Estimation of elastic and modal parameters in composites using vibration analysis

Abstract

This study presents the application of vibration analysis in the determination of elastic constants (i.e. Young’s modulus in fiber direction, transverse Young’s modulus, and shear modulus) and modal damping ratios of a unidirectional composite beam. The frequency domain approach is used for the estimation of material constants, whereas modal damping ratios are predicted by the use of short-time Fourier transform (STFT). An analytical expression of the STFT for the free vibration response of a viscously damped system has been derived using the Hanning window. Analysis of a simulated vibration for a three-degree-of-freedom system has revealed that the STFT is capable of predicting the modal damping ratios accurately even when they are considerably large. For the experimental assessment of modal damping values of a composite beam, longitudinal, flexural, and torsional vibration responses are analyzed by both the STFT and well-known Q-factor approximation, and the obtained results show very good agreement.

Keywords

Modal damping ratio modulus of elasticity shear modulus short-time Fourier transform Q-factor unidirectional composite

1. Introduction

The use of composite materials has gained great popularity, particularly in the aerospace, automotive, and marine industries as they often exhibit a higher strength with low density compared to conventional materials, they have good wear and fatigue properties, and they are able to provide good sound and heat isolations. In general, composites can be studied from two points of view: micromechanics and macromechanics. Micromechanical analyses aim to provide an understanding of the behavior of composites in terms of the properties of fibers such as strength and stiffness. Macromechanics is the approach used to predict the strength and stiffness of composite structures such as the modulus of elasticity in tension (or compression) and bending, in-plane share modulus, Poisson’s ratio, thermal expansion coefficients, and the appropriate strength values (Baker et al., 2004). The use of correct material properties during design and manufacturing improves product quality and reliability, and their incorrect determination may lead to misdiagnosis of structures that could cause catastrophic failures (Gibson, 2000).

The dynamic response of a system is mainly characterized by its mass, stiffness, and damping parameters. Consideration of damping is crucial for accurate prediction of the vibration response and extremely important in analysing vibratory system near resonance. In order to avoid excessive vibration, either excitation (or natural) frequency or the amount of damping of the system should be changed. The presence of persisting excess vibration either causes noise and slight wear on working parts or makes a machine break down much more quickly within a short period of time.

A large number of research studies on the estimation of properties of composite materials have been reported in the literature using mechanical testing (Hazizan and Cantwell, 2002; Quispitupa et al., 2004; Chen and Kam, 2007; Kam et al., 2009; Raghavan et al., 2010), ultrasound or acoustic measurement (Matter et al., 2009; Mylavarapu and Woldesenbet, 2010), and theoretical, numerical and/or experimental vibration analysis (Gibson, 2000; Kostopoulos and Korontzis, 2003; Chandra et al., 2003; Wang and Wang, 2004; Yoon et al., 2004; Choi et al., 2007; Cortes and Elejabarrieta, 2007; Guz and Rushchitsky, 2007; Kyriazoglou and Guild, 2007; Qiao and Yang, 2007; Wang and Yuan, 2007; Mahi et al., 2008; Xiang and Yang, 2008; Matter et al., 2009; He, 2011; Mehrez et al., 2012). Of these methods, vibration analysis has been widely used to estimate the properties of composites as vibration analysis provides the basis for rapid and inexpensive estimation of material and modal properties. Besides, composite fabrication often leads to non-uniform property distributions in the finished parts. Current test procedures based on tests of small specimens typically yield only local properties, but the response of the structure to imposed loads is governed by the globally averaged properties which can be determined by modal testing (Gibson, 2000).

