Analysis of the Vibrational Modes of a Brass Plate and Mellophone

Analysis of The Vibrational Modes of a Brass Plate and Mellophone

A Thesis Presented to The Honors Tutorial College Ohio University

In Partial Fulfillment of the Requirements for Graduation From the Honors Tutorial College With the degree of Bachelor of Science in Engineering Physics

Sophia Medvid

April 2020

Dr. Martin Kordesch Professor, Physics Department Thesis Advisor

Dr. David Tees Professor, Physics Department Director of Studies

TABLE OF CONTENTS:

INTRODUCTION………………………………………………………..…..………….4 BACKGROUND……………………………………………….….…………...………...6 NORMAL MODES AND RESONANT MODES……………….…………….…6 STANDING WAVES……………………………………………………….….…7 OVERTONES AND HARMONICS…………………………………..………….8 CALCULATIONS OF RESONANT FREQUENCIES……………………….….9 QUALITY FACTOR…………………………………………………………….12 AXISYMMETRIC MODES VERSUS ELLIPTICAL MODES…………….…..13 PIEZOELECTRIC SENSORS………………………………………………..…14 SPECKLE INTERFEROMETRY…………………………………………….…17 CHLADNI PATTERNS…………………………………………………………18 PREVIOUS VIBRATIONAL ANALYSIS STUDIES………………………….22 METHODOLGY AND EXPERIMENT……………………………….………….…..29 BRASS PLATE SETUP…………………………………………………………30 BRASS PLATE: FREE EDGE, FREE CENTER……………………………….31 BRASS PLATE: FREE EDGE, CLAMPED CENTER…………………………31 BRASS PLATE: CLAMPED EDGE, FREE CENTER…………………………33 BRASS PLATE: CLAMPED EDGE, CLAMPED CENTER…………………...35 MELLOPHONE SETUP: ARTIFICIAL LIPS…………………………………..38 MELLOPHONE SETUP: SPECKLE INTERFEROMETRY………...…………39 MELLOPHONE SETUP: PIEZOELECTRIC SENSORS………………………41 MELLOPHONE SETUP: CHLANDI PATTERNS……………………….…….42 MELLOPHONE SETUP: DRIVING WITH PIEZOELECTRIC SENSORS.…..44 MELLOPHONE SETUP: PLAY TEST………………………………………....46 DATA AND RESULTS………………………………………………………………...47 BRASS PLATE: FREE EDGE, FREE CENTER……………………………….47 BRASS PLATE: FREE EDGE, CLAMPED CENTER…………………………49 BRASS PLATE: CLAMPED EDGE, FREE CENTER…………………………52 BRASS PLATE: CLAMPED EDGE, CLAMPED CENTER…………………...54 ADDING WEIGHT TO THE BELL WIRE……………………………………..57 MELLOPHONE: CLANDI PATTERN………...……………………………….60 MELLOPHONE: DRIVING WITH PIEZO ELECTRIC SENSOR…………….62 MELLOPHONE: PLAY TEST………………………………………………….67 ANALYSIS……………………………………………………………………………...68 EFFECT OF CLAMPING AT CENTER………………………………………..69 COMPAARING ALL BOUNDARY CONDITIONS…………………………...71 EFFECTS OF WEIGHT OF BELL WIRE………………………………………73 COMPARING THEORETICAL CALCULATIONS WITH EXPERIEMTNAL RESULTS…..……………………………………...…………………………….75 MELLOPHONE…………………………………………………………………78 CONCLUSION…………………………………………………………………………80 ACKNOWLEDGMENTS…………………………………………………………...…82 WORKS CITED………………………………………………..……………………....83 APPENDIX…………………………..…………………………………………………85

INTRODUCTION

Some of the first devices humans created and studied were musical instruments.

For thousands of years humans have developed a better understanding of acoustic physics and how musical instruments work (for example, Kausel et al., 2010 and Richardson,

2011). Many studies have been done to understand all kinds of instruments including stringed instruments, woodwinds, percussion, and brass instruments (for example,

(Gilbert et al., 1998), (Richardson, 2011) and Moore et al., 2005). Studies have been done on common brass instruments such as the trumpet (Kausel et al., 2010) and the trombone

(Gilbert et al., 1998). Very few, if any, studies have been conducted on more peculiar instruments, like the large belled mellophone. The mellophone is a the marching version of a French horn and is recognizable for its big flared brass bell (See Figure 1). Because of its large bell, the mellophone is an interesting instrument to study and to determine its vibrational behavior.

Fig 1. A picture of a mellophone. Notice the large, flared bell. (“Jupiter JMP1100M Quantum Marching Mellophone,” n.d.).

Many brass musicians have a preferred style or brand of horn because they argue that the design of their horn produces a better sound. They prefer a certain make and model of an instrument over another because certain notes sound better on certain horns.

