FEA by nitin gokhale

Crash Anolysis 'roblem Name oof jhell Roof loaded by itr >wn weight q= 90 per unir Conrtraintr : Straight edgei i e e curved edgerldiagram inched cylinder Cylinder Subjected t=0.25 L=90 q=90 :oncentrated loads R=2S E=432x1O6 V=O.O to F ppq fJ7) ci Ii tF F=1.0 R=300 t=3.0 L=600 E=3x105 V=0.3 ends of cylinder diaphragn3 SUDDOlt. lemisphere subjected t o concentrated load Wisted Beam A twisted cantiiever strip iubjectedto two forces It is a good idea if you make comparison of the elements for running a quasi static simulation and compare the results with standards with respect to accuracy and computational cost.This motivates you t o gain a major confidence in full vehicle 1large practical problems. Reduced integration elements, hourglassing and control : Two things are important : 1. Use of lower order elements as opposed to higher order. If you lookcarefully at the stability condition, higher order elements yield a higher maximum frequency than the lower order elements.This is one of the major reason that lower order elements are preferred in al1practical codes.338

=PracticalFiniteElernent Analysis2. Reduced integration against full one. The implications of this are a computationally veryefficientprocedure but a t the cost of increased effort to control the hourglass orchequer boardmodeslzero strain energynon physicalmodes. (alFull lb) ReducedIf you carry out FEA of a single shell eiement by using 4 guass points or solid elernent using 8guass points then you shallget exactly six rigid body modes. By using reduced integrationwe get spurious rigid body modesapartfrom 6 rigidbody modes due to what is calledas \"rankdeficiency': See e.g. Hughes (6). These modes arecalledas Hourglass modes.Although theydon't add any extra energy into the system, they must be avoided as they are non -physical.This is usuallydone by adding extra stiffness or viscosity across the diagonal which adds extrastrain energy into the system and we cal1 this as Hourglass Energy. Now we must be carefulenough to see to it that this is much less as compared to the interna1 energy of the system. .--.-.-... - - - -1,Translationinx 2.Tranriatronin y 3.Translationin z direction4. Rotationabout x 5. Rotationabout z 6.Rotation about y RlgltBodyMadeof ShelElernent %me TypiwlHourglassmodesofa ShelElernenlAthumb ruleis thatthe hourglassenergy should be less than 10% of Internalenergy. Also it isvery important not to stiffen the already will working elernent.The reader isreferred to software

Crash Anolysis manuals for various stiffness and viscous forms and also Belytschko et.al. (1). Following are some of the observations on HourglassControl Methods 1. Default viscous formulations work better. .2. Stiffnessform i s most stable for large crash problems butalso results in a slightly stiffer response 3. Flanagan-Belytschko stiffnesscontrol is best for large rotations associatedwith highly nonlinear problems. 4. One canassign different hourglasscontrol methods to different parts. Better use a refined mesh rather than a coarse mesh for large rotations. 18.8 Contact Impact Algorithms There are three methods used in practical softwares: 1. Kinematic Constraint Method or Lagrange Multipliers 2. Distributed Parameter method 3. Penalty Stiffnessformulation Method Kinematic Constraint Method : Constraints are imposed into global equations by a transformation of the slave node displacement components along the contact interface. The transformation will distribute the slave node normal force component to adjacent master nodes. If the master surface i s finer than the slave surface then master surface nodes can penetrate through the slave surface without any resistance and creates a kink in the slideline. This can leadto hourglass problem. Distributed Parameter Method : In this algorithm. half the mass of each slave in contact is distributed to the master surface area. The interna1 stress in each element determines a contact pressure distribution for the master elements that receives the mass. The acceleration is updatedat the mastersurface.We then impose the impenetrationconstraints on slave node accelerationsand veiocities to make sure the movement along the master surface. Penalty stiffness method :This is probably the widest used method both in implicit and explicit codes. The methoduses normal interfacesprings between al1the nodesofthecontact surface. A stiffness modulus is computed for each master and slave segment based on the elasticity and the thickness property of each ofthecontacting elements.Oneshouldbe careful enough in selecting spring stiffness as this decreases time step.This method is very reliable and the hourglassing problem is lessas compared to the other methods. Some points t o note while running a dynamicsimulation : 1. Always mention the system of units which you have used for simulation. 2. You must have a fairly good idea of what would be the total file size that the output as otherwise if your input is wrong wrt generation of too much output of ascii and binary files, you will run out of space very soon. A thumb rule to remember is: If T i s the termination time, then your binary output file generation t h e step shouldbe T/10 (so that youhaveminimum loframes available) and the

