MODAL ANALYSIS OF A BEAM WITH CLOSELY SPACED MODE SHAPES
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1 ME 164 Senior Captone Deign The Cooper Union Spring 2011 MODAL ANALYSIS O A BEAM WITH CLOSELY SPACED MODE SHAPES Eglind Myftiu The Cooper Union New York City, NY, USA ABSTRACT Thi paper invetigate the peudo-root problem by performing modal analyi on a plate pecially deigned with two cloely paced mode. The firt technique ued i impact teting and the econd technique i ine-weep haker teting. requency Repone unction (R) and coherence plot were obtained uing LMS Tet.Lab. The PolyMAX curvefitting algorithm wa ued to etimate the natural frequencie. The firt twenty mode of the beam were obtained. Two et of cloely paced mode were obtained at 302 and 315 and at 546 and 551 INTRODUCTION The Cooper Union offer a equence of control ytem and vibration clae a part of the required engineering curriculum. Although control ytem and vibration i one of the mot weighed topic of tudy by credit, relatively little reearch i currently being done by undergraduate tudent. Ue of the lab i motly due to mini-project in the ME101: Mechanical Vibration and ME401: Advanced Vibration coure. With the acquiition of more and more hardware, the Vibration Lab will have greater potential for reearch. In 2010 Profeor Baglione (Mechanical Engineering, Vibration Lab) purchaed the SCADAS MOBILE from LMS. Thi device i a DAQ that i more capable than the current NI DAQ in the Lab. The SCADAS ha a frontend with 16 input port (for impact hammer, accelerometer, tranducer, microphone, etc) and 2 output port for ource control. Thi add the poibility for multipleinput-multiple-output (MIMO) experiment which i limited with the current DAQ. The ource output alo facilitate haker experiment away from the lab ince an AC ource can be provided by the SCADAS which ha a couple of hour of battery life. Thi device hould be utilized for reearch at the undergraduate or graduate level becaue of it capabilitie. Effort hould be devoted to learn to ue thi hardware a well a the Tet.Lab oftware that i required for operation of SCADAS Mobile. One objective i to provide a detailed intruction manual for baic operation of the ytem (impact teting, haker teting). A more academic objective i to ue thi ytem to tudy a plate with cloely paced mode hape. Profeor Karen from Michigan Technological Univerity ha provided the Cooper Union Vibration lab with uch a pecially deigned polycarbonate plate. MOTIVATION Thi goal i important becaue finding cloely paced mode hape i a real problem in the vibration world, and it i called the peudo-repeated root problem (Allemang R. J., 2007). One example of thi peudorepeated root problem wa the pogo-effect in the Titan 2 pace rocket. In the Titan 2, two of the component had natural frequencie cloe to that of the reonant frequency of the rocket, cauing unwanted longitudinal vibration along the rocket. Engineer had only meaured one natural frequency cloe to that of the reonant frequency. When they came up with an engineering olution to lower that frequency, they made the vibration wore becaue they did not notice or take into account the effect or another frequency. Thi i a great example of the peudo-repeated root problem. Since engineer did not initially oberve the two frequencie cloe to the reonant frequency of the tructure, their olution took into account only one frequency and therefore did not correct the problem. Thi wa wated effort, time and money which alo could have potentially damaged the rocket (Doton, 2004). 1 Copyright 2010 by The Cooper Union
2 At the Cooper Union we are tudying a imple plate with peudo-repeated root to ee if we can better undertand thi experimental iue. Thi will aid in dicovering the limitation of our SCADAS ytem a well a to improve our experimental modal analyi method. BACKGROUND MODAL ANALYSIS Modal analyi i the tudy of the dynamic propertie of a tructure ubject to ome input excitation force. It i the experimental method of determining the modal parameter of a tructure. Thi include the natural frequencie, damping, tiffne, and mode hape of a ytem. The aumption that thi theory i baed on i that the ytem i linear and time invariant. The reult of modal analyi may be completely invalidated if thoe aumption are not true. EXPERIMENTAL METHODS BACKGROUND Impact Hammer A tated previouly modal analyi i the tudy of the dynamic propertie of