Modal Analysis: A Comparison Between Finite Element .

2017 UKSim-AMSS 19th International Conference on Modelling & SimulationModal analysis: a comparison between Finite Element Analysis (FEA) and practicalLaser Doppler Vibrometer (LDV) testing.Luca PaganoKelvin Lake.Faculty of Architecture, Computing and Engineering.University Of Wales Trinity Saint David.Swansea, Wales (UK).e-mail: luca.pagano@uwtsd.ac.ukFaculty of Architecture, Computing and Engineering.University Of Wales Trinity Saint David.Swansea, Wales (UK).e-mail: kelvin.lake@uwtsd.ac.uk(Sol.103) EIGRL card and an LDV test using discrete pointmeasurements on the actual test piece. The key advantage ofthe LDV is that it is a non-contact measuring device whichuses the Doppler Effect to monitor the component surfacevibration, therefore, the component that needs to be testedwould not be loaded. However, it can be understood thatreplicating a free-in space test, which is the FEA model inreal life could be extremely complex, therefore the methodsare used to try reduce these effects resulting in minimizederrors within the physical test.Abstract — Natural frequencies and modal shapes simulated inFE (Finite Element) environments are not simple to validatewith physical testing or calculous, especially when relating tocomplex shaped components. In this paper a Laser DopplerVibrometer (LDV) system is utilized as an aid in validating thenatural frequencies, modes and mode shapes computed froman FEA model of a motorcycle swing arm. The experimentalerrors observed help to illustrate the difficulty of replicatingthe correct boundary conditions when dealing with this sort ofanalysis, due to the fact that the FEA environment is usuallyideal. In this case the analyzed component was supported witha very basic set-up; however, slight changes in testconfigurations and repeat multiple tests assisted in reducingexperimental errors ensuring continuity in measurements andthe exclusion of non-coherent values which were not relevant.A constant reference surface of the component was used inorder to visualize the LDV computed modal shapes, eventhough the excitation point of the component was varied. Thislast combination was fundamental for validation.II.The LDV working principle is fairly straight forward.This system is used for determining vibration velocity anddisplacement at a fixed point. The technology is based on theDoppler-shift effect; sensing the frequency shift of backscattered light from a moving surface.If this scenario was then considered firstly, the laser willproject a defined wave (f) which will be equal to:Keywords: NDT&E, Laser Doppler Vibrometer, LDV, Modalanalysis, NVH, Nastran, EIGRL, FEA, Motorcycle, swing arm,vehicle Engineering.I.f c/λINTRODUCTION(1)Where (c) will be the velocity which is considered equalto the speed of light in the case of a laser beam and (λ) thewavelength.Modal analysis is one of the most important studies thatare considered when dealing with complex structures.Computing natural frequencies and mode shapes helps toassess and understand the dynamic interaction between acomponent and its supporting structure.The natural frequencies of a structure can be defined asthe frequencies at which a structure tends to vibrate ifsubjected to a disturbance. The deformed shape of thestructure at a specific natural frequency of vibration istermed as mode of vibration. These shapes are functions ofthe structural properties of the component itself and itsboundary conditions and each mode shape is associated to aspecific natural frequency band. However, a variation ofstructural properties could reflect a variation of the bandwhich the structure vibrates but the mode shape will remainthe same. Having said that, if the boundary conditionschange, both characteristic mode shape and naturalfrequencies would change.The paper presents a comparison between modal analysiscarried out through FEA simulation using a NASTRAN978-1-5386-2735-8/17 31.00 2017 IEEEDOI 10.1109/UKSim.2017.27LDV WORKING PRINCIPLEIf the surface is then excited, it would have its ownvelocity, displacement and acceleration. The sourcefrequency will be subjected to a variation which has to beconsidered twice as this variation is also affecting thereflected frequency (frequency shift). Therefore ( f) isrelated to the actual frequency (f) and the relative speeds ofthe source (vs), observer (vo), and the speed of the wave inthe medium (v). Therefore: f ((v vo)/(v vs))f(2)The sign utilised depends on whether the observer andsource are moving towards each other or away from eachother. f is greater than (f), if the source is moving towardsthe observer. f is less than (f) if the source is moving awayfrom the observer. f is equal to (f) if the source is stationarywith respect to the observer. Fig. 1 gives a visual75

