EXPERIMENTAL STUDY AND NUMERICAL ANALYSIS ON VIBRATION CHARACTERISTICS OF SIMPLY SUPPORTED OVERHANG BEAM UNDER LARGE DEFORMATION

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1 Proceengs of the Internatonal Conference on Mechancal Engneerng 0 (ICME0) 8-0 December 0, Dhaka, Banglaesh ICME- EXPERIMENTAL STUDY AND NUMERICAL ANALYSIS ON VIBRATION CHARACTERISTICS OF SIMPLY SUPPORTED OVERHANG BEAM UNDER LARGE DEFORMATION Anrban Mtra, Dpenra Kr. Roy, Prasanta Sahoo an Kashnath Saha Department of Mechancal Engneerng, Jaavpur Unversty, Kolkata 70003, Ina. ABSTRACT The present paper unertakes an expermental an theoretcal ynamc analyss of a smply supporte overhang beam. A set up, consstng of two knfe eges to smulate smply supporte bounary contons an provson for applcaton of transverse loa through two suspene weght pans at the extremtes of the beam, s prepare to carry out expermentatons on a slener beam. The free vbraton expermentaton s carre out by exctng the loae system wth the blow of a soft rubber hammer an capturng the response by a mnature accelerometer. The theoretcal ynamc analyss s carre out on the bass of energy prncples an a technque base on the evelope axal forces s ntrouce to ncorporate the stretchng effect n the present work. The smple supports are replace by two sprngs so that the pont above the support can have some movement epenng on the stffness of the sprngs. Keywors: Overhang Beam, Smply Supporte Beam, Free Vbraton.. INTRODUCTION Slener beams are one of the most common structural elements that can be use separately or n assocaton wth other beams or plates, n many fels of engneerng. They are extensvely use n varous branches of moern cvl, mechancal an constructon engneerng. In these applcatons they are regularly subecte to statc an ynamc loas. Hence, analyss of beams uner fferent loang an bounary contons has always been an area of mmense nterest to researches an research stues carre out n ths fel have been recore n fferent revew papers. Sathyamoorthy [] revewe the works on classcal methos of non-lnear (geometrc, materal an other type of non-lneartes) beam analyss. Sathyamoorthy [] also surveye the evelopments on non-lnear beam analyss uner statc an ynamc contons usng fnte elements methos. Kapana an Ract [3] presente a revew on avances n the analyss of lamnate structures (beams an plates) usng shear eformaton theores, fnte elements methos an also on bucklng of such structures. Me [4] presente a fnte element metho to etermne the nonlnear frequency of beams for large ampltue free vbratons. Km an Dcknson [5] performe a smple analyss of the free vbraton problem of slener beams subect to varous complcatng effects usng Raylegh-Rtz metho, wth orthogonally generate polynomals as amssble functons. Klausbruckner an Pryputnewcz [6] put forwar a theoretcal an expermental stuy of couple vbratons of channel beams. Ther expermental analyss was base on laser hologram nterferometry. Ganapath et al [7] stue the nonlnear free flexural vbratons of orthotropc straght an curve beams utlzng a cubc B-splne shear flexble curve element, base on the fel consstency prncple. They solve the nonlnear governng equatons by employng Newmark's numercal ntegraton scheme couple wth mofe Newton-Raphson teraton technque. Azrar et al [8] evelope a sem-analytcal approach to the nonlnear ynamc response problem of smply supporte an clampe beams base on Lagrange s prncple an the harmonc balance metho. Kapura et al [9] carre out statc an free vbraton response of layere FGM beams expermentally as well as theoretcally. Hollan et al [0] escrbe the behavor of a slener, tapere, cantlever beam loae through a cable attache to ts free en. Large statc eflectons were compute together wth natural frequences an moe shapes for small-ampltue vbratons about equlbrum. Gupta et al [] nvestgate large ampltue free vbraton analyss of unform, slener an sotropc beams through a relatvely smple fnte element formulaton, applcable to homogenous cubc nonlnear temporal equaton. Ths fnte element formulaton was apple to analyze free vbraton of unform sotropc Tmoshenko beams wth geometrc nonlnearty by Guna et al []. Karaagac et al [3] presente theoretcal an expermental free vbraton stues of a slener cantlever beam wth an ege crack. In ths stuy, a fnte element algorthm base on energy metho was evelope an experments were carre out n orer to ICME0

