Rayleigh damping parameters estimation using hammer impact tests

Mechanical Systems and Signal Processing 135 (2020) 106391

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Rayleigh damping parameters estimation using hammer impact tests Fernando Sánchez Iglesias ⇑, Antonio Fernández López Technical University of Madrid, Plaza Cardenal Cisneros, 3, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 16 June 2019 Received in revised form 28 August 2019 Accepted 22 September 2019

Keywords: SHM Damping Composite materials simulation Impacts

a b s t r a c t Structural Health Monitoring (SHM) systems for aerospace applications are becoming more prevalent and its potential cost-benefit relation is rapidly improving as more structures that were traditionally made with metallic materials are gradually being replaced by composites. To support these systems, simulations can play a crucial role; however, in the case of low energy impacts, the structural damping becomes a very significant element in the simulation. This damping is usually not well known, or is based in estimates with very low reliability. Therefore, in order to improve the reliability of these values this paper presents an attempt to estimate the Rayleigh damping parameters with the application of timefrequency analysis methods to transient signals, specifically, the reduced interference distribution. These parameters are estimated based in the results of a large number of structural tests done in a square carbon-fiber reinforced plastic panel, and are then validated by correlation of an explicit Finite Element Method (FEM) simulation. A discussion on how to apply the reduced interference distribution to the signals measured by piezoelectric sensors under an impact scenario and other possible usages of these results is also presented in this document. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The use of composite materials in the aerospace industry is becoming more prevalent in current aircraft developments due to their potential advantages in weight specific strength, stiffness and good fatigue and corrosion resistance and are replacing the classical metallic material structures in many areas [1,2]. However, these materials may require some special attention as they present a very poor impact resistance and a much wider range of failure modes, most of them not easy to be detected by visual inspection [3–6]. For this reason, new and affordable inspection systems must be developed to be able to study the performance of these structures in service. A Structural Health Monitoring process involves the inspection of a system over time using periodical measurements from an array of sensors permanently installed in the structure; the extraction of damage-sensitive features from these measurements and the analysis of these features can be used to determine the current state of the structure health. Additionally, the system could be used to capture events that could possibly damage the structure and assess them in real time [7]. These SHM systems could present many cost advantages when compared with the traditional inspection methods, and could also be used to improve the life prediction of the structure based on its usage [8,9]. ⇑ Corresponding author. E-mail address: [email protected] (F. Sánchez Iglesias). https://doi.org/10.1016/j.ymssp.2019.106391 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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Due to the high cost of physical tests, numerical simulation can be especially helpful to predict the structure behavior under the kind of events to be monitored and may help improve the accuracy of these systems [10]; however, one of the main drawbacks of the simulation is that structural characteristics must be known beforehand, this may be a simple task in the case of density or elasticity modulus that are well known for most aeronautic structures, but may be more challenging in the case of other parameters such as damping. In general, elastic waves in solid materials are guided by the boundaries of the media in which they propagate. Waves in metallic plates were among the first guided waves to be analyzed in 1917 by Horace Lamb; its characteristic equations were established for waves propagating in an infinite plate and describe two modes of propagation: symmetric (S0) and antisymmetric (A0). Lamb waves differ from the standing waves as the particle motion at an arbitrary point in space is not periodic, they are then a particular case of traveling waves. By definition, Lamb waves have no particle motion in the transverse direction. This motion is found in the so-called shear wave modes (Sh), which have no motion in the longitudinal or vertical directions, and are thus complementary to the Lamb wave modes. These waves can be generated either by active interrogation with sensors, or as a result of the environment on the structure, such as an impact. Lamb waves traveling in composite thin plates are very sensitive to defects or damages as shown in [11]; this opens a wide field of applications for SHM, especially for composite structures, as they are prone to damages such as delaminations which are not easily detected by visual inspection techniques; or structures for space applications that cannot be inspected during their service life. There are currently many successful application of these waves for damage detection and SHM [12,13] using a distributed piezoelectric actuator/sensor network based on the pitch-catch method; and an application for crack and corrosion localization is shown in [14]. Computational simulations provide a very effective way to capture and understand the structural behavior in terms of elastic wave response; general purpose computational codes such as time domain spectral finite element methods [15,16] or explicit finite element methods [10,17,18] are commonly used to calculate the wave propagation. However, these codes require a large amount of computational resources, and may become a problem in large and complex structures. For this reason it is important to derive the solutions on simple structures and understand the local behavior of the wave when facing an structural detail before attempting to simulate a complete aerospace structure. 1.1. Damping of a one degree of freedom system Damping mechanisms in composite materials are different, more complex and usually higher than those from conventional metals, and have been studied extensively [19–23]. This is due to the multiple energy dissipation sources that can be present in a composite structure and are not present in metals, such as the viscoelastic nature of the matrix which is the major contributor to the damping of the laminate or the damping due to the interphase between the fiber and matrix. Additionally, composites may also present a special behavior under damages, high vibration or temperature [24–26]. In the case of impact detection, the damping characteristics of a structure become crucial in the response measured in the piezoelectric sensors. An ideal one degree of freedom linear damped system is governed by the following equation:

