ASSESSMENT OF DAMPING PROPERTIES OF LARGE 2-STROKE MARINE DIESEL ENGINES

Transcription

1 Internatonal Conference on Engneerng Vbraton Lublana, Slovena, 7- September 5 ASSESSME OF DAMPIG PROPERIES OF LARGE -SROKE MARIE DIESEL EGIES homas Secall Wmmel*, Domnc Müller, Jürg Meer 3 Wnterthur Gas & Desel Ltd. (former Wärtslä Swtzerland Ltd.) thomas.secallwmmel@wngd.com Zürcher Hochschule für Angewandte Wssenschaften domnc.mueller@zhaw.ch 3 Zürcher Hochschule für Angewandte Wssenschaften uerg.meer@zhaw.ch Keywords: Modal Dampng Rato, Egenfrequency, Mode Shape, Recprocatng Engne, Vbraton Measurement, Vbraton Smulaton. Abstract. When smulatng the vbraton behavour of large -stroe marne desel engnes the dampng propertes are an mportant parameter to be consdered. Smulatons only allow determnng realstc vbraton ampltudes f the engne s dampng characterstc s nown. Modal dampng ratos of recprocatng engnes vary dependng on whch parts of the engne nteract durng oscllaton and on how energy s dsspated whle vbratng. herefore dfferent dampng ratos can be expected for dfferent engne mode shapes. For determnng typcal modal dampng ratos of a desel engne a methodology for an automated analyss of vbraton measurement data was developed: After dentfyng resonance peas n a measured vbraton spectrum a spectral curve correspondng to a smple masssprng-damper system wth degrees of freedom s automatcally ftted onto the measured curve by parameter optmzaton. At the end of ths teratve process the dampng propertes of the substtutng mass-sprng-damper-system, whose curve adusts most precsely to the measured spectrum, are attrbuted to the measured resonances. In addton to determnng modal dampng ratos and egenfrequences the procedure allows separatng overlappng resonance peas. he analyss process was appled on vbraton measurement results of a seres of dfferent marne desel engnes and characterstc modal dampng ratos could be determned for the engne s prncpal mode shapes. Consderng the determned modal dampng propertes when performng forced response smulatons of the engne by modal superposton n the frequency doman allowed to acheve a hgher accuracy n the calculated vbraton ampltudes.

2 homas Secall Wmmel, Domnc Müller, Jürg Meer UMERICAL VIBRAIO ASSESSME O LARGE -SROKE MARIE EGIES Prototypng of large -stroe marne engnes s almost mpossble due to ther large dmensons (Fgure ). Due to the engne s extreme manufacturng costs and tme only mnor changes are allowed on the machne after the frst unt s bult. herefore smulaton plays an mportant role n the engne s desgn process. 3 m 3 m.5 m 9.5 m Fgure : Approxmate dmensons of a Wärtslä 6- (left) and 7-cylnder -stroe engne (rght) for the propulson of large cargo vessels. For assurng the engne s vbraton safety a numercal modal analyss and a harmonc response smulaton are carred out by means of fnte element analyss (FE) []. For the modal analyss the equaton of moton for an undamped system wthout exctaton s consdered: M x K x () In ths expresson M s the mass matrx, K the stffness matrx and x the poston vector. By pre-multplcaton wth the nverse of the mass matrx M the equaton can be reformulated: I x A x () where I s the dentty matrx and A M hs dfferental equaton can be solved by assumng the followng soluton: x X e K t (3) (4)

3 homas Secall Wmmel, Domnc Müller, Jürg Meer Wth Eq. (4) and Eq. () the followng egenvalue problem can be formulated: I X A (5) he relaton between the egenvalues, the angular egenfrequency and the egenfrequency f s: 4 f (6) he egenvalue problem s solved teratvely durng the modal analyss by applcaton of the Lanczos solver. Wth the calculated egenvectors X the procedure reveals the mode shapes that the engne structure can adopt whle t vbrates and allows to understand the vbraton mechansm (Fgure ). Egenvectors permt to dentfy the ponts on the engne on whch to apply a desgn modfcaton for avodng that a determned mode shape occurs. Addtonally the modal analyss supples the egenfrequences whch help to pre-estmate f a mode shape can be excted on a determned engne speed range. [] But a numercal modal analyss does not supply any nformaton about the exctablty of mode shapes durng engne servce. herefore a harmonc response analyss by modal superposton based on the egenvectors that were determned n a prevously performed modal analyss s carred out. [4]-[7] he base for ths analyss s the modal matrx n whch the calculated egenvectors are arranged columnwse: P X, X,, X (7) In [3] the orthogonalty of the egenvectors X wth respect to the mass matrx M and the stffness matrx K s demonstrated. hese propertes mply: P P M M P K K P M K K M where the generalzed mass s where the generalzed stffness s M K X X M X K X (8) 3

