Eg. MSc. Kataria Mokova, PhD. Techical Uiversity of Kosice Faculty of Maufacturig Techologies Departmet of Techological Systems Desig Sturova 31, 080 01, Presov, Slovak Republic e-mail: a.cizikova87@gmail.com e-mail: kataria.mokova@tuke.sk Eg. MSc. Jaromir Markovic, PhD. Slovak Metrological Society Hviezdoslavova 1124/31 97401 Baska Bystrica, Slovak Republic e-mail: slm@slm.sk Eg. MSc. Jakub Cieciala Techical Uiversity of Ostrava Faculty of Mechaical Egieerig Departmet Applied Mechaics 17. listopadu 15, 708 55 Ostrava - Poruba, Czech Republic e-mail: jakub.cieciala@vsb.cz 1236