INVESTIGATION OF THE OPERATIONAL MODAL ANALYSIS APPLICABILITY IN COMBUSTION ENGINE DIAGNOSTICS

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1 INVESTIGATION OF THE OPERATIONAL MODAL ANALYSIS APPLICABILITY IN COMBUSTION ENGINE DIAGNOSTICS Macin Łukasiewicz Univesity of Technology and Life Science ul. S. Kaliskiego 7, Bydgoszcz, Poland tel.: Abstact The pape contains application of opeational modal analysis in use of combustion engine chosen technical state identification. The combustion engine No. 38C.2.48 with.4l. swept capacity, powe 55 kw, geneally applied to Fiat was the investigation object. It makes possible to intoduce geneated vibation signals as well as the investigation of his adjustment influence on the combustion engine vibation signals change. Conducted eseaches of combustion engine depended on delimitations of viboacoustics measues fo fit engine and compaison of this measues, with measues appointed fo damaged engine (e.g. damaged injecto) and accomplishment the assessment of eceived esults influence on engine state by opeational modal analysis. The pesent eseach use vibation methods as Opeational Modal Analysis to ecognize the technical state of the system. Keywods: opeational modal analysis, diagnostic infeence, combustion engine. Intoduction Owing to the complicated chaacte of vibations, technical diagnostics of combustion engines is vey difficult and only a few of poposed methods of diagnosis can have a wide technical application. Application of the modal analysis as one of the viboacoustics tools is the new appoach to the technical analysis of the combustion engine. Following the changes of modal model paametes as a esult of engine maladjustment, waste, damages o its failue is the main idea of opeational modal analysis. The modal paametes ae [4,5,6]: modal fequency, modal damping, mode shape. Modal analysis is the pocess of detemining the modal paametes of a stuctue fo all modes in the fequency ange of inteest. The ultimate aim is to use these paametes to constuct a modal model of the esponse. Opeational modal is the name fo the technique to do modal analysis on opeational data - cases whee we do not excite the stuctue atificially but just allow the natual opeating loads to excite the stuctue. 2. Opeational modal analysis LSCE method theoy In this pape Least Squaes Complex Exponential method was used to detemine the modal model paametes, by which the coelation function is appoximated by the sum of exponentially decaying hamonic functions. This method, applied to impulse esponse of system is a well known method in modal analysis yielding global estimatos of system poles the oot of the tansfe function denominato. It can be poved that the coss coelation function can be used to identify modal system paametes in identical way as does the impulse esponse of the system. [4,5]

2 The dynamic equation of the system motion can be expessed by fomula [4,5]: whee: M & x + Cx& + Kx = F(t), () M,C,K mass, damping and stiffness matices, & x, x, & x acceleation, velocity and displacement vectos, F(t) vecto of exciting foces. The next step was tansfomation of fomula () to pincipal coodinates applying the tansfomation expessed by the fomula [4,5]: whee: n x (t) = Ψq(t) = Ψ q (t, (2) = ) Ψ matix of modal vectos, the columns of which ae eigenvectos coesponding to the given fee vibation fequency, q pincipal (modal) coodinate, n numbe of vibation foms included in the model of vibation foms. Assuming that damping is small and popotional, on substituting elation (2) into fomula () and multiplying by Ψ T, de-coupled equation set is obtained in the fom [4,5]: whee:... 2 T q (t) + 2ξωn q(t) + ωnq (t) = Ψ f (t), (3) m ω n is the -th fee vibation fequency, ξ is the modal damping coefficient fo the -th vibation fom, m is the modal mass. Assuming zeo initial conditions fo abitay excitation, the solution of equation (3) can be witten in the fom of convolution [4,5]: t T q (t) = ψ f ( τ)g (t τ) dτ, (4) whee: g (t) = fo t <, g (t) = m ω d 2 2 d = ωn ( ξ ) exp( ξ ω n t) sin ω n t fo t, ω is the fequency of damped fee vibations.