4 homas Secall Wmmel, Domnc Müller, Jürg Meer If addtonally the modal matrx s mass normalzed by dvdng each egenvector by the square root of the generalzed mass M the resultng weghed modal matrx P has the followng propertes [3]: X n P P K P M P I P (9) By means of harmonc response smulatons the engne structure s answer to stmulatng gas and mass forces f (t) s determned. Also the dampng propertes (descrbed by matrx C ) determne the structure s response ampltude. herefore the equaton of moton () has to be reformulated: M x C x K x f (t) () he orthogonalty s usually not gven wth respect to the dampng matrx C. herefore Ralegh dampng s assumed, so that the dampng matrx s expressed by a mass and by a stffness proportonal term wth the constants and : C M K () Addtonally a coordnate transformaton nto a modal coordnate system wth poston vector y s performed by expressng x as a lnear combnaton of the egenvectors X : x P y () Wth Eqs. () and () the equaton of moton () can be rewrtten as: M P y (3) M P K P y K P y f ( t) After pre-multplyng Eq. (3) wth the transposed mass normalzed modal matrx P and by consderng the propertes that are expressed by Eq. (9) the followng expresson s obtaned: y (4) I y y P f ( t) 4

5 homas Secall Wmmel, Domnc Müller, Jürg Meer Eq. 4 s equvalent wth the followng set of (number of degrees of freedom of the smulated system) uncoupled dfferental equatons: y y y P f ( t) X f ( t) wth, (5) Where X s the mass normalzed egenvector. he modal dampng s defned by the followng equaton: (6) Wth Eq. (6) the set of dfferental equatons (5) can be rewrtten as: y y y P f ( t) X f ( t) wth, (7) Equaton set (7) s the base for the harmonc response analyss. he exctaton forces f (t) correspond to the gas and mass forces actng on the engne structure [8]. he egenvectors X or the modal matrx P respectvely and the egenfrequences are determned by means of the prevously run modal analyss. he values of the modal dampng ratos were experence values up to now. For a large number of degrees of freedom solvng the equaton system (7) can be a cost ntensve computaton. But, f the number of consdered egenvectors n the modal matrx P n equaton () s lmted to only a small subset of modal vectors ths method becomes a very powerful procedure for determnng the dynamc propertes of a vbratng system. When puttng nto practce ths technque only those mode shapes that can be excted by the stmulatng forces f (t) are consdered. Mode shapes wth a consderably larger egenfrequency than the largest exctaton frequency are omtted n the modal superposton. In case of the smulated combuston engnes the experence showed that consderng mode shapes s suffcent for obtanng acceptable results. For the analysed recprocatng combuston engnes the equaton system (7) s computed by usng a commercal FE-solver. At the end of the calculaton the solver performs the transformaton from modal coordnates y bac to real world coordnates x by applcaton of expresson (). Usng an approprate modal dampng rato n the smulaton model s mportant for reproducng vbraton ampltudes correctly. Dampng propertes vary among mode shapes because for each mode dfferent engne parts nteract wth each other. Fgure shows the example of an H-mode (transversal bendng), where most deformaton occurs n the regon of the engne bed. In case of the X-mode (torson around the vertcal axs) the man deformaton occurs on the engne column and on the cylnder acet. herefore the energy that s dsspated by frcton among the nteractng engne regons s also unequal for dfferent mode shapes, whch explans that the modal dampng rato vares for dfferent vbraton shapes. 5