Vibration analysis can be performed in the time, frequency, and combined time-frequency domains. In time domain analysis, the time history of the signal itself can be used to determine the parameters of a vibration including signal energy (Quispitupa et al., 2004; Kyriazoglou and Guild, 2007), displacement or peak value (Botelho et al., 2006; Choi et al., 2007; Kyriazoglou and Guild, 2007; Qiao and Yang, 2007; Xiang and Yang, 2008; Mylavarapu and Woldesenbet, 2010), mode shapes (Xiang and Yang, 2008; Matter et al., 2009), attenuation coefficient (Mylavarapu and Woldesenbet, 2010), strain (Chandra et al., 2003), and damping or loss factor (Chandra et al., 2003; Wang and Wang, 2004; Botelho et al., 2006; Choi et al., 2007; Kyriazoglou and Guild, 2007; Matter et al., 2009). In frequency domain analysis, the amplitude of vibration response is represented against frequency, which provides a better understanding of harmonic content of a signal and which can be used to determine the system parameters including natural frequencies (Gibson, 2000; Kostopoulos and Korontzis, 2003; Kyriazoglou and Guild, 2007; Xiang and Yang, 2008; Matter et al., 2009), damping or loss factor (Gibson, 2000; Kostopoulos and Korontzis, 2003; Cortes and Elejabarrieta, 2007; Kyriazoglou and Guild, 2007; Mahi et al., 2008; Matter et al., 2009; Mehrez et al., 2012), material constants (Gibson, 2000; Kostopoulos and Korontzis, 2003; Wang and Wang, 2004; Botelho et al., 2006; Cortes and Elejabarrieta, 2007; Matter et al., 2009; Mehrez et al., 2012), and wave speed (Yoon et al., 2004).

Conventional time and frequency domain techniques can be applied to stationary signals. However, in many cases spectral composition of a system changes with time, and methods based upon Fourier analysis cannot truly reflect such vibration characteristics. One approach to non-stationary signal analysis is combined time-frequency representation which displays signal energy as a function of both time and frequency simultaneously. A variety of linear and bilinear time-frequency analysis methods such as the STFT (Hlawatsch and Boudreaux-Bartels, 1992; Yesilyurt, 2004), the Wigner-Ville distribution (WV) (Cohen, 1989; Hlawatsch and Boudreaux-Bartels, 1992; Roshan-Ghias et al. 2007), instantaneous power spectrum distribution (IPS) (Hippenstiel and De Oliveira, 1990; Yesilyurt, 1997), and the continuous wavelet transform (CWT) (Yesilyurt, 1997; Lardies and Gouttebroze, 2002; Le and Argoul, 2004; Ozturk et al., 2008) have been introduced for the analysis of vibration signals to extract useful diagnostic information (Yesilyurt, 1997). These methods can exhibit local features of signals and give an account of how energy distribution over frequencies changes from one instant to the next. Of these methods, the WV and IPS transforms are bilinear distributions, which result in interference terms when a multi-component signal is analyzed, and this might make the interpretation of distribution difficult. In contrast, the STFT and CWT perform linear decompositions of the analyzed signal, and therefore do not cause any interference which might detract from the interpretation of the targeted signal.

This study presents the application of vibration analysis in the determination of material properties (i.e. the dynamic longitudinal and flexural Young’s moduli and shear modulus) and modal damping ratios of a unidirectional composite beam. The paper commences with a brief theoretical background to the vibration analysis of beams. The frequency domain approach is used for the determination of elastic constants, whereas modal damping ratios are predicted by the STFT. An analytical expression of the STFT for the free vibration response of a viscously damped multi-degree-of-freedom (m.d.f) system has been derived using the Hanning window. Analysis of a simulated vibration for a three-d.f. system has revealed that the STFT is capable of not only decoupling vibration into its natural modes, but predicting the modal damping ratios accurately even if they are very large. Moreover, the longitudinal, flexural, and torsional vibration responses of unidirectional composite beams are obtained. It has been found that the result for the dynamic modulus of elasticity obtained by the longitudinal vibration analysis shows very good agreement with that found by the mechanical testing. Furthermore, application of the STFT to the resulting vibrations has revealed that the modal damping ratios are close to each other in longitudinal vibration, whereas they exhibit alterations in the flexural vibration such that the higher the modal frequency, the larger the value of damping. In torsional vibration, however, the first mode exhibits a notably large amount of damping compared to those in longitudinal and flexural vibrations, while the other modes have relatively small damping values.

2. Vibration theory of beams

This section provides information about the vibration theory of beams by which material properties are estimated from the measured vibration response. The targeted material properties to be estimated are Young’s modulus in fiber direction, transverse Young’s modulus, and shear modulus of a unidirectional composite beam. Since the mentioned material properties are all frequency dependent, the frequency equations are given as the basis for the determination of elastic constants.