a tructure ubject to ome input excitation force. To obtain R of a tructure the tructure mut be excited at all the frequencie of interet and the output mut be meaured at all thoe frequencie. An impact hammer can impart a force on an object a well a meaure the force a a function of time (with a load cell at the hammer head). An accelerometer (or many) can be placed on the tructure to meaure the output of the tructure at pecific location. An R will be obtained for each impact at a location y and a repone at a location x, labeled H xx (i.e. the repone at location 1 over the force at location 3 i the R H 13 ) Ideally the impact of a hammer act a an impule which excite all the frequencie in the frequency domain (impule in time domain i contant value in frequency domain). Thi would mean that the impule of the impact i infinite in magnitude and lat an infiniteimal amount of time. However thi i not realitically poible. Different tip can be crewed on the hammer (platic, rubber, metal, etc); the harder tip reult in impact cloer to impule function with an impule that i higher in magnitude and horter time. However, it i not alway deired to have uch an impact to excite a many frequencie a poible. In a modal analyi experiment, one hould excite frequencie of which the repone can be meaured. If we excite frequencie higher than the bandwidth, a determined by the ampling parameter, then participation of out of band mode ditort the data obtained at lower frequencie reulting in poor coherence. or thi reaon, different tip are ued depending on what frequencie you want to excite. Ideally when you trike a tructure you want to impact it only once to get clean data. However, becaue the impact can occur o quickly that the tructure may vibrate fat enough to hit the hammer again before the uer pull it away. Thi reult in a double-impact (or more). Thi i a kill that i obtained over time and care hould be taken to try to reduce double-impact when poible. Thi i ometime unavoidable, and averaging of many ample help reduce the error. Shaker Impact teting act to excite all the natural frequencie of a ytem but higher frequencie are attenuated. Which frequencie are attenuated depend on the choice of the tip ued. Thi i not a uer-friendly ytem and require much care to get clean data. A haker however doe away with all of thi by exciting a ytem at pecific frequencie of interet. A tinger that i attached to haker and i crewed into the tet piece force the vibration. The ytem can be excited by different function, uch a a ine curve wept from a lower frequency to higher frequency or a burt random ignal with a pecific frequency range. In thee experiment a ine curve wa wept from 0 to 2048 (in one meaurement period). In LMS uch a function i called a Periodic Chirp and i hown in igure 1 in the time domain. igure 1 LMS Periodic Sine Chirp Curve EXPERIMENTAL Plate The polycarbonate plate i 53 cm by 32 cm. Twenty point were marked on the plate in a 4X5 grid for each of the tet location. However, for haker teting another point wa added ince a pre-drilled hole exited there for the haker tinger to be attached. Therefore 21 tet point were ued for haker teting. or the impact teting, the accelerometer wa placed at Point 7 and each of the firt 20 point were impacted. Thi i called a roving hammer tet ince the hammer move with an accelerometer in place. or the haker teting, the haker wa attached at Point 21 while the accelerometer wa moved to each of the 21 point. 2 Copyright 2010 by The Cooper Union
3 A model A352C65 PCB Piezotronic Accelerometer wa ued to meaure acceleration. The accelerometer wa placed at Point 7 and the beam wa impacted at Point 1 to 20. ive average were taken at each point. igure 3 how the impact hammer etup. igure 2 Plate Meaurement Point Diagram LMS The LMS SCADAS i turned and plugged into an outlet (although it can run on battery if needed). Special cable are ued to connect BNC cable into the frontend of the SCADAS. The accelerometer i attached to the econd input in the frontend of the SCADAS. The reference which i either the impact hammer or the force tranducer in a haker tet i plugged into the firt input. A USB cable connect the SCADAS to a computer with licened Tet.Lab oftware. Tet Setup The plate i upended on two rubber tube which are tied acro two vertical fixture on the optical table in Acoutic Laboratory at the Cooper Union. Thi aim to contrain the