Investigation of noise radiation from tire using experimental modal identification

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1 Investgaton of nose radaton from tre usng expermental modal dentfcaton Atsush KITAHARA 1 ; Takuya YOSHIMURA 2 ; Shnsaku KATAYAMA 3 1 Brdgestone Corporaton, Japan 2 Tokyo Metropoltan Unversty, Japan 3 Brdgestone Corporaton, Japan ABSTRACT To reduce nose from automoble tre, clarfcaton of dynamc characterstcs of tre s mportant. Ths paper focuses on modal dentfcaton. In such cylndrcal shell structures, vbraton modes wth complcated shapes exst densely. Also, structures made of composte materal often have hgh dampng characterstcs. Therefore, the expermental modal parameter extracton s a dffcult task. To solve the dffcultes, a new modal parameter dentfcaton method s appled to nvestgate such structures of dynamcs. Crcumference reducton method s proposed. Ths method can effectvely reduce modal densty for modal dentfcaton, so the dffculty of dentfcaton s mproved. It s verfed that the modal property of the tre can be dentfed usng proposed method. Furthermore, by applyng ths dentfcaton approach to the acoustc transfer functon, we clarfy the vbraton modes contrbutng to the nose radaton. Keywords: Modal dentfcaton, Radaton I-INCE Classfcaton of Subjects Number(s): INTRODUCTION It s mportant to evaluate the modal propertes (1, 2) to reduce nose and vbraton of the cylndrcal shell structures, such as automoble tre (3). The CAE s often used for such purposes. In order to mprove the CAE predcton, t s mportant to verfy the CAE model through the vbraton experment. However, because of couplng of crcumferental and axal bendng n such structures, the vbraton modes wth complcated shapes exst densely (4). It s also understood that structures made of composte materal, such as tre, have hgh dampng characterstcs due to polymer. The resonance peaks are not obvous but dull, especally md and hgh frequency such that the locatons of natural frequences are not clear. Therefore, the expermental modal parameter extracton s somewhat dffcult task. In ths study, we nvestgate modal analyss method whch s able to be appled to structures of hgh modal densty. In partcular a cylndrcal shell structure s selected as bass functon can be explctly defned. By defnng Fourer seres as bass functon, measured frequency response functons (FRFs) are transformed nto spatal Fourer seres. Natural modes n certan crcumferental characterstc can be extracted. Ths pre-condtonng helps to reduce modal densty and to mprove curve-fttng effcency. Also to acqure dynamc characterstcs accurately ncludng crcumferental multple root, FRFs are measured usng multple exctaton. Modal parameters are then extracted from the transformed FRFs by usng NLS (non-lnear least squares) approach, whch had been presented by one of the authors (5, 6). NLS estmates modal parameters by an optmzaton approach and t can take account of non-proportonally damped system. Ths approach s vald especally for hgh damped system. We verfy the valdty of ths proposed method usng numercal smulaton. The modal parameters of FEM modal analyss s consdered to be true value. A cylndrcal shell structure wth non-proportonally dampng modeled by Fnte Element approach s used as an example. Random error s added to the steady state FRFs of the FE model. Then modal parameters are dentfed from these 1 atsush.ktahara@brdgestone.com 2 yoshmu@tmu.ac.jp 3 snsaku.katayama@brdgestone.com 2294