- Practicai Finite Eiement Anaiysisascii file size should be a further ten times lesserasyou would requirean accuraterepresentationof energy, contact forces and stresses data.3. There should be a zero penetration in the model so that there are no initial contact relatedenergies in the model. It is always a good practise to run a O msec run forthe model makingyou understand if there are gross modeling errors in the model.4. Itis always better to set the global element length to 5 mm whichcorresponds to a timestep of 1 e-06 seconds. Try to make your initial runs with coarse mesh so that you know howyour model behaves.5. There are lot of element formulations and different material models and it is no wonderthat a beginner would be too muchconfused with theoverall software understanding. in theinitial stage, it is good idea to understandthe elementformulation performto i t s level best andwhy it does so. This cornes through a lot of verification runs. reading the software manualscarefully and experimental validation of the material models.6. Just defining the contacts and leaving to the software to do the rest of the things is notenough. You must also see whether it is giving youa physically compatible behaviour orresult.7. Usually a crash simuiation software has different cards suitable for rigid bodies anddeforrnablebodies. Carefully study these options sothat you geta full ldea ofthe capabilitiesof the software.8. Try to use restarts to the maximum extent as this is a powerful utility associated withexplicit dynamics code.9. Always check the CAE results with some hand calculations and make a note if it makessome sense. Always check the hourglass energy as if this is large enough than the internalenergy (strainenergylof the system, thenit canmask the physical failure and cangive a totallyerroneous result. A thumb rule is that the hourglassenergy shouldbe a maximum of 10% ofthe internalenergy.Note : By using reduced integration you save a lot of computationaltime which is alwaysnecessary in transient dynamics. (Any philosophy which makes your run fast should beused e.g. lumped mass matrix insteadof consistent mas matrix). The risk of using reducedintegration is that it generates non -physical zero strain energy modes which are to becompulsorily eliminated by using artificial stiffness /viscosity thus generating additionalstrain energy which we cal1as hourglass energy. Another strategy to make a fast run is touse rnass scaling which increases the t h e step but you shouldset reasonable limitations onthe maximum rnass increase.10. There can be many errors and mistakes by a beginner. Always remember that the causeof these errors i s very much present in the deck itseif. Common are wrongly formattedinput, initial contact penetration, improper load curve definitionsand massless nodes due totemporary nodes created not deleted. Floating point exceptions can be caused by parts withzero density or thickness, over constrained nodes. if you are getting a result and get excitedabout it very quickiywithout bothering muchinto theresults,always take a closer look whether

Crash Anolysiryou have inputted the correct information and whether the mesh is of agood quality. Normalrun doesn't necessarily guarantee that you have solved the problem in a correct way as whatis more important is always checking the CAE simulationresult with the test data or physicalbehaviour of the system.11. If the time step is too small due to the presence of a small element, it is always better tocoarsenthe mesh thusallowingan considerable increase in the time step.12. Total energy of the system is always constant. + +Total Energy = lnternalEnergy Kinetic energy External WorkFor most of the impact problems, there will be initial kinetic energy and no extemal workForceForceat time t=O.O. As the simulation advances Kinetic energy will decrease, lnternal energy will1)increase and the total energy of thesystemshould always remainConstant. Ifthe total energy1Oplot shows a very big jump then the model has a error and you must check the definition ofcontactS.The momentum printout also shows whether the bodiesafter hitting are going in theright direction. Always ensure that parts are going in the right direction with right velocitiesat the right time.18.9 Full Dynamic I Impact vs. QuasistaticSimulationsThe distinction between these two is quite obvious as if we see the load application infollowing figure, a suddeniyapplied load (mathematicallya heavy side or step input) then thedisplacementthat we will get will be two times that of the static displacement of the system.The term Quasi Static simulation is used when we use a dynamic code to produce a staticresult. Itmust be remembered that we will get a static or close to static result only if the rampup time is suficiently large (Le. the ramp up dopeis not too steep so as to get a closeresultas sameas of that as Full dynamic simulation). The term Quasi Static is used to representa slow.dynamic process as opposedto a fast dynamic process. 1 -- Time Quasi-Static Simulation lime Full Dynamic or lmpact Simulation UnderstondingFull Dynomlcm.QuosistaticSimulationA Dynamic code always produces oscillations in the result and the remedy is to reduce thisis to use a larger ramping. The following figure depicts the response of an undamped singledegree freedom system to ramp input.

- PracticaiFiniteEiernent Analysis L time (d)ttesparc of nepinputkl~umreapansetormpinput Responseofasingledegreeoffreedornsysfernto rompinpuisIt can be seen that the maximum response ( Rm=2 )accurs for an ideal step input (I.e.forramping time = 0.0 ). For ramps with, there will be a little overshoot and system will undergosmall oscillationsabout the pseudostatic deflectioncurve. Thus a loadcanbe considered tobe\" Slowly applied\" and dynamic effects can generally be ignored if the ramp up time islonger than about 3.18.10 Lagrangian and EuSerian CodesHistorically this differencein the approaches is due to thedifferent approachesused for solidmechanics and fluid mechanics. In solid mechanics, we always follow a particle tracking orlagrangian approach,that is the Finiteelement mesh keeps on deforming with the structure.As Belytschko(1) putsit\" Every material point is compulsorilyalso afinite element node !' Herethe mesh continuously follows the deformed structure and hence this approach has alwayssome limitation with respect to the very large deformation problems. In FluidMechanics, theattention is put on a control volume bound by a control surfaces and then one monitors themass momentum and energy exchange across it. Essentially we solve the sarne equation asthat in a solidmechanicsbut we put themin a solutionsuitable f o m alsocalledasconservationform. The conservationformhas lot of practical applicationsin problems with shockwaves andif used in the integral form is the basis of FiniteVolume Method.This approach is also calledthe Eulerianor Field approach.Thus the mesh remains fixed al1the t h e in spaceand you don'thaveany severe or stringent limitations on the meshquality.

