[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY ANALYSIS OF VIBRATION DAMPING IN PROPELLER SHAFT USING VISCOELASTIC POLYMERS J.Suresh Kumar*, G.V. Pradeep Varma Mechancal Department, Assstant Professor n DIET College, Inda. Mechancal Department, Assocate professor n CITM College, Inda. DOI: 10.581/zenodo.48330 ABSTRACT Ths project work expresses the dfference between the structures wth and wthout dampng materal. The effect of dampng on the performance of sotropc (steel) and orthotropc (Carbon Epoxy) structures s to be analysed by usng Fnte Element Analyss. The values of dampng factor, fundamental natural frequency and the statc deflecton for Steel Shaft, Carbon Epoxy Shaft and are to be compared wth and wthout vscoelastc polymer (Rubber). A new composte dampng materal s to be studed, whch conssts of a vscoelastc matrx and hgh elastc modulus fber nclusons. Ths fber-enhanced vscoelastc-dampng polymer s ntended to be appled to lghtweght flexble structures as a surface treatment for passve vbraton control. A mcro mechancal model s to be establshed and closed form expressons for the effectve storage and loss propertes of the dampng materal are to be derved. An optmal relaton between desgn parameters, such as the length, dameter, spacng, and Young s modulus of fbers and the shear modulus of vscoelastc matrx, s to be derved for achevng maxmum dampng performance. The characterstc value for the maxmum value of Loss modulus s to be found out for the dfferent values of matrx loss factors. KEYWORDS: Vscoelastc polymers, Statc, Model, Transent Dynamc Analyss. INTRODUCTION Composte materals are those contanng more than one bonded materal, each wth dfferent materal propertes. The major advantages of composte materals are that they have a hgh rato of stffness to weght and strength to weght. A prncpal advantage of composte materals les n the ablty of the desgner to talor the materal propertes to the applcaton Vscoelastc materal A Vscoelastc materal sometmes s called materal wth memory. Ths mples that a Vscoelastc materal's behavor depends not only on the current loadng condtons, but also on the loadng hstory. They are characterzed by possessng both vscous and elastc behavor. Fgure 1. shows how varous types of materals behave n the tme doman. A purely elastc materal s one n whch all the energy stored n the sample durng loadng s returned when the load s removed. As a result, the stress and stran curves for elastc materals move completely n phase. For elastc materals, Hooke s Law apples, where the stress s proportonal to the stran, and the modulus s defned at the rato of stress to stran. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [647]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 Fg 1. Stress Stran relatonshp of Vscoelastc Materal w.r.t Tme LITERATURE SURVEY Vscoelastc dampng materals add passve dampng to structures by dsspatng vbraton stran energy and generate heat energy. The ncorporaton of dampng materals n advanced composte materals offers the possblty of hghly damped, lght weght structural components that are vbraton resstant. Concurrng vscoelastc dampng materals n compostes has shown to be successful n greatly ncreasng the dampng of composte structures. The dampng performance, however, s often not as hgh n cocured compostes as n secondarly bonded compostes, where the dampng materal does not undergo the cure process. Substtutng composte structures for conventonal metallc structures has many advantages because of hgher specfc stffness and specfc strength of composte materals. The fber enhanced vscoelastc dampng polymer s ntended to be appled to lghtweght flexble structures as a surface treatment for passve vbraton control. A desrable packng geometry for the composte materal s proposed, whch s expected to produce maxmum shear stran n the vscoelastc dampng matrx. A general method for modelng materal dampng n dynamcal systems s presented and t s prmarly concerned wth a dsspaton model based on vscoelastc assumptons. Dfferent numercal approaches for modelng and analyzng the behavor of structures havng constraned layer dampng. Two numercal studes are presented that reveal the