2 FRFs usng crcumference reducton method and non-lnear least squares approach. We examne ths dentfed result n comparson wth the result whch s solved by FEM modal analyss. Next, as an example of a real structure, the modal property of an automoble tre s dentfed. It s verfed that the modal property of cylndrcal shell structure can be dentfed effectvely usng proposed method. Furthermore, the contrbutng modes for nose radaton are clarfed by dentfyng modal partcpaton factor usng FRFs of acoustc exctaton test. 2. THEORY 2.1 Crcumference Reducton Method The response ponts of FRFs are set on vrtual crcular rngs on the cylndrcal structure surface, as shown n Fgure 1. The response of each vrtual rng can be expressed usng bass functon. The bass functon s superposton of snusodal waves. So, crcumferental Fourer seres of FRFs of each vrtual rng s formulated as follows: N 2 2n A H n cos N n1 N N (1) 2 2n B H n sn N n1 N where A and B are crcumferental Fourer seres, H n s accelerance FRF at response pont n, s crcumferental order and ω s angular frequency. These Fourer seres mean reduced FRFs to crcumferental drecton. If the number of response ponts of one rng s N, the degree of freedom of FRFs used for later curve-fttng s reduced to 2/N of orgnal quantty. And modal densty of each crcumferental order FRF s reduced clearly. Response ponts ( N ponts per rng ) Cylndrcal shell Vrtual crcular rngs Fgure 1 Response ponts on vrtual crcular rngs 2.2 Modal parameter Identfcaton Technque In ths paper, modal parameters are estmated by assumng general vscous dampng system. Therefore, FRF s formulated as follows: N * * rr r r G (2) r1 r j dr r j dr where ω dr s damped natural angular frequency, σ r s modal decay rate, ξ r s modal partcpaton factor and ϕ r s natural mode vector. Complex conjugates are denoted by *. For the modal parameter estmaton of Equaton (2) s reformulated for the mult-reference measurement cases as G n ξ r j r r r r r1 T j dr r where n s a number of modes wthn the frequency band of nterest. Z k s a resdual term whch represents the nfluence of natural mode out of the frequency range. If k s equal to -2, Z k represents term of resdual mass. If k s equal to 0, Z k represents term of resdual stffness. The dentfcaton procedure of the proposed curve-fttng technque results n the non-lnear least * ξ * T dr k Z k j k (3) 2295

3 squares problem whch can be solved by the Gauss-Newton method. Modal parameter vector a s a vector whose elements consst of all the modal parameters n Equaton (3). The parameter estmaton leads to the determnaton of the vector a, whch mnmzes the error functon Γ: m G W G (4) A where G A (ω ) s syntheszed FRF expressed by Equaton (3), G E (ω ) s target FRF. W(ω ) s a weghtng functon such as W E p E 2 1 (5) G Normally, we suggest that parameter p can be set between 1 and 2. In ths paper, p s set to 2. The modal parameters are descrbed by non-lnear parameters and lnear parameters. When the non-lnear parameters are gven, the lnear parameters are determned. Ths can mnmze the error functon Γ on the condton that non-lnear parameters are fxed. Let the modal parameters on the l-th teraton be a (l) and the modfcaton vector be δa, l1 l a a a (6) All the non-lnear parameters are organzed as elements of vector a N, and all lnear parameters as elements of vector a L : a N a (7) a L It s recommended that the lnear parameters a L (l+1) should be obtaned by the lnear least-squares method where the non-lnear parameters a N (l+1) s fxed. Consderng the calculaton effcency, the technque presents teratve parameter estmaton algorthm whch obtans only the non-lnear modfcaton vector δa N out of δa, and determnes the renewed lnear parameters a L (l+1) by the least squares method. After all, the modfcaton vector of the non-lnear terms δa N s found compact, and then, a N (l+1) s obtaned. After a N (l+1) s obtaned, the lnear parameters a L (l+1) s determned by the lnear least-squares method. Ths process s terated untl the error functon Γ s converged to a mnmal value. In ths method, ntal values are need for non-lnear parameters. The ntal values can be obtaned usng complex mode ndcaton functon (CMIF) (7). 3. DESCRIPTION OF ANALYTICAL MODEL 3.1 FE Model The FE model of cylndrcal shell structure used n ths study s shown n Fgure 2. Commercal software ABAQUS (8) s used to calculate FRFs for dentfcaton, and to analyze egenvalue for true modal value. The shell structure s constructed of carbon steel. A specfed regon has proportonal vscous dampng expressed as C=βK, and other regon s undamped. Therefore, the whole modal has non-proportonal dampng. Both edges of the cylnder are smply supported. To dentfy modal parameters of ths model usng proposed method, FRFs are calculated n case of three exctaton ponts. The number of Response ponts s 160. Exctaton drecton s perpendcular to the shell surface, and FRFs n drecton perpendcular to the shell surface are acqured. So we obtan 480 (=3x160) FRFs. 2296

