Study of Active Vibration Control using the Method of Receptances

Transcription

1 Louisin Stte Univesity LSU Digitl Commons LSU Doctol Dissettions Gdute School 05 Study of Active Vibtion Contol using the Method of Receptnces Amit Mh Louisin Stte Univesity nd Agicultul nd Mechnicl College Follow this nd dditionl wos t: Pt of the Mechnicl Engineeing Commons Recommended Cittion Mh, Amit, "Study of Active Vibtion Contol using the Method of Receptnces" (05). LSU Doctol Dissettions This Dissettion is bought to you fo fee nd open ccess by the Gdute School t LSU Digitl Commons. It hs been ccepted fo inclusion in LSU Doctol Dissettions by n uthoized gdute school edito of LSU Digitl Commons. Fo moe infomtion, plese contctgdetd@lsu.edu.

2 STUDY OF ACTIVE VIBRATION CONTROL USING THE METHOD OF RECEPTANCES A Dissettion Submitted to the Gdute Fculty of the Louisin Stte Univesity nd Agicultul nd Mechnicl College in ptil fulfillment of the equiements fo the degee of Docto of Philosophy in The Deptment of Mechnicl & Industil Engineeing by Amit Mh B.S., Louisin Stte Univesity, 000 M.S., Louisin Stte Univesity, 005 August 05 i

3 I would lie to dedicte this dissettion to my fmily Smt. Jyotsn Mh, Shi. Ajit Mh nd Su. Amut Mh fo thei unconditionl suppot thoughout my cdemic pusuits. ii

4 Acnowledgements My sincee nd het filled ppecition fo my Advisos D. Yitsh Rm nd D. Su-Seng Png. I owe lot of espect, pssion, nowledge nd sills fo thei countless hous of spending time nd enegy in poviding me with get lening nd teching epeiences. I would lso lie to etend my ppecition to my dviso D. Mcio de Queioz, fo poviding notes, feedbc, nd discussions to impove my teching sills. I wnt to etend my ppecition to my committee membes D. Gouqing Li, D. Kemin Zhou, nd D. Pdmnbhn Sund fo evluting my esech. I wnt to thn ll the fculty nd stff membes of the Deptment of Mechnicl nd Industil Engineeing fo thei suppot. In ddition to my pents, my wife Amut Mh hs been get suppot in my pusuit of this degee. iii

5 Tble of Contents Acnowledgements.iii List of Figues....v Abstct....vi Chpte Intoduction.... Oveview of the Dissettion.... Litetue Review..... Chpte Bcgound nd Theoy..4. Intoduction to Receptnce.4. Ptil Pole Plcement fo Discete Systems using Receptnce Method Pole Plcement fo Continuous Vibting Rod without spillove...7 Chpte 3 Receptnce Method fo Continuous System Open Loop System Objectives nd Poblem Definition Receptnce of the Rod 3.4 Solution fo Continuous System Multiple Actutos...5 Chpte 4 Ntul Fequency Modifiction nd Numeicl Emples Ntul Fequency Modifiction Unifom Rod with Sinusoidl Ecittion Eponentil Rod Emple : Ntul Fequency Modifiction of Unifom Rod Emple : Ntul Fequency Modifiction of Eponentil Rod..4 Chpte 5 Concluding Rems nd Futue Wo Conclusions Recommendtions...7 Refeences...8 Appendi.. 3 Vit...44 iv

6 List of Figues Figue.: Closed loop system Figue.: Two degee of feedom mss sping dmpe system...4 Figue 3.: Open loop system..9 Figue 3.: Ptil pole plcement...0 Figue 3.3: Closed loop system...0 Figue 3.4: Eponentil ecittion.. Figue 3.5: Contolled od.. Figue 3.6: Rod with multiple ctutos.5 Figue 3.7: () Tnsvesely vibting bem, nd (b) vibting plte....6 Figue 4.: Ntul Fequency Modifiction..8 Figue 4.: Fequency function of the closed loop unifom od 3 Figue 4.3: Fequency function of the closed loop eponentil od..5 Figue 4.4: Fequency function of the open nd closed loop eponentil od..5 v

7 Abstct Using the Method of Receptnce, it is possible in genel to ssign poles (i.e. n conjugte pis) of n illy vibting od of vible coss section by sensing the stte t points long the od. Using the senso dt, closed loop feedbc contol gins cn be clculted by solving simple set of line equtions. The feedbc gins cn be pplied to genete the equied contol foce using n ctuto. This ptil pole ssignment cn be done without continuous o discete nlyticl modeling. Theefoe, the physicl pmetes such s the igidity, the density, nd the vible coss-sectionl e of the od my be consideed unnown fo clculting the gins fo the contol foce. These contol gins e puely detemined by the mesuement of eceptnces between the point of ctution nd the points of sensing, which my be conducted epeimentlly. In the nlyticl en, the estimtes fo the eceptnces e ect nd theefoe the ssignment of the desied poles is lso ect nd suffes fom no discetiztion o model eduction eos. Howeve, in pcticl implementtion, the contol foce ffects the poles of the closed loop system. In pcticl implementtion, the contol foce my lso shift some of the poles which e not intended to be desibly plced o to emin unchnged. Although in this dissettion, the nlysis is cied on n illy vibting od fo simplicity, the Method of Receptnces is genelly pplicble to othe line elstic stuctues of highe dimensions. vi

8 .. Oveview of the Dissettion Chpte : Intoduction This dissettion involves the ppliction of the Method of Receptnces to continuous stuctues. In pticul, the Method of Receptnce hs been pplied to illy vibting od. This method utilizes dt fom senso mesuements to povide suitble set of closed loop feedbc gins to the ctutos, which in tun pply contol foce to pomote ptil pole plcement. It is shown tht genelly we my ssign set of n poles (i.e. n conjugte pis) by using one ctuto nd n sensos mesuing the displcement nd velocities t n points long the od. Using this method thee is no need fo ny discetiztion o nowledge the model. The Method of Receptnces ws initilly developed fo discete systems. This study hs been implemented fo the fist time on continuous system. This is n inteesting technique to povide stte feedbc contol to continuous system without nowing physicl pmetes (density, igidity, coss section e) o pefoming ny model eduction nlysis. The closed loop feedbc contol of n illy vibting od unit length od with vible coss sectionl e s shown in Figue. hs been consideed fo pplying the Method of Receptnce. Appendi A shows the deivtions fo non-dimensionl nlysis of govening equtions of motion. Figue.: Closed loop system The eigenvlues nd the coesponding eigenfunctions of the fee vibtion motion is given v, espectively. Suppose tht concentted stte feedbc by { λ ± } nd { },,... ±,,... contol foce u ( t) is pplied to the end of the od such tht the some of the undesied w, eigenvlues nd the coesponding eigenfunctions e eplced by { µ } ± nd { } ± espectively. In nlyticl emples, whee the eceptnces do not suffe fom epeimentl eos, the ssignment of the desied poles is ect. Howeve, the contol ffects the loctions of the othe poles of the od. This document is ognized s follows. Section. coves litetue suvey poviding some bcgound on meits of using the method of eceptnces fo discete systems. In Chpte, the discussions fo the bcgound nd theoy e povided. They contin bief intoduction to the concept of Receptnce, simple illusttion fo discete system nd n oveview of pole plcement fo continuous vibting od developed by Rm [8] nd ll the useful equtions fom this impotnt ppe. In chpte 3, the poblem objective hs been lid out nd the nlysis of eceptnce method fo illy vibting od cn be seen. In ddition, the ppoch fo two ctutos, plced t two distinct loctions, long the length of the od hs u ( t )