accuracy lmts of the dfferent fnte element modelng approaches for addtve and ntegrally damped plate type structures. Now through the use of mproved computatonal based approaches (.e. fnte element method) along wth the avalablty of relable dampng materals wth accurate thermal and dynamc property characterzatons, t s possble to ncorporate dampng treatments as part of the ntal structural desgn process, thereby vrtually elmnatng sharp resonant peaks. Cylndrcal shells wth a constraned dampng layer treatment are studed usng three theores. Constraned layer dampng n structures s a very popular method to control resonant ampltudes of vbraton. Shells of revoluton (e.g., cylndrcal and concal) fnd wde applcaton n the aerospace ndustry. An effcent method s descrbed for fnte element modelng of three-layer lamnates contanng a vscoelastc layer. Modal dampng ratos are estmated from undamped normal mode results by means of the Modal Stran Energy (MSE) method. The soluton for a radally smply supported shell has been obtaned and the procedure for determnng the dampng effectveness n terms of the system loss factor for all famles of the modes of vbraton n a multlayered shell wth elastc and vscoelastc layers s reported. Approxmately 85% of the passve dampng treatments n actual applcatons are based on vscoelastc materals. The soluton for the vbraton and dampng analyss of a general multlayered cylndrcal shell consstng of an arbtrary number of orthotropc materal elastc and vscoelastc layers wth smply supported end condtons has been reported. DESCRIPTION OF PROBLEM The torque transmsson capablty of the propeller shaft for passenger cars, small trucks, and vans should be larger than 3,500 Nm and fundamental natural bendng frequency of the propeller shaft should be hgher than 6,500 rpm to avod whrlng vbraton. The outer dameter of the propeller shaft should not exceed 100 mm due to space lmtatons. The propeller shaft of transmsson system s desgned for followng specfed desgn requrements as shown n Table 4. 1. Due to space lmtatons the outer dameter of the shaft s restrcted to 90.4 mm. The one-pece hollow composte drve shaft for rear wheel drve automoble should satsfy three desgn specfcatons, such as statc torque transmsson capablty, torsonal bucklng capacty and the fundamental natural bendng frequency. For http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [648]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 gven specfcaton, the dampng factor for Steel, carbon Epoxy are to be calculated and compared wth and wthout dampng materal (Rubber). Table 4. 1 Problem Specfcaton Sl. No. Parameter Notaton Unts Value 1. Torque T N-m 3500. Max Speed N RPM 6500 3. Length L m 1.50 Table 4. Materal Propertes. Sl. No. Propertes Unts Steel Carbon Epoxy Rubber 1 Young s Modulus E11 N / m.068e 11 1.34 e 11 3.107e 6 Young s Modulus E N / m.068e 11 7 e 9 3.107e 6 3 Densty kg / m 3 7830 1600.466 4 Posson Rato - 0.3 0.3 0.49 5 Shear Modulus G N / m - 5.8e 9 1.379e 11 A new composte dampng materal s to be studed, whch conssts of a vscoelastc matrx and hgh elastc modulus fber nclusons. Ths fber-enhanced vscoelastc-dampng polymer s ntended to be appled to lghtweght flexble structures as a surface treatment for passve vbraton control. A mcro mechancal model s to be establshed and closed form expressons for the effectve storage and loss propertes of the dampng materal are to be derved. An optmal relaton between desgn parameters, such as the length, dameter, spacng, and Young s modulus of fbers and the shear modulus of vscoelastc matrx, s to be derved for achevng maxmum dampng performance. STATIC ANALYSIS Statc analyss calculates the effects of steady loadng condtons on a structure, whle gnorng nerta and dampng effects, such as those caused by tme-varyng loads. A statc analyss, however, ncludes steady nerta loads (such as gravty and rotatonal velocty), and tme-varyng loads that can be approxmated as statc equvalent loads (such as the statc equvalent wnd and sesmc loads commonly defned n many buldng codes). Statc Analyss of Steel Shaft wthout vscoelastc Dampng Materal by Beam and Shell Element In ths part statc deflecton of