€ þ c
u_ þ k
u ¼ 0; m
u

€ þ 2fxn u_ þ x2n u ¼ 0 u

ð1Þ

where:  k; c; m Are the stiffness, damping and mass of the structure respectively pffiffiffiffiffiffiffiffiffiffi  xn Is the natural frequency of the system, defined as xn ¼ k=m  f Is the critical damping ratio, defined as f ¼ c=ccr , with ccr ¼ 2k=xn . The critical damping ratio f can have a significant influence in the response of a system; it is said that a system is under damped when f < 1, critically damped when f ¼ 1 and over damped when f > 1. A typical response of each of these cases is shown in Fig. 1. Systems used for structural applications tend to have a very low value of structural damping and the signals measured by piezoelectric sensors under low energy impacts usually behave as an under damped system. For these kind of systems the solution of Eq. (1) can be written as:

  u_ 0 þ fxn u0 uðtÞ ¼ exn ft u0 cos xD t þ sin xD t

xD

ð2Þ

 1=2 where xD is the damped natural frequency, such as: xD ¼ xn 1  f2 A simple model for implementing structural damping in an explicit simulation could be the one proposed by Rayleigh in 1877, also studied in [27,28] for lamb wave simulations, that defines the critical damping ratio f as a function of the mass and stiffness matrix in the following manner:

½C  ¼ a½M  þ b½K 

ð3Þ

F. Sánchez Iglesias, A. Fernández López / Mechanical Systems and Signal Processing 135 (2020) 106391

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Fig. 1. Effect of the critical damping ratio on the response of a 1 degree of freedom system.

where ½C  is the damping matrix, ½M  is the mass matrix, ½K  is the stiffness matrix and a and x are two parameters of the damping model, that in the case of a one degree of freedom system can be defined as:

f¼

a 2x

þ

bx 2

ð4Þ

Therefore, by knowing an average frequency of the oscillation x, both the f and the Rayleigh parameters a and b, can be obtained with a numerical fitting of this model on the test results. 2. Time-frequency distributions theory The frequency content of a given signal as a function of time can be studied with the help of time-frequency distributions. These distributions can describe how the spectral content of a signal varies during time, representing the energy or intensity of said signal simultaneously in time and frequency. A very detailed review of time frequency distributions can be found in [29], and an application of the pseudo Wigner-Ville distribution on signals produced by Lamb Waves is shown in [30]. A general equation for a time frequency distribution wðt; xÞ for an input time signal sðt Þ was derived by Cohen [29] as:

Pðt; xÞ ¼

1 4p2

ZZ Z

 s  s eihtisxihu /ðh; sÞs u  s u þ dudsdh 2 2

ð5Þ

where the integrals are evaluated from 1 to 1;, s is the complex conjugate of the input signal and /ðh; sÞ is an arbitrary function that is referred as kernel. t and x represent time and frequency respectively. An particular case is the Wigner-Ville Distribution (WVD); as described in [30] it may offer a significant advantage over the Short Time Fourier Transform and the Wavelet Transform as it has a more general application range and it is not limited by the uncertainty relationship on simultaneous time and frequency resolution. However this distribution presents a number of limitations, the most significant ones being:  Sampling frequency must be four times that of the highest frequency of interest of the signal as opposed to the discrete Fourier transform which requires it to be only two times.  When more than one frequency component is contained in a signal it may result in a noisy distribution or the appearance of signal content at frequencies and times not actually contained in the waveform. This shape noise is mostly contained in the cross-terms from the multiple frequency components and it could complicate the interpretation of results.  The distribution may return negative values. These values do not have a physical meaning and most often appear as a result of the interference or noise discussed in the previous point. 2.1. Reduced interference distribution A possible solution to overcome some of the Wigner-Ville distribution limitations may be the shape Reduced Interference Distribution (RID), used extensively in the field of civil engineering for the analysis of earthquakes signals (31). In general, a RID refers to any distribution that reduces the cross-terms expression relative to the auto-terms in a quadratic time frequency representation, significantly reducing the interference or noise caused when more than one frequency component is present in the signal. A particular class of such distributions, RIDðt; xÞ) with kernel Rx ðt; sÞ based on a time series sðtÞ with analytic associate xðtÞ can be written as:

