Using and Comparing Two Vibration Methods in ...

Using and Comparing Two Vibration Methods in Determining the Acoustical Properties of White Mulberry Wood (Morus alba. L) Aida Se Golpayegania*, Kambiz Pourtahmasib a

PhD holder, Department of Wood and Paper Science and Technology, Faculty of Natural Resources, University of Tehran, Karaj, Iran. P.O.Box 31585-1356

b

Associate Professor, Department of Wood and Paper Science and Technology, Faculty of Natural Resources, University of Tehran, Karaj, Iran. P.O.Box 31585-1356

* Corresponding author e-mail: [email protected]

Abstract Different methods of non destructive testing had been used to determine the acoustical properties of musically important woods or tone woods. From those, vibration tests are found to be accurate and relatively fast in configuring damping and specific modulus. Wood used in Middle Eastern inments are rarely been investigated for their acoustical properties. In this study, white mulberry wood (Morus alba L.), a leading material used in fabrication Iranian long necked lutes (Tar, Setar and mancheh) was tested using forced and free vibration methods installed in two international laboraries (Mechanic and genie civil laboratory of university of Montpellier II, and Biomass, Wood, gy, Bioproducts laboratory of CIRAD, Montpellier, France). The two methods found to be related in calculating E'/ρ (specific modulus) and tanδ. However the homogeneity of samples, the origin of calculated damping and the source of vibration seemed to affect the credibility and comparability of two methods. White mulberry wood was characterized with low specific modulus and even lower than expected tanδ. This finding put the species closer to maple – used in back and ribs of the violin– than to spruce which is actually used as a sound board in western musical instruments. It was concluded that a different criteria, one that take cultural and regional traditions and preferences into account, should be used in classifying the Middle Eastern tone wood in comparison to western ones.

Keywords: vibration method; damping; white mulberry wood; musical instrument.

1.

Introduction

Wood as a biological, hygroscopic, quasi-orthotropic and viscoelastic material has been used for centuries for structural and aesthetical purposes by human. From its first applications as a sound maker for communication between tribes, the acoustical purposes of this material were disguished. Its usage as a strong insulating material in recent centuries goes hand in hand with its more ancient application as the main vibratory agent in most musical instruments. 1

4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 The determination of mechanical properties of wood using non destructive methods has been around for some time [1], [2], [3], [4], [5], [6], [7], [8]. It first started as a method for grading timbers in site and progressed to be widely used for observing the material condition in use in structures and establishments [1]. Furthermore the NDT methods have been also in application for determining the fungi and termites decays hidden inside the timbers and trees. The characteristics of musically important woods or “tone woods” are mostly described by ing two factors: specific modulus or E'/ρ, that is the elasticity modulus divided by specific gravity and tanδ, that is damping or loss tangent. These two factors are found to be related between musically important soft and hardwoods [9], [10]. Amongst different methods used for determining wood elastic properties, the vibration methods are found effective and desirable [8]. They are specific, easy and are not destructing the sample specimen. This method has already been used in determining the biological corrosion in wood [11], the mechanical properties of fiber and particle board [8], and that of raw wood plate [12]. However in recent years vibration methods has been widely used to characterize, classify and distinguish the different tone woods [13], [14], [15], [16], [17], [18], [19], [20]. The physico-mechanical properties of wood used in Western musical instruments have already been the subject of some scientific reports. The Middle Eastern ones however are rarely studied. White mulberry wood (Morus alba L.), the leading material in fabricating Iranian lutes (i.e. Tar, Setar and Kamancheh), is one of those. Thus in this study vibration methods are used to measure acoustical properties of this species. Furthermore, while measuring the acoustical characteristics of any particular tone wood the accuracy and similarity of different methods play a crucial role in comparing and classifying the results. Because of that two vibration methods, forced vibration of LMGC (Mechanic and genie civil laboratory of university of Montpellier II, France) and free vibration or Bing of CIRAD (UR BIOWooEB, Biomass, Wood, Energy, Bioproducts, Montpellier, France), were applied simultaneously to measure the target characteristics of white mulberry. The resulting data were then compared to decide on the relativity of two methods.

2.

