Energy Dissipation Mechanism and Its Application of to a Slotted Disc Waveguide Vibration Absorber

Transcription

1 Iteratioa Joura of Materia ad Mechaica Egieerig (IJMME), Voume 5 16 doi: /ijmme Eergy Dissipatio Mechaism ad Its Appicatio of to a Sotted Disc Waveguide Vibratio Absorber Xi Li, Dafeg Feg* Huazhog Uiversity of Sci. & Tech. Schoo of Mechaica Sci. & Eg., Wuha 4374, Chia *Correspodig author, i.daw@mai.hust.edu.c Abstract This paper studies the characteristics of cyidrica wave propagatio ad its eergy dissipatio i thi sotted disc viscoeastic waveguide absorber. Based o the dyamic aaysis of uiform-beam viscoeastic waveguide absorber uder cotiuous excitatio, we discussed the reatios betwee vibratio eergy dissipatio rate ad other variabes of the systems, such as sotted disk (SD) geometry parameters, materia parameters ad vibratio wave frequecy. The eergy dissipatio aw of sotted disk viscoeastic waveguide vibratio absorber (SDWVA) was further obtaied. The simuatio ad vibratio experimet show that viscoeastic circuar pate waveguide absorber has good ihibitio effect o ow-frequecy vibratio. This ca be further ivestigated as vibratio or vibratio suppressio compoet of arge machie toos ad other processig machiery. Keywords Viscoeastic Materia; Sotted Disc; Waveguide Vibratio Absorber; Vibratio Dampig Itroductio Atteuatio ad suppressio of vibratio i mechaica devices are aways amog popuar research topics i mechaica fied. The use of absorbig devices for coductio ad dissipatio of machie vibratio is a simpe ad ecoomica method [1-3]. The core compoet of the absorbig device is the eergy dissipatio device which has a direct impact o the performace of vibratio absorptio. The appicatio of viscoeastic materias to absorbig device provides a ew research method. It shows great scietific sigificace i the improvemet of the performace of vibratio absorptio, ad therefore soo becomes the iteratioa research focus. I additio, some schoars [4-8] adopted fiite eemet aaysis to cacuate wave propagatio ad eergy cosumptio, through which a reativey compete theoretica system was estabished. I cocusio, further research of the properties of waveguide vibratio absorber usig viscoeastic materia is i urget eed. At preset, eergy dissipatio devices usig viscoeastic beam are difficut to ista ad icoveiet. This paper adopts a arrow sotted disc to dissipate eergy from a waveguide vibratio absorber. This ca be cosidered as combiatio of a puraity of o-uiform beam. By usig the o-uiform beam as viscoeastic damper, it is possibe to perform further aaysis ad cacuatio. The vibratio of pate is three-dimesioa itrisicay. However, as the thickess of the thi pate is much smaer tha the sizes i the other two dimesios, the 3D probem ca be reduced to a two dimesioa probem. Therefore, the mechaica properties of the thi pate ca be simpified because of this assumptio. Hypothesis 1: The straight ie remais perpedicuar to the surface before ad after deformatio. This assumptio meas that the etire deformatio withi the pae is oy depedet o the shape of the defectio surface, as the probem ad its soutio have bee sigificaty simpified from 3D to D. From the mechaica poit of view, it is assumed that the orma ie is aways perpedicuar to the surface, amey the trasverse shear deformatio is zero ad the trasverse shear stress is much smaer tha the bedig stress of the pae directio. Hypothesis : The orma stress aog the atera surface is assumed to be egigibe. 58