Z

RIDðt; xÞ ¼

1

1

hðsÞRx ðt; sÞeixs ds

ð6Þ

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And its kernel Rx :

Rx ðt; sÞ ¼

  Z jsj=2 gðmÞ 2pm  s  s x t þ m þ x t þ m  dm 1 þ cos s 2 2 jsj=2 jsj

ð7Þ

where hðsÞ is a time smoothing window, and gðmÞ is a frequency smoothing window. Both the RID and the standard Wigner-Ville distribution are time-frequency distributions of the Cohen’s general class but the RID may present some advantages over a WVD, as discussed in 31. The RID can be interpreted as a smoothed WVD by applying a boxcar smoothing filter to reduce the oscillatory interference that tends to appear in the cross-terms. The study presented in this paper uses a RID formulation for a Hanning window of ½ð1 þ cosð2pm=sÞÞ=2, although other common smoothing kernels could include binomial, Bessel and triangular (or Bartlett) windows. 3. Experimental work The experimental test campaign is performed on a square composite panel with a side length of 726 mm. The panel has a symmetric, quasi-isotropic, 7-ply laminate of AS4/8552 with the following layup: (+45, 45, 90, 0)$ resulting in a total thickness of 1.288 mm. The impacts are done using a Brüel and kJær instrumented hammer model 8206-003; it is equipped with two interchangeable tips of aluminum and rubber with a diameter of 10 mm. The choice of tip can determine the impact amplitude, duration and bandwidth. The piezoelectric sensors signals and hammer load cell are recorded with a National Instruments data acquisition card model NI-USB-6366 with a sampling frequency of 1 MHz and continuous acquisition over a time of 0.5s using zero span after the impact is detected. The plate is instrumented with an array of eight piezoelectric sensors distributed evenly. A schematic of the test set-up is shown in Fig. 2 and a picture of the test is shown in Fig. 3. Signals measured on the hammer and piezoelectric sensor 2 are shown in Fig. 4 for all the valid tests performed with the aluminum tip. The impact velocity v i , required for the FEM simulation, has been estimated from the impact force measured by the hammer (Fig. 4 left), as follows:

vf  vi ¼

Z 0

ti

V h ðtÞ dt qm

ð8Þ

where V h is the signal measured by the hammer, m is the mass of the hammer, ti is the time of the impact, q is the sensitivity of the hammer, equal to 1.14 mV/N (as per the impact hammer datasheet) and v f is the hammer velocity at the end of the impact, assumed zero, to conservatively simulate a perfect energy transfer. An average impact velocity of 0.5 m/s is estimated applying this method on all the valid impact tests and is used for the FEM simulation. 4. Results and discussion 4.1. Rayleigh damping parameters adjustment An approximation of the antisymmetric wave for low frequencies is proposed in [32]. This approximation can be compared with the time frequency diagram as follows:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 Eh 1 12qð1  m2 Þ d 2 cB ¼ ð2pf Þ ) f ¼ 2 2 2 2p 12ð1  m Þq t Eh

Fig. 2. Experimental set-up schematic.

ð9Þ

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Fig. 3. Experimental set-up picture.

Fig. 4. Hammer force and piezoelectric sensor 2 signal for all valid tests performed with the aluminum hammer tip.

where cB is the antisymmetric wave speed (cB ¼ d=t), E is the plate stiffness in the direction of the wave to the sensor, m is the Poisson ratio, q is the material density, h is the plate thickness, d is the distance from the impact point to the sensor, t is the time and f is the wave frequency. The RID applied to an example signal obtained from one of the hammer impact tests (5a) is shown in Fig. 5b. As shown in [30] the initial part of the RID corresponds to the dispersion curve of the material for the anti-symmetric wave, and a comparison with the theoretical curve is shown in Fig. 5b. After a short amount of time the frequency content of the signal stays at a constant value, rather than becoming zero, this is due to the reflections of the wave returning to the sensor and the response of the structure approaching a normal mode, and its this average frequency the one that dominates the damping of the one degree of freedom system described in Section 1.1. After a time of 0.003 s the theoretical dispersion curves predict an arrival frequency of less than 0.1 kHz and its energy or intensity in the RID becomes negligible compared with the energy of other frequency components present in the signal. Therefore the average frequency is calculated from this point; this effect is shown in Fig. 5b. The signal measured with the piezoelectric sensors under an impact with the rubber hammer tip (Fig. 6a), and its distribution, shown in Fig. 6b, presents a similar behavior as the impacts done with the aluminum tip, although the measured response frequency is lower.

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Fig. 5. Example of piezoelectric sensor 5 signal results with aluminum impact hammer tip.