9 been shown. Chpte 4 povides nlysis on ntul fequency modifiction long with some numeicl emples of how to pply the Method of Receptnce. In ddition, the effect of nominl ndom eo hs been consideed in the numeicl emples to show povide confidence fo Receptnce Method. Concluding ems s well s ecommendtions e discussed in Chpte 5... Litetue Review Vibtion contols cn be clssified into pssive nd ctive contols. The uses of pssive contols e vey limited especilly when the dynmic systems show undesible vibtions. Theefoe, etensive studies within the engineeing communities wee stted fom 970s by using ctive contol method to the poblem of eigenvlue ssignment. If given vibting system is contollble, then Wonhm [4] in 967 hd shown tht the system s poles o eigenvlues could be ssigned by using n ppopite choice of stte feedbc system. It ws lso detemined by Dvison [5] tht the eigenvlue ssignment could be chieved unde the conditions of which output feedbc could be pplied. A numeicl method of detemining pole-ssignment to the stte-feedbc system ws demonstted by Kutsy et l. [9] using line lgeb. In ddition, n effot to minimize the sensitivity of the pole ssignment ws lso cied out. Lte, the engineeing community stted to focus on invese eigenvlue poblems. An effot is being cied out to ssign ntul fequencies s well s nti-esonnces with the id of stuctul modifictions [3,6]. Epeimentl mesuements wee cied out by Motteshed et l. [7,] to mesue the ottionl eceptnces of stuctue nd pplied stuctul modifictions to ssign set of the ntul fequencies nd ntiesonnces. Additionl eseches in stuctul modifictions wee cied out by Motteshed nd Tehni [] on helicopte tilcone to mesue the ottionl eceptnces. An invese eignevlue poblem fo vibtion bsoption due to stuctul modifiction nd using ctive contol ws studied by Motteshed nd Rm [3]. The min dvntge pefoming stuctul modifiction is to gunteed the stbility of the system. Howeve, the disdvntges of the mtices of being symmety, positive-definiteness, nd the ptten of non-zeo mti tems e estictive. In ddition ottionl eceptnces e vey difficult to mesue ccutely s well s the eigenvlues needed to be ssigned must mtch the n modifiction. Agin ecently, thee hs been suge in the study of eigenvlue ssignment fo second ode systems. A second-ode mti pencil elevnt to the ntul fequencies nd dmping of engineeing stuctues wee studied by Tisseu nd Meebegen [40] descibing vious methods. Additionlly, numeiclly obust solutions to the qudtic eigenvlue poblems wee eseched by Chu nd Dtt [4]. Howeve, method fo ptil pole plcement ws chieved by Dtt et l. [5]. New methods e being developed nd estblished fo eigenvlue ssignment o pole plcement in vibting stuctues. The poles nd zeos of vibting system cn be ssigned by stte-feedbc system with only single input-stte ws shown in [3] nd output feedbc []. These methods use equtions fo contolle gins fo single-input stte feedbc. The output feedbc method genelly offes eigenvlue plcement t educed

10 contol cost. These methods hve been developed to te into ccount of contolle timedelys [30]. It ws lso shown tht the closed-loop system poles my be ssigned obustly s well s being insensitive to uncetinties in the contol gin tems [37]. The method of eceptnces fo vibtion contol ws intoduced in [3] by implementing the Shemn-Moison fomul. This method ws bsed entiely on mesued vibtion dt, hving significnt modelling dvntges ove conventionl mti methods, including no equiement to now o to evlute the M, C nd K ( n n symmetic mtices), M is positive definite, C nd K e positive semidefinite mtices. Thus thee is no need fo the estimtion of the unmesued stte nd no need fo model eduction. Howeve, the method did not lend itself ntully to multi-input multi-output contol, s highlighted in []. The efomultion in [33], which ws cied out by diect ppoch, i.e., without using the Shemn-Moison fomul, emoved this limittion nd llowed etension to the cse of multi-input multi-output contol. Implementtion of the method in lbotoy epeiments ws epoted in [8], [9] nd [36]. The theoeticl developments descibed bove wee cied out on finite dimensionl systems. In this dissettion, the use of the method of eceptnces ws pplied to the continuous model of n illy vibtion od. The etension of the nlysis fo othe line elstic systems in two o thee sptil dimensions is stight fowd. The detils contined in this dissettion hve lso been published in [43]. 3

11 .. Intoduction to Receptnce Chpte : Bcgound nd Theoy The bsic theoeticl definition of eceptnce is tht it is mti deived by the invese of the dynmic stiffness mti. In genel, this is mesue of tnsfe function between input/output dt including the dynmics of the sensos nd ctutos. [ Receptnce mti] H( s ) ( s M + sc + K) In pctice, eceptnces e the mesuble quntities vilble fom modl tests. The eceptnce eqution is displcement-type eqution mde complete by the non-zeo foce tems. The dvntge is the bility to hve displcement mesuements t discete loctions nd coelting this esponse to the input foce. The eceptnce mti H ( s) my be epessed in tems of eigenvlues nd eigenvectos (ntul fequencies nd mode shpes). The following nlysis shows the deivtion of the eceptnce mti fo discete system. Conside two degee of feedom mss sping dmpe system s shown in Figue.. Figue.() shows the ppliction of foce on the fist mss nd Figue.(b) shows the ppliction of foce on second mss. Then the eceptnce mti fo this system cn be shown s follows: e st e st h h h () (b) Figue.: Two degee of feedom mss sping dmpe system h Fom Figue. () m 0 0 h&& m h && c + c c c h& h & + h h e 0 st o M h & + & e st Ch + Kh e st ( t) e st st h ( M + sc + K) p e e e p s M C K p e ( s + s + ) 4