2.1. Longitudinal vibration with free-free end conditions

The equation of free longitudinal vibration of a homogeneous beam is governed by the following equation (Rao, 1995):

c_{l}^{2} \frac{\partial 2 u}{\partial x 2} = \frac{\partial 2 u}{\partial t 2}

(1)

where

u = u (x, t)

is the longitudinal displacement at a distance x, t is time, and c_l denotes the speed of longitudinal wave which is expressed as follows:

c_{l} = \sqrt{\frac{E}{ρ}}

(2)

where E and ρ denote Young’s modulus of elasticity and density of the material respectively. If the wave equation given in equation (1) is solved with free-free end conditions, the frequency equation can be obtained as follows:

f_{n}^{l} = \frac{{nc}_{l}}{2 L}

(3)

where

n (n = 0, 1, 2, 3, \dots,)

denotes the wave number,

f_{n}^{l}

is the frequency of the n^th mode of longitudinal vibration in Hz, and L represents the length of beam. Equation (3) signifies that the frequency spectrum of the longitudinal vibration consists of well-separated frequency components meaning that the higher natural frequencies should be at multiple integers of the first natural (or fundamental) frequency.

2.2. Transverse vibration with free-free end conditions

From the elementary theory of bending of beams, the equation of motion for the free transverse vibration can be expressed as follows (Rao, 1995):

c_{b}^{2} \frac{\partial 4 v}{\partial x 4} + \frac{\partial 2 v}{\partial t 2} = 0

(4)

where

v = v (x, t)

is the transverse displacement at a distance x and the symbol c_b denotes the speed of the transverse wave, which is expressed as follows:

c_{b} = \sqrt{\frac{EI}{ρ A}}

(5)

where

EI

denotes the flexural rigidity and A is the cross-sectional area. The solution of equation (4) with free-free end conditions yields the following frequency equation.

f_{n}^{b} = \frac{1}{2 π} (β_{n} L) 2 \sqrt{\frac{EI}{ρ AL 4}}

(6)

where

n (n = 1, 2, 3, \dots,)

denotes the wave number,

f_{n}^{b}

and

β_{n}

represent the frequency in Hz and eigenvalue of the n^th mode of transverse vibration.

2.3. Torsional vibration of a cantilever beam with a torsional inertia at the free end

The governing equation of free torsional vibration of a beam can be expressed as follows (Rao, 1995):

c_{t}^{2} \frac{\partial 2 θ}{\partial x 2} = \frac{\partial 2 θ}{\partial t 2}

(7)

where

θ = θ (x, t)

denotes the torsional displacement at a distance x and c_t is the torsional wave speed, which can be expressed as follows:

c_{t} = \sqrt{\frac{G}{ρ}}

(8)

where G denotes the shear modulus. During the experimental phase of this work, the free torsional vibration response of the beam was detected using fixed-free end conditions. In addition, a small plate with a mass moment of inertia, I₀, was fixed to the free end of the beam not only to apply torsional impulse excitation, but to attach accelerometers by which the torsional vibration response is obtained. When the effect of I₀ is taken into account, the frequency equation of free torsional vibration can be derived as follows:

\frac{ω L}{c_{t}} \tan \frac{ω L}{c_{t}} = \frac{ρ I_{p} L}{I_{0}}

(9)

where ω denotes circular frequency

(rad / \sec)

, L is the length of the beam and I_p represents the polar moment of inertia.

