Modal analysis: a comparison between Finite Element Analysis (FEA) and practical Laser Doppler Vibrometer (LDV) testing.

Luca Pagano Faculty of Architecture, Computing and Engineering.

University Of Wales Trinity Saint David. Swansea, Wales (UK).

e-mail: luca.pagano@uwtsd.ac.uk

Kelvin Lake. Faculty of Architecture, Computing and Engineering.

University Of Wales Trinity Saint David. Swansea, Wales (UK).

e-mail: kelvin.lake@uwtsd.ac.uk

Abstract — Natural frequencies and modal shapes simulated in FE (Finite Element) environments are not simple to validate with physical testing or calculous, especially when relating to complex shaped components. In this paper a Laser Doppler Vibrometer (LDV) system is utilized as an aid in validating the natural frequencies, modes and mode shapes computed from an FEA model of a motorcycle swing arm. The experimental errors observed help to illustrate the difficulty of replicating the correct boundary conditions when dealing with this sort of analysis, due to the fact that the FEA environment is usually ideal. In this case the analyzed component was supported with a very basic set-up; however, slight changes in test configurations and repeat multiple tests assisted in reducing experimental errors ensuring continuity in measurements and the exclusion of non-coherent values which were not relevant. A constant reference surface of the component was used in order to visualize the LDV computed modal shapes, even though the excitation point of the component was varied. This last combination was fundamental for validation.

Keywords: NDT&E, Laser Doppler Vibrometer, LDV, Modal analysis, NVH, Nastran, EIGRL, FEA, Motorcycle, swing arm, vehicle Engineering.

I. INTRODUCTION

Modal analysis is one of the most important studies that

are considered when dealing with complex structures. Computing natural frequencies and mode shapes helps to assess and understand the dynamic interaction between a component and its supporting structure.

The natural frequencies of a structure can be defined as the frequencies at which a structure tends to vibrate if subjected to a disturbance. The deformed shape of the structure at a specific natural frequency of vibration is termed as mode of vibration. These shapes are functions of the structural properties of the component itself and its boundary conditions and each mode shape is associated to a specific natural frequency band. However, a variation of structural properties could reflect a variation of the band which the structure vibrates but the mode shape will remain the same. Having said that, if the boundary conditions change, both characteristic mode shape and natural frequencies would change.

The paper presents a comparison between modal analysis carried out through FEA simulation using a NASTRAN

(Sol.103) EIGRL card and an LDV test using discrete point measurements on the actual test piece. The key advantage of the LDV is that it is a non-contact measuring device which uses the Doppler Effect to monitor the component surface vibration, therefore, the component that needs to be tested would not be loaded. However, it can be understood that replicating a free-in space test, which is the FEA model in real life could be extremely complex, therefore the methods are used to try reduce these effects resulting in minimized errors within the physical test.

II. LDV WORKING PRINCIPLE

The LDV working principle is fairly straight forward.

This system is used for determining vibration velocity and displacement at a fixed point. The technology is based on the Doppler-shift effect; sensing the frequency shift of back scattered light from a moving surface.

If this scenario was then considered firstly, the laser will project a defined wave (f) which will be equal to:

f=c/λ (1)

Where (c) will be the velocity which is considered equal

to the speed of light in the case of a laser beam and (λ) the wavelength.

If the surface is then excited, it would have its own velocity, displacement and acceleration. The source frequency will be subjected to a variation which has to be considered twice as this variation is also affecting the reflected frequency (frequency shift). Therefore (∆f) is related to the actual frequency (f) and the relative speeds of the source (vs), observer (vo), and the speed of the wave in the medium (v). Therefore: ∆f=((v∓vo)/(v∓vs))f (2) The sign utilised depends on whether the observer and source are moving towards each other or away from each other. ∆f is greater than (f), if the source is moving towards the observer. ∆f is less than (f) if the source is moving away from the observer. ∆f is equal to (f) if the source is stationary with respect to the observer. Fig. 1 gives a visual

2017 UKSim-AMSS 19th International Conference on Modelling & Simulation

978-1-5386-2735-8/17 $31.00 © 2017 IEEE

DOI 10.1109/UKSim.2017.27

75

representation of how the waves stretch or compress depending on stationary and moving source [1].