3 Making use of solution fomula (4) fo modal coodinates to detemine the solution in genealised coodinates x(t), we obtain the fomula [4,5]: whee: n ψ T x (t) = ψ f ( τ)g (t τ) dτ, (5) = n numbe of vibation foms taken into account in the solution. t Coss coelation function fo two esponse signals at point i and j, esulting fom an excitation applied at point k in the fom of white noise has fom [4,5]: whee: E o denotes the expected value opeato. R (T) = E [x (t T)x (t)], (6) ijk o ik + If we know elation between coss coelation function, having the fom of a sum of exponentially decaying hamonic functions and impulse tansition function fo diect application to do modal analysis tests, coelation function can be tansfomed to the fom [4,5]: whee: ϑ the New phase angle, G j constant. 3. Model of diagnostics signal geneation n i j ij (T) = exp( ξ ωnt)sin( ωdt + ϑ ) = m ωd ik ψ g R, (7) The investigations object was a combustion engine no. 38C.2.48 applied to the Fiat and Lancia cas that is shown on figue. Basis on this system duing investigations was ceated model of diagnostics signal geneation [2,3]. The poposed model of combustion engine diagnostic signal geneation is shown on figue 2. The eceived signals in the any point of engine body ae the sum of the answe at all elementay events u n (t,, outputs in individual patial dynamic aangements with the pulse function of input h n (t,. These influences afte passing by pope dynamic aangements ae sum up on the engine body, on chosen points was measued by the vibation tansduces. As a esult of conducted measuements output signals was used to estimation. By n (t, was maked accidental influence stepping out fom pesence of dynamic mico effects such as fiction [2,3,6]. Conducted investigations of combustion engine depended on delimitations of viboacoustics measues fo fit engine and compaison them with measues appointed fo damaged engine (eg. damaged injecto) and accomplishment the assessment of eceived esults influence on engine state by opeational modal analysis methods.

4 Fig.. The investigation object combustion engine No. 38C.2.48 Natual event daw Patial dynamic aangement Possible places of measuement Engine body Mate of kinematics couple h(t,θ x(t,θ x2(t,θ u2(t,θ Adjustments of moto opeation h2(t,θ x3(t,θ u3(t,θ Engine accessoies with otatos h3(t,θ System of engine mount h4(t,θ x4(t,θ u(t,θ n(t,θ 2 3 u4(t,θ un(t,θ Anothe hn(t,θ xn(t,θ n- Measuement System n Fig.2. Combustion engine diagnostics signal geneation model[2,3] 4. The investigations esults The modal model of combustion engine was ceated fo put dynamic states on the basis of eceived measuing esults. Duing investigations have been done viboacoustics measues fo fit engine and fo engine with damaged injecto and spak plug fo each cylinde. As a esults of engine modal tests was ceated the stabilisation diagams fo each technical state. Basis on the stabilisation diagams was ceated the modal model includes modal ode, natual fequency and damping [2,4,5]. Figue 3 display the window with the stabilization diagam of engine in fit state. Table pesent the esults of modal investigations modal model fo put engine technical states. Basis on modal model paametes and estimatos of viboacoustics signal eceived duing investigations in table 2 was shown the main obsevation matix fo engine pefomance. The final obsevation matix of engine pefomance descibed 3 symptoms. The matix have six modal symptoms (ω - fist natual fequency, ząd - modal ode of fist natual fequency, ξ - modal damping coefficient of fist natual fequency, ω2 - second natual