the steel shaft s calculated and compared wth ANSYS results. The specfcaton for the shaft s gven n the Table 5.1. For calculatng the deflecton, the cantlever boundary condton s taken by consderng ts self weght. Table 5.1 Specfcaton of steel shaft Sl. No. Parameters Values 1 Outer Dameter 0.0904 m Thckness.1 e -3 Steel Shaft wthout Dampng Materal for Shell 99 In ths case shell element s taken to calculate the deflecton value for steel shaft. Here Shell element s taken due to specfy the number of layers to nclude the dampng polymer. Here steel shaft wthout dampng materal s consdered and specfcatons are tabulated n table 5.1. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [649]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 Fg 5. Statc Deflecton for Steel Shaft Usng ANSYS 7.0 the deflecton value s calculated. The value s 0.116*10-3 m. The deformed shape of the shaft s shown n the Fg 5.. Steel Shaft wth Dampng Materal In ths type a dampng materal (.e.) Rubber s nserted between the two layers of shaft and the deflecton value s calculated usng ANSYS. The specfcaton of the shaft wth dampng materal s shown n the Table 5.3 Table 5. 3 Specfcatons for Steel Shaft wth Rubber Sl. No. Parameters Values 1 Outer Dameter 0.0904 m Thckness of each layer 1.05 e -3 m 3 Number of layers 3 4 Dampng Materal Rubber 5 Element Shell 99 Fg 5.3 Stackng Sequence for Steel Shaft wth Rubber Fg 5.4 Deflecton of Steel Shaft wth Rubber Usng ANSYS 7.0 the deflecton value s calculated. The value s 0.91 e -4 m. The deformed shape of the shaft s shown n the Fg 5.4. Carbon Epoxy Shaft wthout Dampng Materal http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [650]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 In ths case Carbon Epoxy shaft s modeled wth 13 layers by consderng the shell element. The specfcatons are shown n the Table 5.4 Table 5.4 Specfcaton for Carbon Epoxy Shaft Sl. No. Parameters Values 1 Outer Dameter 0.0904 m Thckness of each layer 1.5 e -4 m 3 Number of layers 13 4 Element Shell 99 Fg 5.5 Stackng Sequence for Carbon Epoxy Shaft Fg 5.6 Statc Deflecton for Carbon Epoxy Shaft Usng ANSYS 7.0 the deflecton value s calculated. The value s 0.84 e -4 m. The deformed shape of the shaft s shown n the Fg 5.6. Carbon Epoxy Shaft wth Dampng Materal In ths case Carbon Epoxy shaft s modeled wth dampng materal (Rubber) and t s ncorporated n between the layers. The specfcaton of the shaft s shown n the Table 5.5. Table 5.5 Specfcaton for Carbon Epoxy Shaft wth Rubber Sl. No. Parameters Values 1 Outer Dameter.0904 m Thckness of each layer 1.5 e -4 m 3 Number of layers 14 4 Dampng Materal Rubber 5 Element Shell 99 http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [651]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 Fg 5.7 Stackng Sequence for Carbon Epoxy Shaft Fg 5.8 Statc Deflecton for Carbon Epoxy Shaft wth Rubber The stackng sequence of the Carbon Epoxy shaft wth dampng materal (Rubber) s shown n the fg 5.7. Here the 7 th layer s the rubber. Usng ANSYS 7.0 the deflecton value s calculated. The value s 0.71 e -4 m. The deformed shape of the shaft s shown n the Fg 5.8. MODAL ANALYSIS Any physcal system can vbrate. The frequences at whch vbraton naturally occurs, and the modal shapes whch the vbratng system assumes are propertes of the system, and can be determned analytcally usng Modal Analyss. Modal analyss s the procedure of determnng a structure's dynamc characterstcs; namely, resonant frequences, dampng values, and the assocated pattern of structural deformaton called mode shapes. It also can be a startng pont for another, more detaled, dynamc analyss, such as a transent dynamc analyss, a harmonc response analyss, or a spectrum analyss. Modal analyss n the ANSYS famly of products s a lnear analyss. Any nonlneartes, such as plastcty and contact (gap) elements, are gnored even f they are defned. Modal analyss can be done through several mode extracton methods: subspace, Block Lanczos, Power Dynamcs, Reduced, Unsymmetrc and Damped. The damped method allows you to nclude dampng n the structure. Modal Analyss of Steel Shaft usng Beam Element wthout Rubber Consder the free-body dagram of an element of a beam shown n fgure. Where M(r, t) s the bendng moment, V(r, t) s the shear force and f (r, t) s the external force per unt length of the beam. Fg 6.1 Beam n Bendng Euler -Bernoull beam equatons wth external force, external moment s gven by followng equatons. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [65]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 4 y(r,t) y(r,t) EI A f(r,t) (1) 4 r t Usng Assumed modes approach, by solvng the above equaton the Egen values and mode shapes can be calculated. Usng the varable separable method y(r, t) may be expressed by followng equatons. 1 y(r, t) (r) q (t) () '''' '' EI (r)q (t) A (r)q (t) Ca 1 1 Va ( r,t) r (3) EI L 0 1 '''' (r)q (t) dr A L 0 1 '' (r)q (t) dr C L a 0 Va ( r,t) r By multplyng (r) on both sdes then apply the Orthogonalty prncple. EI L 0 1 '''' (r) q (t) (r) dr '''' EI (r)- (r) 0 A 4 4 (r)- (r) 0 A L 0 1 '' (r) q (t) (r) dr C L a 0 dr (4) Va (r, t) (r) dr r. (5) (6) (7) The complementary soluton of the above equaton s gven by (r) A sn ( r) B cos ( r) C snh ( r) D cosh ( r) (8) Boundary condtons of cantlever beam, y(r,t) At the fxed end: y(0, t) 0, EI 0 r (9) 3 y (L,t) y (L,t) At the free end: EI 0, EI 0 3 r r (10) Applyng these boundary condtons the soluton of dfferental equaton transformed cosh L cos L to (r) L cosh r-cos r- snh r sn r (11) snh L sn L Shear force at free end s zero, by applyng ths boundary condton n the above equaton the above equaton converted to followng equaton. (1) 1 cos L cosh L 0 From the soluton of the equaton the value can be calculate. The natural frequency of the system s gven by substtutng can be calculated. EI (13) A http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [653]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 Table 6.1 Modal Frequences for Steel Shaft usng BEAM 3 Mode Shapes Analytcal Value Theoretcal Value Usng ANSYS 7.0 ( HZ) (HZ) 1 58.9 57.89 359.68 355.98 3 1007.04 983.61 4 197.94 189 5 360 311.1 Fg 6. Modal Analyss for Steel Shaft usng Beam 3 Fg 6.3 Comparson of Results The fundamental natural frequency of Steel shaft usng Beam 3 element s shown n the Fg 6.. The value of the frequency s 57.89 Hz.The theoretcal and FEA values are compared and shown n the Fg 6.3. Modal Analyss of Steel Shaft usng Shell Element wthout Rubber and wth Rubber In ths case Steel shaft s modeled usng Shell 99. The specfcatons used are same as n the Statc Analyss. Fg 6.4 Modal Analyss for Steel Shaft usng Shell 99 Rubber Fg 6.5 Modal Analyss for Steel Shaft wth http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [654]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 The fundamental natural frequency of the steel shaft usng Shell 99 s shown n the Fg 6.4. The value s 57.587 Hz.The fundamental natural frequency of the Steel Shaft wth Rubber s shown n the Fg 6.5. The value s 64.937 Hz. Modal Analyss Of Carbon Epoxy Shaft Wthout rubber and Wth Rubber Fg 6.6 Modal Analyss for Carbon Epoxy Shaft Fg 6.7 Modal Analyss for Carbon Epoxy Shaft wth Rubber The fundamental natural frequency of the Carbon Epoxy Shaft s shown n the Fg 6.6. The value s 67.601 Hz. The fundamental natural frequency of the Carbon Epoxy Shaft wth s shown n the Fg 6.7. The value s 7.443 Hz TRANSIENT DYNAMIC ANALYSIS Transent dynamc analyss s a technque used to determne the dynamc response of a structure under a tmevaryng load. The tme frame for ths type of analyss s such that nerta or dampng effects of the structure are consdered to be mportant. Cases where such effects play a major role are under step or mpulse loadng condtons, for example, where there s a sharp load change n a fracton of tme. If nerta effects are neglgble for the loadng condtons beng consdered, a statc analyss may be used nstead. It should be noted that a transent analyss s more nvolved than a statc or harmonc analyss. It requres a good understandng of the dynamc behavor of a structure. Therefore, a modal analyss of the structure should be ntally performed to provde nformaton about the structure's dynamc behavor. In ANSYS, transent dynamc analyss can be carred out usng 3 methods. The Full Method: Ths s the easest method