Good Vibrations: a study of sound pressure as a function of strum force in acoustic guitars

Transcription

1 Good Vibrations: a study of sound pressure as a function of strum force in acoustic guitars Jacob Robinson Department of Mechanical Engineering (Undergraduate) Brigham Young University Provo, UT jacobrobinson4@gmail.com Acoustics research has focused on the vibrational response of stringed instruments for many centuries to better understand how to manufacture and play them. Every note a guitar plays corresponds to a specific frequency of string vibration which is transferred into the body. Many studies have been conducted in order to understand how the construction and strings of a guitar contribute to its resonant frequencies, however little work has been done to resolve how the plucking forces and resonant frequencies contribute to higher or lower sound levels. This study measured three specific variables that contribute to the sound levels produced by guitars: plucking force, decibel levels produced from a pluck, and the body resonant frequencies. In order to understand how guitar construction effects the decibels produced from plucking, three guitars with different body styles were used. Two methods were employed to measure the decibel level as a function of the plucking force; one using a pick and force gauge, the other using a finger pluck pulling the string to a specified distance. The resonant frequency of each guitar also needed to be determined in order to analyze the relationship between the sound level produced for guitars with different resonant frequencies. The results from method one for three plucks of increasing forces produced inconsistent decibel levels. The second method showed an s-shape curve of decibels as the pull back distance increased. When the resonant frequencies of each guitar were compared it was found that they all had similar values and thus the decibel levels produced were also similar. This research has shown that a guitar can produce a certain range of decibel levels depending on the force with which the string is plucked. Introduction Acoustics research has focused on the vibrational response of stringed instruments for many centuries to better understand how to manufacture and play them. However, the study of guitars is still a relatively new field with many variables yet to be investigated. It is well known that construction and body type are two of these variables that affect the sound produced by a guitar [1]. When a guitar string is plucked, vibrations propagate through the string and into the neck and the bridge. These vibrations cause a coupling of air between the sound board and back plate and an audible sound is heard [2]. Changing the body shape and construction will affect the sound level produced since it is the structure of the guitar body that determines which frequencies of vibrations will resonate. Every note a guitar plays corresponds to a specific frequency of string vibration which is transferred into the body. Each guitar will respond differently to these vibrations depending on the driving frequency of the note and the construction of the body. Most guitars have three strong resonances in the Hz range [3]. When the driving frequency matches a resonant frequency of the body the vibrations are amplified and higher decibel levels are be produced. This leads the musician to question how best to play the guitar and the manufacturer to question how to make a guitar with optimal resonant frequencies for a desired sound. Many studies have been conducted in order to understand how the construction and strings of a guitar contribute to its resonant frequencies. For example, the amplitude, frequencies, and decay rates of a given string, instrument body, and player s plucking gesture were explored by J. Woodhouse [4]. However, the paper is concluded with a number of questions raised by [the] results about the auditory significance of the phenomena seen in measurements. This leads one to question how the plucking force affects the sound levels produced by a guitar. This paper presents the results found by measuring decibel levels produced by varying plucking forces. By comparing these results to the resonant frequencies of each guitar tested, a relationship between the plucking force, resonant frequency, and sound level was found. The significance of these findings will help musicians 1

2 and manufacturers to better understand the way to play and construct an acoustic guitar that will produce the desired sound levels. held flat on the surface of the guitar, thus ensuring consistent plucking direction perpendicular to the string. Experimental Methods This study measured three specific variables that contribute to the sound levels produced by guitars: plucking force, decibel levels produced from a pluck, and the body resonant frequencies. Each of these variables was measured using two different methods. Both methods employed three guitars with different body styles, as seen in Figure (1), in order to observe how the difference in construction effects the sound levels produced. Two of the guitars were strung with steel strings while the classical guitar used bronze coated nylon strings. Each guitar was plucked with three levels of increasing force, ten plucks per force. All plucking was performed on the low E string. Figure (2) The force gauge-plucking device engaged over the low E string for method one. The force gauge was grasp and pulled directly horizontal away from the string In order to measure the force of each pluck, the string was inserted into the plucking device and placed so that it was just touching the pick. The plucking device was then pulled away from the low E string, while holding the force gauge, until the pick released. The force gauge output real time data to a Vernier hand held LabQuest device, which recorded the force over time as shown in Figure (3). The pluck force was changed using three different picks of increasing thickness. (a) (b) (c) Figure (1) The three guitars used for testing were: (a) Yamaha GC Series, acoustic, bronze coated nylon string, classical body style (b) Dean Exotica FM, acoustic-electric, steel string, classic body style (c) Ovation Applause, acoustic-electric, steel string, plastic round back body style Force (N) Plucking Force Time (s) Method One: For the first method each guitar was laid down on its back and plucked using a custom made plucking device which could be screwed into a force gauge, as seen in Figure (2). This configuration allowed the plucking device to be -2 Figure (3) A sample of the forces produced in one pluck. The pluck force was determined by taking the peak force just before the curve drops. Good Vibrations 2 Jacob Robinson