plate a little a poible to imulate a freefree etup. Note that the rubber tube hould both be a equally taut a poible. igure 3 how the etup. Shaker A model K2007E01 Modal Shop Shaker and model 208C01 PCB Piezotronic orce Tranducer were ued for haker teting. A imilar etup wa ued for haker teting a for impact hammer teting. The tinger wa attached at Point 21. A model A352C65 PCB Piezotronic Accelerometer wa ued to meaure acceleration. The accelerometer wa placed at Point 1 to 21. Ten average were taken at each point. igure 4 how a picture of the hakeretup. igure 4 Shaker Teting Setup RESULTS AND DISCUSSION Impact Hammer Reult The R magnitude and phae are hown in igure 5 below for the drive point. The coherence i alo diplayed on the magnitude graph / B R PlateA:Node7:-Z/PlateA:Node7:-Z Coherence PlateA:Node7:-Z/PlateA:Node7:-Z igure 3 Plate Boundary Condition Impact Hammer A model 086C03 PCB Piezotronic Impact Hammer wa ued to impact the beam. A hard platic tip wa ued becaue it had the bet impact. The force data obtained when uing a hard platic tip wa much cleaner than with the other tip ued R PlateA:Node7:-Z/PlateA:Node7:-Z Phae igure 5 R Magnitude (top) and Phae (bottom) or Drive Point Meaurement in Impact Tet 3 Copyright 2010 by The Cooper Union
4 / Our magnitude plot indicate that our ytem i highly damped ince our natural frequencie do not exhibit harp peak and neither do the anti-reonance. Our magnitude hift by -180 degree and by 180 degree at anti-reonance, a expected for a drive point meaurement. The coherence i very high which mean that there i little noie or leakage in the ytem, except for higher frequencie greater than Thi i expected becaue our reference function attenuate at higher frequencie. In igure 6 we can ee the coherence zoomed in between 0.99 to 1. We ee that for the mot part the coherence i higher at anti-reonance than at the nearby reonance. Thi i not the cae for the firt mode hape at around 108. The anti-reonance ha a much lower coherence than the reonance in that region. Thi may be due to leakage. However there i no indication of the ource of thi leakage igure 6 Coherence igure 7 how the real and imaginary component of the R. We ee that the imaginary component reache relative minima at the ame frequencie that the real component croe the x-axi. Thi again i expected for a drive point meaurement. B R PlateA:Node7:-Z/PlateA:Node7:-Z Coherence PlateA:Node7:-Z/PlateA:Node7:-Z Real Imag R PlateA:Node7:-Z/PlateA:Node7:-Z igure 7 R Real (top) and Imaginary (bottom) or Drive Point Meaurement In Impact Tet Table 1 Natural requencie and Damping for Impact Teting Damping % % % % % % % % % % % % % % % % % % % % R PlateA:Node7:-Z/PlateA:Node7:-Z The tabilization diagram can be een in igure 8 along with the elected pole. The red curve i the um of all the R and the green curve i the R for the drive point meaurement. The value for the natural frequencie and damping ratio are tabulated in Table Copyright 2010 by The Cooper Union
5 40.8 v v Linear 2.05e+3 igure 8 Stabilization Diagram and Selected Pole for Impact Tet Shaker Reult igure 9 below how the R function magnitude and phae. The coherence i alo hown on the magnitude graph table wa being excited at thee frequencie which would caue unwanted vibration on the table that could have made it way to the plate. During my lat few run I noticed my laptop making a rattling noie, coming from within, at higher frequencie. Thi wa not oberved until the very lat run and o the laptop wa not moved o that the data would all be conitent N Phae igure 9 R Magnitude (top) and Phae (bottom) or Drive Point Meaurement In Shaker Tet We oberve the ame propertie a before in the phae hift. However, the haker tet give much cleaner data. There eem to be le noie for both magnitude and phae. Thi i alo manifet in the coherence function which i much higher than for the impact tet. The only exception are the two region circled above. However if we look at the force pectrum in igure 10 we ee that the output of the haker i noiy at thee exact ame region. One poibility i that thee may frequencie may excite the natural frequencie of the tinger. It i alo poible that omething ele on the igure 10 orce Spectrum or Drive Point Meaurement in Shaker Teting igure 11 how a cloer look at coherence from to 1. The coherence i relatively low at the two region dicued above, however it value i till very high for uch a mey portion on an R. If there i ome unmeaured input to the ytem a dicued above then the coherence hould be much lower. Coherence hould be further analyzed igure 11 Coherence 1 A lit d 5 Copyright 2010 by The Cooper Union