The difference can be clearly observed in the following figure. In lagrangian meshes, elementboundaries remain coincident with boundaries and material interfaces. ln Eulerian, they don'tand then you will have to use other approximate or tracking methods for the treatment ofmoving boundaries.The physics thus dictates the following requirements. Lagrangian meshes for slight to moderate deformation. Eulerianmeshes for large deformation.It is also possible to use another type offormulation called Arbitrary Lagrangian Eulerian (ALE)formulation in which the nodes can be programmed to move arbitrarily. The nodes on theboundariesare moved on the boundaries itself while the interior nodesaremoved to minimizethe mesh deformaiion. Before Deformation After Deformation ..+ (a) LagrangianMesh : ..............Control UStyle -,Control Iv a i u m ë.l.................. Before Deformation After Deformation (b) EulerianMesh CornparisonofLogrongian & EulerianMesh fora blocksubjectedto sheor forceor o fluidrweepingporta computationdomoin. ................., Influx -Net convectivecharge (Efflux - Influx) ................... (CI Conservationlaw for a control volume fora quantity, N: Demonstrationof Reynold'sTransportThereon ++DN aN V.VN............ (1) Mathematics -= - Dt atTotal = Local change Convective/Transport ............ (2) PhysicsChange with CV change dt the control surface +

Mesh rezoning is also another approach which can be used with the projection of variablesbeîween the meshes (similarto submodeling technique used in static analysis). But this isalso a t h e consuming procedure and intmduces spurious jumps in the state variable historiesas it violates the conservation laws. Zukas(1992)gives a detailed account of the applicationsof various Lagrangian and Eulerian codes available for impact dynamics along with theirdetailedcapabilities. Some common codes are as follows:Lagrangian : HEMP3D, EPIC-3, LSDYNAEulerian : HULL,TRIOL, METRICHybrid : CELFEMost of the Eulerian codes are availablein research laboratoriesand we mostly use Lagrangiancodes for practical problems.18.11 Effect of Process and Residual Stress on Crash Analysis: MappedPlarricStminr PhyricolTest 51 -FERerultswtrhwton the CrashComponenr Rerulrs formlngeffects rifecrr olFormingon Roll(chmrisJComponenr (imoye 5ource:Alralr (iilcnd~r2007,Courtery: Tara Mofarr lrd.1The image from Tata Motors, represents, correlation between physical crash test and FEsimulated crash result of rail (chassis) component without and with forming effect includedin the model. The work hardening effect and formulation of residual stresses-strain duringmetal forming, causes change in subsequent yield stress and deformation behaviour offormed components. Thus the mechanicat properties of formed components can significantlydifferent from those of the blank. Consideration of these material property changes in crashsimulation is vital for achieving good simulation accuracy. In this exercise, one step formingsimulation technique from Hyperform is used improve accuracy of crash simulation result bymapping residual stress-strain data predicted by Hyperform ont0 crash simulation model. Inthe aboveimage the simulation results from configuration S2 CO-relates with the physical test.18.12 Typical Application of Crashworth/n= Shulations in VariousIndustriesAutomotive :Automotive industry has probably the widest application of crash simulation. Simulating the

Crash Anolysis crashworthinessof the vehiclein terms of very simple modelsbasedon the spring massdamper systems was the focus when the computers were very slow and the breakthrough occurred when LSTC was formed. Nowadays softwares such as LS DYNA and others have very wide practical aspects such as use of special seat belt elements and development of dummies for occupant safety. Several standards have been otiginated in various countries and it is impossible togive a detailed account of these. In the following we discuss some of the most essential applications related to the automotiveindusîry. There are very detailed procedures such as five star ratings and no attempt will be made here to take a detailed account of these. Thisislmogedlrpl~y~tnhge vonMrsesstrersconrourpiol ofoninstrumentpanelforaMarcilicoru, singHypeNiew. The analysis carrled out wos for head form impact in occardance wrth the Eumpeonsafety standordECE R- 21. The mode1waspreparedusingAltairHyperMeshand this non-fineor dynarnic annfysis war periormedusing LSDyno. Safety of the occupant is determ~nedfrom O HICunluewhichcon becaiculatedusingHyperVw.tieodform lmpadAnalyslsoflnstnimenf p n e l Ilmagesaurce: AllairCal@ndar2OO5, Courtesy :Marut1Udyog Lrd.JMost commonstandardsinUSAareFMVSS (Federal MotorVehicleSafetyStandards) regulationsand ECE ( Economic Commission of Europe)regulations in Europe. In Our country now we haveARAI testing standards.FMVSS standards can be obtained free of cost from the website and there are three 3 serieslisted in thevol 49 Code of Federal regulations. The 100 series deals with active safety or crashavoidanceand 200 series forms the most importantaspects of crashworthiness tests. Listedin series 200 are several standards and an attempt has been made to give the reader a briefsurveyof them. One can refer to the information available oniineonNCAP (Newcar assessmentprogram), IIHS[Insurancelnstitute for Highway Safety ) and NHTSA (NationalHighway Trafficsafety association).OccupantpHead restraintsImpact protection for the driver frorn steering controlsysternsSteeringcontrol rearwarddisplacementDoor locksand door retentioncomponentsSeatlng systemsOccupant crash protectionSeat belt anchorage systernsChild restraint systemsSidelmpa