5 fequency, ząd2 - modal ode of second natual fequency, ξ 2 - modal damping coefficient of second natual fequency) and the last seven symptoms ae vibation pocess (H(f) eal pat of tansfe function, H(f)L imagine pat of tansfe function, γ 2 xy coheence function, A RMS(t) Root Mean Squae in time domain, β kut Kutosis, C s - Cest facto, I - Impulse facto). Fig. 3. Opeational Modal Analysis stabilization diagam of investigated engine in fit technical state: s stable pole, v the fequency of vibation and modal vecto is stabilized, d the fequency of vibation and the stifling is stable, f only the fequency of the vibation is stable, o the pole is unstable Table. Paametes of modal model eceived duing investigations fo put of 9 technical states of combustion engine: ω is the fee vibation fequency, Ode ode of the model, ξ is the modal damping coefficient Technical state - fit engine 2 - damaged injecto on 4 th cylinde 3 - damaged injecto on 3 th cylinde 4 - damaged injecto on 2 th cylinde 5 - damaged injecto on th cylinde 6 - damaged spak plug on 4 th cylinde 7 - damaged spak plug on 3 th cylinde 8 - damaged spak plug on 2 th cylinde 9 - damaged spak plug on th cylinde Paametes of modal model ω (Hz) 23,27 46,96 Ode 8 7 ξ (%),67,34 ω (Hz) 6,62 2,82 38,9 Ode ξ (%) 4,8,68 4,33 ω (Hz) 7,8 22,57 27,94 39,74 Ode ξ (%) 4,8,47 4,82 2, ω (Hz) 6,33 22,3 27,99 38,59 49,3 Ode ξ (%) 7,5 3, 7,3 4,9 5,5 ω (Hz) 7,36 22,82 29,24 4,3 5,87 9,64 Ode ξ (%) 6,69,8 6,7 3,2 2,9 2,24 ω (Hz) 2,3 22,5 39,8 49,6 Ode ξ (%),93 7,93 6,98 4,8 ω (Hz) 6,52 2,7 25,5 4,43 47,43 Ode ξ (%), 2,48 6,73 4,6 4,7 ω (Hz) 6,5 2,89 37,74 46,34 Ode ξ (%),47,2 6,33,78 ω (Hz) 7,59 23,58 45,93 Ode ξ (%) 3,83,7,27

6 Table 2. The main obsevation matix fo engine pefomance State ω ząd ξ ω 2 ząd 2 ξ 2 H(f) H(f)L γ 2 xy A RMS(t) β kut C s I 23,27 8,67 46,96 7,34 68,56-2,8 8,8,277,5567,7239, ,82 9,68 38,9 2 4,33 47,8 3,59,22,392,8989 2,24 2, ,57 7,47 39,74 8 2, 36,42 8,84 4,4,24,7532,8656 2, ,3 8 3, 38,59 7 4,9 3,34-5,28 9,,769,9245 2,762 2, ,82 9,8 4,3 27 3,2 46,6-75,94,5,232,748 2,982 2, ,3 8,93 39,8 23 6,98 42,24-8,5 83,73,72 2,525 2,857 3, ,7 7 2,48 4, ,6 38,76 22,77 82,34,363 2,2943 2,2926 2, ,89 8,28 46,34 2,78 4,5-9,29 83,29,726,74 2,76 2, ,58 7,7 45,93 8,27 9,45-23,34 99,63,94,644,826 2, Results validation As a esults of investigation in this pape is shown pesentation of singula value decomposition (SVD) method usage fo combustion engine technical state esults validation. The SVD method is the appopiate tool fo analyzing a mapping fom one vecto space into anothe vecto space, possibly with a diffeent dimension []. The fist step of SVD pocedue is to cente and nomalization all symptoms given in table two elative to the initial value of symptom vecto. The obsevation matix of tansfomate symptoms elative to the initial value is shown on figue 4. Matix of symptoms Amplitude 5 S imptoms R ealtive amplitude S tate Matix of tansfomate symthoms elative to the inicial value S tate Fig.4. Matix of symptoms befoe and afte tansfomation The second step of SVD pocedue is to calculate the fist genealized damage and evolution of damage. Gaphical intepetation of this calculations is given in figue 5. Making data analysis in SVD method as a esult we got the line up of symptoms togethe with the popotional desciption of given individual symptom of combustion engine technical state. Thanks to SVD methods we could decide which symptom given in obsevation matix is the best to ecognize a set of combustion engine technical state [].