2 Iteratioa Joura of Materia ad Mechaica Egieerig (IJMME), Voume It is assumed that the orma stress aog the atera surface is much smaer tha the bedig stress. Accordig to this poit, the assumptio becomes vaid whe the thickess of the pate is much smaer tha its size. Hypothesis 3: Oy the movig iertia force is cosidered, whie the rotatio iertia momet is egected. It is assumed that the movig iertia force is proportioa to the thickess of pate. However, the rotatio iertia momet is proportioa to the iertia momet of the cross sectio, amey third power of the pate thickess. So, this assumptio is justified compared to the very sma thickess of the thi pate. Dyamic Aaysis of Variabe Cross Sectio Beam Based o the above assumptio, a viscoeastic wave-guide absorber based o a thi sotted disk is desiged. The structure is show i Figure 1(a). The base is fixed o the vibratig surface. Oe ed of the meta guide rod is coected to the osciatig body, whie the other ed is boded to the geometrica ceter of the thi sotted disk. Whe the body is osciatig, the basic support disk is movig correspodigy with certai acceeratio. So the disc geerates dyamic respose through iertia force with respect to mass. FIGURE 1 SLOTTED DISC, GUIDE ROD AND SINGLE BEAM WITH VARIABLE CROSS SECTION DIAGRAM Figure1 (b) is the structure of sotted disc cotais a viscoeastic materia disc ad a guide rod. Figure 1(c) is a cross sectio of sige beam cross sectio diagram. Accordig to the sige variabe cross-sectio geometry, i variabe cross sectio beams with arbitrary positio x its cross-sectioa area: A x d ( xtg w) Kx A (1) Where is haf of the o-uiform beam age ad is beam umber, obviousy A dw, K dtg. Sice the structure of disc is symmetrica ad guide rod is istaed i its geometric ceter, this eads to symmetrica ifuece of vibratio ad disturbace o disc. Assume that basis motio is vertica trasatio. Its acceeratio is a simpe harmoic motio, so the differetia equatio of disc motio is: ( K x A ) G ( j ) v ( x, K G ( j ) v( x, ( K x A ) v( x, q( x, () Where v( x, v x, v( x, v x, v( x, v t,ρ is desity of materias,ν is Poisso's ratio of materias. So the compex shear moduus ca be described as: 1 1 G ( j) G ( ) jg ( ) G ( ) G ( ) e j The foudatio of acceeratio is be expressed as: arctg G ( ) G ( ) 1 j t V e motio of variabe cross sectio beam. The oad of mass iertia force ca The dispacemet soutio form ca be assumed as: j t q( x, A( x) hv e (3) v jt x t (, ) CV ( x) e (4) 1 Where V ( x ) is the vibratio mode fuctio. The moda fuctio is seected accordig to referece [1]. 59

3 Iteratioa Joura of Materia ad Mechaica Egieerig (IJMME), Voume 5 16 ( j ) x ( j ) x () 1 V x c e c e 1 1 (5) I which (4 J1 K ) ( K 4 J1) 16 J K 1 8 ( K 4 J1) ( K 4 J1) 16 J Where K K ( Kx A ), J ( j) 1 G ( j) J1( ) jj( ). It is evidet that, Eq. (5) is Besse equatio, ad its boudary coditios are costat. We empoyed the zero order of Besse fuctio of the first kid ad the zero order of modified Besse fuctio of the first kid. Substitutig Eq. (4) ad Eq. (5) ito Eq. (), oe obtais: P D C M ( ) (6) The coefficiets c 1 ad c i Eq. (5) are determied by boudary coditios V ( ), V (), ad the get the atura frequecy. Accordig to Eq. (5), two pae waves i opposite positio occurs i the eergy dissipater of o-uiform beam. Geeraized force ad geeraized quaity I Eq. (6) are cacuated by the foowig two equatios respectivey: ( ) ( ) P A x hv V x M A( x) V ( x) The dispacemet soutio of Eq. (4) ca be determied by: ( j ) x ( j ) x jt v( x, C [ c e c e ] e (7) 1 v ( x, iv( x, The the strai is: v ( x, v( x, ( ) ( x, v( x, x ( x) e j t (8) I which ( j ) x ( j ) x arctg 1 1 ( x) C [ ( j ) c e ( j ) c e ] ( j ) x ( j ) x Im{ C [ ( j ) c e ( j ) xc e ]} 1 ( j ) ( j ) x Re{ C [ ( j ) c e ( j ) c e ]} Therefore, the expressio of stress i variabe cross sectio beam becomes: 1 ( x, G ( j) ( x, G G ( x) e j t (9) ( ) 1 For further iustratio, the above discussio is based o the semi-ifiite og beam, whie i a fiite beam the refectio of waves exists. Whe the wave propagates, its ampitude expoetiay decays. For istace, whe the wave traves a short distace, its ampitude is cose to zero. This pricipe ca be appied to simuate a fiite beam simuatio by usig a semi-ifiite og beam whe the beam is far greater tha the vibratio of wave egth. Accordig to the above aaysis, the dissipative eergy per uit voume of o-uiform beam i oe cyce is: 6