2 valate the results obtane from the numercal metho. Gunta et al [4] propose a unfe formulaton of one-mensonal beam moels for the free vbraton analyss of functonally grae beams. It s event from the revew of exstng lterature that theoretcal an expermental nvestgatons of a sngle beam uner fferent loang an bounary contons have been carre out extensvely. The present paper puts forwar an expermental an theoretcal free vbraton analyss of a smply supporte overhang beam uner transverse loang at ts extreme ens. Fg. shows such a smply supporte thn rectangular beam havng two overhang portons of equal length, wth ts sgnfcant mensons an the coornate system use n the present analyss. As can be seen from the fgure, the span between the two supports s l an the two overhang portons are of equal length (a). The total length of the beam s represente by L, whch s relate to the span length through the relaton, L = l + a. Such experments on a beam have not been performe prevously. Also a technque base on the evelope axal forces s ntrouce to ncorporate the stretchng effect n the present work. Although the present work s lmte wthn the elastc regme of materal behavour but the analyss metho can be easly reconstructe for post elastc behavour havng multlnear materal characterstcs. Fg. Smply supporte overhang beam wth sgnfcant mensons an the coornate system. EXPERIMENTATION Fg. Photograph of the expermental set up A basc expermental set up, consstng of two knfe eges at a fxe stance apart that smulate smply supporte contons, s prepare to carry out free vbraton expermentaton on a slener beam. Fgures an 3 present the photograph an schematc agram of the above mentone expermental set up. The set up conssts of a sol base wth knfe eges, an accelerometer, coupler an osclloscope. The free vbraton expermentaton s carre out at the eflecte confguraton by exctng the system wth the blow of a soft rubber hammer. The followng secton proves a bref escrpton of the set up an test proceure. Fg 3. Schematc agram of expermental setup for free vbraton experments on smply supporte overhang beam.. Expermental Set-up A slener beam wth rectangular cross-secton s carefully postone over the two knfe eges of the base n such a way that the overhang lengths at the two ens of the beam are equal. The base tself s rgly fxe to a heavy base wth C-clamps to prove further stablty. Two weght pans are suspene from the extremtes of the slener beam as loang arrangement for applcaton of transverse loa at those locatons. A shear moe pezoelectrc accelerometer (Manufacturer: Kstler Instrument Corporaton, Type: 878A500, acceleraton range: 500g (g = m/s ), frequency range: Hz 0 khz ( 5%)) s mounte on the beam at sutable locatons usng Petro-Wax ahesve. The postons of the accelerometer are carefully selecte to avo any noal ponts. The mass of the accelerometer s.6 grams, whch s sgnfcantly less than the mass of the beam. Hence t can be assume that system response s not sgnfcantly altere by the effect of mass loang of the accelerometer. Constant current power supply to the mpeence converter of the accelerometer s prove by a coupler (Manufacturer: Kstler Instrument Corporaton, Type: 54, Frequency response: 0.07 Hz 60 khz ( 5%)) connecte to the accelerometer. It also ecouples the DC bas voltage from the output sgnal. The connecton between the accelerometer an the coupler s through a two-wre cable. The coupler proves the electrcal nterface between the accelerometer an the splay evce, a gtal storage osclloscope (Manufacturer: Tektronx Inc., Moel: TDS 0) wth the followng specfcatons: peak etect bath: 50MHz, sample rate range: 50 samples/s Ggasamples/s, recor length: 500 samples, an lower frequency lmt: 0 Hz. It has the capablty to transform a tme oman sgnal nto frequency oman through a Fast Fourer Transform (FFT) moule.. Test Proceure Experments are carre out to etermne the free vbraton characterstcs of the slener beam at ts eflecte confguraton uner transverse loang. The set ICME0

