Measurement of the damping properties of carboncomposite plates by the power input method

M. Jaber 1, H. Schneeweiss 1, J. Bos 2, T. Melz 2

1 BMW Group, Structural Dynamics and Analysis,Knorrstr. 147, 80788, Munich, Germanye-mail: mariam.jaber@bmw.de

2 TU Darmstadt, Research group System Reliability and Machine Acoustics SzM,Magdalenenstraße 4, D-64289, Darmstadt, Germany

AbstractCarbon composite structures are widely replacing metallic ones in the automotive sector because of theirmany advantages over conventional structural materials, e.g., their high stiffness and strength-to-mass ratio.Moreover, their potential high inherent material damping can improve the dissipation of vibrational energyin the vehicle and, consequently, enhance its NVH performance. The main objective of the present investi-gation is to examine how the damping properties of carbon composite plates vary with the laminate stackingsequence and boundary conditions. Free and simply supported boundary conditions are investigated. In orderto measure the damping in a carbon composite plate, a 2-D laser doppler vibrometer is used. The loss factoris calculated by the power input method. In this paper, the frequency-dependent loss factor is studied forvarious laminate stacking of carbon composites (orthotropic, symmetric) with and without damping layers.A comparison between the power input method and the power bandwidth method is done for the differentcases. An application of the power input method to measure the loss factors of a car’s roof is performed.

1 Introduction

Passive control of noise plays an important role in the NVH (noise,vibration, and harshness) performance ofvehicles. For this reason, damped panels are used in the interior of vehicles in order to acoustically isolate thecabin. In addition, carbon composite structures are widely used in the automotive sector due to the challengeto produce light structures. However, the composite structural components can undergo a high level ofvibrations because of their low weight. Therefore, it is important to understand the composite dampingproperties, which play an essential role to reduce unwanted vibrations. Consequently, it is necessary toperform accurate measurements of material damping in order to predict the dynamic behavior of mechanicalsystems in the design stage correctly and have the chance to precisely model and optimize them. In orderto describe the panel damping, the damping loss factor η is used. The loss factor of a structure representsits capability to damp and dissipate the interior vibration coming from the outside environment. In thispaper, the power input method and the power bandwidth method are used to estimate the loss factor. Thesemethods are based on experimental data measured by means of a scanning laser doppler vibrometer. Acomparison between the two methods is done. With these methods the damping of carbon composite plateswith and without additional damping layers is measured for free and simply supported boundary conditions.Moreover, an application on the roof of a car covered with various types of viscoelastic layers is performed.

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2 Damping

2.1 Damping of composites

Damping describes the dissipation of mechanical energy in a vibrating system, usually by conversion of thisenergy into heat. Damping helps to control the steady-state resonant response and to reduce traveling wavesin the structure. There are two types of damping: material damping and structural damping. Material damp-ing is the damping inherent in the material while structural damping includes the damping at the supports,boundaries, joints, and interfaces in addition to the material damping.

Generally, the damping in welded metal structures is low, while it is higher in built-up components joined bymechanical fasteners such as rivets or screws. For fiber reinforced composite laminates, the material dampingcan be even higher and it depends on their assembly and the micro and macro-mechanical characteristics ofthe constitutive materials. In these composites, the energy dissipation process depends on the viscoelasticityof the matrix, the damping at the fiber-matrix interface, the layer orientation, and inter-laminar mechanisms.

2.2 Notations

A variety of nomenclature exists to denote damping. These include: logarithmic decrement δ, damping ratioε, specific damping capacity ψ, loss factor η, and quality factor Q.

• Logarithmic decrement [5] is the logarithm of the ratio between the subsequent free oscillations, x1

and x2

δ = lnx1

x2. (1)

• Damping ratio [6] is the ratio of the damping constant to the critical damping constant

ε =c

2mω. (2)

with m being the mass of the system, c being the critical damping, and ω being its resonance frequency.

• Specific damping capacity [6] is a measure between the energy dissipated per cycle (∆ W) and themaximum of the energy of the system W

ψ =∆W

W. (3)

• Structural damping loss factor [6] is a measure of hysteretic damping in a structure and is related to εas follows

η = 2ε. (4)

• Quality factor [5], describes the acuity of the resonance and is related to η as follows

Q =1

η. (5)

Moreover, in viscoelastic materials the damping is described by the imaginary part of the complex modulus.

3 Experimental techniques

The various damping test methods can be classified into three groups: a) time domain decay-rate methods,b) frequency-domain modal analysis curve-fitting methods, c) energy and wave propagation methods. Eachmethod has its own set of advantages and disadvantages [1, 3]. In this paper the power bandwidth methodfrom group (b) and the power input method from group (c) are studied.