7 Contibution of genealize faults 5 4 Fist fault genealize % R e altiv e c oe lation Genealize faults (S ingula values ) Coelation (S G,S ymptoms) Paametes % Time θ Fig.5. Gaphical intepetation of fist genealized damage and evolution of damage In SVD pocedue as a esult we got a line up of five best symptoms given in table 3 that ae most impotant in desciption of set technical state of combustion engine. Table 3. Results of SVD method with five best symptoms fo set of engine technical state State symptom 2 symptom 3 symptom 4 symptom 5 symptom ξ ω A RMS(t) β kut C s 2 H(f)L ząd ξ 2 H(f) γ 2 xy 3 ξ H(f)L ząd ω H(f) 4 ząd 2 H(f)L ω 2 ξ 2 ξ 5 ξ ząd β kut A RMS(t) γ 2 xy 6 H(f)L ξ A RMS(t) H(f) ząd 2 7 ξ C s I ząd γ 2 xy 8 H(f)L ξ ξ 2 γ 2 xy ω 9 H(f)L ξ β kut C s I Relationships cause - consecutive expessing quantitative elation between studied vaiable symptoms esults in this wok wee qualified using the function of the multiple egession. Basis on SVD esults as a best symptoms in multiple egession wee given: ω fist natual fequency, ξ - modal damping coefficient of fist natual fequency, ξ 2 - modal damping coefficient of second natual fequency, H(f)L imagine pat of tansfe function, γ 2 xy coheence function. The equation of multiple egession is obtained in the fom: y = -,44923ω -,6558ξ -,35989ξ 2 -,652H(f)L +,4424γ 2 xy + 35,9994, (8) Gaphical intepetation of this calculations fo fist dependent vaiable ω is given in figue 6. The ed line pesent eal data eceived duing investigations, the blue line estimated model fo dependent vaiable.

8 .5 Dependent vaiable.5 w Real data Model R 2 = F =7.569 F stat= Pvalue= α=.5 J Btes t: OK Residuals Obsevations Residual Case Ode Plot Histogam of esidual Nomal Pobability Plot R esiduals Case Numbe R esiduals Case Numbe F equency Residuals Pobability Data 6. Conclusion Fig. 6. Gaphical intepetation of multiple egession fo fist dependent vaiable ω Received in the expeiment modal paametes and numeical estimatos of viboacoustics signal unambiguously show that the peviously assumed conditions of the combustion engine's state eflect themselves in modal as well as othe paametes chaacteising the vibations and they ae possible to be identified. The use of the opeational modal analysis in diagnostic investigations finds its use as one of many methods of making the actual technical state of studied object. To complete the analysis pocess a SVD method and multiple egession wee used. SVD methods maked most impotant symptom in desciption of engine technical state. On the basis of the esults, it is possible to detemine the actual technical state of an object of the same type by means of compaison of the achieved esults with the model ones and assigning them to the paticula model's state, which answes to a paticula damage, o its loss, in the object. The intoduced in pape esults of investigations ae the pat of ealized investigative poject and they do not descibe wholes of the investigative question, only chosen aspects. Refeences [] Cempel, Cz., SVD Decomposition Of Symptom Obsevation Matix As The Help In A Quality Assessment Of A Goup Of Applications, Diagnostyka v.35, PTDT Waszawa 25. [2] Łukasiewicz, M., Póba odwzoowania modelu modalnego stanu technicznego silnika spalinowego w zastosowaniu do badań diagnostycznych, Diagnostyka v.33, PTDT, Waszawa 25. [3] Łukasiewicz, M., Badania napędu zasilania awayjnego eksploatacyjną analizą modalną, InŜynieia i Apaatua Chemiczna /27, pp.83-84, Bydgoszcz 27. [4] Uhl, T., Lisowski, W., Eksploatacyjna analiza modalna i jej zastosowanie, AGH, Kaków 999. [5] Uhl, T., Kuowski P., Vioma instukcja uŝytkownika, AGH, Kaków 22. [6] śółtowski, B., Badania dynamiki maszyn, ATR, Bydgoszcz, 22.