3. Short time fourier transform (STFT) analysis

The Fourier transform of a signal $x (t)$ is expressed as follows:

X (f) = \int_{- \infty}^{\infty} x (t) e - j 2 π ft d t

(10)

where f is the frequency variable. It is known that spectral representation is a stationary measure, which indicates the frequency content of the signal, but it does not reveal any time information about these frequency components. When the amplitude and frequency content of the signal change in time, combined time-frequency methods can be used effectively. The STFT of a time signal

x (t)

can be expressed as follows (Hlawatsch and Boudreaux-Bartels, 1992):

STFT (t, f) = \frac{1}{T} \int_{t - T / 2}^{t + T / 2} x (τ) w (τ - t) e - j 2 π f τ d τ

(11)

where t indicates the position of the window, τ is time variable, T is window length, and

w (τ)

is the windowing function which satisfies

w (τ) = 0 for | τ | > T / 2

The STFT has a simple interpretation and, by choosing appropriate windows, the physical parameters of a signal can be measured or estimated. The use of the window function not only permits local observation of the signal, but minimises the effects of spectral leakage and smearing. It should be kept in mind that the time and frequency resolutions of the STFT depend upon the type and length of the time window being used. High resolution in time requires the use of a short time window; however, high time resolution is achieved at the expense of high frequency resolution, and vice versa. As a consequence, there exists a trade-off between the time and frequency resolutions. Once fixed, however, the length of the time window remains constant during the STFT analysis which results in a constant time-frequency resolution.

4. Determination of modal damping ratios using STFT

The free vibration response of a viscously damped n d.f. system can be expressed as follows (Lardies and Gouttebroze, 2002):

x (t) = \sum_{i = 1}^{n} X_{i} e - ξ_{i} w_{ni} t \sin (ω_{di} t)

(12)

where X_i denotes the magnitude,

w_{ni}

and

w_{di} (w_{di} = w_{ni} \sqrt{1 - ξ_{i}^{2}})

represent undamped and damped natural frequencies associated with the i^th mode of vibration. For the STFT analysis of

x (t)

, a Hanning window, which is an excellent general purpose window, is used to weight the signal in the time domain and can be expressed as follows:

w (t) = 0.5 (1 + \cos \frac{2 π t}{T})

(13)

Substitution of equations (12) and (13) into equation (11) yields the single-sided STFT representation of the free vibration of a viscously damped multi-d.f. system, which can be expressed as follows:

STFT (t, ω) = \frac{1}{T} \sum_{i = 1}^{n} X_{i} e - R t (\frac{1}{R} - \frac{R}{R 2 + (2 π / T) 2}) \sinh (ξ_{i} ω_{ni} \frac{T}{2})

(14)

where

R = ξ_{i} ω_{ni} + (ω - ω_{di}) j

. It can be deduced from equation (14) that the STFT analysis exhibits maxima when

ω = ω_{di}

. In another word, the frequencies which maximise the magnitude of the STFT correspond and are related to the analyzed modal frequencies of the system. Similarly, the magnitude of the STFT at

ω = ω_{di}

gives the free decay curve (or ridge plot) corresponding to the modal frequency, which can be expressed as follows:

| STFT (t, ω_{di}) | = \frac{1}{T} \sum_{i = 1}^{n} X_{i} e - ξ_{i} ω_{ni} t [\frac{1}{ξ_{i} ω_{ni}} - \frac{ξ_{i} ω_{ni}}{(ξ_{i} ω_{ni}) 2 + (2 π / T) 2}] \sinh (ξ_{i} ω_{ni} \frac{T}{2})

(15)

where

| . |

denotes magnitude. In order to estimate the modal damping ratio from the STFT, it is sufficient to consider the ratio of the values of the

| STFT (t, ω_{di}) |

for two arbitrary time occasions. Let t₁ and t₂

(t_{1} < t_{2})

denote the considered time instances, the modal damping ratio can be expressed as follows:

ξ_{i} = \frac{1}{ω_{ni} (t_{2} - t_{1})} \ln \frac{| STFT (t_{1}, ω_{di}) |}{| STFT (t_{2}, ω_{di}) |}

(16)

It is known that the frequencies corresponding to the peak amplitudes can be identified as the natural frequencies of the system. Estimates of modal damping ratios can also be computed from the location and half-power bandwidth (i.e. Q factor) of resonant peaks. For small values of damping $(ξ < 0.05)$ , the damping ratio is as follows:

ξ = \frac{Δ f}{2 f_{n}}

(17)

where Δf denotes half power bandwidth corresponding to modal frequency of f_n.