Material and Methods

2.1 Material White mulberry wood was selected by a professional instrument maker, making sure his ria for a final excellent instrument were met in the wood. Several rods (Longitudinal (L) =50×Radial(R) =5×Tangential (T) =5cm³) were prepared from the heartwood exclusively. The rods were then kiln dried for two weeks before moving to a climatic chamber (20°C±2°C, 65%±5%RH, 3 weeks) to reach EMC (Equilibrium moisture content). Vibration tests were planned and performed along the grain, thus the specimens were prepared accordingly. 180 specimens of following dimensions were cut: L=150± 0.04 ×R=15± 0.03 ×T=2 ±0.01mm3 (Fig.1)

Figure 1. Detailed cutting plan for vibration specimens

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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 2.2 Method The following two models establish a relationship between vibration frequencies and mechanical properties. Both models are based on the assumption of a homogeneous material. They differ by the accounting or not for shear in the cross-section. 2.2.1 Bernoulli equation Bernoulli method (first developed in 1748) is based on the hypothesis that if the ratio of length to width of a beam is very high, the beam can be considered as a thin line and the shear (as well as transverse deformations) can be ignored. The effect of elastic support is also ignored [21]. Based on Bernoulli, Ex (Elastic modulus in longitudinal direction) can be calculated as follows: ρ AL4 fn 2 Ex = 4π 2 (1) I GZ Pn where; Ex= Elastic modulus fn= resonance frequency for the nth mode of vibration π Pn = solution of Bernoulli equation: n Pn = (2n + 1) (2) 2 A= cross sectional area L= specimen length I= cross sectional inertia 2.2.2 Timoshenko equation Bernoulli's model does not take into account the shear between the cross sections; in other words the cross sections of the beam remain perpendicular to its neutral line. Timoshenko's model takes into account a shear angle that reflects the fact that cross sections do not remain perpendicular to the neutral line. Timoshenko's beam theory constitutes an improvement over Euler-Bernoulli theory, in that it incorporates shear and rotational inertia effects. The basic formula for an unsolicited homogenous orthotropic beam in bending would be [22]:  Ex  ∂ 4 v ρ I GZ ∂ 4v ∂ 4v ∂ 4v E x IGZ 4 − ρ I GZ 1 + + + S =0 ρ (3)  2  KG  ∂x 2∂t 2 KG ∂t 4 ∂x ∂ t xy xy   where: Ex = Longitudinal Elastic modulus IGZ = cross sectional inertia ρ = specific gravity K = Timoshenko shear factor dependent on the specimen geometry, equal to 5/6 for a rectangular cross section. Gxy = shear Modulus S = cross sectional area v =transversal displacement The original Timoshenko equation had ignored the effect of elastic supports. More information on the methodology and their theoretical origin can be found in [23], [24], [21] and [16]. 2.2.3 Non-contact forced bending vibrations of free–free bars (Mechanic and genie civil laboratory of Montpellier II University) In this method, the specimen is suspended on two very thin elastic supports. At one end a tiny rectangular shape steel piece is glued facing an electronic magnet (15–20 mg, a negligible addi3

4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 tional weight), which imposes the vibration. The specimen movement is measured using a contactless displacement sensor. E′/ρ is calculated by the frequency resonance according to Bernoulli method (equation 1). Damping factor (tanδ) is determined by the logarithmic decrement of the amplitude of vibration after stopping the excitement. Meanwhile, in frequency domain, quality factor (Q) is measured by bandwidth at the half power (-3dB). The two factors, tanδ calculated from logarithmic decrement and 1/Q from bandwidth, are theoretically the same, but each of them is subjected to errors of different kinds which would eventually cause them not to be exactly similar. When starting with the test, the resonance frequencies of the specimens are unknown (as they have never been tested with this method). For that reason, the first test begins with default setting of the program: 10 s frequency sweep (0.1 Hz resolution): scan between 150 and 750 Hz (which covers E′/ρ of 3 to 43 GPa) (Fig.2).

Figure 2. Schematic plan of the non contact forced vibration

From the detected resonance frequency, band width at half power (-3dB) is measured. According to Bremaud[16], the uncertainty of Q-1 depends either on bandwidth or on resonance frequency The specimen is then excited at the resonance frequency determined in the previous step. After some seconds enough for vibration to be stabilized (less than 10 seconds), it is stopped manually. A signal is produced and appears on the screen. If the signal is not completely regular (due to unevenness of the surface or other problems); one can redo the excitation and get another one. If the signal seems correct, the biggest regular part is chosen and the lateral parts – which represent the weak frequencies-, are avoided. From there an exponential curve is produced and its regularity is determined by a coefficient (R2) recorded along with tanδ. If all factors obtained seems reasonable: Q-1 and tanδ sufficiently close and high R2, the results are recorded and the program is ready for a new measurement. 2.2.4 Free vibration of floating beam (BING®) (UR BIOWooEB: Biomass, Wood, Energy, Bioproducts, CIRAD) The free vibration of floating beam (BING® method) is based on studying the induced vibration in a piece of wood. The ratio of the modulus to ρ of the material or specific modulus (E′/ρ) is equal to the square of the propagation speed of a signal in the material [25]. This relationship indicates the existing connection between mechanical properties and vibratory behavior. By analyzing 4