4 Undamped regon Vscous dampng regon Fgure 2 Cylndrcal shell FE model 3.2 Error Addton and Modal Parameter Identfcaton We add random complex error to the calculated FRFs. The maxmum magntude of absolute error s 5 % of each real and magnary part of FRF, at each frequency. The response ponts of FRFs are postoned on 8 vrtual crcular rngs on the structure surface. Focusng on partcular crcumferental order, FRF groups of each rng are transformed nto Fourer seres every each exctaton case. Then the reduced FRFs of the partcular crcumferental order are used for modal dentfcaton usng NLS. 3.3 Results of Crcumference Reducton Fgure 3 shows a sample of accelerance FRF wth random error, and crcumferental 2nd order Fourer seres. It s clear that modal densty s reduced. The dfference between orgnal FRF and 2nd order FRF means component of other crcumferental orders. Ths s unnecessary component for modal dentfcaton of crcumferental 2nd order. Crcumference reducton s performed at each vrtual rng, at each exctaton case. The number of rngs s 8, and the number of exctaton ponts s 3. And Fourer seres conssts of a par shown n Equaton (1). So we use 48 FRFs per one crcumferental order curve-fttng. Phase deg Ampltude m/s2/n Accelerance wth random error Crcumferental 2nd order Fgure 3 FRF 3.4 Results of Modal Parameter Identfcaton Focusng each crcumferental order, modal parameters are extracted by NLS approach usng a total of 48 FRFs smultaneously. We evaluate the curve-ft results by observaton of comparson of syntheszed FRFs expressed by Equaton (3) and orgnal FRFs. A sample FRF s shown n Fgure 4. It s easy to understand that precson of the synthess result s good. A result of dentfcaton below 3000 Hz s shown n Table 1. And some mode shapes are 2297

5 summarzed n Fgure 5. The modes below 3000 Hz are fully extracted. Egen frequences and mode shapes are dentfed accurately. Modal dampng ratos are dentfed wth a lttle error. From the above results, t s understood that the modal parameters are well extracted n frequency range of hgh modal densty, and hgh modal dampng rato up to 4 %. And ths proposed method seems to be tolerant of random error addton. Phase deg Crcum. 2nd, FRF no.7, exctaton # Ampltude m/s2/n 10 0 Orgnal FRF Synthess Fgure 4 Example of curve-fttng result (Crcumferental 2nd order) Crcumrerental order Table 1 Result of dentfcaton, below 3000 Hz Number of modes Average of dentfcaton error [%] Maxmum of dentfcaton error [%] FEM Identfed Egenfrequency Dampng rato Egenfrequency Dampng rato

6 FEM: f r =1272Hz, ζ r =0.94% Identfed: f r =1272Hz, ζ r =0.77% (a) Crcumferental 1st, axal 1st mode FEM: f r =2795Hz, ζ r =4.00% Identfed: f r =2796Hz, ζ r =4.04% (b) Crcumferental 3nd, axal 5th mode FEM: f r =628Hz, ζ r =0.85% Identfed: f r =628Hz, ζ r =0.82% (c) Crcumferental 4th, axal 2nd mode FEM: f r =1443Hz, ζ r =2.12% Identfed: f r =1444Hz, ζ r =2.14% (d) Crcumferental 6th, axal 5th mode Fgure 5 Example of mode shapes 4. Analyss of Automoble Tre 4.1 Exctaton Test We carry out the modal parameter dentfcaton of an automoble tre. The test tre s shown n Fgure 6. The tre and wheel s fxed at bolt-hole of the wheel. The number of exctaton pont s 15. The number of exctaton ponts on one vrtual crcular rng s 40. The number of the vrtual rng s 18, so the total number of exctaton ponts s 720. The number of FRF s Sdewall area of tre can be regarded as a part of cylndrcal shell structure. Therefore, the response ponts are set on the whole outer surface of tre. The drecton of exctaton and response s vertcal drecton of the surface. We carry out the modal parameter dentfcaton usng these measured FRFs by proposed dentfcaton method. At frst, the FRFs are transformed to the spatal Fourer seres of each crcumferental order. For one crcumferental order, the number of FRF s reduced to 540. Next, the modal parameter dentfcaton of these reduced FRFs s carred out usng NLS approach. 2299