5. Numerical example

In an attempt to identify the feature extraction capabilities of the STFT in damping estimation, a simulated vibration signal for a linear system with three-d.f. (whose modal parameters are summarized in Table 1) was considered. The generated vibration signal was sampled at 1024 Hz and had a length of 512 samples. A Hanning window of length 256 samples was used to weight the signal in the time domain and the center locations of successive windows were chosen with a separation of one sample.

Table 1.

Modal parameters and estimated damping values of the simulated linear system.

Mode	Amplitude	Natural frequency (Hz)	Damping ratio ξ (%)
Exact	By the STFT	By the Q factor
1	1	50	15.00	15.04	14.87
2	1	100	5.00	5.01	4.91
3	1	150	2.00	2.00	2.03

STFT: short-time Fourier transform.

Figure 1 shows both the time and frequency domain representations of the simulated vibration signal. It can be seen that spectral representation reveals the frequency content of the signal (i.e. mode frequencies), but it does not provide any time information about these frequency components. Figure 2 illustrates the mesh and contour plots of the $| STFT |$ together with the free decay curves generated at mode frequencies. It can be seen from both the mesh and contour plots that the STFT decomposes vibration into natural components according to its natural frequencies. Modal damping ratios are calculated using equation (16) by considering the 1^st and 100^th samples of the ridge plots and presented in Table 1 together with the Q-factor results. It can be seen that the STFT analysis is quite successful in accurately estimating damping ratio compared to the Q-factor approximation even when the value of the damping ratio is quite large.

6. Experimental results

6.1. Test specimen and experimental setup for vibration analysis

A 4800 tex glass-fiber reinforced unidirectional composite beam (whose behavior is expected to be isotropic in a cross-section perpendicular to the fibers) was used for the experiments. It consisted of 62.4% of glass-fiber, 37.6% of resin, and had a density of 2048 $kg / m 3$ . In addition, some additives and materials such as polyester, filling materials, accelerators, hardeners, and pigments were also used during its production.

A typical vibration response measurement system whose schematic representation is given in Figure 3 mainly requires an excitation mechanism, a transducer to measure the response, a signal conditioner to strengthen the weak signal, and a dynamic signal analyzer to extract and store desired information. A steel-tipped PCB impact hammer of mass 0.17 kg was used as an excitation mechanism whose mass and impacting tip are mainly influential in determination of energy transmitted to the structure and frequency range of analysis. Attached to the impact hammer was a PCB 21775 force transducer and the vibration response of the system was detected by a Dytran 3200B6 accelerometer which has a mass of 6 gr, an axial sensitivity of 2 mV/g, and a natural frequency of 100 kHz.

6.2. Longitudinal vibration analysis

The composite beam used for the longitudinal vibration analysis had a length of 2998 mm and a diameter of 17.65 mm. During testing, the test specimen was suspended longitudinally and freely in space by three elastic cords (whose stiffnesses are considered to be too small to account for) as shown in Figure 4. All tests were performed by exciting the beam longitudinally from one end and the resulting vibration response was detected by an accelerometer attached to the other free end of the composite bar. Both the resulting vibration response and the force signal were sampled at 40 kHz and collected for 1 second. In order to minimize random errors and to improve the accuracy of estimated parameters, an average vibration response from eight separate measurements was obtained and this was used for the time, frequency, and STFT analyses.

Figure 5 shows both the time and frequency domain representations of the averaged impact force. It can be seen that the spectrum of this excitation decreases gradually with frequency, giving a bandwidth of nearly 0–5 kHz for the analysis, and for this reason the upper frequency of the analysis was set to 4.2 kHz.

Figure 6 illustrates both the time and frequency domain representations of the averaged longitudinal vibration for the composite beam. It can be seen that the amplitude of the vibration lessens with time due to damping and nearly dies out after 0.4 seconds. From the vibration spectrum, the first (or fundamental) natural frequency of the system is seen to be 793.9 Hz and the others are presented in Table 2. The applications of equations (3) and (2) consecutively yield both the longitudinal wave speed and dynamic modulus of elasticity in fiber direction, which are $c_{l} = 4760 m / s$ and $E = 4.64 \times 1010 N / m 2$ respectively.

Table 2.

Estimated modal frequencies and damping ratios of the composite beam for the longitudinal vibration.