12 Fom Figue. (b) p M C K e ( + s + ) s (.) m 0 0 h&& m h && c + c c c h& h & + h h 0 e st M h & + & e st Ch + Kh e st ( t) e st st h ( M + sc + K) p e e e p s M C K p e ( s + s + ) p M C K e ( + s + ) Equtions (.) nd (.) cn be witten in the fom s (.) [ p p ] ( s M + sc + K) [ e ] o [ Receptnce mti] ( s M + sc + K) e.. Ptil Pole Plcement fo Discete Systems using Receptnce Method Conside the fee vibtion of the finite dimensionl system M & y + Cy& + Ky 0, (.3) whee M, C nd K e n n symmetic mtices, M is positive definite, C nd K e positive semidefinite mtices nd y ( t) is the fee esponse. The dynmics of the open loop system () my be egulted by stte feedbc contol leding to with whee ( t) ( t) M & z + Cz& + Kz bu, (.4) T T ( t) f z g z u & +, (.5) z is the esponse of the closed loop system, b is el constnt input vecto nd f nd g e el constnt contol gin vectos. Substituting λt ( t) ve y (.6) 5

13 in (.3) gives the qudtic eigenvlue poblem fo the open loop system λ M + λc + K v, (.7) ( ) 0 whee v is constnt vecto. Substituting µ t ( t) we z (.8) in (.4)-(.5) gives the eigenvlue poblem fo the closed loop system T ( M + µ C + K) w b( µ f T + g )w µ. (.9) whee w is constnt vecto. The poles of (.7) e the oots { λ } n of the open loop system, nd the poles of (.9) e the oots{ µ } n of the closed loop systems. The contol gin vectos f nd g cn be evluted such tht the poles of the closed loop system (.9), { µ } n e ssigned desibly. Since the numbeing of poles is bity we my, without loss of genelity, equie tht m poles of the open loop system { λ } m e elocted by the using the desied contol to the set { µ } m while eeping the est of the poles unlteed, i.e., µ,,..., m µ. (.0) λ m +, m +,...,n In ode to hve el solution fo f nd g, tht cn be elized by the contol, the sets { λ nd { µ hve to be closed unde conjugtion. } m The Receptnce mti fo discete system is defined s nd let ( ) ( s ) M + sc + K H s (.) H( )b. (.) µ Then, it hs been shown in [7] tht with f nd g stisfying P f e, (.3) Q g 0 will ssign the new desied poles (.0), whee } m 6

14 7 T m T m m T T T T P µ µ µ M M, T n T n n T m T m T m T m v v v v v v Q µ µ µ M M (.4) nd whee ( ) m T R L e. (.5).3. Pole Plcement fo Continuous Vibting Rod without spillove The detils povided in this section e fom [8], which gives theoeticl method fo obtining ptil pole plcement without spill ove fo continuous od vibting illy. The following e the impotnt equtions fom tht povide gin functions to chieve pole plcement If { } m ± µ be self-conjugte set whee m,...,, µ µ. (.6) Then, it hs been shown in [8] tht with the contol gin functions ( ) m m v f 0 β (.7) ( ) m m v g 0 β λ (.8) the poles of the continuous od e + + ± ± ±,...,,..., m m m λ µ µ. (.9) whee ( ) ( ) m d bv m m,...,, 0 0 ± ± ± ± ± ± λ λ µ λ µ λ β. (.0)

15 Howeve, pcticl implementtion of this method is cumbesome s mesuements fo evey il loction long the length of the od is not possible. Theefoe it is difficult to evlute the eigenfunction nd use it to estimte the gins fo the contolle. In ddition thee e eos fom instuments used fo ny pcticl ppliction nd this mes implementtion pocess to be vey chllenging. Theefoe this poblem hs been evisited with the intension of poviding stte feedbc contol using the Method of Receptnces. 8

16 3.. Open Loop System Chpte 3: Receptnce Method fo Continuous System Conside the fee il vibtion of unit length od with vible coss sectionl e ( ) nd constnt modulus of elsticity E nd density ρ. The ptil diffeentil eqution govening the il motion is & & y c y ( y ) 0 < < ( 0, t) 0 y (, t) 0 t > 0, (3.) whee E ρ t fom its sttic equilibium position, nd dots nd pimes denote ptil deivtives with espect t nd, espectively. c is the speed of sound in the od, y (, ) is the displcement of ( ) Figue 3.: Open loop system Seption of vibles y λct ( t) v( ) e, (3.) Applied to (4) leds to the eigenvlue poblem ( v ) λ v 0 v( 0) 0 v ( ) 0 0 < <, (3.3) We denote the eigenvlues nd the coesponding eigenfunctions of (3.3) by { ± },,... { v ± }, espectively.,, Objectives nd Poblem Definition Thee e situtions when some the ntul fequencies of the od { λ± },,.. m. λ nd e undesible nd we wish to eliminte these fequencies. In pctice it is difficult to completely emove the undesible fequencies. Howeve, the ppoch is to shift the undesied poles to diffeent loction such tht the new loctions dmp out the system fste thn the open loop 9

17 system. This is chieved by pefoming stte-feedbc ppoch to induce feedbc foce to set the system to desied fequencies. Im Im Re Re Figue 3.: Ptil Pole Plcement The poblem is defined s how to chieve ptil pole plcement to move the undesied fequencies to pefeed loction s shown in Figue 3.. u ( t ) Figue 3.3: Closed loop system Suppose tht concentted stte feedbc contol foce ( z) ( t) E( ) c f ( ) z g( ) u & + d (3.4) 0 is pplied to the end of the od s shown in Figue 3.3. Then the motion of closed loop od is govened by z && c z ( z ) 0 < < t > 0 ( z) ( 0, t) 0 z (, t) c f ( ) z& + g( ) 0 d. (3.5) Seption of vibles 0