There are obvious reasons for this, such as some horns are made with higher quality material or are handmade. But another, more scientific, reason they prefer certain horns is that some notes that are played on the instrument align with the vibrational modes of the bell. This only occurs at certain frequencies and changes depending on the design of the horn. But if these vibrational modes occur at the same frequencies of the played notes, it changes how the musician hears and interprets the sound of the note. However, the normal modes of a trumpet bell were found and the amplitude of vibration near the bell was discovered to be ~200 micrometers which was not significant enough to affect the resultant spectrum. (Moore et al., 2005). With this competing evidence, I conducted research to better determine if the vibrational modes make a noticeable difference in how the horn plays. Because the mellophone bell is larger than the trumpet bell, it will give more definitive evidence on whether the size of the bell affects the amplitude of vibration.

My project used speckle interferometry, piezoelectric detection, and Chladni patterns to analyze the vibrational modes and behavior of the mellophone bell. It was easiest to compare the behavior of the mellophone bell to a circular brass disk of the same size and to see how alike the bell behavior was to that of the brass disk.

BACKGROUND

In this section I discuss the background necessary to understand the vibrational analysis I conducted as well as the science involved with acoustic physics. For this purpose I will have background for each of the vibrational analysis methods I used to study brass circular plates and the mellophone bell. Following this, I will discuss previous studies of vibrational analysis involving brass circular plates and brass instruments.

NORMAL MODES AND RESONANT FREQUENCIES

The tendency of any system is to return to equilibrium once a perturbation is applied. This tendency is the reason why there are so many vibrations and oscillations that naturally occur. The technical definition of a normal mode is a pattern of motion in which all parts of the oscillating system move sinusoidally with the same frequency and with a fixed phase relation. This frequency is known as a resonant frequency and surfaces and objects can have more than one resonant frequency. Normal modes require the condition that the motion is not coupled meaning that the modes move independently.

That is to say that an excitation of one mode will never cause motion of a different mode.

In layman’s terms, any oscillating system has a preferred way it likes to vibrate. These preferred vibrations are the resonances of the system. (“Vibrations and normal modes,” n.d.).

Bells of brass instruments are known to have oscillatory behavior and produce normal modes at certain frequencies. Moore, at Rollins College, found the resonant frequencies of a trumpet bell by driving the vibrations of the instrument with a small speaker and observing the bell with an electronic speckle pattern interferometer. (Moore

6 et al., 2005). Moore found that the ﬁrst detectable resonance of the bell coincided with the Bflat 4 of the air column. This is an important finding because the resonance aligns with a common note played by the musician. If the vibrations of the normal mode are strong enough, it may affect the note that the musician is playing. (Kausel et al., 2010;

Moore et al., 2005; Nief et al., 2008).

STANDING WAVES

Each of the natural frequencies at which an object vibrates is associated with a standing wave pattern. When an object vibrates at a resonant frequency, it vibrates in such a way that a standing wave is formed within the object. A standing wave is a vibrational pattern created within a medium when the vibrational frequency of a source causes reflected waves from one end of the medium to interfere with incident waves from the source. (“Physics Tutorial: Standing Wave Patterns,” n.d.). This interaction results in specific points along the objects’ surface to appear to be standing still while other points are vibrated back and forth. The pattern is referred to as a “standing wave pattern”.

Nodal points are points in the pattern that are standing still and experiencing no displacement. In between each nodal point, there are antinodes that vibrate back and forth between a large upward displacement and a large downward displacement. A standing wave pattern has an alternating pattern of nodal and antinodal points. Figure 2 shows a diagram of a standing wave with the nodes and antinodes labeled.

Fig 2. Diagram of a standing wave. (“Physics Tutorial: Standing Wave Patterns,” n.d.).

OVERTONES AND HARMONICS

In acoustics, an overtone is a tone sounding above the fundamental tone when a string or air column vibrates as a whole, producing the fundamental or first harmonic.

Someone listing can hear the fundamental frequency clearly and if they have a good ear the overtones can be heard as well. (“Overtone | acoustics | Britannica,” n.d.).

Harmonics are a series of overtones that result when the frequencies are exact multiples of the fundamental frequency. The upper harmonics occur at frequencies that form simple ratios with the frequency of the first harmonic (e.g., 2:1, 3:1, 4:1). Figure 3 shows the overtones of an ideal string. Some musical instruments, specifically those whose sound is produced from the vibration of metal or wood, also produce nonharmonic overtones, meaning that the frequencies of the overtones are not multiples of the fundamental frequency. (“Overtone | acoustics | Britannica,” n.d.).

Fig 3. Shows the vibrational modes of an ideal string. Dividing the string into integer divisions produces the overtones shown. The first string shown is at the fundamental frequency or first harmonic. The ones following it are the second, third, fourth, fifth and sixth overtones. (“Overtone,” 2019).