7 Fgure 6 Automoble tre used for the test 4.2 Results of Modal Parameter Identfcaton The example of measured FRF s shown n Fgure 7. The blue lne s measured FRF. Resonance peaks can be seen clearly less than 200 Hz. However, n the case of more than 200 Hz, resonance peaks are nvsble because of hgher modal densty and hghly dampng. Modal parameter dentfcaton becomes dfferent n such stuaton. Meanwhle, the red lne s the 1st order Fourer seres component. The reducton of ampltude denotes dsappearance of unnecessary component. Because of reducton of modal densty, the dffculty of dentfcaton seems mproved. Phase deg Ampltude m/s2/n 10 0 Orgnal Extracted Fgure 7 Example of accelerance FRF The CMIF of ths crcumferental 1st order FRF s shown n Fgure 8. Each lne shows sngular value component. The multple roots and neghborng modes can be detected. Intal values of modal parameter can be determned from ths result accurately CMIF Fgure 8 Example of CMIF value (Crcumferental 1st order) The modal parameter dentfcaton s carred out by NLS approach usng these ntal values. The example of curve-fttng result s shown n Fgure 9. The blue lne s an orgnal FRF (crcumferental 1st order) and the red lne s a syntheszed FRF usng dentfed modal propertes. The curve-fttng has been done wth good accuracy. 2300

8 Phase deg FRF no.12, p006-n Ampltude m/s2/n 10 0 Orgnal FRF Synthess Fgure 9 Example of curve-fttng result (Crcumferental 1st order) Example of dentfed modal property s shown n Table 2. Examples of dentfed mode shapes are shown n Fgure 10. Here, x-y mode denotes crcumferental x-th and axal y-th order mode. The mode shapes whch are smlar to the cylndrcal shell s mode shapes are dentfed. From the dentfed modal property, t can be seen that modal densty s hgh and dampng rato s also hgh. Although t s such dffcult condton, modal property s dentfed well usng proposed dentfcaton method. Table 2 Identfed modal property usng proposed method (a) Crcumferental 1st order modes (b) Crcumferental 2nd order modes (c) Crcumferental 3rd order modes No. f r [Hz] ζ r [%] Axal order * * * * * * * * * * * * No. f r [Hz] ζ r [%] Axal order * * * * * * * * * * No. f r [Hz] ζ r [%] Axal order * * * * * * * * * * * f r : damped natural frequency, ζ r : modal dampng rato * denotes crcumferental multple root. 2301