to use. All types of non-lneartes are allowed. It s however very CPU ntensve to go ths route as full system matrces are used. The Reduced Method: Ths method reduces the system matrces to only consder the Master Degrees of Freedom (MDOFs). Because of the reduced sze of the matrces, the calculatons are much qucker. However, ths method handles only lnear problems (such as our cantlever case). The Mode Superposton Method: Ths method requres a prelmnary modal analyss, as factored mode shapes are summed to calculate the structure's response. It s the quckest of the three methods, but t requres a good deal of understandng of the problem at hand. In ths project the Reduced Method s used for conductng the transent analyss. Usually one need not go further than revewng the Reduced Results. However, f stresses and forces are of nterest than, we would have to Expand the Reduced Soluton. Transent Dynamc Analyss of Steel Shaft wthout dampng materal by beam element and shell element In ths part Transent Dynamc analyss of the shaft s calculated usng the same specfcatons as gven n the statc analyss. Transent Dynamc analyss s needed because the output of ths analyss s used n calculatng the dampng factor usng Log Decrement method. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [655]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 Transent Dynamc Analyss of Steel Shaft usng Beam Element Fg 7.1 Transent Dynamc Analyss of Steel Shaft usng Beam 3 The Transent Dynamc Analyss of Steel Shaft usng Beam 3 Element s shown n the Fg 7.1. From the above Fg 7.1 dampng factor s calculated usng Log Decrement method and the value s 0. 01396 Transent Dynamc Analyss of Steel Shaft usng Shell Element wthout and wth Rubber In ths case Steel shaft s modeled usng Shell 99. The specfcatons used are same as n the Statc Analyss. Fg 7. Transent Analyss for Steel Shaft wthout rubber Rubber Fg 7.3 Transent Analyss for Steel Shaft usng The Transent Dynamc Analyss of Steel Shaft usng Beam 3 Element s shown n the Fg 7.. From the above Fg 7. dampng factor s calculated usng Log Decrement method and the value s 0. 016766.The Transent Dynamc Analyss of Steel Shaft usng Shell 99 wth Rubber s shown n the Fg 7.3. From the above Fg 7.3 dampng factor s calculated usng Log Decrement method and the value s 0. 0195. Transent Dynamc Analyss of Carbon Epoxy Shaft wthout and wth Rubber http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [656]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 Fg 7.4 Transent Analyss for Carbon Epoxy Shaft Fg 7.5 Transent Analyss for Carbon Epoxy Shaft wth Rubber The Transent Dynamc Analyss of Carbon Epoxy Shaft usng Shell 99 s shown n the Fg 7.4. From the above Fg 7.4 dampng factor s calculated usng Log Decrement method and the value s 0. 0657.The Transent Dynamc Analyss of Carbon Epoxy Shaft usng Shell 99 wth Rubber s shown n the Fg 7.5. From the above Fg 7.5 dampng factor s calculated usng Log Decrement method and the value s 0. 09005. Optmzaton Of Dampng Propertes It s noted that means proposed for mprovng free layer treatment, and the lghtweght of fbers wll beneft applcatons where weght s an mportant consderaton. Fber aspect rato, fber tp spacng, fber angle and a number of other parameters were vared to mprove the dampng performance of a structural composte materal. In ths chapter, a mcro mechancal model for the fber-enhanced vscoelastc-dampng polymer s establshed, and closed form expressons for the effectve complex modul are shown. Based on ths model, dampng performance of the enhanced dampng polymer s optmzed by establshng an optmal relaton between the desgn parameters, such as length. Vscoelastc Dampng Treatment Vscoelastc dampng treatment performances nclude the followng: 1. Hgh loss modulus vscoelastc dampng materals. Hgh extensonal stffness constranng layers 3. A multplcty of layers 4. Optmal secton length constranng layer segments Due to the hgh elastc modulus, usually twce and three tmes as much as that of steel and alumnum, respectvely hgh elastc modulus fbrous materals, for example Kevlar or graphte fber, are desrable canddates for constrant materals n vscoelastc dampng treatments. By