-Requlatian Number 1 Descrimion I WU\". .\".... \"..\" \"\"\". .->. ...\"...>\"...\"\",.+ -. - Protection of driveragainst steering mechanismin impact - . Safety beit anchorages - . Child rertraint systems Seat anchorage rystemsand head restraintr lnterim fittings (darhboard )R-25 Head restraints-~ 9 4 L - Rear end impact Behaviouroflmpactedstructurein headon collisionR95 Front and rear protectivedevices Childrestraint syrtemr Occupant protectionin frontalcollision Occupant protection in lateral impact isldelmpacFMVSS 202 -Head RestralntsFirstestablishthedisplacedtorsoreference linebyapplyingarearwardmoment of 373 Nmaboutthe R-point, perpendicular to the design torso line, to the top of the back frame. Then apply arearwardmomentof 373 Nm about the R-point, perpendiculartothe displacedtorso line, 64mmbelow the top of the head restraint. Now increase the load to 890 N or failure, whichever occursfirst.The maximum displacement of the headrest, measured perpendicular to the displaced torsoline, should be no more than 102 mm, for the load. Furthermore there should be no failure atthis load.FMVSS 207 - SeatingSystemsIn al1 adjustable positions, a force equal to 20 times the weight of the seat, in the forwardlongitudinaldirection, applied throughthe seat centre of gravity by a suitablerigid fixture. In al1adjustablepositions, aforceequal to 20timestheweightoftheseat,in the rearwardlongitudinaldirection, applied through the seat centre of gravity by a suitable rigid fixture. ln its rearmostposition, a force that producesamoment of 373 Nm about the R-point, appliedto the upper seatback,for each designated seat position.The force is applied in therearward directionforforward-facing seats, and in a forward direction for rearward-facing seats. Permanent deformation orruptureof a seat belt anchorage or its surrounding areas i s not considereda failure, providedthe

Crash Anolysis required force is sustained for the specifiedtime. FMVSS 208-Occupant Crash Protection Frontal Impact impact a vehicle with a seated test dummy 50\" percentilemale hybrid III dummy in the front driverseat, travelling longitudinallyforwardat 30 mph, intoa fixedcollisionbarrierthat isnormal to the line of travel. All portionsof the test dummy should be contained within the outer surface of the vehicle.The HIC value, measured at the c.g. of the head, should not exceed 1000.0. The HIC value should be calculated by the following formula :Where \" a,\"is the resultant acceleration measured at in\"gn units. a= Ja:+ayZ+a;.The time interval prescribedin FMVSS 206is 36 msec Accelerationa is measured at the headcentre of gravity in terms of g units. The resultant acceleration, measured a t the c.g. of theupper thorax, should not exceed609, except for intervals whosecumulativeduration is not morethan 3 sec.FMVSS 210 - Seat Belt AnchorageSystemA longitudinal force of 13345 N (3000 Ib)in the direction the seat faces, applied simultaneouslyto the upper torso body block and the peivic body block.The force is ramped up to i t s maximumvalue in not more than 30 sec. and held constant till 100 sec. The force application rate shouldnot be more than 133447 Nlsec. of a seat belt anchorage test for a Marut; car uslng HyperVlew.the anolysls wai carried ouf ln accordonce to AIS 15/2LWSofety belt anchorage tert standard1The FEmodel wascreatedurlngAitalrHyperMeshandsolver SeatBeltAnchorage Test(;mageSourre :Altoi, CaIendarM06. Courtesy: Maruti Udyog Ltd)

-FlWVSS214 Side ImpactPlace the side of the vehicle opposite to the sidebeing tested against a rigid vertical surface. Fixthevehicle rigidiyin position.Theloading deviceis arigid steelcylinderof 305 mm diameterwithan edge radius of 13 mm. Apply the load continuously such that the loading device travel ratedoes not exceed0.5 inlsec, until the devicetravels 18 in.Theloading device shouldbe preventedfrom being rotated or displaced away from its line of travel. Record the applied load versusdisplacement of the loading device, for the entire crush distance of 18 in. Determine the initialcrush resistance, intermediate crush resistance and the peak crush resistance.Todetermine thecrush resistance, obtain the integrals under the force displacement curve within the specifiedlimitsand divide them by the crush distance. Requirements for side impact are as follows: 33.5mph crabbed impact. lmpactor mas 3015 Ib.ThoraxTraumaIndex TTl(d) i s the meanof the maximumacceleration of the abdominal spine(12* spine) and higher of the two values between the maximum acceleration of the upper (8\"rib) and the lower (4'hrib).TTl(d) c 85 g for LTVs and 4 door passenger carsTTi (d) s 90 g for 2 door passenger cars.Pelvic acceleration < 130 gThe interested reader is referred to the NHTSA website for a detailed discussion of occupantsafety related crash criteriadescription.Consumer Goods lndustry /Communications Industry :Crashworthiness also finds a lot of applicationsin drop test of components suchas television,plastic buckets and mobile phone. Points of interest here are to check the structural integrityof the component and monitor any damage caused to the system. In mobile industry anydamage caused to the antenna and the LCD display are very important as they make thedevicetotally useless.Applications in other industries:Although developed mainly for automotive applications, crash simulation softwares havealso found applications in train , ship and aircraft crashworthiness. The two main standardsassociated with FAA (Federal Aviation Administration) requirements are those of bird strikeimpact and engine blade containment. Other applications in defence sector are simulating.the explosive detonation process and design of weapons Computational Biomechanics alsoiscontinuously evolving with thedevelopment of finite element models closely following theactuai physical models.