9 1-2 mode 1-12 mode 2-7 mode 2-11 mode 2-15mode 3-3 mode 3-9 mode Fgure 10 Example of dentfed mode shapes 4.3 Consderaton of Nose Radaton The vbraton modes of the tre are dentfed by usng proposed dentfcaton method. Meanwhle, to reduce of nose radaton from tre, mprovng all vbraton modes s not always necessary. It s mportant to clarfy contrbutng modes for nose radaton. Therefore we consder contrbuton of vbraton modes for nose radaton. We examne contrbuton by acoustc exctaton test. Acoustc source s set at the nose evaluaton pont. When the tre s excted by acoustc source, tre vbraton s generated by acoustc force. At ths tme, hghly contrbutng modes for nose radaton wll be excted strongly. Therefore, the contrbuton rato of excted modes n acoustc exctaton test s equal to the contrbuton rato for nose radaton. It s explaned from recprocty prncple of vbraton and acoustc pressure (4), such as po vs (8) FS QO where p O s acoustc pressure at an observaton pont O when a structure s excted at pont S by harmonc mechancal force F S. v S s vbraton velocty produced n the structure at pont S when acoustc source Q O s located at pont O. From ths prncple, contrbuton of each vbraton mode n nose radaton can be calculated by curve-fttng of FRFs of the acoustc exctaton test. The modal parameters σ r, ω dr, ϕ r are known by above-mentoned mechancal exctaton test. Other parameters, modal partcpaton factor ξ r and resdual term Z k are dentfed usng least-squares method of Equaton (3). 4.4 Results of Nose Contrbuton Evaluaton The nose evaluaton pont s set at the poston of 0.5m from the tre surface. For an acoustc source, LMS Md Hgh Frequency Volume Source (Q-MHF) s used. The response ponts of acoustc exctaton test are set on the vrtual crcular rngs on tre surface, same as Fgure 1. Measured FRFs are transformed to the spatal Fourer seres of each crcumferental order. The modal parameter dentfcaton of these reduced FRFs s carred out by curve-fttng. Then the modal partcpaton factor s calculated. The evaluated results of contrbuton for nose radaton are shown n Fgure 11. They are the results of crcumferental 1st, 2nd and 3rd order components. The blue lne s the measured radaton nose usng volume source. The red lne s the result of curve-fttng usng dentfed modal property. The brown dashed lnes show contrbuton of each mode n curve-fttng. By comparng blue lne and red lne, resonance peaks are dentfed mostly. For example, n crcumferental 1st order component, the peak around 600 Hz has contrbuton of No. 15 mode, and the peak around 700 Hz has contrbuton of No. 22 mode. No. 15 and No. 22 modes shows mode shape wth few number of axal nodes comparng to other close frequency modes. Therefor t seems to be vald that these modes show hgh contrbuton. Accordng to the above results, t s shown that the contrbutng modes for nose radaton are clarfed by dentfyng modal partcpaton factor usng FRFs of acoustc exctaton test. 2302

10 Radaton nose FRF Synthess Contrbuton of each mode No. 15 No. 22 No. 20 No. 9 No. 10 No. 21 Ampltude Pa/N 10-3 Ampltude Pa/N 10-3 Ampltude Pa/N (a) Crcumferental 1st (b) Crcumferental 2nd (c) Crcumferental 3rd Fgure 11 Vbraton mode contrbuton for nose radaton 5. CONCLUSION Non-proportonally hghly damped cylndrcal shell structure s used for the nvestgaton of modal parameter dentfcaton. FRFs of ths structure wth random error are performed crcumference reducton usng spatal Fourer seres as bass functon. Modal parameters are extracted usng these reduced FRFs by NLS approach. We obtan hgh accuracy results n spte of hgh modal densty and hgh modal dampng rato. And ths proposed method seems to be tolerant of random error addton. Vbraton modes of an automoble tre n md-frequency range are dentfed usng proposed method. Furthermore, the contrbutng modes for nose radaton are clarfed by dentfyng modal partcpaton factor usng FRFs of acoustc exctaton test. REFERENCES 1. Nagamatsu A. Modal Analyss. Tokyo, JAPAN: Bafukan; p. 99. (n Japanese). 2. Modal Analyss Handbook Edtoral Commttee. Modal Analyss Handbook. Tokyo, JAPAN: Corona Publshng; p. 68. (n Japanese). 3. Kndt P, Berckmans D, Connck FD, Sas P, Desmet W. Expermental analyss of the structure-borne tyre/road nose due to road dscontnutes. Mechancal Systems and Sgnal Processng. 2009; 23(8): p Fahy F, Gardono P. Sound and Structural Vbraton. 2nd ed. : Academc Press; p , p Yoshmura T. A mult-reference modal parameter dentfcaton technque based on a weghted least squares prncple. Proc 9th Internatonal Modal Analyss Conference; Aprl 1991; Florence, Italy Part 1. p Yoshmura T, Sato M, Maruyama S, Iba, S. Modal analyss of automotve cabn by multple acoustc exctaton. Proc 2012 Internatonal Conference on Nose and Vbraton Engneerng; September 2012; Leuven, Belgum p Shh CY, Tsue YG, Allemang RJ, Brown DL. Complex mode ndcaton functon and ts applcatons to spatal doman parameter estmaton. Proc 7th Internatonal Modal Analyss Conference (IMAC VII); 30 January - 2 February 1989; Nevada, USA p ABAQUS 6.11 Analyss User's Manual. Rhode Island, USA: Dassault Systèmes Smula;