The change of the optical path length per unit of timerepresents the Doppler frequency shift of the measurementbeam. This means that the modulation frequency of theinterferometer pattern determined is directly proportional tothe velocity of the object relating to Δf.To compare the two beams, an optical interferometer is used.The LDV uses a heterodyne interferometer which has gooddirectional and amplitude sensitivity, the main component ofwhich is the Bragg cell (Fig. 3).representation of how the waves stretch or compressdepending on stationary and moving source [1].Figure 1: Doppler-effect representation [1].This can be then correlated to the frequency shift withinthe laser beam which will be: f 2*(v/λ)(3)Figure 3: Bragg cell visual representation [3].Where v will be the object's velocity and λ is thewavelength of the emitted wave. It can be then understoodthat to be able to determine the velocity of an object, theDoppler frequency shift has to be measured at a knownwavelength. This was done in the LDV which works on thebasis of optical interference (interferometer) [2].Fig. 2 shows the schematic of the LDV used for the test. AHe-Ne laser beam is split by the beam splitter (BS1) intoobject beam and reference beam. The object beam will thenbe the measurement beam. This beam then is focused on tothe vibrating or moving object through another beam splitter(BS2). The object reflects then the incident beam. Thereflected beam striking the second beam splitter is furtherdeflected to another beam splitter (BS3) which directs it tothe detector through a Bragg cell coupled with an oscillator.The detector measures then the difference between thereference beam and the measurement beam.The Bragg cell is basically an acousto-optic modulator(AOM) which introduces an optical frequency shift to obtaina virtual velocity offset. The Bragg cell is driven atfrequencies of 40 MHz acoustic waves which, inside the cellaffect the beam passing through the cell to yield a frequencyshift for that beam of 40MHz [3].This works using a piezoelectric transducer usuallymechanically bonded to a transparent material which usesradio frequencies that then are transformed into sound wavesthrough the acousto-optic material. This acousto-optic Bragginteraction is basically the coupling of the incident beam to adiffracted and un-diffracted beam (offset) made by a setdisturbance generated from the passage of the acoustic wave[4]. If the movement of the object frequency modulates thecarrier signal, the positive or negative velocity determinessign and amount of frequency deviation with respect to thecentre frequency. If the object moves towards theinterferometer, the modulation frequency is reduced and if itmoves away from the Vibrometer, the detector will receive afrequency higher than 40MHz. This helps to detect not onlythe amplitude of movement but also the direction ofmovement.III.MODAL ANALYSIS OVERVIEWThe input deck that was used (NASTRAN sol.103,EIGRL) assumed that the damping was negligible, which,for such a stiff component (swing arm torsional stiffness 2500 Nm/deg) was assumed to be a suitable condition. Tocompute natural frequencies, in this case, the FEenvironment will not consider applied loadings, due to thefact that the component was in a ‘free in space’ situation.The equation of motion can then be written in matrixreduced form as follows.Figure 2: LDV schematic [2].The path length of the reference beam is constant overtime. The vibrating or moving object will then generate atypical interferometer fringe pattern on the detector. Eachfringe corresponds to an object displacement of exactly halfof the wavelength of the light used.76

[M]{ü} [K][u] 0(4)It can be understood then that mass and stiffness(respectively [M] and [K] represented in matrix form, {u}and {ü} refer to the displacement and acceleration vector)were the parameters which would have then determineddirectly the response. Modes shapes, however, arefundamental characteristic shapes of the structure and aretherefore relative quantities. In the solution of the equationof motion, the form of the solution would be represented asa shape with time-varying amplitude. Therefore, the basicmode shape of the structure does not change while it isvibrating; only its amplitude changes.Although the scaling of normal modes is arbitrary, forpractical considerations mode shapes should be scaled (i.e.,normalized) by a chosen convention. [5]From a simple analysis, the mode shapes were seen asrelated to sine waves. Fig. 4 shows a lateral view of the firstthree bending and first two torsional valid mode shapes thatwere computed on a simple planar rectangular surface justfor visual purposes.Figure 5: Swing arm positioning.Due to the way the component was constrained to thesupporting structure the exciter was decided to act whitindifferent axis (Fig. 6), in order to avoid possible sistematicinterference of the supporting springs axial movements withthe LDV readings.Figure 4: Visual representation of mode shapesIt was seen then how all the modes are related by sametrigonometric functions. The modes were then stated to beorthogonal one to another, and this sets the fundamentals ofmodal analysis. Accordingly to this, the system could havebeen as complex as possible, therefore have n-th degrees offreedom and still be solved singularly as n-th one degree offreedom oscillator systems. Therefore, it was understoodthat to describe the motion of a complex system it isnecessary to include only one natural mode.IV.LDV SET UPThe component chosen for this test was a motorcycleswing-arm. The set-up was very basic however multipletests with slightly different configurations were carried out,in order to ensure the user sufficient data to compare. Themain positions in which the swing-arm was tested areshown in Figure 5. The component was then excited indifferent positions, these were considered accordingly tohow the component would be excited when mounted on theactual motorcycle (for example, frame mountings, wheelmountings etc.).Figure 6: Different axis excitation positions adopted.A reference surface (Fig. 7) was then set within the LDVsoftware, and kept as a constant. This specific surface waschosen because the component had quite a complex shapeand, ideally, the laser should be projected perpendicular.Therefore, a planar surface is the best in order to have aneat mode representation and accurate beam feedback.77