3 up s reae by makng the electrcal connectons for osclloscope, coupler an accelerometer. The accelerometer s mounte on the test specmen at a preefne locaton usng ahesve. A two-wre cable between the accelerometer an the coupler s use an the sgnal an power share the same lne. Output from the coupler s connecte to one of the channels of the osclloscope. Equal transverse loa at the two ens of the beam s apple by placng ea weghts on the weght pans attache at the extremtes of the beam. Uner loang the beam assumes a eflecte confguraton. The osclloscope s set to math moe an auto trgger moe s kept on. It s then kept reay by pressng the RUN button as the system s hammere to prove sturbance. Osclloscope captures the sgnal from the vbratng beam an plots the sgnal n frequency oman. The natural frequences of the beam at the eflecte confguraton can be calle loae natural frequency. The osclloscope captures an plots the sgnal from the vbratng system n frequency-ampltue plane. Usng vertcal cursors frequency (n Hz) of the sgnal s rea from the splay an the ata s tabulate. The proceure s repeate for the next loa level after ang ea weght to the pan to graually ncrease loa. 3. THEORETICAL ANALYSIS The theoretcal ynamc analyss s carre out n two steps. Frst, eflecton s statcally mpose by applyng transverse concentrate loa on the two ens of the beam an then the free vbraton analyss s performe as an egen value problem to entfy the natural frequences of the system at eflecte confguraton. The statc analyss yels the ntal eflecton profle, whch s use n the subsequent free vbraton analyss. As the ynamc problem s solve on the bass of the soluton of the statc splacement fel, the stretchng effect of statcally mpose splacement s ncorporate nto the ynamc system. The orgn of the coornate system s taken at the left en of the beam, as shown n fgure. In the present work cross-secton of the beam has been taken as rectangular an the wth an thckness are enote by b an t, respectvely. It s assume that the thckness of the beam s suffcently small to gnore the effects of shear eformaton. The mathematcal formulaton s further base on the assumpton that beam materal s sotropc, homogeneous an lnearly elastc. It shoul be mentone here that n the present formulaton axal splacement s not consere rectly an stretchng s taken nto account through axal forces. The mathematcal formulaton of both the statc an ynamc analyss s base on energy formulaton. 3. Statc Analyss It s known from the prncple of mnmum potental energy that, U V 0. () Here, V s the work functon or potental of the external forces an U s the total stran energy store n the system. The expresson of the stran energy can be erve from the followng expresson, E U ( x ) V () vol where, x z ( w x ) s the axal stran of a fber ue to benng acton. w enotes transverse splacements of m-plane an x enotes axal coornate. It s to be note that all the computatons are carre out n normalze coornate ξ ( = x / L), L beng the overall length of the beam. However, n the present case, an extra term contanng the axal force strbuton (N x ) along the length of the beam s nclue n the total stran energy (U) expresson. Ths term (N x ) takes nto account the stretchng effect arsng out of the change n length of the beam. So, the complete expresson of the stran energy (U) of the system s gven as, U EI 3 L w 0 L 0 N x ( ) w where, E an I are the elastc moulus of beam materal an moment of nerta of the cross-secton, respectvely. The external concentrate loang of magntue P s apple at the two extreme ens of the overhang beam to