4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 the spectral vibration in different directions, the natural frequencies of beam from its responses to the excitation pulse can be identified. This method was developed by Bordonné [23] and has been used in CIRAD to expand the database by experimenting E' of different woods. It calculates mechanical properties (E' and tanδ) by using two different models relating vibrational and mechanical properties. By recording the audio signal and the obtained frequencies and then analyzing them in the software written in Matlab® [24] eventually the mechanical factors would be achieved. Generally the first signal mode of frequency (which is the highest one and is called the fundamental mode/frequency) is used for the calculations. The dimensions of the specimens for this test can vary, but the length to height and width ratio should be enough to bear the impact causing the vibration. Specimens of (L=360× R=20× T=20 mm³) have already been successfully tested for their E′/ρ with same method but bigger dimensions are also being used in CIRAD. However to be able to compare both forced and free vibration methods, the same (L=150× R=15× T=2mm³) specimen dimensions were used for BING test. The specimen is put on two elastic supports to allow the free propagation of the vibration (Fig.3). A simple percussion at one end causes a vibration recorded by a microphone placed at the other end. In this case the first mode of frequency is manually chosen. These recorded vibrations are transmitted via an anti-aliasing filter to an acquisition card which in turn converts the analog signal and provides the computer with digitized signal. Fast Fourier Transform (FFT) is used for processing and interpreting the information in frequency domain. tanδ is determined according to first mode of frequency by a software written in Matlab® [24], while E′ is also calculated using each specimen geometrical dimensions as well as its weight.

Figure 3. Schematic description of BING® method

3.

Results and discussion

3.1 Comparison of two vibrational methods: forced and free vibration. The results of these methods were going to be used alternatively to understand the effect of different treatments/extractions on the white mulberry specimens [26], [27]. Because of that it was important to know whether (and how) the calculated damping factors of these methods relate together. In the case of tanδ, it seems reasonable to relate the damping of same origins. This means relating tanδ by bandwidth from forced vibration to the tanδ of free vibration as they were both based on frequency domains. The vibration tests were carried out in the course of two years during which time the instrument for testing vibration of the floating beams (BING) was modified. These modifications (calculating additional results, replacing microphone with a laser beam…) had most definitely played a role in relating the two methods. Furthermore, getting more experienced on how to test with fewer errors resulted in more accurate data in either method. 5

4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 It should be also noted that the specimens were cut and prepared based on the requirements of forced vibration method (LMGC), while the free vibration method (BING) was demanding bigger specimens to deliver a trustworthy result. Having said that, the relationship between tanδ of two methods seemed reliable when comparing the results obtained for bigger samples [16], [28]. This study was the first attempt to try to relate the two methods for smaller samples. Fig.4. shows the relationship between two specific moduli as well as tanδ obtained by free and forced vibration. Measurements were done on 80 specimens in the first year of the study. As it can be seen in Fig.4.b, tanδ shows large dispersion especially in the mid range. Moreover, the error bars show large variations between repetitions in both vibration measurements, which could be caused by different reasons (lack of stability of specimen on the elastic holders, disoriented specimen, inexperienced technician…). Specific modulus, on the other hand (Fig.4.a) exhibits a better relationship between two methods (R2= 0.72).

Figure 4. Relationship between specific modulus (E'/ρ) (a) and tanδ (b) obtained by forced and free vibration. The error bars in 1b are calculated based on three repetitions of measurements done on the same specimen. Nb: 80.

Fig.5 shows the same factors for 62 measurements done on the same vibration test machines in the second year of the study. A stronger relation between the two methods is clearly shown. E′/ρ values are related significantly together (R2=0.99) and tanδ values even though still dispersed in low and middle range, are better related to one another (R2=0.68).

Figure 5. Relationship between specific modulus (E'/ρ) (a) and tanδ (b) obtained by forced and free vibration. Nb: 62. (Both values are collected from running the tests in same range of frequency-i.e. 360 Hz in average).