18 z µ ct ( t) w( ) e, (3.6) pplied to (3.5) leds to the eigenvlue poblem of the closed loop system ( w ) µ w 0 w 0 < < ( 0) 0 w ( ) ( µ fw + ) 0 gw d (3.7) µ nd We denote the eigenvlues nd the coesponding eigenfunctions of (3.7) by { } { w } ± ±, espectively. In ode to chieve ptil pole plcement, the poles of this closed loop system must be s follows µ,,... m µ (3.8) λ m +, m +,...,n to move the fist m poles to desied loction nd thus the conjugte pi of the these poles will lso be elocted Receptnces of the od β t ξct Let α be genel point on the od nd suppose tht concentted foce F( t) E( ) e β ecites the od s shown in Figue 3. α F ξ ct ( t ) E ( β ) e β Figue 3.4: Eponentil ecittion The govening diffeentil equtions fo this cse my be witten in the fom y && c y && c ( y ) ( y ) 0 < < β. (3.9) β < < with boundy conditions (, t) 0 y (, t) 0 0 y, (3.0) nd mtching conditions

19 ξct ( β t) y ( β, t) ( y ( β, t) y ( β, t) ) e y,. (3.) The solution fo (3.9)-(3.) tes the fom y ξct (, t) h (,, ξ ) e, Substituting (3.) in (3.9)-(3.) gives β. (3.) ( h ) ξ h 0 ( h ) ξ h 0 h h 0 < < β β < < ( 0) 0 h ( ) 0 ( β ) h ( β ) 0 ( h ( β ) h ( β )) (3.3) which detemines the eceptnce functions ( α, β, ξ ) Fo convenience we denote ( α, β, ξ ) ( α, β, ξ ) h,,. h α < β h ( α, β, ξ ). (3.4) h α > β In pticul, when β the eceptnce (,,ξ ) h is detemined by the solution of (,, ξ ) ξ h(,, ξ ) ( 0,, ξ ) 0 h (,, ξ ) ( h h 0 0 < < (3.5) 3.4. Solution fo Continuous System n u ( t ) Figue 3.5: Contolled od Suppose tht n sensos e locted t the positions { } n 3.5. Let us define the vecto of sensing positions s follows long the od s shown in Figue ( L ) T (3.6) n

20 whee, nd denote nd n ( ( ) v ( ) v ( )) T v (3.7) v L ( ( ) w ( ) w ( )) T w (3.8) whee v ( ) nd ( ) coesponding to λ nd Let w L w e the eigenfunctions of the open nd closed loop systems µ, espectively. nd n ( ) f ( ) f δ (3.9) n ( ) g( ) g δ (3.0) be the gin functions of the closed loop system whee elements of the vectos f nd g, espectively. f nd g,,,..., n, e the th Lemm If T T ( f g ) v 0 + µ (3.) then { w } { λ v } µ (3.) p p fo some intege p. The physicl mening is tht if (3.) is stisfied then the eigenpi { λ v } is common to the open nd the closed loop systems. Poof With g ( ) nd ( ) f s epessed by (3.9) nd (3.0) we hve T T ( f + g) w d ( µ f + g ) w 0 0 µ (3.3) 3

21 by vitue of (3.3). It thus follows tht the closed loop eigenvlue poblem (3.7) educes to ( w ) µ w w ( 0) 0 w ( ) < < (3.4) which is the eigenvlue poblem ssocited with the open loop system (6). Suppose tht { } ± nd denote µ λ. Then we define the eceptnce mti (,, µ ) h(,, µ ) L h(, n, µ ) (,, µ ) h(,, µ ) L h(,, µ ) h ( ) h n H µ, (3.5) M M MMM M h(,, µ ) (,, µ ) (,, µ ) n h n L h n n ( ) e n H µ, (3.6) whee e is the Lemm If th unit vecto. T T ( f g ) + µ (3.7) then µ is pole of the closed loop system (3.7). Poof Since w ( ) is n eigenfunction of (3.7) it my be scled bitily. We choose scling such tht 0 ( f + g) wd µ, (3.8) so tht the closed loop eigenvlue poblem (3.7) educes to ( w ) µ w 0 w( 0) 0 w ( ) Note tht 0 < < (3.9) 4

22 h (,, ) w( ) i,,..., n, m by vitue of (3.5). With g ( ) nd ( ) µ (3.30) i i,..., f s epessed by (3.9) nd (3.0) we hve T T ( f + g) w d ( µ f + g ) w 0 µ. (3.3) The poof is completed by noting tht w (3.3) by vitue of the definitions (3.5), (3.6) nd eqution (3.30). Note tht (3.3) nd (3.7) my be witten equivlently s nd f v (3.33) g T T ( λ p p v p ) 0 µ λp f. (3.34) i g T T ( µ ) { } µ λ± i It thus follows tht the desied gin vectos f nd g e detemined by the set of line equtions obtined fom eqution (.3) Multiple Actutos Conside the cse wee the contol is done by moe thn one ctuto. Fo the se of simplicity we choose the two concentted contol foces F ( t) b ( ) u( t) F ( t) b δ ( ) u( t) δ, (3.35) n n F ( t ) n F n ( t ) Figue 3.6: Rod with multiple ctutos 5

23 shown in Figue 3.6, whee b nd n u t is given by (3.4). The etension of the nlysis to contol by moe thn two foces is self-eplntoy. The eigenvlue poblem of the closed loop system is b e some constnts nd ( ) ( w A) µ w ( w B) µ w wa wa B A B < < < < T T ( 0) 0 w B( ) bn ( µ f + g ) w T T ( ) w ( ) 0 ( w ( ) w ( )) b ( µ f + g ) A B w (3.36) whee f, g nd w e given by (3.9), (3.0) nd (3.8), espectively, nd w ( ) w w A B ( ) ( ) 0 < <. (3.37) < < Lemm holds fo this cse since fo ech eigenpi { µ w } which stisfies (3.3) the eigenvlue poblem of the closed loop system (3.36) educes to tht of its countept ssocited with the open loop system (3.3). Suppose now tht { } ± p p µ λ. Then thee eists eceptnce mti H ( ). Let ( ) [ h h L ] h n H µ (3.38) be the column ptitioning of H, nd denote b b e + b e. (3.39) n n µ senso ctuto () (b) Figue 3.7: () Tnsvesely vibting bem, nd (b) vibting plte With the nomliztion of w ccoding to (3.3), the system of equtions (3.36) yields w ( µ ) + b h ( µ ) H( )b b h µ, (3.40) n n by vitue of (3.3), (3.5), (3.3) nd the pinciple of supeposition. It thus follows tht Lemm, in conjunction with (.), holds fo this cse. We my lso conclude tht it is 6

24 necessy to mesue only the eceptnces between the points of ctution nd the points of sensing. It is cle fom eqution (3.40) tht thee is no unique solution fo stte feedbc contol using multiple ctutions using the Method of Receptnces. The coefficients b nd b n cn be scled bitily depending on the weightge povided to the senso dt. Theefoe thee e mny combintions tht eist tht cn povide equied feedbc to contol the system. 7