prouce the benng eformaton. The total potental energy for the apple external loang s gven as follows. V P w P w (4) 0 where, w 0 an w are the eflectons of the two extreme ens of the beam. For the theoretcal analyss the smple supports are replace by two sprngs of stffness k, as shown n fgure 4. It mples that the pont above the support can have some movement epenng on the stffness of the sprngs. Ths s one n orer to replcate the actual stuaton prevalent urng the free vbraton analyss. Fg 4. Schematc representaton of uneforme an eforme beam. (3) The above mentone energy functonals (U an V) can be etermne from the assume transverse (w) splacement fel or whch can also be calle as the eflecton. Ths splacement fel can be approxmate as lnear combnatons of orthogonal amssble functons an unknown coeffcents as shown below. w( ) ( ) (5) Here, ( ) s a set of numbers of orthogonal functons, an these functons are to be selecte n such a ICME0 3

4 way that they satsfy the flexural bounary contons of the beam. In orer to generate the start functon for the approxmate splacement fel a 5 th orer polynomal of the form shown n Eq. (6) s assume an ths equaton s subecte to the geometrc bounary contons of the beam as mentone below n Eq. (7). Polynomal: Bounary Contons: ( ) 5 0 C, where, C 0 = (6) w 0 at = 0 an, w = F/k at = a/l an ( - a/l) (7) w an 0 at = 0.5. Applyng the bounary contons on the assume polynomal a set of lnear smultaneous equatons s obtane. Solvng ths set of equatons the coeffcents (C ) of the polynomal can be etermne rectly. It s worth pontng out that epenng on the locaton of the support (a) an stffness of the sprngs, coeffcents of the polynomal woul change an fferent start functons woul be obtane. The hgher-orer functons are generate through a numercal mplementaton of the Gram - Schmt orthogonalzaton proceure. To cater to the nee of the numercal scheme, all the functons are escrbe numercally at some sutably selecte Gauss ponts. Substtuton of the complete energy expressons an approxmate splacement fel n Eq. () gves the set of system governng equatons n matrx form, K f (8) [K], {} an {f} are stffness matrx, vector of unknown coeffcents an loa vector, respectvely. The fferent elements of the stffness matrx an the loa vector are gven below. k f EI L 3 L 0 0 P N x ( ) 0 P (9) (0) The stffness matrx [K] contans the axal force strbuton (N x ) whose values are not known a pror wthn ts elements. Hence at the frst step of the teraton N x s assume as zero an the unknown coeffcents are calculate on the bass of ths assumpton. From the compute coeffcent values the transverse splacement (w) of the beam are etermne an from ths splacement fel the axal stran values along the beam are compute. Corresponngly, the stress an axal forces are calculate an process s repeate wth new values of axal force. The calculate splacement values are compare wth those from the prevous teraton. If the error s outse the preefne lmt, the process s repeate wth new values of N x, untl error becomes less than the specfe value an convergence s acheve. The mofcaton of axal force strbuton values are acheve through the followng expresson, {N x } = {N x } ol + λ({n x }-{N x } ol ), where λ s the relaxaton parameter. Once convergence s acheve for a partcular loa step, an ncrement s prove to the concentrate loa an the next loa step starts by agan ntalzng the axal force strbuton to zero. 3. Dynamc Analyss The governng set of equatons for the ynamc analyss s erve followng Hamlton s prncple, whch s expresse as, T U 0 () Accorng to Hamlton s prncple a ynamc system can be characterze by two energy functonals, knetc energy an potental energy. In the mathematcal expresson T an U represent the total knetc energy of the system an total stran energy store n the system. The expresson for total knetc energy (T) s gven by, T where, b t L 0 w s ensty of the beam. () The ynamc splacement, w (, ), s assume to be separable n space an tme an can be approxmately represente