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3.1 Power bandwidth method (PBM)

With this method the damping is measured only around resonance frequencies. The loss factor is given by

η =ω2 − ω1

ωn, (6)

where ω1 and ω2 are those frequencies on either sides of ωn that have corresponding amplitude values equalto the amplitude at ωn multiplied by 1√

2. This method requires the use of a high frequency resolution in order

to obtain precise results. However, it only works for damping mechanisms independent from the responseamplitude, i.e., viscous and hysteretic damping [2].

3.2 Power input method (PIM)

The PIM is based on a comparison of the dissipated energy of a system to its maximum strain energy understeady-state vibration. The loss factor of a structural system can be defined as

η(ω) =∆E

ESE, (7)

where ESE is the strain energy, ∆E is the energy dissipated from damping, and η is the loss factor at thefrequency ω considered. Assuming a stationary input energy at a fixed location, ∆E can be replaced withEinbecause the input energy must be equal to the dissipated energy under steady-state conditions [3]. However,neither ESE nor Ein can be measured directly. The input energy can be calculated with a simultaneousmeasurement of the force and the velocity at the input point of energy. The numerator of Eq. (7) can then becalculated as

Ein =1

2ωRe[hff (ω)]Gff (ω), (8)

where hff is the driving point mobility function, and Gff is the power spectral density of the input force.The following assumptions are made in order to estimate ESE. Firstly, since the strain energy cannot becalculated directly from velocity and force measurements, then ESE must be replaced by twice the kineticenergy which holds true at the natural frequencies of the system. The kinetic energy can be evaluated by

EKE =1

2

n∑k=1

miGii(ω), (9)

where EKE is the system kinetic energy, n is the number of measurement locations, mi is the mass ofthe discrete portion of the system, and Gii is the power spectral density of the velocity response at eachmeasurement location [4]. Assuming that the system is linear allows

|hif (ω)|2 =Gii(ω)

Gff (ω), (10)

where hif is the transfer mobility function. With the given assumptions, all measurement points uniformlyspaced throughout the system, and equal mass portions, Eq. (7) can be approximated by combining Eqs. (8),(9), and (10) as

η(ω) =Re[hff (ω)]

ωm∑n

k=1 |hif (ω)|2. (11)

Accurate measurements of the mobility function at the driving point are essential to obtain accurate lossfactors estimations, otherwise large errors can be introduced.

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4 Experimental setup and procedure

Figure 1 shows the acoustic used to obtain the simply supported boundary conditions, and how the shakerwas connected to the plate by opening the cover in the front of the box. The box is made from differentmaterials glued together. The inside dimensions of the enclosure are 0.85 m, 0.62 m, and 0.55 m in x, y, andz direction, respectively. The validation of this simply supporting process was established by the repetitiveresults obtained when plates were unset and set successively and by the agreement between the experimentalresults and the results deduced from models.

Figure 1: Acoustic box used to obtain simply supported boundary conditions (left), shaker connected to theplate inside the box (right).

An impedance measuring head was used to measure the force input to the plate surface. The shaker waspositioned to make the excitation direction as orthogonal as possible to the plate because the impedance headonly measures the orthogonal components. Without orthogonality, the shaker applies forces with nonnullcomponents parallel to the plate, thereby causing the underestimation of the forces. The shaker is connectedto the impedance head by a connecting rod. The input signal of the amplifier to the shaker is provided bythe data acquisition system software. During the experimental approaches in this study, a periodic chirpand white noise signals are used. A laser vibrometer is used to measure the frequency response of theplate. The primary advantage of using a laser vibrometer is the non-contact nature of the accelerometer,which eliminates mass loading of the structure due to response measurement accelerometers. Moreover,laser measurements can be taken more quickly than with accelerometers because of the reduction in setuptime and time spent moving accelerometers between different measurement locations. Figure 2 shows themeasurement setup used in this study to measure the damping.

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Figure 2: The measurement setup used to measure the damping.

The excitation point position is chosen to be different than the symmetry axes, along which the nodal lines ofnumerous vibrating modes are encountered [4]. It was found that the mass of the mobile part of the excitationsystem composed of the shaker, the connecting rod, and the mobile section of the impedance head did notaffect the system [4].

Besides, the loss factors are also measured for square and rectangular plates for free boundary conditions.In this set-up the plate is suspended from two points. A rubber band is inserted in the hole and attachedto the suspension frame. In this case, the loss factor includes the damping added by the rubber band inaddition to the material damping. Finally, a practical application to measure the damping of the roof of a caris performed.