Figure 1: Doppler-effect representation [1].

This can be then correlated to the frequency shift within the laser beam which will be:

∆f=2*(v/λ) (3)

Where v will be the object's velocity and λ is the wavelength of the emitted wave. It can be then understood that to be able to determine the velocity of an object, the Doppler frequency shift has to be measured at a known wavelength. This was done in the LDV which works on the basis of optical interference (interferometer) [2]. Fig. 2 shows the schematic of the LDV used for the test. A He-Ne laser beam is split by the beam splitter (BS1) into object beam and reference beam. The object beam will then be the measurement beam. This beam then is focused on to the vibrating or moving object through another beam splitter (BS2). The object reflects then the incident beam. The reflected beam striking the second beam splitter is further deflected to another beam splitter (BS3) which directs it to the detector through a Bragg cell coupled with an oscillator. The detector measures then the difference between the reference beam and the measurement beam.

Figure 2: LDV schematic [2].

The path length of the reference beam is constant over

time. The vibrating or moving object will then generate a typical interferometer fringe pattern on the detector. Each fringe corresponds to an object displacement of exactly half of the wavelength of the light used.

The change of the optical path length per unit of time represents the Doppler frequency shift of the measurement beam. This means that the modulation frequency of the interferometer pattern determined is directly proportional to the 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 good directional and amplitude sensitivity, the main component of which is the Bragg cell (Fig. 3).

Figure 3: Bragg cell visual representation [3].

The Bragg cell is basically an acousto-optic modulator

(AOM) which introduces an optical frequency shift to obtain a virtual velocity offset. The Bragg cell is driven at frequencies of 40 MHz acoustic waves which, inside the cell affect the beam passing through the cell to yield a frequency shift for that beam of 40MHz [3].

This works using a piezoelectric transducer usually mechanically bonded to a transparent material which uses radio frequencies that then are transformed into sound waves through the acousto-optic material. This acousto-optic Bragg interaction is basically the coupling of the incident beam to a diffracted and un-diffracted beam (offset) made by a set disturbance generated from the passage of the acoustic wave [4]. If the movement of the object frequency modulates the carrier signal, the positive or negative velocity determines sign and amount of frequency deviation with respect to the centre frequency. If the object moves towards the interferometer, the modulation frequency is reduced and if it moves away from the Vibrometer, the detector will receive a frequency higher than 40MHz. This helps to detect not only the amplitude of movement but also the direction of movement.

III. MODAL ANALYSIS OVERVIEW

The 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. To compute natural frequencies, in this case, the FE environment will not consider applied loadings, due to the fact that the component was in a ‘free in space’ situation. The equation of motion can then be written in matrix reduced form as follows.

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 determined directly the response. Modes shapes, however, are fundamental characteristic shapes of the structure and are therefore relative quantities. In the solution of the equation of motion, the form of the solution would be represented as a shape with time-varying amplitude. Therefore, the basic mode shape of the structure does not change while it is vibrating; only its amplitude changes. Although the scaling of normal modes is arbitrary, for practical considerations mode shapes should be scaled (i.e., normalized) by a chosen convention. [5] From a simple analysis, the mode shapes were seen as related to sine waves. Fig. 4 shows a lateral view of the first three bending and first two torsional valid mode shapes that were computed on a simple planar rectangular surface just for visual purposes.

Figure 4: Visual representation of mode shapes

It was seen then how all the modes are related by same trigonometric functions. The modes were then stated to be orthogonal one to another, and this sets the fundamentals of modal analysis. Accordingly to this, the system could have been as complex as possible, therefore have n-th degrees of freedom and still be solved singularly as n-th one degree of freedom oscillator systems. Therefore, it was understood that to describe the motion of a complex system it is necessary to include only one natural mode.

IV. LDV SET UP

The component chosen for this test was a motorcycle swing-arm. The set-up was very basic however multiple tests with slightly different configurations were carried out, in order to ensure the user sufficient data to compare. The main positions in which the swing-arm was tested are shown in Figure 5. The component was then excited in different positions, these were considered accordingly to how the component would be excited when mounted on the actual motorcycle (for example, frame mountings, wheel mountings etc.).

Figure 5: Swing arm positioning.