3 While the force was being measured, a decibel meter, resting on the sound board of the guitar, recorded the peak decibel level produced by the pluck. To cancel effects of reverberation, each guitar was tested in a 3.00 x 2.38 x 2.59 m anechoic chamber located at Brigham Young University. This was done because sound pressure depends on the enviornment and reverberant quality of the space which the source and listener are situated [3]. The maximum force and decibel level for each run was recorded in an Excel spreadsheet to be analyzed later. Ten runs were performed for each of the three picks all on the low E string of each guitar. Method Two: Simliar to the first, the second method measured decibel levels for three different forces of increasing magnitude using the same decibel meter. However, this method used a finger instead of a pick in order to reduce any effects the pick could have on the sound levels produced. I found in the first method that when a pick releases from a string it makes a loud clicking noise which effects the decibels measured. A finger pluck produces no residual noise as it releases from the string and thus ensures more accurate decibel measurements. Each guitar was laid on its back with the decibel meter resting on the sound board as shown in Figure (4). Figure (4) Set-up used in the second method. The string was pulled back with a finger to the specified distance and then released. Instead of measuring the plucking force directly, this method measured decibel levels relative to the distance the string was pulled back. This was done by pulling each string back to a specific distance measured by markings on the sound board as shown in Figure (5). The string was then released and the maximum decibel level was recorded. Ten runs were perfomed with three different pull-back distances for each guitar, all on the low E string. Figure (5) Markings used to measure pull-back distance for each run. Every line is 1/8 apart. Resonant Frequencies: The resonant frequency of each guitar needed to be determined in order to analyze the relationship between the sound level produced for guitars with different resonant frequencies. This was done by attaching an accelerometer to the sound board of each guitar. The accelerometer was connected to a National Instruments DAQ and programmed using their LabView 2011 software. The program recorded the amplitudes of vibrations produced by the sound board and converted this data into a frequency spectrum plot using a Fast Fourier Transform. Each guitar was excited by tapping the sound board repeatedly. This caused the guitar body to vibrate at its resonant frequencies which were recorded by the LabView program. Since the low E string has a resonant frequency of 82 Hz [3] it was expected that a guitar body which resonates at or near 82 Hz would amplify the sound levels. However, a guitar that does not resonate near 82 Hz was expected to produce lower sound levels at the same plucking force. This is because each guitar has specific resonant frequencies depending on the constuction of the body. When the driving frequency of a note matches the resonant frequency of the guitar, the sound is amplified. Results Plucking: Method One and Two The results from method one for three plucks of increasing forces produced inconsistent decibel levels. Plots of the sound level responses from Good Vibrations 3 Jacob Robinson

4 each guitar are shown in Figure (6). The decibels produced from a given force show ranges of up to a 16 db difference. For example, in ten runs on the classical guitar the highest decibel level was measured at 101 db at a plucking force of 6.55 N while the lowest vaule of 84 db was produced by a measured pluck of 6.61 N with a standard deviation of 6.36 db. Averaging these spread out vaules seems to tell very little about the relationship between plucking forces and the sound level produced. The second method was expected to have more variation in sound level since the pull-back distance was only measured visually. However, when the E string was plucked using a finger, the results turned out to be more consistant than method one, especially for the smaller pull-back distances (< 0.3 in). Of these smaller pull-back distances, the largest standard deviation was measured to be 3.71dB Pluck Force (N) (a) Classic Style Pluck Force (N) (b) Good Vibrations 4 Jacob Robinson

5 Round Back Pluck Force (N) (c) Figure (6) Method one decibel responses from varying plucking forces for the: (a), (b) Classic Style, (c) Round Back guitar. The wide range in decibel levels for a given force may be a result of clicking from pick release. 90 1/8 1/4 3/ (a) 90 Classic Style (b) 1/16 1/8 1/4 Round Back 1/16 1/8 1/ (c) Figure (7) Method two decibel responses from varying string pull back distances for the: (a), (b) Classic Style, (c) Round Back guitar. Notice the smaller range of decibel levels when compared to Figure (6). Good Vibrations 5 Jacob Robinson