5 Tests results

In this section an analysis of the measurement parameters is performed.

5.1 Validation of simply supported boundary conditions

In order to validate the simply supported boundary conditions, the frequency response function of an alu-minum plate is measured in simply supported boundary conditions. The mode shapes were compared withnumerical simulations. A good agreement was found. The maximum of difference of the natural frequenciesvalues between numerical simulations and measurement was 7%. Moreover, in order to verify the repeata-bility of the test procedure, one of the plates was tested twice. Both times, the plate and the shaker wererepositioned and all the FRF measurements were carried out. The loss factor was calculated by the powerinput method for the two measurements. A good agreement is found between the two loss factor curves, andthe error is 1%.

5.2 Effect of the spatial resolution

The effect of the sizes of the portions associated to each measurement point were examined. The standardtest procedure was repeated with the plate divided into 51, 81 and 165 equal portions. By increasing thenumber of points the curves stabilize asymptotically as expected and there are no appreciable variations ofthe loss factor. This procedure yields the optimal number of points 165 points which gives accurate resultsin a reasonable time.

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Figure 3: Effect of measurement points spatial density on the loss factor.

5.3 Effect of the frequency resolution

The effect of the frequency resolution associated to each excitation signal was examined. The standard testprocedure was repeated with the plate divided into 165 portions for different frequency resolutions namely,0.5, 0.25, 0.125, and 0.0625 Hz. By increasing the frequency resolution the curves stabilize, and the variationof the loss factor determined by the power input method becomes very small. The resolution at 0.0625 Hzwas applied for the following tests.

Figure 4: Effect of frequency resolution on the loss factor.

5.4 Comparison between PIM and PBM

The results of the damping at the resonance frequencies were compared with the power bandwidth methodfor the sample Prob 3b, see Section 6.1. The results are illustrated in Figure 5.4.

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Figure 5: Comparison of damping results between power input method and power bandwidth method.

An agreement was found between the two methods. The power bandwidth method can only be used on astructure in the lower frequency ranges where the natural frequencies are widely spaced, and it cannot beused in higher frequency ranges where the structure is modally dense and the peaks of the FRF might be soclose that the response does not decrease to half power levels in the vicinity of a peak. The use of the powerbandwidth method also requires a high frequency resolution so that the peak point and half power pointscan be measured accurately. This method is also dependent on a high quality FRF which typically requiresthe time domain input and output signals to be averaged. However, the power input method can be usedto calculate loss factors at all frequencies. Moreover, it calculates the loss factor for the whole frequencydomain. These averaged loss factors can be used in the models based on finite element method (FEM). Inthe rest of this paper, the power input method is used.

6 Damping results

This section analyzes the damping results of different carbon composites samples.

6.1 Dependance of the damping on laminate stacks

Figure 6 shows that the damping in carbon composites is dependent on the orientation of its constitutivelayers. Two samples were used, namely, prob3b [-45,90,90,90,90,90,90,45] and prob4b [-45,45,-45,45,-45,45,-45,45]. The loss factor curves show that prob 3b has higher damping than prob4b, which is expectedbecause prob4b is stiffer than prob3b in the various directions, consequently, prob4b has lower dampingcapacity than prob3b.

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Figure 6: Effect of laminate stack on the damping.

In order to study the influence of adding damping layers on the damping of the whole composite, fourdifferent types of damping layers embedded in the middle of carbon composites are tested. Table 1 showsthe components of the various samples used in this test. The names are not mentioned in this paper, butthe goal is to show that the damping layer type can have a very big influence on the damping of the wholecomposite. The results show also that prob 1 with a layer oriented at 90 degrees in the middle has higherdamping than prob 5, which has the middle layer oriented at 0 Degrees. The high amount of damping in thecase of prob 1 comes from the inter-laminar dissipation, which is higher in the case of [0,90,0] than [0,0,0].Figure 7 shows the effect of adding damping layers on the carbon composites.

composite lay-upprob 1 [45, -45, 0, 0, 90, 0, 0, -45, 45]prob 2b [0, 0, 0,0, 0, 0]prob 3b [-45, 90, 90, 90, 90, 90, 90, 45]prob 4b [-45, 45, -45, 45, -45, 45, -45, 45]prob 5 [45, -45, 0, 0, 0, 0 , 0, 0,-45, 45]prob 7 [45, -45, 0, 0, damping layer A, 0, 0, -45, 45]prob 8 [45, -45, 0, 0, damping layer B, 0, 0, -45, 45]prob 9 [45, -45, 0, 0, damping layer C, 0, 0, -45, 45]prob 10 [45, -45, 0, 0, damping layer D, 0, 0, -45, 45]

Table 1: composite lay-ups

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Figure 7: Effect of embedded damping layers.