6 homas Secall Wmmel, Domnc Müller, Jürg Meer Fgure : H-mode (transversal bendng) and X-mode (torson around the vertcal axs) of a large marne engne DEVELOPME OF A MEHODOLOGY FOR DEERMIIG DAMPIG PROPERIES FROM VIBRAIO MEASUREME RESULS Up to now the value of the modal dampng rato was always uncertan and had to be estmated. herefore vbraton ampltudes that were obtaned by smulaton often dffered from the magntudes that were obtaned by measurement. For ganng experence regardng dampng propertes of large -stroe marne engnes a methodology was developed for determnng modal dampng ratos from measured vbraton spectra [9] and []. In the present case a modal based system dentfcaton method s most approprate. For a measured response spectrum wth resonance peas a substtutory sprng-mass-damper-system wth degrees of freedom, whose vbraton propertes are equvalent wth those of the measured structure, s determned by applcaton of a heurstc algorthm []. he followng approach functons are ntroduced nto equaton system (7) whch corresponds to the equatons of moton for a sprng-mass-damper system wth degrees of freedom n modal coordnates: y Y e ( t ) f ( t) F e t Y e e t (8) In Eq. (8) s the phase angle between exctaton and response and the stmulaton frequency. After applcaton of the approach the equaton system (7) transforms nto: Y e X F (9) 6

7 homas Secall Wmmel, Domnc Müller, Jürg Meer 7 Beng Eq. 9 equvalent to: arctg e F X Y () After transformng Eq. () from the modal coordnate system bac to real coordnates by applcaton of expresson () the followng expresson s obtaned: arctg e P F X () It s assumed that a spectrum wth resonance peas that was obtaned wth a measurement sensor attans to Eq. (), whch can be smplfed as: arctg e a e Q A () he ampltudes n the sensor s spectrum should be equvalent wth the magntude of A: sn cos a a A (3) For fttng the curve of the substtutory system nto the measured frequency response functon the 3 parameters,,,,,,,, Q, Q,, Q are optmzed by the method of the least squares, so that the functon A adapts best to the sensor s spectrum. he algorthm s target s mnmzng the resdual functon R:,,,,,,,,,,, M Q Q Q A m w R (4) Where w s a weghng factor that s large n regons of small measurement pont densty and small n regons where measurement ponts are very close. m are the measured spectrum ponts. Fgure 3 shows the mportance of consderng the correct phase relatons among the summands a e n Eq. (). If erroneously the frequency response functon s bult by addng the magntudes of the summands (dashed curve) ampltudes mght be overestmated between overlappng resonance peas. Due to a phase shft of 8 when trespassng resonances the summands are not n phase anymore. hs explans why the phase correct superposton spectrum (green curve) always shows less ampltude between overlappng resonance peas. Wth ncreasng modal dampng ratos ths effect s more sgnfcant because the 8 frequency shft occurs over a larger frequency range.

8 homas Secall Wmmel, Domnc Müller, Jürg Meer Fgure 3: Spectrum for a degree of freedom sprng-mass-damper system wth egenfrequences =Hz and =5Hz and modal dampng ratos =. and =.5 wth ampltude (top) and phase (bottom). he proposed method for determnng modal dampng values from measurement data was coded and automatzed for effcently beng appled on large -stroe engnes. Fgure 4: he developed method allows to reproduce the measured spectrum by means of a substtutory sprngmass-damper system and allows to separate overlappng resonance peas. Fgure 4 demonstrates that the developed tool successfully reproduces the measured spectrum curves. In the shown example (left) a curve wth 3 resonance peas s dentfed by means of a 3 degree of freedom substtutory sprng-mass-damper-system. As output the egenfrequences and the dampng ratos for ths best fttng system are obtaned. It can also be seen that overlappng resonance peas can easly be separated and dentfed wth the procedure. In case of the depcted frequency response functon wth overlappng resonance peas (rght) the substtutory degree of freedom system allows to vsualze the sngle peas that compose the spectrum. 8