Crash Analysis References : Ted Belytschko, Wlng Kam Liuand BrianMoran (2000): NonllnearFinite Elements for continua and structures, John Wiieyand Sons LS- DYNA Theoretical Manuai(2006).Livermore SoftwareTechnologyCorporation, USA LS DYNA Keywords Users Manual V970: Livermore SoftwareTechnologyCorporation, USA (The abovemanualscan be downioadedfree of cast from lot ofwebsites.At the timeofwriting, alatestversion of the documentation corresponding to 971 canalso be downloaded) Y. CFung (1965 ): Foundations of SolidMechanics, PrenticeHall of India J N Reddy (2W6):An introduction totheFinite ElementMethod,Tata McGrawHlll .T. J R .Hughes ( 2000 ): The FiniteElement Method, Linear Static and dynarnic Finie ElemenïAnalysis. Dover Publications JonesAZukas,TheodoreNlcholas.HallockFSwifi, LonginB Greaczuk, DonaidRCurran(1992):ImpactDynamics, Krieger PublishingCompany, USA R. D .Cook, D. S. MaikusandM. E. Plesha(1989) : Concepts and applications of Finite ElernentAnalysis, John WileyandSons R.D.Cwk ( 1995): Fin'k Ekment Modeling for Stress Analysis, John WileyandSons Ray A Craigi1981 ): StructuralDynamics, An introduction to computerMethods, JohnWiley and Sons RaoV Dukkipati,M A Rao. Rama Bhat i20W): Computer Aided AnalysisandDesign ofmachineelements .New Age International Publlshers CodeofFederal Regu1at:ons :Title49 Transportalion. PubIlshedby the office of the Federal Register.Nationd Archiver and Records Admlnistration, sale by US Govl. Print:ng office Warh:ngton Maurice Petyt (1990) :Introduction to FiniteelementVibrationanalysis, CambridgeUniversity Press Websltes : The following webrites glvea lot of additional information a) httpJ/~l~~~nhtsa.dot.gov ( FMVSS Standards can be downtoadedfrom this website) b) httpJlcrash-nenvork.comlRegulations/ECE-Reguaitions (ECE regulations can bedowniwded from this website) cl httpd/www.faa.gov (nie Federal Aviation Administration regulationscanbeobtainedfrom This website) d) httpY/wwwWWWeuroncap,com e) httpJIwww.iihrorg f ) hnpYIwww.lstc.com (This websitegives lotof Information on practicalapplicationsofLSDYNA including theexamples of variour courses offered and some course notes also have appearedin the monthly newsletter which can be obtainedfromwww.fealnformation.com)

NVH Analysis (By FE Method)Virtualprototypeconcepts enablemodeling ofvarious problemsrelating to noiseandvibrationtoa great detaii.The analyses concerned are very different from that of static analysis.Theconceptscan become coniplex due to structural-acoustic interaction.The aim i s to assess what numericalmodels, like finite element discritization, can be used to obtain approximate solutions.Thereareconcerns such as how the NVH targets are understood in terms of results from finite elementanalysis.In this chapter an attempt is made to relate NVH concepts to output from finite elementmodeling. In achieving this some matters involved in dynamic analysis that have implications onaccuracy of the analysis will be discussed.The acceptability of noise and vibration levels generated by vehicles is one of the importantcriteria determining the competitiveness of different public or personal transportation modes.There are two aspects to it: a) government legislations that regulate the noise produced bymanufactured products (like European Community, EC directives)and b) customer satisfaction,some products makesound that give a good feeling to thecustomer than others likedoorclosingfirmness, although ioud can give a more secure feeling.The regulations generally define targetsto achieve in terms of dB levels of sound. Products are categorised and dB levels are decided, forexamplefor lawn mover thedB level will be different from passenger vehicles etc.Thetargets canbe like pass-by noise levels for passenger vehicles. Table 1 gives some of the targets for pass-bynoise levels.Theseare measured on specified track with specified conditions at 7.5m away fromthe vehicle and the vehicle reaching certain specified speed. These legislations are for externalnoise level limits. Each product then has to conform to these targets.The targets for customersatisfaction are very complex to define. It can depend on frequency content and relative levels.of different freauencies. The wercewtion of sound level is also freauenc,v de~endenat s is wellknown.The sensitivity of humanear changes with the frequency.To account, to some extent, forthis variation in sensitivity concepts like A-weighting are defined. Some details of this are givenin the next section.Table 1.Typicalpass-by noise levels as given in EC directive forvehicles [Il.In the industry this issue of noise and vibration are encompassed in the term NVH (Noise, 351