Moreover, the effect of adding cutouts of viscoelastic layers was studied via prob 2b [0,0,0,0,0,0] in Figure8 for free boundary conditions. This plate shows high variation in the damping ratio. For those mode shapesthat have high amplitudes in the parts without viscoelastic layer the damping is low, and for those modeshapes that have high amplitudes in the parts with viscoelastic layers the damping is high.

Figure 8: Prob 2b: square carbon composite plate with rectangular cutouts of material damping.

The results of the loss factor as a function of frequency are illustrated in Figure 9.

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Figure 9: Loss factor of prob 2b as a function of frequency.

6.2 Effect of boundary conditions

The mode shapes of a vibrating structure and the amount of the energy stored and dissipated in the differentdirections are affected by the boundary conditions. Moreover, a big amount of energy can be dissipateddue to the friction at the boundaries. Therefore, the boundary conditions affect hugely the total structuraldamping. Figure 10 shows the loss factor curves of prob 4b measured in free and simply supported boundaryconditions.

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Figure 10: Effect of boundary conditions on the damping

The test shows that the damping in simply supported boundary conditions in average is two times higher thanthe damping in free boundary conditions. Moreover, the two samples prob3b and prob4b were measuredwith the same measurement parameters, excitation position, frequency resolution, and mesh size for free andsimply supported boundary conditions. A comparison between the results shows that prob 3b has higherdamping than prob 4b in both boundary conditions. The results are shown in Figures 11 and 6.

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Figure 11: Prob 3b and prob 4b: free boundary conditions.

This test shows that the boundary conditions do not affect the loss factor of the various composites similarlyover the whole frequency range. For instance, the maximum of damping of prob3b is at 290 Hz for freeboundary conditions and it is at 350 Hz for simply supported boundary conditions. However, the average lossfactor of prob3b is approximately two times higher than the average of prob 4b in both boundary conditions.

6.3 Application case: car’s roof

A small shaker was connected to the roof of a car. The frequency response function was measured by thelaser vibrometer and the loss factor curves were calculated by the same procedure mentioned in the previoussections. Figure 12 shows the shaker connected to the roof of the car.

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Figure 12: Shaker connected to the roof of the car in order to measure the damping.

A damping layer was added inside the roof, and the mass of the roof is estimated to be 6.2 kg via Catiasoftware. These measurements can help to get an idea about the loss factor values from 100 to 3000 HZ.These loss factors can be used in the finite element models. Figure 13 shows the loss factor curve obtained.

Figure 13: Loss factor curve of the car’s roof in the frequency domain.

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7 Conclusions

The damping plays crucial role in the response of a structure subjected to dynamic load. The measurement ofthe damping by the power input method can give an idea of the structural damping as a function of frequency,laminate stacks, and boundary conditions. This method is more effective than the power bandwidth methodbecause it gives an idea of the damping over the whole frequency range and not only for the mode shapes.These averaged loss factors can be used in finite element simulations to predict the response function of avibrating structure. The power input method can be used for composite structures and for composites withinterleaved viscoelastic layers in order to determine which layer has the best capacity of dissipation. Thepower input method and the power bandwidth method measure the structural damping. The power inputmethod is more effective than the power bandwidth method for high frequencies, where it is difficult todetermine the peak points and the half power points.

Acknowledgments

The authors gratefully acknowledge the European Commission for its support of the Marie Curie programthrough the ITN EMVeM project (GA 315967).

References

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[2] M. Martarelli, C. Santolini, Scanning laser doppler vibrometry for the characterization of the dampingloss factor in honeycomb panels, Proceedings of the IMAC XXVII, Orlando, Florida, USA (2009)

[3] M. Carfagni, M. Pierini, Determining the Loss Factor by the Power Input Method (PIM), Part 1: Nu-merical Investigation., Journal of Vibration and Acoustics, Vol. 121, 1997, pp. 417-421.

[4] M. Carfagni, P. Citti, M. Pierini, Determining loss factors using the power input method with shakerexcitation, in M. Carfagni, editor, Proceedings of The International Society for Optical Engineering,Vol. 1, 1998

[5] W.T. Thomson, Vibrazioni meccaniche. Teoria ed applicazioni, Tamburini editore,1997, Milano, Italy.

[6] D.J. Ewins, Modal testing 2, Research Studies Press Ltd.,2000.

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