25 Chpte 4: Ntul Fequency Modifiction nd Numeicl Emples 4.. Ntul Fequency Modifiction In this section, we will te loo t the modifiction of ntul fequencies fo illy vibting od. Conside the cse whee µ + µ 0,,..., m, (4.) i.e., the m ssigned eigenvlues e puely imginy coesponding to undmped systems. Im Im Re Re Figue 4.: Ntul Fequency Modifiction Denote the ntul fequencies of the open nd closed loop systems by nd ω λ λ λ,,..., (4.) σ µ µ,,..., (4.3) µ espectively. Then eigenvlue poblems fo the open nd closed loop systems e given by ( vˆ ) + ω vˆ 0 vˆ ( 0) 0 vˆ ( ) 0 0 < <, (4.4) nd 8

26 ( wˆ ) + σ wˆ 0 ˆ w ( 0) 0 wˆ ( ) ( σf + g) 0 0 < <. (4.5) wd ˆ With f m ( ) g( ) 0 ω β v, (4.6) whee m ω σ ω σ β,,..., m, (4.7) bv d ω ω 0 the ntul fequencies { } m ω e ssigned to { } m 4.. Unifom Rod with Sinusoidl Ecittion σ by vitue of (3.0)-(3.3). Conside the il vibtion of unifom od with constnt coss sectionl e ( ) A. Fo simplicity we ddess the cse whee the self conjugte set { µ } n ± consists of puely imginy poles. In such cse it is suitble to epess the open loop eigenvlue poblem in tems of the ntul fequency ω the thn the pole λ. With such fomultion the ithmetic involved in computing the eigenvlues nd the eigenfunctions is el. Substituting ω λ in (3.3) gives the open loop eigenvlue poblem vˆ + ω vˆ 0 vˆ ( 0) 0 vˆ ( ) The eigenpis of (4.8) e el, 0 < <. (4.8) 0 { ( ) ˆ sin } π v ω ω. (4.9) The closed loop eigenvlue poblem (3.7) is simplified to w ˆ + σ wˆ 0 σ µ ( ) ( ) wˆ 0 0 wˆ gwd ˆ 0 (4.0) since the closed loop is undmped. With the scling 9

27 0 g wˆ d, (4.) the eigenvlue poblem (4.0) educes to the poblem of solving diffeentil eqution with non-homogeneous boundy conditions wˆ + σ wˆ 0 wˆ ( 0) 0 wˆ ( ) 0 < <. (4.) We use the fct tht in ou poblem the desied fequency σ is given nd obtin fom (4.) sinσ w ˆ. (4.3) σ cosσ To complete the nlysis of the ntul fequency ssignment in tems of el ithmetic we conside the sinusoidl ecittion Fˆ sinσct (4.4) t β the thn the eponentil ecittion used in Section 3.3. The ssocited fequency esponse function is since ˆ sinσ h (,, σ ) (4.5) σ cosσ (,, ) wˆ ( ) hˆ σ. (4.6) The fequency eqution fo the closed loop system is ( σ ) σ cosσ g sinσ 0 n φ (4.7) 4.3. Eponentil Rod In this section we will conside the il vibtion of n eponentil od with coss sectionl e. By using ω λ the eigenvlue poblem (3.3) gives e ( ) 0

28 ( e vˆ ) + ω e vˆ 0 vˆ ( 0) 0 vˆ ( ) 0 0 < <. (4.8) Let γ be the Then th oot of tn γ γ. (4.9) ω + + 4,,... (4.0) γ th is the ntul fequency of the od. Note tht ω 4 coesponding to γ 0 in (4.0) leds to tivil solution nd it is theefoe not n eigenvlue of (4.8). We theefoe pplied the shift of inde fom in the left hnd side of eqution (4.0) to + in its ight hnd side. The eigenfunctions of (4.8) coesponding to ω e vˆ ( ) e sin ω. (4.) 4 The closed loop eigenvlue poblem fo the undmped eponentil od with sinusoidl ecittion (4.5) is ( e ˆ w wˆ ) + σ e ( 0) 0 wˆ ( ) wˆ 0 0 < < 0 gwd ˆ (4.) With the scling (4.) we obtin ( e wˆ wˆ ) + σ e ( 0) 0 wˆ ( ) wˆ 0 0 < <, (4.3) which gives whee wˆ ( ) hˆ (,, ) s s e e σ (4.4) θ s s s s ( e + e ) e + e θ 4 (4.5) σ

29 nd ± σ s,. (4.6) The fequency eqution fo the closed loop system is ( ) n φ σ e ( sinψ + ψ cosψ ) ge sinψ 0 (4.7) whee ψ σ 4 (4.8) 4.4. Emple : Ntul Fequency Modifiction of Unifom Rod Suppose tht we wish to ssign the ntul fequency ω π of unifom od with to the ntul fequency σ 3π 4 by using the contol (3.4), while leving ω σ 3π nd ω 3 σ 3 5π unchnged. This objective could be done by using thee sensing points. We choose nd obtin by (4.5) h ˆ whee ( σ ) 4 3π 4 3π ( σ ) hˆ ( σ ) hˆ ( σ ) ĥ is the j ij (3.6) tht 4 3π i enty of the sinusoidl tnsfe mti Ĥ ( σ ) ( ) T 4 ˆ. 3 π Fom (3.7) nd (4.9) we hve which gives T ( sinω sinω sinω ), v ˆ 3. It thus follows fom

30 T v ˆ ( ) ˆ 0 v3 ( 3 ) T. Eqution (.3) educes to T ˆ g T Ag v ˆ g 0 κ A T ˆ v g ( ). 88 which gives π ( ) g , κ is the condition numbe of A. The system of line equtions fo detemining g in this cse is well conditioned. whee ( A) φ ( σ ) Noise-fee dt σ σ σ 3 Dt with noise σ π Figue 4.: Fequency function of the closed loop unifom od To emine the sensitivity of the esults to eos ndom noise ws dded to g of noml distibution, with zeo men nd stndd devition 0., esulting in Using the fist thee numbes geneted by MATLAB pseudo ndom numbe geneto fte invoing the softwe. 3