by fnte lnear combnatons of orthogonal amssble functons an a new set of unknown coeffcents c as, w(, ) c e (3) Here, s the natural frequency of the system an {c} represents the egenvectors n matrx form whch ncates the contrbuton of the nvual space functons n a partcular vbraton frequency moe. The spatal functons, ( ), are entcal to those for the statc analyss an are known completely. Substtuton of the knetc (T) an stran energy (U) expressons along wth the ynamc splacement fel gves the governng equaton of the ynamc system n the followng form. M c K c 0 (4) where, M s the mass matrx an ts fferent elements are prove below. M bt L 0 (5) The soluton of the stanar egen value problem of Eq. (4) s obtane numercally through IMSL routnes. The square roots of the calculate egen values represent the free vbraton frequences of the beam at the statcally eflecte confguraton. The plot of these frequences aganst the corresponng ampltues n non-mensonal plane represents the backbone curve of the system. 4. RESULTS AND DISCUSSION The present work has the obectve of stuyng free vbraton of smply supporte overhang beam uner concentrate loang, both expermentally an theoretcally. For the theoretcal analyss, the number of functons () for the transverse splacement (w) s taken as, whereas, the number of Gauss ponts s taken as 4. The tolerance value of the error lmt ( ) for the numercal teraton scheme s taken as 0.0% an the ICME0 4

5 relaxaton parameter ( ) s Experment s carre out for a beam of followng mensons: L = 0.75 m, b = m, t = m an a = 0.05 m. The materal of the beams s ml steel. Table : Natural frequences of smply supporte overhang beam uner fferent loa levels Sl. No. Loa (Kg.) Natural Frequency (Hz.) 3 W P W P W P W P W P the extreme ens of the beam. The moeshapes for the vbratng system can be etermne from the egenvectors corresponng to the egenvalues. The moeshape plots for the frst three vbraton moes of the beam uner pont loang are shown n fgure 6. Changes n the frst moeshape ue to change n locaton of the support has also been stue an presente n Fg. 7. The effect of varaton of overhang lengths s qute clear from the fgure. From the fgure t s clear that for a/l = 0.0 an 0.5 the maxmum vbraton ampltue occurs at the extremtes of the beam; whereas, for the other two cases t occurs at the m-pont of the span. It can be sa that as the overhang length ncreases,.e., as the two supports move closer to each other the maxmum vbraton ampltue takes place at the two ens of the beam. The natural frequences for the frst three moes of vbraton of the system are rea from the osclloscope an presente corresponng to fferent loa levels n Table, where, W P refers to the weght of the pan. The natural frequency of the beam uner no loang conton s obtane as 08 Hz. However, the theoretcal analyss proves the value as Hz. It can be seen that the theoretcal an expermental results ffer from each other. The fference can be attrbute to nsuffcency n replcatng the bounary contons of the system. The theoretcal ynamc behavour of the smply supporte overhang beam s shown graphcally as the backbone curves for the frst three moes n the mensonless ampltue-frequency plane. The rato of the maxmum beam eflecton to beam thckness s taken as the mensonless ampltue (w max /t) whle the mensonless frequency (ω /ω) s obtane by normalzng the nonlnear frequency (ω) wth the corresponng funamental lnear frequency (ω ). Fg 6. Moe shape plots for smply supporte overhang beam uner concentrate loang at ts ens. Fg 7. Varaton n frst moeshape for change n support locaton. Fg 5. Backbone curve for smply supporte overhang beam. Fgure 5 shows the backbone curve corresponng to the frst three moes of vbraton for a smply supporte overhang beam uner transverse concentrate loang at 5. CONCLUSION The present paper unertakes an expermental an theoretcal ynamc analyss of a smply supporte overhang beam uner transverse concentrate loang at the ens. A set up smulatng smply supporte bounary contons s evelope an expermentatons are carre ICME0 5