As shown in Eq. (2) the peaks of the response of a one degree of freedom system follow an exponential curve. Because it is assumed that the piezoelectric sensor behaves in its linear range during the whole time frame, the following curve is fitted to the area measured in the tests: 0

a ¼ A0 exD f t þ C

ð10Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi where A0 is defined as a function of the average absolute voltage, V 0 , as A0 ¼ V 0 =ðxD f0 Þ; f0 ¼ f= 1 þ f2 and C is a constant. The parameters A0 ; f0 and C are adjusted by means of a numerical fitting to match the experimental results. An example of this fitting is shown in Fig. 7a. The time for the fitting is limited to 0.3s sufficient to develop the majority of the impact energy in every case. Taking its derivative the curve can be plotted alongside the time history of the impact resulting in: v ¼ V 0 exp ðxD f0 tÞ, The Rayleigh model, described in Eq. (4), is adjusted assuming the frequency x obtained with the RID as the average oscillation frequency for each piezoelectric sensor signal and the critical damping factor f obtained with the previous method for each sensor and impact performed in the tests. The measured frequency of the impacts can provide a satisfactory estimation of the Rayleigh a parameter, however, in order to obtain a good accuracy for the b parameter additional points at high frequency are taken from [33], taking f ¼ tan ðdÞ=2as proposed in [34]. The fitting of the Rayleigh model is shown in Fig. 8. The resulting parameters for the Rayleigh damping model are a ¼ 11:62 Hz and b ¼ 2:21109 s.

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Fig. 6. Example of piezoelectric sensor 5 signal results with rubber impact hammer tip.

4.2. Numerical simulations The finite element analysis is done with Abaqus/Explicit version 2017. The composite solid panel is represented with continuum shell elements (SC8R) and conventional 2D shell elements (S4R) are used for the piezoelectric sensors/actuators. The simulation must be able to represent the elastic wave behavior as it progresses through the panel therefore, the maximum element length is limited to 1mm; this ensures that an antisymmetric wave up to a frequency of 200kHz can be captured by the model using at least 10 elements per wavelength [35,36]. The size limitation to model the elastic wave results in a very large mesh, of around 600 000 elements and 1 500 000 nodes. The mesh has been generated approximately uniform for most of the panel, however, because of the rounded shape of the piezoelectric sensor, it is distorted at those points as shown in Fig. 9a. The impact hammer tip is modeled as a semi-sphere meshed with solid reduced integration hexahedral elements C3D8R as shown in Fig. 9b. A frictionless contact is defined between the impact hammer tip and the panel. An initial velocity, estimated from the tests, of 0.5 m/s is applied to all the nodes of the impact hammer tip. To ensure a smooth contact force distribution the element size of the impact hammer tip is kept consistent with the panel, with a side length of an average of 1.0 mm on the surface.

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Fig. 7. Exponential fitting adjustment of the piezoelectric sensor signals.

In order to obtain the signal in the piezoelectric sensors, the stress measured in the finite element model elements must be converted to voltage. The following relationship can be applied to obtain the voltage contribution that could be generated by each element i, assuming that the sensor is working on its linear range [37,38]:

Vi ¼

d31 ðT 1 þ T 2 Þ

0


h

ð11Þ

were T 1 and T 2 are the element stresses in directions 1 and 2, h is the piezoelectric sensor thickness, 0 is the electric constant

that has a value of 8:854  1012 F/m and d31 is the piezoelectric constant with a value of -274 C/N ([39]). It is assumed that the voltage contribution of each element, V i , representing the piezoelectric sensor is equal per unit of P area, therefore the final voltage level, V, is obtained as V ¼ ðAPZT Þ1 V i
Ai , where Ai is the area of element i and APZT is the piezoelectric sensor area. The finite element model results are compared with the test in Fig. 10.

F. Sánchez Iglesias, A. Fernández López / Mechanical Systems and Signal Processing 135 (2020) 106391

Fig. 8. Rayleigh damping model fitting.

Fig. 9. FEM model mesh detail.

Fig. 10. Results from FEM model and Tests at piezoelectric sensor 2.

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5. Conclusion Time-Frequency distributions have recently become a key tool to understand temporal signals as they can provide a vast amount of information that may originally be hidden in the signal, making it visible graphically at a first glance. Although the methodology presented may be very costly in terms of computational time and storage, the damping parameters obtained, when applied in the simulations present an excellent agreement with the test results and therefore a smaller amount of tests may be needed in future applications greatly reducing time or cost to adjust this model. Additionally, with this methodology, the dispersion curves of the anti-symmetric mode appear very clearly and they can be used as an alternative method to estimate or validate the material dispersion curves. 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