Due to the way the component was constrained to the supporting structure the exciter was decided to act whitin different axis (Fig. 6), in order to avoid possible sistematic interference of the supporting springs axial movements with the LDV readings.

Figure 6: Different axis excitation positions adopted.

A reference surface (Fig. 7) was then set within the LDV software, and kept as a constant. This specific surface was chosen because the component had quite a complex shape and, ideally, the laser should be projected perpendicular. Therefore, a planar surface is the best in order to have a neat mode representation and accurate beam feedback.

77

Figure 7: Reference surface.

The modal FE analysis was carried out within the interval 0-2500 Hz. Therefore, the LDV had to be set up in order to cover this bandwidth. The LDV software allowed to vary many parameters, Table 1 shows how this was set up for the different bandwidths. The sampling rate was crucial due to the fact that it could help ‘dampening’ down noise related to the way the structure was constrained, however this could affect measurements accuracy due to the fact that the resolution also depends on this.

TABLE 1: LDV SETTINGS. LDV Settings

Bandwidth Sample Freq.

Sample time. Resolution

FFT Lines used

1kH 2.56 kHZ 3.2 sec 312.5 mHz 3200 3200

2KHz 5.12 KHz 3.2 sec 312 mHz 6400 6400

3KHz 12.8 KHz 160 msec 6.25 Hz 800 480

5KHz  12.8kHz 1.28 sec 781.25 mHz 6400 6400

V. RESULTS COMPARISON.

The results were firstly compared analytically, then, in order to validate the modes the exaggerated FEA transition of eigenvectors was used in comparison with the LDV computed mode shape regarding the reference surface.

A. LDV Results.

TABLE 2: NATURAL FREQUENCIES RESULTS GAINED WITH LDV.

78

B. FEA results and visual LDV comparison.

TABLE 3: VISUAL COMPARISON OF SIGNIFICANT MODE SHAPES.

C. Numerical comparison. TABLE 4: COMPARISON BETWEEN FEA AND LDV.

79

VI. CONCLUSIONS.

It can be seen from Table 4 that the error percentage was reduced at higher frequencies. This was affected obviously by the different resolution, however the results were very consistent; also, it had to be considered then that the supporting springs would probably have reacted to lower frequencies rather than high ones, being not as stiff as the actual test piece; therefore the intrinsic dampening of the springs may have been helpful. Also, the mass of the FE model was slightly different from the mass measured on a scale (10-20 grams difference) and, as mentioned, the modal analysis is strictly related to the mass and stiffness matrix; therefore some error could be attributed to this. The components material itself obviously may have had some dampening, which was not computed when simulating with Sol. 103. This kind of simulation, as mentioned, assumed that the component was free in space, therefore the set-up of the actual LDV test would have needed some special rig built and still the result would not be 100% accurate. The mode recognition was not easily accomplished particularly in the case of close natural frequencies ranges where complex mode shapes were displayed. In fact it can be seen from Tables 5 and 5.1 that the reference surface had to evolve in order for the actual mode shapes to be identified. A high number of different trials, adopting differing configurations ensured consistency within the LDV measurements.

AKNOWLEDGMENT

I would like to thank in the first place all staff from the workshops for the support throughout this work. Infinite gratefulness goes then to Dr. Owen Williams and Prof. Peter Charlton for making this work possible.

REFERENCES

[1] Warwick University, 2014, Research projects, ‘Bone Investigation

using Optical Detection’, IMRC Warwick, Laser Doppler vibrometry (LDV), pp 26-30.

[2] Polytech Gmbh, 2015, Official website, ‘Basic principles of vibrometry’, viewed 25-nov-2015, available online at <http://www.polytec.com/us/solutions/vibration-measurement/basic-principles-of-vibrometry>.

[3] Di John G. Webster, ‘The measurement, instrumentation, and sensors: handbook’, CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, FL 33431, pp 43.

[4] Uzi Efron, ‘Spatial light modulator technology: materials, devices, and applications’, Marcel Dekker Inc. Publications, 270 Maddison Avenue, NY, 1995, pp-416.

[5] MSC software, Nastran,‘Real eigenvalue method’ User Manual, 2013, Chapter 3, pp-33.

80