embeddng fber nto a vscoelastc polymer matrx to make a sngle composte dampng product, all the features cted above can be readly acheved. The optmal segment lengths of the fbers can be controlled n fabrcatng processes, and the effectve number of dampng layers can be ncreased to a great extent wthout dffculty. Furthermore, the proposed fber enhanced vscoelastc dampng polymer can be convenently nstalled on structural surfaces as a dameter, spacng, and Young s modulus of the fbers and shear modulus of the vscoelastc matrx. Effectve Complex Module Based on the fundamental prncples of constraned vscoelastc layer treatments, a packng geometry wth staggered layers as shown n the fg. 1 s selected, as t makes effcent use of the materals. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [657]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 In ths Fg 8.1, the rods wth square cross secton and fnte length represent fbers, whch are embedded n a vscoelastc matrx. The square fber cross secton s used n the model for convenence. The spacng between neghborng fbers s assumed to be the same and the gaps between fber ends are neglected n the present analyss. The Fg 8. also show a selected representatve volume element whose geometrcal characterstcs and stress stran relatons are the same for any of such elements, regardless of the poston n the composte materal. The average stress and stran of the representatve volume element therefore are same for any such elements and also the same as that of the entre composte materal under a unform loadng. Fg 8.1 Fg 8. Fg 8.1 Schematc representaton of the fber enhanced vscoelastc polymer under stretchng Fg 8. Schematc representaton of fnte element model for representatve volume element Dsplacement Feld Under Axal Loadng Fber enhanced Vscoelastc dampng materals appled to the surface of flexble structures, wll experence perodcally axal loadng when the base structure vbrates. In the development of the dsplacement feld wthn the composte under ths axal loadng, the followng assumptons are beng made: a. Due to the much hgher stffness of the fbers, usually orders of magntude hgher than that of the vscoelastc matrx, t s assumed that the fber sustans extensonal stress only and the vscoelastc matrx transmts shear stress only. b. Unform normal stress feld and unform shear stress feld are assumed through the cross sectons of the fbers and the vscoelastc matrx, respectvely due to ther very small dmensons. c. Lnear materal propertes are assumed for the fbers and the vscoelastc matrx, as antcpated strans are small and well wthn the lnear range. d. Posson s rato effects are neglgble, agan due to small strans. e. Fbers are elastc and dsspate very lttle energy relatve to the vscoelastc matrx. The front and backsdes of the fbers are subjected to shear stresses due to exstence of the vscoelastc matrx. Due to the symmetrc confguraton of the composte, the soluton of the dsplacement feld can be reduced to a two dmensonal problem. It s well known that the complex modulus approach s an effcent method to descrbe the dynamc behavor of homogenous vscoelastc materals. Based on the above assumptons and by applyng boundary condtons, the fnal expressons for the effectve storage modulus E, loss modulus E and loss factor the propertes of the fber enhanced vscoelastc dampng ' C polymers under unaxal loadng are gven as: '' N EC E f v f D ' N1 EC E f v f D N E N 1 '' C E () (3) (1) http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [658]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 N N (5) 1 v v v v v v 1 coth 1 cot cot cot 1 coth (6) coth cot v v D 4 4 ' (7) L f Gv 1 Where, v v v v v coth 1 cot cot 1 coth t v d f E f 4 v 1 cot coth v 1 v 1 coth 1 cot E Here, represents the characterstc value, represents the matrx loss factor, and V f represents the fbre volume fracton. It can be seen that all these effectve propertes are functons of the packng geometry of the composte, and the materal propertes of the consttuents. These propertes wll be employed n the analyss, whch follows. Table 8.1 Desgn Specfcatons Sl.No Quantty Symbol Unt Value 1. Dstance between fbers t v n 0.0001. Dameter of fbers d f n 0.0005 3. Half length of fbers L f / n 0.05 4. Normal stress lb/n 1.44 e 5 5. Shear modulus of matrx G v ' lb/n 130 6. Tensle modulus of fbers E f lb/n 6.4 e 7 Table 8. Calculated Values for α N1 N D v =0.75 '' E C /VfEf 0. 