6 The consistency in decibel levels produced through method two can also be observed by the general trend seen in the data, whereas method one produced no visible trend. When the sound levels from each pull-back distance were averaged, the results showed a trend that was expected. As the pull back distance increased, thus increasing the force, the decibel levels also increased, as seen in Figure (8). However, it must be noted that since the steel string guitars operate at a higher string tension than the nylon string guitar, the same pullback distance required a higher force than the nylon string guitar. Therefore, the nylon string decibel responses represent a lower plucking force while the steel string guitars show decibel responses for higher plucking forces. Body Style Classic Style Round Back Resonant Frequency 114 Hz 120 Hz 122 Hz Table A A list of the resonant frequencies nearest to 82 Hz for each guitar used in the study Classic Style Round Back Figure (8) Plot of the average decibel levels produced for ten runs at three different pull back distances for each guitar. Resonance As expected, each guitar resonated at many different frequencies between Hz. The resonant frequencies were taken from the frequency spectrum plots, like the one in Figure (9), generated by the accelerometer data. When a certain frequency of vibration resonated with the guitar body it would show a much higher amplitude than any of the other frequencies. This is seen as a spike in the spectrum plot at the resonant frequencies. Since the E string plays at 82 Hz I was interested to see which resonant frequency was closest to that on the E frequency. Table A shows the resonant frequencies nearest to 82 Hz for each guitar. These results were much closer together than expected for the difference in body style. Figure (9) Frequency spectrum plot from the classical guitar. The plot has been magnified to show the highest amplitude frequency near 82 Hz is at 114 Hz for this guitar. Discussion and Conclusions The first method proved to have more trouble measuring decibels than expected. The data from the first method was so inconsistent it was difficult to see any trend in the sound levels. This suggests the noise produced from the pick itself was causing interference with the measurement of decibels. One can easily observe this by damping the strings of a guitar and strumming with a pick. A loud clicking noise can be heard which was presumably the maximum sound level measured by the decibel meter. The results of the second method show very clearly that increasing the distance a string is pulled back before release will continue to increase the decibels produced. It was expected that there is a limit of sound level which an acoustic guitar can produce. However, while performing the second method I noticed that pulling the string back too far, especially with the steel strings, caused the string to vibrate erratically and hit the frets or other strings upon release. This interference produced radical decibel levels that made it difficult to continue increasing the Good Vibrations 6 Jacob Robinson

7 plucking force in order to observe the sound levels plateau. Since the two acoustic electric guitars were equipped with steel strings, which have a higher tension, a smaller pull-back distance resulted in a higher plucking force. Figure (8) shows how higher forces in the steel strings led to plateauing of the decibel levels produced. This confirms that the there is a maximum decibel level that can be produced from a given guitar. It also shows that even though the two acoustic electric guitars have different body styles and constructions, they produce almost exactly the same decibel levels for a given force. The resonant frequencies of the guitars were expected to have more variation than was observed. The resonant frequencies for the classic style and round back guitars differed in this range by only 2 Hz. Figure (8) shows how these guitars have very similar decibel responses for the same pull-back distances. Since both guitars had steel strings, the force to pull back should be equivalent. This shows that two guitars with simliar resonant frequencies and string tension will produce the same decibel response when played on the same note. However, since the guitars resonant frequencies are so similar, it is impossible to tell whether the correspondence of resonant frequency to the driving frequency of a string produces higher decibel levels. So it is clear from the data that two guitars with corresponding resonant frequencies will produce similar decibel levels for a given plucking force. Figure (8) also reveals an interesting trend when comparing the profiles of the two steel string guitars to the classical guitar response. As mentioned before, it was difficult to pull the strings too far back without causing interference from the frets or other strings upon release. With the strings pulled back to the maximum distance without interference, the decibel response seemed to level out around 76 db for both steel stringed acoustic-electric guitars. The lower-tensioned nylon string on the classical guitar required less force to pull back to the same distances and could be pulled back further with out any interference. From the classical guitar data in Figure (8) we can see that there is a minimum force required before an increase in plucking force has a significant impact on the decibels produced. If the nylon string were pulled back even further it is expected that there would be a similar plateauing effect as seen with the steel strings. This means that as plucking force increases the decibels produced will follow an s-shape curve where the greatest change in decibels occurs over a small change in plucking force. It also means there is a minimum and maximum decibel level that can be produced by a given guitar. This research has shown that a guitar can produce a certain range of decibel levels depending on the force with which the string is plucked. For the guitars used in this study the maximum sound level for the two steel stringed guitars did not exceed db, while the nylon string did not go above db. The range of decibel levels jumps from the low to high over a small difference in plucking force along an s-curve. Understanding where this significant area of plucking force occurs will help musicians to better understand how to play their acoustic guitar. Further study could be conducted to better understand how the maximum and minimum sound levels are affected by the resonant frequencies of the guitar. This information would help manufacturers to better understand how the guitar construction and subsequent resonant frequencies effect the decibels which an acoustic guitar is capable of producing. References [1] Richardson, B., The acoustical development of the guitar. Journal of the Catgut Acoustical Society, Vol. 2, No. 5 (Series II). [2]Elejabarrieta, M., Ezcurra, A., and Santamaria, C., Evolution of the vibrational behavior of a guitar soundboard along successive construction phases by means of the modal analysis technique. J. Acoust. Soc. Am., Vol. 108, No. 1, pp [3]Fletcher, N., and Thomas, R., The Physics of Musical Instruments. 2 nd ed. New York: Springer-Verlag, 1998, pp [4]Woodhouse, J., Plucked Guitar Transients: Comparison of Measurements and Synthesis. Acta Acoustica United with Acoustica, Vol. 90, pp Good Vibrations 7 Jacob Robinson