4 Iteratioa Joura of Materia ad Mechaica Egieerig (IJMME), Voume T G ( ) Wd ( x) ( x, dt G( ) ( x) (1) The egth of o-uiform beam dissipatio eergy i oe cyce is: WD ( ) G ( ) ( Kx A ) ( x) (11) Therefore, the eergy cosumptio power of the sotted disc is averaged as: P W D D ( ) T G ( ) ( Kx A ) ( x) (1) The egth of variabe cross sectio beam of the potetia eergy for storage i oe cyce is: 1 1 W ( ) 1 ( K x A ) G ( x) 1 G ( K x A ) ( x) (13) s Ad the egth of variabe cross sectio beam of the kietic eergy i oe cyce is: W ( ) ( K x A ) ( x) ( K x A ) ( x) (14) E Because of the sotted disc ca be cosidered to be parae to o-uiform beam, so the eergy dissipatio rate R i vaue is equa to a sige o-uiform beam of eergy dissipatio rate: R WD () W ( ) W ( ) W ( ) D s E SDWVA Performace of the Simuatio Aaysis (15) FIGURE SDWVA PARAMETERS IN TERMS OF ENERGY LOSS AND ENERGY DISSIPATION RATE The discussio above demostrates cyidrica waves i thi disc viscoeastic waveguide absorber with respect to eergy oss, disc geometry parameters, materia parameters ad wave frequecy. Figure (a)-(d) show the resuts of computer simuatio o eergy cosumptio ad eergy dissipatio rate (i.e., eergy oss ad vibratio eergy tha) versus the SDWVA parameters, where type ZN disc is made of viscoeastic materia. From the chart it ca be oted that: 61

5 Iteratioa Joura of Materia ad Mechaica Egieerig (IJMME), Voume ) With icreasig frequecy, the eergy cosumptio of viscoeastic wave-guide absorber icreases, but the dissipatio rate drops. This is because the vibratio eergy decreases with icreasig frequecy ampitude, as show i Figure (a). Nevertheess, the dissipatio rate is oticeabe, which goes up to 8% at f = Hz, as show i figure (b). ) With the icrease of the disk radius, the eergy cosumptio of the viscoeastic waveguide absorber icreases, as i Figure (c). However, the dissipatio rate does ot vary with radius variatio, as i Figure (d), which shows the ifuece of radius o vibratioa eergy ad eergy savig is equa. Therefore, with a sma radius of the disc istead of a ifiite oe for exampe, h = 1mm, f = 5Hz, the reaches 3mm ad the dissipatio rate is 8.8%. Furthermore, the proportio o oger chages with the icrease of the radius. 3) Varyig thickess ifueces eergy ad dissipatio rate oy i a sma area (d = 1mm ~ 5mm), as show i Figure 4 (due to proportio betwee dissipatio rate diagram it is very difficut to distiguish). This shows circuar thi pate waveguide absorber maiy reies o the radia spread of cyidrica wave eergy dissipatio. The theoretica aaysis ad simuatio show that the desig ca be optimized i terms of eergy cosumptio, wave frequecy, materia parameters ad structure parameters. From the above cacuatio, the dissipatio rate is higher at ower frequecy. Therefore this effect of waveguide vibratio absorber heps to suppress ow-frequecy vibratio. The discussio above, from egieerig practice, idicated that smaer geometric size of circuar viscoeastic waveguide absorber is favorabe to the dampig effect. I additio, the disk shape of the viscoeastic waveguide absorber provides theoretica foudatio for optimizatio desig. Experimet of Vibratio for Machiig Processig The sotted disk wave-guide absorber is appied to a sma machiig ceter i this experimet. The measurig poit ad absorber are istaed o the pate ad the machie too coum separatey, i order to verify the characteristics of the ow frequecy dampig. The work oad excitatio is adopted i the experimet. The vibratio siga is studied whe the machie is workig. Repeated siga has bee coected, ad cacuated, ad spectrum aaysis has bee performed to the data. Repeated measuremet ad aaysis resut are averaged to re duce the iterferece of extera siga. Figure 3(a) ad (b) show the spectrum curve whe the ampitude of vibratio acceeratio is measured with o absorber istaed o the pate ad coum. As ca be see, the performace of the pate ad coum vibratio is i the ow frequecy rage. Figure 3(c) ad (d) provide the spectrum curve whe the ampitude of vibratio acceeratio is measured with absorber istaed. Tabe 1 is the averaged aaytica data before ad after the sotted disk viscoeastic wave-guide absorber is istaed o the pate ad coum. The atteuatio of the vibratio acceeratio ampitude ad dissipatio rate of vibratio eergy is cacuated. A represets the average ampitude before vibratio. A 1 represets ampitude after the vibratio atteuatio. A g A A 1, E A A1 A 1. The uit of the ampitude is1 ms. TABLE 1 EXPERIMENTAL DATA OF THE MACHINE TOOL AND THE COLUMN DAMPER Site Frequecy (Hz) A (ms - ) A1 (ms - ) ΔA(dB) ΔE(%) too tabe coum From the above experimet data of the sotted disk viscoeastic waveguide absorber o the machiig process, it ca be oted that ow frequecy vibratio has good dampig effect. 6