Vibration and Harshness).Theterm NVH is in common usage in the automotive industry. In a way this influences the choice of using vehicle example in the chapter, but the problems addressed have relevancein other fields.The first two letters in NVH define themselves:the harshness term, in general, means roughness and may include fatigue due to vibration and cover both noise and vibration. Before giving any further details, some typical terminology used in noise and vibration control is introduced in the next section. 1.1 Some NVHTerminology Sound pressure Sound is sensed as the pressure that is created.This pressure is over above the meanpressure of surrounding media, for example atmospheric pressure.Therefore, the total pressure is given by p,=p,+p; wherep is the pressure that is of interestThe timeaveragevalueofthesoundpressure, taken over a longer duration, tends to zero. To overcome this sound pressure is quantified in terms of mean square values.The audible frequency range is 20 to 20000Hz. The lowest sound pressure a human can hear is a root mean square (RMS) value 2 ~ 1 0P.a~ and at the other end to threshold of pain it is 200 Pa. The range is so large that it is usual practice to have log scale representation. As in other fields dB scales are adopted. For defining dB scale a reference has to be fixed. In the case of sound pressure it is the lowest audible sound pressure i.e. Z X ~ OP-a~. Therefore, in dB scale sound pressure is given by In practice one uses a term sound pressure level (SPL).When sound pressure is quantified by dB scale, as done here, it is referredto as a sound pressure level. Thesound pressureof a sound source (likeacar) isdependent on where the caris, like on an open road or in a street surrounded by buildings. The sound pressure also depends on the distance between the source and the receiver. The sound source can also be directional i.e. at one angle from the source the pressure could be perceived larger than in other directions. This makes it a difficult parameter to quantify a source of sound for cornparison, for regulation purpose etc. This also means if sound pressure is quantified, one has to notify under what conditions sound pressure is measured. Vibration level The response of a vibrating system can be characterised by either RMS values or peak values. The representation is based on if one is interested in steady state response (RMS values) or short duration response (peak values).The response can be in the form of displacement, velocity or acceleration. Quite often dB scales are also used. However, the reference values chosen are not universally accepted. For example, the reference 10~m9 /s is used in many situations but it is not a standard practice.Thismakes it important to indicate the reference values when dB scales are352

Practical Fiiiitc Eleimeiit Aiiaivsisused.Frequency contentThe time history of sound pressureorvibration responsecontainsinformation of both systemandexcitation. In practice, al1 the information may not be clearly seen in the time history. However,when the time history is converted to contributions from different frequencies one can start tosee the system and excitation behaviour clearly.Thus, the frequency domain representation canbe very useful. In converting the time history to frequency domain representation (spectrum),one can use Fourier coefficientsor Fourier transforms.Fourier coefficient and Fourier TransformThe Fourier series operates on a time signal that is periodic, i.e., a time signal whose waveformrepeats again and again to infinitetime. Such a signal is a collection of sine and cosine functionswhose frequencies are multiples of the reciprocal of the period of the time signa1,Therefore. anyperiodic motion canbe represented by a series of cosines and sinesthat are harmonically related.The amplitudes of these harmonics are called the Fourier coefficients.The complex exponentialre~resentationof Fourier series iswhere Fourier coefficientsareIt can be noted that the units of the Fourier coefficientsare same as the original time history.The original time history can be reconstructed from the Fourier coefficients using the inverseprocess.The signals/responses that are encounteredin practice are not generaliy periodic.The concept ofFourier coefficients can then be extended by the Fourier lntegral Transform, or more simply theFourierTransformand let the coefficient here be U(f).This integral will transform any continuoustime signal of arbitrary shape into a continuous frequency spectrum extending to infinitefrequency. The important difference is that Fourier transform (the amplitude of harmonics) hasunits of (units oftime historyIHz)rather than the units of time history as with Fourier coefficients.For example, if one estimates the Fourier transform of velocity response, units wiil be m/SHz.Thepractical data will not be continuous and will be sampied at some time instants. Discrete FourierTransform (DFT)i s then used to estimatethe harmoniccomponents.Fast FourierTransform (FFT)is an algorithm to estimate DFT.Octave band representationThe frequency domain analysis using FFT is on a linear frequency scale and the frequencyresolution is constant.Thistype of representation results in a narrow band frequency spectrum.

There are applications where one requires different forni of frequency representation viz., broadband spectrum. For example, in acoustics the hunian hearing niechanism i s responsive tofrequency ratios rather than actual frequencies.The frequencies with a ratio of 2:1 are separatedby an octave band. Each of these bands will have a centre frequency. These octaves can bedefined with referenceto 1000 Hzcentrefrequency.Therearesmaller bandwidth representationslike 113 octave, 1/12 octave (simply, third and twelfth octave) etc. The approximate bandwidthand the centre frequencies can be calculated as in Table 2. Some applications, however, requireonly narrow band representation, for example fan noise which is tonal and contains dominantfrequenciesrelating to blade passing. One more example is gear noise where gear mesh tonesare dominant.Table 2. Approximate octaveband centre frequency and bandwidth. n is the frequency band.A weightingOnce the sound pressure is represented in frequency domain, the question arises of how toobtain overall or effective sound pressure. Simple addition of frequency components may notbe of much help as the perception and sensitivity of a human auditory system is frequencydependent. Based on the sensitivityone can weight the spectra to obtain equivalent effect. Laterthe contributions can be added to obtain an overall dB level.The most widely used weighting istermed as A-weighting.The A-weighting curve is based on the loudness curve that gives equalloudness of sound at each octave centre frequency. Table 3 lists the A weighting values whichwhen added to actual spectra result in what i s known as dB(A) levels.Table 3. A-weighting dB levels [21.