417.7 3379.17 11707.4 0.0776 0.81865 0.4 315.71 16.98 475. 0.08766 0.6876 0.6 81.34 43.79 33.58 0.13534 0.5384 0.8 34.14 13.87 9.37 0.150 0.40636 1.0 18.43 5.51 39.65 0.13896 0.990 1. 11.53.47 1.51 0.11478 0.141 1.4 7.94 1.17 13.47 0.087 0.14764 Table 8.3 Calculated Values for 1 1 v v coth cot 4 (4) v =1 α N1 N D '' E /VfEf 0. 083.8 59.94 67781.1 0.03733 1.144 0.4 169.4 161.88 1359.5 0.11909 0.95651 0.6 46.9 3.59 178.6 0.185 0.704 0.8 0.37 10.36 5.9 0.19808 0.50858 1.0 11.39 4.18 3. 0.18009 0.36703 1. 7.34 1.96 13.03 0.1506 0.6694 1.4 5.18 1.03 8.44 0.118 0.19831 C E E http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [659]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 Table 8.4 Calculated Values for α N1 N D v =1.5 '' E C /VfEf 0. 1140 019.39 484.99 0.04715 1.77139 0.4 10.19 18.6 843.7 0.1545 1.5867 0.6 30.4 5.8 111.94 0.3068 0.84939 0.8 14.19 8.4 34 0.439 0.58083 1.0 8.8 3.4 15.8 0.157 0.41085 OPTIMIZATION OF DAMPING PROPERTIES The closed form expressons of the effectve materal propertes developed above allow optmzaton of the dampng propertes for the fber enhanced vscoelastc polymers. The followng dscusson consders the establshment of the approprate crteron for dampng desgn optmzaton. Intutvely, a large value of loss factor s a desred property for dampng materals. Consder the dampng mechansm n a vscoelastc materal. In order to dsspate energy, a dampng materal must store some energy frst. The more energy t can store, the more energy t wll dsspate. The ablty to store energy s measured by the storage modulus of materals. It s clear that the best quantty reflectng the dampng performance of a vscoelastc materal s the loss modulus of the materal,.e., the product of the loss factor and the storage modulus. It may not always be desrable, though, to use a very hgh storage modulus to acheve a hgh loss modulus. The reason s that a hgh storage modulus of a dampng materal could sgnfcantly change the stffness of the base structure on whch t wll be bonded. Therefore, a good desgn nvolves proper balance between loss factor and storage modulus. The proposed strategy for optmzaton of the fber-enhanced vscoelastc-dampng polymer s gven below. Examne the varaton of the composte loss factor, η E, wth the characterstc value. Based on the results of the step one, select a narrowed range of to maxmze the loss modulus, E " c After all the parameters are determned, evaluate the storage modulus, η E, and loss factor, E' c. The desgn specfcaton for the model s shown n the table 8. 1. The calculated values are shown sn table 8., table 8. 3 and table 8. 4. Loss Modulus 0.3 0.5 0. 0.15 0.1 0.05 0 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 Characterstc Value (α) v v v Fg 8.3 Effect of Characterstc value on loss modulus ( E /VfEf ) Loss Factor =1 '' C value on loss factor = E Fg 8.4 Effect of Characterstc DISCUSSIONS For dfferent values of matrx loss factors, η v, maxmum value of E " c occurs at the value of around 0.75. It can be concluded that for maxmum dampng performance, the characterstc value of the composte,.e.,, should be set to 0.75. To satsfy ths optmum condton, one can vary the parameters used to calculate the characterstc value.5 1.5 1 0.5 v v =1 v 0 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 Characterstc Value http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [660]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 (.e., E f, G v ', t v, d f and L f ). For example, f E f, G v ', d f and L f are prescrbed by a specfc desgn, satsfacton of ths condton wll yeld an optmal dstance between fbers, t v. A large value for the fber Young s modulus E f, s a desred property for the fber enhanced vscoelastc dampng polymer. CONCLUSIONS The dampng factor has been found out for Steel Shaft, Carbon Epoxy Shaft wth and wthout Vscoelastc polymer (Rubber). The Statc, Modal and Transent Dynamc Analyses have been carred out usng Fnte Element Analyss. The followng observatons were made by embeddng the Vscoelastc polymer (Rubber) nto the structure. Dampng factor ncreased by 16.3%, 9.