6 igure 12 how the real and imaginary component of the R. Thi data although it look le noiy than the impact data, it doen t how the ame behavior of a drive point that the impact data doe. The real component i not alway zero where there are minima in the imaginary component Real Imag igure 12 R Real (top) and Imaginary (bottom) or Drive Point Meaurement in Shaker Tet Table 2 Natural requencie and Damping for Shaker Teting Damping % % % % % % % % % % % % % % % % % The tabilization diagram can be een below in igure 13. along with the elected pole. The red curve i the um of all the R and the green curve i the R for the drive point meaurement. The value for the natural frequencie and damping ratio are tabulated in Table v v v v v v v v v v v v v v v v v v v v v o v v v v v v v v v v v v v v v v v v v v v v v v v o v v v o o v v v v v v v v v v v v Linear 2.05e+3 The natural frequencie of both method are tabulated in Table 3 for comparion. Table 3 Natural requencie Obtained rom Both Method igure 13 Stabilization Diagram and Selected Pole for Shaker Tet Impact Shaker Copyright 2010 by The Cooper Union
7 The two method give very imilar reult for the mot part. However there are four mode that are not detected by both method. urthermore we ee two et of frequencie that are very cloe to each other. Each method detect only one of the frequencie in thee two pair that have been highlighted in blue. CONCLUSIONS AND UTURE WORKS Two method have been ued to find the natural frequencie of a plate. The method agree for the mot part, however they do not both give exactly the ame reult. Each method mie ome of the natural frequencie. Two pair of cloely paced mode hape have been identified at 302 and 315 and at 546 and 551. The data preented here i much cleaner than data acquired with the tet-rig ued previouly in Vibration Lab. Therefore, ue of the optical table, in the Acoutic Lab i recommended. It i alo recommended that the optical table be cleared of other object when running haker experiment, becaue the haker might excite other object which can caue vibration on the urface of the table that would be only lightly damped by the metal fixture. The tet et up can further be improved by uing new rubber tube of the ame length, which would enure that the rubber tube holding the plate are have the ame tenion. Reult may alo improve if the haker i placed on ome rubber mat on the urface of the optical table o that haker vibration would not be tranmitted to the table and up along the optical fixture. urther tudie can be done by varying the bandwidth. The current bandwidth may be too high to provide much detail on cloely paced mode hape. Shaker teting on a maller frequency range may reveal more information in the frequency domain. In addition, the coherence hould be analyzed. Although a high coherence implie that our data i good, the data i upect ince high coherence reult at place with ignificant noie in the R and at anti-reonance. Latly, the curve-fitting function in Tet.Lab are decribed in the help file. An undertanding of thee curve-fitting technique hould be achieved to truly be confident in the data obtained. Reearch hould be done on curve-fitting technique and on the pecific one ued by Tet.Lab to fully analyze thi data. ACKNOWLEDGMENTS I would like to thank profeor Baglione for giving me the opportunity to work on thi project and for all the knowledge I have gained thi year in her Advanced Vibration cla and during weekly enior deign meeting I would alo like to thank LMS Cutomer Service for their upport with Tet.Lab quetion. REERENCES Allemang, R. J. (2007). Vibration: Experimental Modal Analyi. Univ of Cincinnati. Allemang, R. J. (2008). Vibration:Analytical and Experimental Modal Analyi. Univ of Cincinnati. Baglione, M. (2011). ME401:Advanced Vibration Cla Note. NY: The Cooper Union. Doton, K. (2004, Winter). Mitigating Pogo on Liquid- ueled Rocket. Crolink. Inman, D. J. (2008). Engineering Vibration: 3rd Edition. New Jerey: Pearon Education, Inc. Modal Analyi. (n.d.). Retrieved 2011, from LMS Site: VanKaren, C. MEEM470 Coure Note. Michigan Tech, Copyright 2010 by The Cooper Union
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