31 ~ ~ g g g g The fequency functions (4.7) coesponding to the noise fee nd noise coupted dt e plotted in Figue 4.. The fequencies σ,,, 3 hve been ssigned s desied by the noise fee contol. It is visully cle fom this figue tht the chnge in the ssigned fequencies due to the noise is mginl in this cse. The net fou ntul fequencies, which wee not intended to be contolled, e ω 4,5 σ We note tht ω nd ω Hence, the contol ffected the loctions of some of the unssigned ntul fequencies Emple : Ntul Fequency Modifiction of Eponentil Rod We wish to ssign the lowest thee ntul fequencies { } 3 coss sectionl e The sensing points e e to 3π Using (63), (37) nd (38) we hve ˆ ˆ ˆ 3 ( ) ( T.6644) ( T.6658) Eqution (.3) fo this cse is T ˆ g T Ag ˆ g κ A T ˆ 3 3 g ω of the eponentil od with σ, σ 3 nd σ 5, s in Emple. π T ( ) π 4

32 which gives ( ) T g. φ ( σ ) 6 4 Noise-fee dt 0 - σ σ σ 3-4 Dt with noise σ π Figue 4.3: Fequency function of the closed loop eponentil od φ ( χ ) λ σ λ σ 3 σ λ 3 φ ( χ ) λ 4 σ 4 λ 5 σ 5 σ 6 λ χ π Figue 4.4: Fequency functions of the open nd closed loops eponentil od The sme noise to g s in Emple hs been intoduced, esulting in 5

33 ~ g ~ g g g 0.40 The fequency equtions of the closed loop system coesponding to g nd the noise coupted ~ g e plotted in Figue 4.3. It shows tht the fequencies σ,,, 3 hve been ssigned s equied by the noise fee contol. It is visully cle fom this figue tht the chnge in the ssigned fequencies due to the noise is mginl in this cse. The fequency equtions φ nd φ coesponding to the open loop nd the closed loop systems, espectively, shown in Figue 4.4 indicte tht the fist si ntul fequencies of the eponentil od wee influenced by the contol. 6

34 Chpte 5: Concluding Rems nd Futue Wo 5.. Conclusion The method of eceptnces is nothe technique to chieve ptil pole plcement using stte feedbc contol of line elstic system by using mesued epeimentl dt, the thn nlyticl modeling. The oiginl method ws fomulted in [3], nd efomulted in [33], in the fmewo of discete n degees of feedom system whee ll n poles ( n conjugte pis) e ssigned desibly. It would be theefoe ntul to ssume tht implementing the method of eceptnce to continuous line elstic systems, the ssignment of poles would ccue some esidul eos. This my be due to ssocition with the incompleteness of the unmesued points. It hs been shown tht in theoy, whee the eceptnces e ect, the ssignment of the desied poles is lso ect nd suffes fom no discetiztion o model eduction eos. It hs been shown tht in genel we my ssign n poles ( n conjugte pis) of n illy vibting non-unifom od by sensing the stte of n points without nowing the density, igidity nd the vible coss-sectionl e of the od. The equied input is the mesuements of eceptnces between the points of ctution nd the sensing points. These functions my be mesued epeimentlly. 5.. Recommendtions The nlysis eveled tht this esult is genel s pplicble to othe line elstic stuctues such s the tnsvesely vibting bem o the plte shown in Figue 3.7. The nlysis pesented in this dissettion does not ddess ny optimiztion fo multiple ctution. It is dvised tht futue study my povide insights nd help to fully elize the implementtion of this method fo el systems. 7

35 Refeences. M.J. Bls, Tends in Lge Spce Stuctue Contol Theoy: Fondest Hopes, Wildest Dems, IEEE Tnsctions on Automtion Contol, 7, No. 3, pp , 98. R.E.D Bishop, G.M.L. Gldwell, S. Michelson, The Mti Anlysis of Vibtions, Cmbidge Univesity Pess, New Yo, I. Buche nd S. Bun, The stuctul modifiction invese poblem, Mechnicl Systems nd Signl Pocessing, 7, pp. 7-38, E.K. Chu nd B.N. Dtt, Numeiclly obust pole ssignment fo second ode systems, Intentionl Jounl of Contol, 64, pp. 3-7, B.N. Dtt, S. Elhy, nd Y.M. Rm, Othogonlity nd ptil pole plcement fo the symmetic definite qudtic pencil, Line Algeb nd Its Applictions, 57, pp. 9-48, E.J. Dvison, On pole ssignment in line systems with incomplete stte feedbc, IEEE Tns. Automtic Contol, AC-5, pp , G.R. Dun nd L. Hung, Robust Pole Assignment in Descipto Second-ode Dynmicl Systems, Act Automtic Sinic, Vol. 33, No. 8, pp , C.S. Guzzdo, S.S. Png nd Y.M. Rm, Optiml ctution in vibtion contol, Mechnicl Systems nd Signl Pocessing, 35, pp , J. Kutsy, N.K. Nichols nd P.V. Dooen, Robust pole ssignment in line stte feedbc, Intentionl Jounl of Contol, 4(5), pp. 9-55, K. Kelel. "Geen's function nd eceptnce fo stuctues consisting of bems nd pltes", AIAA Jounl, 5, No., pp , 987. A. Kypinou, J.E. Motteshed, H. Ouyng, Stuctul modifiction. Pt : ssignment of ntul fequencies nd ntiesonnces by n dded bem, 84, pp. 67-8, 005. T. Li, E.K.-W. Chu, Pole ssignment fo line nd qudtic systems with time-dely in contol, Numeicl Line Algeb with Applictions, 0, pp. 9 30, L. Meiovitch, Dynmics nd Contol of Stuctues, John Wiley nd Sons, NewYo, H.K. Milne, On the eceptnce functions of unifom bem, Jounl of Sound nd Vibtion, 46, pp , 99 8