Mode	f_d (Hz)	Estimated damping ratios ξ (%)
Mode	f_d (Hz)	By the STFT	By the Q factor
1	793.9	0.154	0.188
2	1588.0	0.118	0.129
3	2382.8	0.087	0.092
4	3176.3	0.161	0.162
5	3969.5	0.156	0.156

STFT: short-time Fourier transform.

In order to verify the correctness of the value of the modulus of elasticity in fiber direction obtained by the vibration analysis, a test specimen produced from the same composite on which longitudinal vibration was carried out was subjected to a tensile testing as shown in Figure 7. During the tensile testing, an extensometer whose active measuring length was 48.87 mm was used to determine the resulting elongation. When the tensile load was increased gradually until the specimen was damaged, a maximum tensile stress of $354 N / mm 2$ was reached. At this stress, an elongation of $0.357 mm$ was measured, which corresponds to a tensile strain of 0.73%. By considering the basic relationship between the stress and strain, the modulus of elasticity can be calculated as $E = 4.84 \times 1010 N / m 2$ . It can be seen that the results obtained by the tensile testing and vibration analysis show a close correlation, validating the use of vibration analysis in the determination of the modulus of elasticity for composite materials.

The STFT of the averaged vibration signal for the composite bar is shown in Figure 8(a) and (b) as the mesh and contour plots respectively. For the STFT analysis of the experimental result, the first 20,000 samples (corresponding to a time interval of 0–0.5 seconds) of the averaged vibration signal were considered. The Hanning window of length 1024 was applied to the signal and the successive window locations were chosen with a separation of four samples. It can be seen from the mesh and contour plots that the vibration of the composite bar is decomposed into natural components at its mode frequencies (i.e. 793.9 Hz and integer multiples of this value). Besides, the mesh plot clearly shows the effect of damping which makes the magnitudes of the vibration components diminish gradually with time. In order to quantify the amount of equivalent viscous damping, the free decay curves at the mode frequencies are obtained and displayed in Figures 8(c) to (g). The application of equation (16) to each of these ridge plots permits the assessment of the modal damping ratios which are presented in Table 2.

6.3. Transverse vibration analysis of composite beam

During the flexural vibration testing, the same instrumentations used for the longitudinal vibration analysis were employed. The test specimen with a length of 500 mm and a diameter of 17.65 mm was suspended longitudinally and freely in space by two elastic cords. The accelerometer was attached onto the surface close to the free end in the direction perpendicular to the longitudinal axis and the hammer impact was applied to the other free end of the beam as seen in Figure 9. Both the force and vibration signals were sampled at 5 kHz and collected for one second. During testing, eight sets of measurements were obtained and their average was considered for the analyses.

Figure 10 shows both the time and frequency domain representations of the averaged flexural vibration of the unidirectional composite beam. The effect of damping can be seen clearly as a steady reduction in the amplitude of vibration with time. Three frequency peaks are observed at 279.8 Hz, 776.8 Hz, and 1479.0 Hz, and the applications of equations (6) and (5) successively yield both the dynamic modulus of elasticity and wave speed in flexural vibration, which are $E = 4.14 \times 1010 N / m 2$ and $c_{b} = 19.8 m / s$ .

Figure 11 illustrates the STFT representations of the averaged flexural vibration of the composite beam in the forms of mesh and contour plots together with the free decay curves generated at the natural frequencies. During the STFT analysis, the Hanning window of length 256 samples was used and successive center locations were chosen with a separation of two samples. The effect of damping can be clearly seen on the mesh plot representation, which makes the magnitudes of the vibration components diminish gradually with time. Moreover, the application of equation (16) to each of these free decay curves permits the assessment of the modal damping ratios which are presented in Table 3. It can be seen when the results presented in Tables 2 and 3 are compared that the tested unidirectional composite beam exhibits much higher modal damping ratios in flexural vibration than those in longitudinal vibration, confirming the results obtained by Mahi et al. (2008).

Table 3.

Estimated modal frequencies and damping ratios of the composite beam for the transverse vibration.