9 homas Secall Wmmel, Domnc Müller, Jürg Meer 3 APPLICAIO OF HE MEHOD FOR A MARIE EGIE he developed code was appled for determnng dampng propertes of large -stroe marne desel engnes from a seres of measurement spectra for dfferent engnes (able ). As expected, the modal dampng ratos dffer among the 3 man engne mode shapes H-mode (transversal bendng), L-mode (longtudnal mode) and the X-mode (torson around the vertcal axs). On many engnes the L-mode cannot be dentfed. Specally on long engnes wth a large number of cylnders t s dffcult to stmulate ths mode shape. herefore not many measurement values are avalable for the shape and all of them are almost dentcal. Mode shape type Modal dampng rato H-mode L-mode.39 X-mode able : Example of the constructon of one table. he obtaned values for the engne s dampng rato n able were used n a harmonc response smulaton by modal superposton n a commercal FE-solver (Fgure 5). he measured ampltude spectrum for the H-mode of a 6-cylnder engne s compared wth two calculated frequency response curves. he frst one was obtaned by performng a harmonc response smulaton wth the lower lmt for the modal dampng rato for the H-mode n able and the other wth the top border value. he satsfactory results show that the measured ampltude s nsde the nterval that s delmted by the two extreme numercal curves. he egenfrequency that was obtaned by smulaton s only approxmately % lower than the measured one. Fgure 5: Measured spectrum for the H-mode (transversal bendng) of a 6-cylnder marne -stroe engne and frequency response curves that were obtaned numercally by usng the lmt dampng rato values for the H- mode n able n a harmonc response smulaton. 9

10 homas Secall Wmmel, Domnc Müller, Jürg Meer 4 COCLUSIOS For vbraton assessment of large marne engnes a modal analyss and a harmonc response smulaton by modal superposton are performed. Modal dampng has to be nown for obtanng correct vbraton ampltudes from a harmonc response smulaton. herefore a method to estmate dampng propertes from measurement data was developed. he algorthm allows to dentfy egenfrequences and the modal dampng rato from measurement data of a vbratng system and allows to separate overlappng mode shapes. Modal dampng ratos for the man engne mode shapes could be determned and tabularzed for beng used n harmonc response smulatons. REFERECES []. Secall Wmmel, umercal Vbraton Assessment on Large -stroe Marne Desel Engnes. th Internatonal Conference on Vbraton Problems, Lsbon, Portugal, 9- September 3. [] C. Pestell, F. Pallon, L. Bergant and F. Castellan, Dynamc Propertes Assessment and Updatng of Large Desel Shp Engnes, he hrteenth Internatonal Conference on Sound and Vbraton, Venna, Austra, -6 July 6. [3] W.. homson, heory of Vbraton Wth Applcatons, Prentce Hall [4] J. Zhang and J. Han, CAE process to smulate and optmse engne nose and vbraton, Mechancal Systems and Sgnal Processng,, 4-49, 6 [5] I. Kunnor, S. obua and K. Kench, Vbraton Reducton of Recprocatng Engnes, 7 th Internatonal Symposum on Marne Engneerng, oyo, Japan, October 4-8, 5 [6] H. enhaara and H. Monaho, Engne Dynamcs and Vbraton Control, Consel Internatonal des Machnes a Combuston, Venna, Austra, 3, 7 [7] R. ordr and H. Solbaen, Vbraton characterstcs of a V medum speed gas engne. Smulaton and measurement, Consel Internatonal des Machnes a Combuston, Bergen, orway, 5, [8] W. Schffer, Advanced methods for statc and dynamc shaftng calculatons, ISME Internatonal Symposum on Marne Engneerng, oyo, Japan, 5 [9]. Araawa and K. Yamamoto, Frequences and Dampng Ratos of a Hgh Rse Buldng Based on Mcrotremor Measurement, 3 th World Conference on Earthquae Engneerng, Vancouver, B.C., Canada, August -6, 4 [] J. Butterworth, J. H. Lee and B. Davdson, Expermental Determnaton of Modal Dampng from Full Scale estng, 3 th World Conference on Earthquae Engneerng, Vancouver, B.C., Canada, August -6, 4 [] D. Müller, Vbraton Analyss of -Stroe Marne Engnes, Master hess (Mechancal Engneerng), Zürcher Hochschule für Angewadte Wssenschaften, Wnterthur, Swtzerland, 3..5