36 5. H.K. Milne, The eceptnce functions of unifom bems, Jounl of Sound nd Vibtion, 3, pp , J.E. Motteshed, Stuctul modifiction fo the ssignment of zeos using mesued eceptnces, Tnsctions of the Ameicn Society of Mechnicl Enginees, Jounl of Applied Mechnics, 68(5), pp , J.E. Motteshed, A. Kypinou nd H. Ouyng, Stuctul modifiction, pt : ottionl eceptnces, Jounl of Sound nd Vibtion, 84, pp , J.E Motteshed, M.G. Tehni, S. Jmes nd Y. M. Rm, Active vibtion suppession by pole-zeo plcement using mesued eceptnces, Jounl of Sound nd Vibtion, 3, pp , J.E. Motteshed, M.G. Tehni, S. Jmes, P. Cout, Active vibtion contol epeiments on n AgustWestlnd W30 helicopte ifme, Poceedings of the Institution of Mechnicl Enginees, Pt C: Jounl of Mechnicl Engineeing Science, 6, pp , 0 0. J.E. Motteshed, M. Lin, M. Fiswell, The sensitivity method in finite element model updting:a tutoil, Mechnicl Systems nd Signl Pocessing, 5, pp , 0. J.E. Motteshed, M.G Tehni, Y.M. Rm, Assignment of eigenvlue sensitivities fom eceptnce mesuements, Mechnicl Systems nd Signl Pocessing, 3, pp , 009. J.E. Motteshed, M.G.Tehni, D. Stncioiu, S. Jmes, H. Shhvedi, Stuctul modifiction of helicopte tilcone, Jounl of Sound nd Vibtion, 98, pp , J.E. Motteshed nd Y.M. Rm, Invese eigenvlue poblems in vibtion bsoption: Pssive modifiction nd ctive contol, Mechnicl Systems nd Signl Pocessing, 0, 5-44, H. Ouyng, L. Bez, S. Hu, A eceptnce-bsed method fo pedicting ltent oot snd citicl points in fiction-induced vibtion poblems of symmetic systems, Jounl of Sound nd Vibtion, 3, pp , H. Ouyng, D. Richiedei, A. Tevisni, G. Zndo, Eigen stuctue ssignment in undmped vibting systems: A conve-constined modifiction method bsed on eceptnces, Mechnicl Systems nd Signl Pocessing, 7, pp , 0 6. U. Pells, J.E. Motteshed, M.I. Fiswell, On Pole Zeo Plcement by unit-n Modifiction, Mechnicl Systems nd Signl Pocessing, 7(3), 6 633, 003 9

37 7. J. Qin, S. Xu, Robust ptil eigenvlue ssignment poblem fo the second-ode system, Jounl of Sound nd Vibtion, 8, pp , Y.M. Rm, Pole ssignment fo the vibting od, Qutely Jounl of Mechnics nd Applied Mthemtics, 5, pp , Y.M. Rm, Optiml mss distibution in vibting systems, Mechnicl Systems nd Signl Pocessing, 3, pp , Y.M. Rm, A.Singh, J.E. Motteshed, Stte feedbc contol with time dely, Mechnicl Systems nd Signl Pocessing, 3, pp , Y.M. Rm, J.E. Motteshed nd M. G. Tehni, Ptil pole plcement with time dely in stuctues using the eceptnce nd the system mtices, Line Algeb nd its Applictions, 434, pp , 0 3. Y.M. Rm nd J.E. Motteshed, The Receptnce Method in Active Vibtion Contol, AIAA Jounl, 45, pp , Y.M. Rm nd J.E. Motteshed, Multiple-Input Active Vibtion Contol by Ptil Pole Plcement using the Method of Receptnces, Mechnicl Systems nd Signl Pocessing, 40, , Y. Sd, Pojection nd Defltion Methods fo Ptil Pole Assignment in Line Stte Feedbc, IEEE Tnsctions on Automtion Contol, 33, No. 3, pp , T.L. Schmitz, G.S. Duncn, Receptnce coupling fo dynmics pediction of ssemblies with coincident neutl es, Jounl of Sound nd Vibtion, 89 pp , M.G. Tehni, R.N.R. Elliott, J.E. Motteshed, Ptil pole plcement in stuctues by the method of eceptnces: Theoy nd epeiments, Jounl of Sound nd Vibtion, 39, pp , M.G. Tehni, J.E. Motteshed, A.T. Shenton, Y.M. Rm, Robust pole plcement in stuctues by the method of eceptnces, Mechnicl Systems nd Signl Pocessing, 5, pp. -, M.G. Tehni, H. Ouyng, Receptnce-bsed ptil pole ssignment fo symmetic systems using stte-feedbc, Jounl of Shoc nd Vibtion, 9, No. 5, pp. 35-4, M.G. Tehni, L. Wilmshut nd S.J. Elloit, Receptnce method fo ctive vibtion contol of nonline system, Jounl of Sound nd Vibtion, 33, pp , 03 30

38 40. F. Tisseu nd K. Meebegen, The qudtic eigenvlue poblem, SIAM Review, 43(), pp , K.N. Vessel, Y.M. Rm, nd S.S. Png. "Sensitivity of Repeted Eigenvlues to Petubtions", AIAA Jounl, 43, No. 3, pp , W.M. Wonhm, On pole ssignment in multi-input contollble line systems, IEEE Tns. Automtic Contol, AC-, pp , A. Mh, nd Y.M. Rm, The method of eceptnces fo continuous ods, Jounl of Applied Mechnics, 8, Aticle Numbe: 07009, pp. -7, 04 3

39 Appendi A Non-Dimensionl Eqution of Motion of Ailly Vibting Rod Conside non-unifom coss-section od vibting illy. Let be the il loction of ny coss section vibting bout its sttic equilibium position with n mplitude of ( t ) z,, whee t epesents the time evolution of the motion z. In ddition, let E, ρ epesent the elstic modulus nd the density of the od espectively. The subscipt epesents vibles coesponding to ctul o physicl od unde study. The vibles without the subscipt will epesent nomlized o non-dimensionl vlues fo sclbility. d ρ ( ), ( ) F L ( L t ), Figue A.: Non-Unifom Rod Conside n infinitesiml element d, then we cn epesent the foces on fee body digm nd deive the eqution of motion of the element. E ε ( ) ( ) E ε ( + d ) ( + d ) Repesenting stin s ( ) Newton s second lw: E Figue A.: Fee Body Digm of the element z ( ) d ε we cn then wite the eqution of motion using d ( + d ) ε ( + d ) E ( ) ε ( ) ρ ( ) ( ) d t Applying Tylo s Seies Epnsion nd neglecting highe ode tems, we get the finl fom of the solution s z 3

40 E z ρ () ( ) ( ) z t 33

41 34 Then we cn scle the bove vibles to epesent them in non-dimensionl (ND) tems by: ND il loction, L d d L d d L. () ND il vibtion, L z z (3) ND coss-section e, ( ) ( ) L (4) Speed of sound, E c ρ (5) ND time scle, E L t t E L t t ρ ρ (6) ND Foce, L E F F (7) ND fequency, E L ρ ω ω (8) ND eigenvlue, L λ λ (9) Substituting fo the vibles in tems of non-dimensionl vibles, t b z,,,, in : ( ) ( ) t z L L E L z L L L E L ρ ρ (0) Simplifying the eqution leds to: ( ) ( ) t z z () The boundy conditions will be tnsfomed to: z z () The second boundy condition is tnsfomed to:

42 35 ( ) ( ) ( ) ( ),,, L L t F t z t F z E (3)