6 Iteratioa Joura of Materia ad Mechaica Egieerig (IJMME), Voume FIGURE 3 THE VIBRATION CHARACTERISTICS OF THE MACHINE TOOL UNMOUNTED AND MOUNTED ON SDWVA Vibratios of the workbech ad pedesta iduced by the machie too tur out to be of ow frequecy. It is feasibe to dissipate the ow frequecy vibratios. Iitia studies show that the viscoeastic wave guide absorber has better performace at ow frequecy tha it does at high frequecy i dissipatig vibratios. This is aso verified by the dissipatio ratig deduced i the experimet. Secody, there is differece betwee the theoretica mode ad the actua system. Machie too aways teds to be buky. These ucertaity factors expai the differece betwee measuremet resuts. The temperature of machie too rises durig operatio, ad the temperature of the wave-guide vibratio absorber istaed at differet ocatios wi aso chage. The properties of viscoeastic materia are strogy depedet o its temperature. However, i the theoretica aaysis the temperature is assumed to be costat, which resuts i differeces betwee theory ad experimet. Lasty, the theoretica aaysis ad cacuatio are based o the ifiite pate. I some coditios, the i- fiite pate ca be used to simuate wave propagatio without cosiderig the refectio. However, i practice, refectio exists o the iterface of the pate. The dissipatio rate i theoretica aaysis is based o the assumptio that the dampig of guide-bar is zero, which is ot true i rea situatio. Therefore, there are differeces betwee theoretica aaysis ad experimeta measuremet. Cocusio The performace of waveguide absorber is good. It ca be widey ad effectivey used i ow frequecy rage to suppress the vibratio. The above theoretica research ays foudatio for future optimizatio o sotted disk waveguide absorber. The experimeta resuts show that, the sotted disk waveguide vibratio absorber is especiay suitabe for vibratio cotro i ow frequecy. ACKNOWLEDGEMENT The fiacia support of the Natioa Natura Sciece Foudatio (Grat o ) ad Natioa Sciece ad 63

7 Iteratioa Joura of Materia ad Mechaica Egieerig (IJMME), Voume 5 16 Techoogy Major Project (Grat o. 1ZX41-1 ad 1ZX41-1) is ackowedged. REFERENCES [1] Hettema, C.D. ad Kim, K.S.: Viscoeastic circuar pate wave guide absorber, Joura of vibratio, acoustics, stress, ad reiabiity i desig, 1991, 113(3), [] Kame Amichi, Noureddie Ataa.: A waveguide absorber based o aaytic Gree's, 5th Europea Microwave Coferece, 1995, 1, [3] J. W. Tu, Y.Yu, L. Huag, B.Tu ad J. Y. Xu.: Research o ew type viscoeastic damper based o MRE smart materia: desig, experimet ad modeig, Materias Research Iovatios, 14, 18(S), S-43-S-49. [4] Spreger, H., Rama, S.R. ad Gau, L.: Absorbig boudary coditios for soid wave guides, Mechaics Research Commuicatios, 11, 38(3), [5] White, Brady J. ad Harmo, H. James.: Ezyme-based detectio of sari (GB) usig paar waveguide absorbace spectroscopy, Sesor Letters, 5, 3(1), [6] Froufe-Perez, L.S., Garcia-Mochaes, P., Serea, P.A. ad Saez, J.J.: Trasmittaces distributios at the diffusive- ocaized crossover i disordered wave-guides with absorptio Noise ad Fuctuatios, 18th Iteratioa Coferece o Noise ad Fuctuatios-ICNF 5, Spai, Saamaca, 5, [7] Kame Amichi, Noureddie Ataa.: A ew 3D fiite eemet for sadwich beams with a viscoeastic core, Joura of Vibratio ad Acoustics Trasactios of the ASME, 9, 131, 11-1~11-9. [8] Shorter, P.J.: Wave propagatio ad dampig i iear viscoeastic amiates, J. Acoust Soc.Am, 4, 115(5),