6 out. The mathematcal formulaton s base on energy methos an axal stretchng s ncorporate through an teratve proceure nvolvng successve relaxaton of axal forces. Soluton of the governng set of equatons s acheve through IMSL routnes. 6. ACKNOWLEDGEMENT The frst author acknowleges the research support receve from AICTE, Ina, ve Fle No.: -0/RID/NDF/PG/(7) Date : The author also acknowleges the help of Kunal Dutta, uner grauate stuent from NIT, Durgapur, urng expermentaton. 7. REFERENCES. Sathyamoorthy, M., 98, Nonlnear analyss of beams, Part-I: A survey of recent avances, Shock an Vbraton Dgest, 4: Sathyamoorthy, M., 98, Nonlnear analyss of beams, Part-II: Fnte-element methos, Shock an Vbraton Dgest, 4: Kapana, R.K. an Ract, S., 989, Recent avances n analyss of lamnate beams an plates, Part-I: Shear effects an bucklng, Amercan Insttute of Aeronautcs an Astronautcs Journal, 7: Me, C., 973, Fnte element splacement metho for large ampltue free flexural vbratons of beams an plates, Computers & Structures, 3(): Km, C.S. an Dcknson, S.M., 988, On the analyss of laterally vbratng slener beams subect to varous complcatng effects, Journal of Soun an Vbraton, (3): Klausbruckner M. J. an Pryputnewcz R. J., 995, Theoretcal an expermental stuy of Couple vbratons of channel beams, Journal of Soun an Vbraton, 83(): Ganapath, M., Patel, B. P., Saravanan, J. an Tourater, M., 998, Applcaton of splne element for large-ampltue free vbratons of lamnate orthotropc straght/curve beams, Compostes Part B, 9B: Azrar, L., Benamar, R. an Whte, R. G., 999, A sem-analytcal approach to the non-lnear ynamc response problem of S S an C C beams at large vbraton ampltues part I: general theory an applcaton to the sngle moe approach to free an force vbraton analyss, Journal of Soun an Vbraton, 4(): Kapura, S., Bhattacharyya, M. an Kumar, A. N, 008, Benng an free vbraton response of layere functonally grae beams: A theoretcal moel an ts expermental valaton, Composte Structures, 8(3): Hollan, D. B., Vrgn, L. N. an Plaut, R. H., 008, Large eflectons an vbraton of a tapere cantlever pulle at ts tp by a cable, Journal of Soun an Vbraton, 30(-): Gupta, R. K., Guna J. B., Janarhan, G. R. an Rao, G. V., 009, Relatvely smple fnte element formulaton for the large ampltue free vbratons of unform beams, Fnte Elements n Analyss an Desgn, 45(0): Guna J. B., Gupta, R. K., Janarhan, G. R. an Rao, G. V., 00, Large ampltue free vbraton analyss of Tmoshenko beams usng a relatvely smple fnte element formulaton, Internatonal Journal of Mechancal Scences, 5(): Karaagac, C., Öztürk, H. an Sabuncu, M., 009, Free vbraton an lateral bucklng of a cantlever slener beam wth an ege crack: Expermental an numercal stues, Journal of Soun an Vbraton, 36(-): Gunta, G., Crsafull, D., Belouettar, S. an Carrera, E., 0, Herarchcal theores for the free vbraton analyss of functonally grae beams, Composte Structures, Artcle n Press. 8. NOMENCLATURE Symbol Meanng Unt a Length of overhang m b Wth of beam m C Coeffcents of Polynomal - c Unknown coeffcents for ynamc analyss - Unknown coeffcents for statc analyss - E Elastc moulus of beam materal N/m F Force at support locatons N {f} Loa vector N I Moment of nerta of beam cross-secton m 4 [K] Stffness matrx N/m k Stffness of sprngs N/m L Length of beam m l Length between two smple supports m [M] Mass matrx kg N x Axal force strbuton N Numbers of orthogonal functons - P Magntue of concentrate loa N T Knetc energy of the system N-m/s t Thckness of beam m U Stran energy store n the system N-m/s V Potental energy of external forces N-m/s w Transverse splacement fel m x Axal coornate m ( ) Set of orthogonal functons - Natural frequency Hz. Tme coornate s Normalze axal coornate - Varatonal operator - Densty of beam Kg/m 3 Relaxaton parameter - ICME0 6

7 9. MAILING ADDRESS Anrban Mtra Department of Mechancal Engneerng, Jaavpur Unversty, Kolkata 70003, Ina E-mal: ICME0 7