43 B Ptil Pole Plcement of Unifom Rod b e st Figue B.: Unifom Rod Open loop system The diffeentil eqution of motion u t u, 0 < < t > 0, () The two boundy conditions nd ( 0, t) 0 u, () u, t 0 (3) Seption of vibles pplied to () gives u ( t) v( ) sinωt, (4) v + ω v 0 (5) with the boundy conditions v ( 0 ) 0 (6) ( ) 0 v (7) Sinusoidl fom: The genel solution to (5) tes the fom v ( ) Asin ω + Bcosω (8) 36

44 The boundy condition (6) gives ( 0 ) B 0 v (9) so tht (8) educes to v ( ) Asinω (0) Substituting (0) in (7) gives ( ) Aω cos 0 v ω () so tht the chcteistic fequency eqution is Its oots e cos ω 0 () ( ) π ω,,,3,... (3) with coesponding eigenfunctions v ( ) sinω (4) Eponentil fom: Seption of vibles ( t) v( ) e st u, (5) pplied to () gives v s v 0 (6) with boundy conditions v ( 0 ) 0 (7) ( ) 0 v (8) The genel solution to (6) is v s s ( ) Ae + Be (9) 37

45 The boundy condition () gives ( 0 ) A + B 0 v B A (0) so tht (9) educes to v s s ( ) A( e e ) () Substituting () in (3) gives s ( ) As( e + e ) 0 s v () so tht the chcteistic fequency eqution is e s + e s 0 (3) With s ( ) iπ,,,3,... (4) the oots of (3) e v s s ( ) ( e e ) ± s with eigenfunctions (5) Closed loop system The diffeentil eqution of motion is u t u, 0 < < t > 0, (6) nd the two boundy conditions e ( 0, t) 0 u, (7) u, t ( fu gu) & + d. (8) 0 Seption of vibles ( t) v( ) e st u, (9) 38

46 pplied to (6) gives v s v 0 (30) nd the boundy conditions e v ( 0 ) 0 (3) ( sf g) v '() + vd (3) 0 The genel solution of (30) is v s s ( ) Ae + Be Fom (7) we hve (33) ( 0 ) A + B 0 v (34) so tht (33) educes to v s s ( ) A( e e ) (35) Fom (8) we detemine the chcteistic fequency eqution s s s s s ( e + e ) ( sf + g)( e e ) 0 d (36) with eigenfunctions v s s ( ) Ae + Be (37) whee s is the -th oot of (36) The Poblem: find f ( ) nd ( ) g such tht the oots of (36) e: 3πi s ±, 4 ( ) πi s ±,,3,4,... (38) 4 f then the oots of (36) e Comment: If ( ) g( ) 0 39

47 ( ) πi s ±,,,3,... (39) 4 Solution: f ( ) 0 Chcteistic fequency eqution g : 5 6 π sin π eq : s ( e s + e ( s) ) π s ( + e ( ) ) 4 s + π s e ( s) Figue B.: Fequency function of Unifom Rod Sinusoidl chcteistic eqution φ ω ( ) ( 9π 6ω ) ω 4( π 4ω ) cosω 40

48 4 cos( w) w ( 9 π 6 w ) π 4 w Figue B.3: Fequency function of Unifom Rod 4

49 C MATLAB Code fo Emple poblems C. MATLAB code: Emple Using ndn function of MATLAB, the noise level ws geneted NOISE: 0.*ndn(3,) cle ll /3;/3;3; s[0:0.0:0]; g-3/4*3^(/)/(3*^(/)+*3^(/))*pi; gg+0.*ndn(size(g)); g-9/4/(3*^(/)+*3^(/))*pi; gg+0.*ndn(size(g)); g3-3/4*3^(/)/(3*^(/)+*3^(/))*pi; g3g3+0.*ndn(size(g3)); g[g,g,g3]'; g[g,g,g3]'; wsin(s*); wsin(s*); w3sin(s*3); fs.*cos(s)-(g*w+g*w+g3*w3); fs.*cos(s)-(g*w+g*w+g3*w3); plot(s/pi,f,'b',s/pi,f,'') gid on is([ ]) P[-4/3/pi -4*sqt()/3/pi -4/3/pi; 0 -; / -sqt(3)/ ]; b[;0;0]; 4

50 C. MATLAB code: Emple Using ndn function of MATLAB, the noise level ws geneted NOISE: 0.*ndn(3,) cle ll /3;/3;3; s[0.5:0.0:8]'; g ; g ; g ; g[g,g,g3]'; wep(-/*)*sin(/*(4*s.^-).^(/)*); wep(-/*)*sin(/*(4*s.^-).^(/)*); w3ep(-/*3)*sin(/*(4*s.^-).^(/)*3); f-/*ep(-/)*sin(/*(4*s.^-).^(/))... +/*ep(-/)*cos(/*(4*s.^-).^(/)).*(4*s.^-).^(/)... -(g()*w+g()*w+g(3)*w3); f-/*ep(-/)*sin(/*(4*s.^-).^(/))... +/*ep(-/)*cos(/*(4*s.^-).^(/)).*(4*s.^-).^(/); plot(s/pi,f,'b',s/pi,f,'') gid on P[ ; ; ]; 43

51 Vit Amit Mh ws bon in Hydebd, Indi. He completed his schooling fom Si Vidy Senio Secondy School, Hydebd nd then finished studying t MES Indin School, Doh, Qt. He then joined Louisin Stte Univesity, Bton Rouge, LA, in 996 nd ened Bchelo of Science degee in Mechnicl Engineeing in Decembe 000 long with mino in Compute Science. Afte his gdution, he joined the Gdute Pogm in the Mechnicl Engineeing Deptment t Louisin Stte Univesity nd ened his degee s Mste of Science in Mechnicl Engineeing in Decembe 005 with concenttion in the e of Themo-Het Tnsfe nd Fluid Mechnics. He joined s Resech Associte t Southen Univesity, Bton Rouge LA. In 007, he joined Louisin Stte Univesity to wo towds Docto of Philosophy degee in Mechnicl Engineeing in the e of Mechnicl Systems. He lso woed fo ye s n Enginee t Mim Wtemes LLC, Shevepot, LA (fom My 0 Aug 03). He etuned bc to Louisin Stte Univesity in fll 03 to continue with this Ph.D. degee pogm. Amit is epected to gdute with his Docto of Philosophy degee in Mechnicl Engineeing in August 05 nd pusue cee in the multidiscipliny spects of Mechnicl Engineeing involving Mchine Design, Vibtions, Contols, Fluid Mechnics, Themodynmics nd Het Tnsfe. 44