Mode	f_d (Hz)	Estimated damping ratios ξ (%)
Mode	f_d (Hz)	By the STFT	By the Q factor
1	279.6	0.190	0.202
2	776.6	0.297	0.301
3	1494.0	0.612	0.628

STFT: short-time Fourier transform.

6.4. Torsional vibration analysis of composite beam

During the torsional vibration testing, the composite beam with an effective length of 480 mm and a diameter of 17.65 mm was fixed as a cantilever beam as seen in Figure 12. A small plate, made of aluminium, was fixed to the free end not only to apply torsional excitation, but to attach accelerometers. In order to apply an impulsive torque to the composite beam about its longitudinal axis, an impulse excitation (delivered by the impulse hammer) was applied to the upper side of the plate close to its corner. Two accelerometers (which are located at points A and B on the end plate) were used for the direct estimation of the resulting torsional vibrations (i.e. $\overset{··}{θ} = ({\overset{··}{x}}_{B} - {\overset{··}{x}}_{A}) / a$ ). When the accelerometers were attached, the plate had a total mass moment of inertia of $I_{0} = 3.07 \times 10 - 5 kgm 2$ about the longitudinal axis of the beam. The excitation and resulting vibration signals were both sampled at 10 kHz and detected for 3 seconds. During testing, eight sets of measurements were obtained and their average was considered for the analyses.

Figure 13 illustrates both the time and frequency domain representations of the resulting averaged torsional vibration. In contrast to the longitudinal and bending vibrations, the amplitude of the torsional vibration diminishes quickly and nearly dies out after 25 msec, which is a clear indication of a large amount of damping. From the spectrum of torsional vibration, the first three natural frequencies are observed at 323 Hz, 2042 Hz, and 3996 Hz. By considering the first natural frequency, both the torsional wave speed and dynamic shear modulus can be calculated using equations (9) and (8) successively which are $c_{t} = 1876 m / s$ and $G = 7.21 \times 109 N / m 2$ respectively. By considering these values, torsional frequencies of the system are also estimated using equation (9) and presented in Table 4. It can be seen that the error between the theoretical and experimental results are quite acceptable (i.e. 1.24%, 3.96%, and 1.42% for the first, second, and third modes respectively) confirming the trueness of the experimental results.

Table 4.

Estimated modal frequencies and damping ratios of the composite beam for the torsional vibration.

Mode	Frequency (Hz)		Estimated damping ratios ξ (%)
Mode	Theoretical	Experimental	By the STFT	By the Q factor
1	326.6	323	9.510	10.241
2	1960.5	2042	0.772	0.796
3	3938.9	3996	0.777	0.797

STFT: short-time Fourier transform.

Figure 14 shows the STFT representations of the averaged torsional vibration for the composite beam in the forms of mesh and contour plots together with the ridge plots generated at the natural frequencies. In order to quantify the amount of equivalent viscous damping, equation (16) was applied to each of these ridge plots and the results are presented in Table 4. It can be seen that the modal damping ratio for the first mode exhibits a notably larger value (i.e. nearly 10%), whereas the other two modes have the same amount of damping (i.e. 0.77%).

7. Summary and conclusions

In this study, determination of elastic constants and modal damping ratios of a unidirectional composite beam have been investigated from the measured vibration responses. Estimations of elastic constants are based upon the determination of modal frequencies and the results obtained by the tensile testing and vibration analysis show a close correlation, validating the use of vibration analysis in the determination of the modulus of elasticity for composite materials.

The STFT has been employed for the estimation of modal damping values. An analytical expression of the STFT for the free vibration response of a viscously damped m.d.f system has been derived using Hanning window. Analysis of the test signal for a three-d.f. system has revealed that the STFT is more successful than Q-factor approximation in predicting the modal damping ratio even it is considerably large.

Impact responses of the composite beams in longitudinal, flexural, and torsional vibrations have been used and modal damping ratios are predicted using the STFT. It has been found that the modal damping ratios are close to each other in longitudinal vibration, whereas they differ in flexural vibration such that the higher the modal frequency, the larger the value of the modal damping ratio. In torsional vibration, however, the first mode exhibits a significantly large amount of damping while the higher modes exhibit small damping values.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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