The improved relation found between the values obtained from two vibration methods can be explained by several factors: - The technician got more experienced (more aware of error origins on a particular instrument).

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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 - The microphone was replaced with a laser beam in free vibration (which caused a more precise registration of the vibration). - Changes in the specimens themselves: coming from another tree, not only they were slightly different in properties (higher specific modulus and lower tanδ) but also easier to measure (less repetitions before acquiring a correct result) which might have reduced the errors. Smaller error bars of the forced vibration in Fig .4.b (with the exception of two samples) confirm the above statement: tests of second year were more repeatable. In the end, considering the results above, and still noting that the specimens size were not “suitable” for BING method, the correlation between the two methods seems correct. 3.2 Brief on the acoustical results on white mulberry The full results and discussion on acoustical properties of white mulberry and the effect of treatments could be found at [29], [26], [27] and [30]. In an attempt to compare the white mulberry – as a leading Middle Eastern tone wood- to European ones, Fig.6 shows the relationship between tanδ and E'/ρ for all the samples used in the entire study (Nb.207). According to Ono and Norimoto [9], [10], there is always a “standard trend” between specific modulus and damping. For white mulberry the same trend was observed, although the E'/ρ is quite low compared to tone wood suitable for violin [31]. Also the tanδ is lower than expected for such specific modulus (6%-11% lower than trend). In the same manner, the specific gravity of white mulberry (ρ) is ≈ 0.55, a moderate ρ, and it is also lower than those of the species known to be used in fabrication of European soundboards [16].

Figure 6. Relationship between E'/ρ and tanδ for all the samples used in the study (Nb. 207). The standard trend was added from Ono and Norimoto (1983) [9].

These remarks are putting white mulberry closer to maple (used for back or sides of western musical instruments) than the resonance spruce (used for top plates) [32]. However, it should be noted that criteria for classifying a “good” wood for making instrument is highly different in various regions and is strongly affected by cultural and historical backgrounds. In other words a desirable sound/ wood quality in one place may not be so convenient in other regions. Thus strong care should be taken in classification and comparison of Iranian tone woods with western ones.

4.

Acknowledgements

Authors are grateful to SCAC (Service de Coopération et d’Action Culturelle) of the French embassy in Tehran and CISSC (Iranian Center for International Scientific Studies & Collaboration) for supporting this project (Gundishapur project, Egide nb 20714UJ). Authors also thank Samad Zareh, the Iranian Tar maker and Luic Brancheriau, at CIRAD for their valuable help through this study.

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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014

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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 24. B. David, PhD thesis, ''Caractérisations acoustiques de structures vibrantes par mise en atmosphère raréfiée. Méthodes d'estimation relatives aux fréquences et amortissements des modes propres : Applications en acoustique musicale''. Université Paris 6. Paris, France, 233 (1999). 25. V. Bucur, Acoustics of wood, Springer, 2006. 26. A. Se Golpayegani, PhD Thesis, Caractérisation du bois du Mûrier blanc (Morus alba L.) en référence à son utilisation dans les luths Iraniens, University of Montpellier II. Montpellier, France, 288 (2011). 27. Se Golpayegani, I. Brémaud, J. Gril, MF. Thévenon, O. Arnould, K. Pourtahmasi, "Effect of extractions on dynamic mechanical properties of white mulberry (Morus alba L.) ", Journal of wood Science, 58 (2), 9 (2012). 28. M. E. El Mouridi, Master thesis, "Caractérisation mécanique (module d'élasticité - coefficient d'amortissement) du matériau bois, Cas du pin d'Alep et de chêne vert", Université Mohammed V. Maroc, 44 (2007). 29. K. Pourtahmasi, A. Se Golpayegani, "Introducing Mulberry’s wood (Morus alba L.) used in bowl shaped musical instruments of Iran". Le bois : instrument du patrimoine musical, Eds. Leconte S, Vaiedelich S. 29 May 2009 Cité de la Musique, Paris (2008). 30. A. Se Golpayegani, K. Pourtahmasi, S. Zareh, I. Bremaud, J. Gril, M-F. Thevenon, "White mulberry for Tar: Basic acoustical properties and the effect of traditional treatments", (In Farsi), Art Research Journal, 3 (2013). 31. U. G. K. Wegst, "Wood for sound", American Journal of Botany, vol. 93(10), 9 (2006). 32. E.V. Jansson, Acoustics for Violin and Guitar Makers, 4th ed. Royal Institute of Technology, Stockholm, 2002.

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