- Practical FiniteEiement AnaiysisFrom the aboveTableone cansee that, humanear is very sensitiveto noise around 2OOOHz andleast sensitiveto below, say, 1OOHz. For a sound pressure levelof 100 dB, it will be perceived as if83.9 dB at 125Hzand 101.2 dB a t 2000Hz.Sound intensitySound intensity is defined as time rate of sound energy passing through unit area. It is a vectorquantity,itis usedmostwidelytodiagnoseorhelpin quantifyingsound sources.As thedefinitionimplies it has units of W h 2 .Mathematically, it is a t h e average of product of particle velocityand sound pressure. Particlevelocity comes into focus because vibrating structures disturb thesurrounding media, particles around structures oscillateabout equilibriumposition (there is nonet movement of particles). The pressure builds-up due to this disturbance. Furthermore theenergy transferis some productof particlevelocity andpressure. In dB scales,for soundintensity,the reference intensity used is 1 x 10\" W h 2 .Sound powerSound power is the product of average sound intensity and area through which energy passes.Itis a quantity which isgenerallynot affectedby where the sound source i s located.The locationhere means, like an engine in an anechoic test cell (anechoic test cell as shown below, is a roomwherethereisvery littlereflectionof soundwavesfromthesurroundingwalls; most oftheenergygetsabsorbedby theabsorptionmaterialonwalls) or whether engine is placedon a groundin afactory.Hence, it is generally used for comparative purposes, for diagnostic purposesetc. Whennumericalsimulationsare performed, sound power can be easily estimated. ln measurement ofthis one can use anechoic room or reverberationroom measurements or sound intensity. In dBscales, for sound power, the reference power used i s a \"\"W. - AnechoicRoom ICourr@syT:he AutomotiveResea~hAssociation oflndialFrequencyresponse functionsIt is of interest to know the vibration response of structures to point harmonic force. Thisresponse can give insight into what happens when vibrationcontrol measures suchas, stiffenersare added, mass is added or damping is added. One can define receptance of the structure asdisplacement response to unit harmonic point force. It is a compiex quantity i.e. it containsinformation about both amplitude and phase. If the response i s velocity then it is mobility and

when the response is accelerationit is calledas accelerance.Some times term impedance is used,which is a reciprocal of mobility. In practice these responses to unit harmonic force are groupedas Frequency Response Functions (FRFs).There are laws that govern the influence of any designmodifications (like adding mass) on FRFs. These work in a very similar way to how series orparallel connections in electrical systems work, for details see [191. These rules allow assessingthe influence of addition of mass, stiffness, damping or combinations of tliese.A special type of FRF can be defined which is basically the response measuredat the point whereforce is applied. This FRF is called as drive point frequency response function. The drive pointmobility, for example, should have positive real part. The real part in mobility is an indicatorof damping. Generally in a drive point FRFs the resonances and anti-resonances are clearlydefined.1.2. Airborne and structure-borne noiseThere are numerous mechanisms that can result in vehicle noise and vibration. Over the yearsnoise and vibration from dominant sources have been reduced significantly.For example, due todevelopments in the automotive vehicle design, contributions from dominant sources such asthe engine have reduced [3].Therefore, noise and vibration tend to be made-upof contributionsfrom multiple sources, al1of which play important role, instead of single dominant source.These-sources transmit noisetothe interiorofthevehicle throuahvarious oathswhich mavbe seoaratedinto airborne and structure-borne paths [41. In general the concept of source, transfer path andreceiver as shown below, can be used in representing any noise and vibration problems.Source Transmission Receiver paths Source tronrferpath conceptFollowing figure shows two distinctive paths for an example of a car. In airborne paths, acousticexcitation of various panels can result in acoustic and vibration response on the other side.airborne pallis Slriicture-borne pallis Structure-borne o,idairbornepoths in a car The acousticexcitation in turn is due todisturbance introduced by the sourceon thesurrounding air. On the other hand, structure-borne sound and vibration is transmitted as vibrational energy356

from the source, for example the engine, through mounts and other structural connections, andpasses throughthe structure to interiorpanels which can radiate soundinto the cabin.The samepath can result in, for example, chassis vibration transrnittedto steering column or dash boardetc. In both the cases the source could be at some distance from the receiver and the energytransmission could be through complicated paths. Transmission paths can be represented byFrequency Response Functions (FRFs).These canbe structural, acoustical or structural-acoustic.The source can be mechanical forces or volume velocity in case of acoustical excitation. Theknowledgeof both sourceand the transmissionpaths i s essential in understanding the problemand in designing appropriate modifications.In vehicles the dominance of either airborne path or structure-borne path is dependent on thefrequency. For a typical case, below 500 Hz, structure-borne noise is dominant. At frequenciesabove 1000Hzairbornenoisebecomesdominant.Typicalzonesofdominancecanbe representedas below.100 500 lm 3m Frequency[HrlNoise insidein a fypical uehickThestructuralandacousticalfrequencyresponsefunctionsaredependenton naturalfrequencies,mode shapes and associated damping models. The structural acoustic response function apartfrom dependingon the above also dependson the interactionbetween structural elementsandsurrounding media.Thisinteraction can be described in terms of sound radiation efficiency.The natural frequencies and mode shapes can be estimated for simple shapes of structuresand acoustic cavities by analytical methods i.e. dynamic responses can be obtained by solvingtheir equations of motion. As in static analysis, in many situations either the geometrical ormaterialproperties Vary, or the shape of the boundaries cannot be described in terms of knownmathematical functions. Also, practical structures consist of an assemblage of components ofdifferent types, namely beams, plates, shells and soiids. In these situations it is impossible toobtain analytical solutions to the equations of motion.This can be overcome by approaches likefinite element methods.