% for Steel, Carbon Epoxy shafts respectvely. The fundamental natural frequency ncreased by 1.76%, 7.% for Steel, Carbon Epoxy shafts respectvely. The deflecton value decreased by 1.38%, 1.71% for Steel, Carbon Epoxy Shafts respectvely. The ncrease n dampng factor results n further suppresson of vbratons and hence results n ncreased structural lfe. An optmal relaton between desgn parameters such as the length, dameter, spacng, and Young s modulus of fbers and the shear modulus of vscoelastc matrx has been derved for achevng maxmum dampng performance. It has been found that for maxmum dampng performance, the characterstc value of the composte should be set to 0.75. BIBLIOGRAPHY [1] Autar K. Kaw, "Mechancs of Composte Materals", CRC press, 1997. [] Ahd D. Nashf, Davd I. G. Jones and John P. Henderson, Vbraton Dampng, John Wley & Sons Publcaton, 1985, Newyork. [3] C. T. Sun and Y. P. Lu, "Vbraton Dampng of Structural Elements", Prentnce Hall PTR, New Jeresy, 1995. [4] K. L. Napoltano, W. Grppo, J. B. Kosmatka and C. D. Johnson, A comparson of two cocured damped composte torson shafts, Composte Structures, Vol. 43, 1998, pp. 115-15. [5] J. M. Bggerstaff and J. B. Kosmatka, Dampng Performance of Cocured Composte Lamnates wth Embedded Vscoelastc Layers, Journal of Composte Materals, Vol. 3, No.1/ 1998. [6] Jn Kook Km, Da Gl Lee, and Durk Hyun Cho, 001, Investgaton of Adhesvely Bonded Jonts for Composte Propeller shafts, Journal of Composte Materals, Vol.35, No.11, pp. 999-101. [7] T. E. Alberts and Houchun Xa, Desgn and Analyss of Fber Enhanced Vscoelastc Dampng Polymers, Journal of Vbraton and Acoustcs, Vol. 117, October 1995, pp. 398-404. [8] K. J. Buharwala and J. S. Hansen, "Dynamcs of Vscoelastc Structures", AIAA Journal, Vol. 6, February 1988, pp 0-7. [9] J. B. Kosmatka and S. L. Lguore, Revew of Methods for Analyzng Constraned Layer Damped Structures, Journal of Aerospace Engneerng, Vol.6, No.3, July 1993, pp. 68-83. [10] T. C. Ramesh and N. Ganesan, Vbraton and Dampng Analyss of Cylndrcal Shells wth Constraned Dampng Treatment- A Comparson of Three Theores, Journal of Vbraton and Acoustcs, Vol. 117, Aprl 1995, pp. 13 19. [11] Conor D. Johnson and Davd A. Kenholz, Fnte Element Predcton of Dampng n Structures wth Constraned Vscoelastc Layers, AIAA Journal, Vol. 0, No. 3, September 198, pp. 184-190. [1] N. T. Asnan and Nayar Alam, " Vbraton and Dampng Analyss of a Multlayered Cylndrcal Shell, Part I: Theoretcal Analyss", AIAA Journal, Vol., No. 6, June 1984, pp 803-810 [13] N. T. Asnan and Nayar Alam, " Vbraton and Dampng Analyss of a Multlayered Cylndrcal Shell, Part II: Numercal Results", AIAA Journal, Vol., No. 7, July 1984, pp 975-981 [14] D. A. Saravanos and J. M. Perera, Dynamc Characterstcs of Specalty Composte Structures wth Embedded Dampng Layers, Journal of Vbraton and Acoustcs, Vol. 117, January 1995, pp. 6-69. [15] C.D. Johnson, Desgn of Passve Dampng Systems, Transactons of the ASME, Vol. 117, June 1995, pp. 171-176. [16] H. V. Panossan, "Structural Dampng Enhancement Va Non- Obstructve Partcle Dampng Technque", Journal of Vbraton and Acoustcs, Vol.114, January 199, pp.101-105. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [661]
[Kumar*, 5(3): March, 016] ISSN: 77-9655 (IOR), Publcaton Impact Factor: 3.785 [17] M. A. Trndade, A. Benjeddou and R. Ohayon, Modelng of Frequency- Dependent Vscoelastc Materals for Actve- Passve Vbraton Dampng, Journal of Vbraton and Acoustcs, Vol.1, Aprl 000, pp.169-174. [18] John F. Baldwn and Stanley G. Hutton, " Natural Modes of Modfed Structures", AIAA Journal, Vol. 3, No. 11, November 1985, pp.1737-1743. [19] M. L. Son and F. K. Bogner, "Fnte Element Vbraton Analyss of Damped Structures", AIAA Journal, Vol. 0, No. 5, May 198, pp. 700-707. [0] Y. P. Lu, B. E. Douglas and E. V. Thomas, Mechancal Impedance of Damped Three- Layered Sandwch Rngs, AIAA Journal, Vol. 11, No. 5, March 1973, pp. 300-304. http: // www.jesrt.com Internatonal Journal of Engneerng Scences & Research Technology [66]