The analysis can be carried out either in frequency domain or time domain. The domain of analysis is dependent on the source characteristic.The source can be periodic in nature (engine unbalance) or an impact or a random input. In obtaining the required response one has to consider damping models. The damping models can decide which approach is followed in obtaining the responses. The proportional damping results in the simplest of solutions and for very sniall non-proportional damping values this approximation still works well. The FE model can be very similar to that used in static analysis except that the frequency limit of analysis decides the element density.There is equivalence between wavelengths and frequency of vibration. In structural vibration tliere are three types of waves that can convey energy: flexural waves (bending waves), longitudinal waves and shear waves. At joints these waves can interact and transfer energy from one wave t o another. In acoustics, however, only one wave type is considered i.e. compressional wave. Only compressional waves transport energy through acoustic media.Except for bending waves, relation between wavelengthsand frequency is linear. The relation for bending waves is called dispersive and this relation decides structural acoustic interaction. As the frequency of analysis increases the uncertainties in the properties become important.The uncertainties involved can be significant as compared to wavelengths thus deciding the limit for deterministic analysis based on finite element discritization. However, FEA can still used in combination with probabilistic approachest o quantify uncertainties in responses. In whatfollows, in the rest ofthechapter, initially frequencylimitsforapplication offinite element modeling is discussed.This section shows the effect of uncertainties on the accuracy of results obtained from deterministic FE analysis. An overview is given about FEA for structural dynamics. Both free vibration and forced vibration are covered. Later FEA for acoustics is considered. In structural-acoustics interaction, definition of radiation efficiency is given. The frequency dependence of interaction is shown. Sincethe modeling difficulties can be much complicated as conipared with static analysis, generally the models are validated before use. A brief discussion is given on the methods used in validation and the associated instruments. As in its basic form the finite element modeling approach for structural dynamics and acoustics is a deterministic analysis tool.This means for a single model we get a single answer, for examplea particularfinite element model will havefixed frequencybehaviour.However, practicalstructures show uncertainties in both geometrical and material parameters. A mass produced thin plate used in a car body can show thickness variation.The finite element model, however, belongs to one such tliickness.The results of natural frequencies, mode shapes and forced responses are therefore for a single product from mass production. One can use probabilistic approaches to obtain mean and variances associated with the uncertainties. How sensitive are the dynamic paranieters to these uncertainties can decide the frequency range for finite element modeling. Generally, this frequency range, where finite elenient modeling is very effectively applied, is called as'low frequencies:The low frequency here does not mean absolute value, but depends on the size of the structure and the wavelengths. For a car typically this frequency is below 200 Hz. Small uncertainties in parameters can have significant effect at higher frequencies.At higher frequencies the wavelengths are smaller and the uncertainties associated can be coniparable358

to wavelengths. Under these circumstances the individual frequencies or a response at a lpoint do not give any information therefore finite element modeling cannot be used for thesefrequencies. A simple example as shown below shows this effect very well. The figure shows ldrive point mobility of a beam.Thepeaks in the plot are resonances; the first resonance occurs at23 HzThere are in al150 samples of beams shown in the plot.These beams are almost identical iin size except that they differ slightly in their length. From the mobility plots, it is very difficult tovisualize what is happening abovearound 200Hz.Thefirstfew frequencies are seen less sensitiveto uncertainties. A single analysis of a finite element model can give a reasonable estimate ofniodal properties here. Drivepoint mobility of0 beom wifh 2% uncertainryin leogrh onddoniping !It is not that finite element modeling cannot be used above certain frequencies, it is possible ito combine finite element modeling approaches with probabilistic approaches in quantifyinguncertainties (mean and variance at least) to a large extent. However, the computational effortcan become enormous and errors increase as frequencies increase. The fact that a certainnuniber of elements are required within a wavelength of resonance frequency, to estimatenatural frequencies and mode shapes accurately restricts this extension. At higher frequenciesenergy methods 151 that neglect phase information and work with average parameters canbe used effectively for structural andlor acoustical analysis. The starting frequency for usinghigher frequency method depends on so called modal overiap, M. The modal overlap can bedefined as average number of modes in damping bandwidth qw, ( qis daniping loss factor andw is radian frequency). A parameter called niodal density can be defined for this purpose. Themodal density i s defined as number of modes per unit frequency and generally represented byn. For frequencies where modal overlap becomes greater than unity it is difficult to associatepeaks in the response to any resonances.The response will have large contribution from severalmodes and the average contribution can be estimated by number of modes contributing. Atthese frequencies statisticalapproacheslike statisticalenergy analysis (SEA) [51are accurateandare very efficient. The average modal density can be obtained simply by knowing some averageparameters of structures. There are mainly two types of waves that need consideration: out-of-plane waves and in-plane waves. Flexural or bending waves belong to first category. Shearand longitudinal waves belong to in-plane waves group.Table 4 lists modal densities for typicalstructures 161.