CALCULATIONS OF RESONANT FREQUENCIES

The equation of free motion for small out of plane deformations 푤 of a flat uniform elastic plate of density 휌, in 푥, 푦 Cartesian coordinates has the form (Mrozek et al., 2018):

(Eq. 1) where 퐸 is the modulus of elasticity of plate material, 푣 is the Poisson’s ratio, ℎ is the plate thickness, 푡 is the time, 훾 is the mass per unit area of the plate (훾 = 휌ℎ).

Using a separation of variables solution for Eq. (1) allows us to obtain the associated formulas of resonant frequencies and mode shapes for rectangular and annular plates. (Mrozek et al., 2018). Resonant frequency 푓푖푗 is expressed in a form:

2 휆푖푗 퐷 푓 = √ (Eq. 2) 푖푗 2휋푎2 ϒ where D is the flexural rigidity defined by the equation:

퐸ℎ3 퐷 = (Eq. 3) 12(1−휈2)

푖 is the number of half-waves in mode shape along 푥 axis (rectangular plate) or number of nodal diameters (round and annular plate), 푗 is the number of half-waves in mode shape along 푦 axis (rectangular plate) or number of nodal circles (round and annular plate), 휆푖푗 is the dimensionless frequency parameter, 푎 is the linear dimension of the plate (length of a rectangular plate or outer radius of round or annular plate).

The mode shapes of rectangular plates are combinations of sinusoidal and hyperbolic functions. The mode shapes of circular plates are Bessel functions. (Mrozek et al., 2018). In my project I will only be concerned with the calculations resonant frequencies of circular plates.

The dimensionless frequency parameter, 휆푖푗, is a unitless parameter that depends on the geometry, nodal number, and boundary conditions of the plate or object being studied. The equation for calculating the dimensionless frequency parameter is:

2 2 휌 휆 = 휔푎 √ (Eq. 4) 푖푗 퐷

where ω is the angular frequency. In my research I studied circular brass plates under various boundary conditions so I will share the frequency parameters for these cases in

Tables 1-3 that were calculated using equation 4 by Leissa. (Leissa, 1969) .

Table 1. Values of 휆푖푗 for a circular plate clamped at the outside edge. s refers to the number of nodal circles and n refers to the number of nodal diameters. For my study, only the first column will be used, n = 0. (Leissa, 1969).

Table 2. Values of 휆푖푗 for a simply supported circular plate. s refers to the number of nodal circles and n refers to the number of nodal diameters. For my study, only the first column will be used, n = 0. For these calculations ν = 0.3. (Leissa, 1969).

Table 3. Values of 휆푖푗 for a completely free circular plate. s refers to the number of nodal circles and n refers to the number of nodal diameters. For my study, only the first column will be used, n = 0. For these calculations ν = 0.33. (Leissa, 1969).

QUALITY FACTOR

The quality factor, denoted as ‘Q’, is defined as the ratio of a resonator’s center frequency to its bandwidth when subject to an oscillating driving force. Resonators with high quality factors have low damping, so they ring or vibrate longer. Sinusoidally driven resonators have a higher quality factor and resonate with greater amplitudes at resonant frequencies but they have a smaller range of frequencies around the frequency at which they resonate. In other words, for a high Q resonator, the resonant frequency occurs over a very short frequency range. For a low Q resonator, the resonant frequency occurs over a very broad frequency range. High quality factor oscillators oscillate with a smaller range of frequencies and are more stable. (“Q factor,” 2020).

In the context or resonators, the quality factor is defined as

푓푟 푄 = (Eq.5) ∆푓

where Q is the quality factor, 푓푟 is the resonant frequency, and ∆푓 is the full width at half maximum.

In my study, modes that occur at a resonant frequency over a broad range of frequencies will have a small Q. Modes that occur at a resonant frequency over a narrow range of frequencies will have a large Q.

AXISYMMETRIC MODES VERSUS ELLIPTICAL MODES

Axisymmetric vibrations are those in which the mode shape is characterized exclusively by the number and position of the nodal circles in the structure. As the name suggests, the modal pattern is perfectly symmetric around the axis. Modes are denoted by

(i, j). ‘i’ represents the meridian or the circumferential index (number of diameters) and

‘j’ represents the longitudinal index (number of nodal circles). For axisymmetric modes

‘i’ is zero. Elliptical modes are characterized by i > 0. (Balasubramanian et al., 2019) .

Figure 4 shows a (2, 2) elliptical mode (a) and (0,1) axisymmetric mode (b) found by

Kausel using three-dimensional modeling software. (Chatziioannou and Kausel, n.d.) .

Fig 4. Plot of a (2, 2) elliptical mode (a) and a (0, 1) axisymmetric mode found by Kausel using three-dimensional modeling software. Reproduced with permission from

Chatziioannou. (Chatziioannou and Kausel, n.d.) .

Elliptical modes will not be considered in my study because previous studies have found the acoustic effects of elliptical modes to be insignificant. Kausel did a vibrational analysis of brass instruments and found that the elliptical motion of the walls occurs over a narrow frequency range and elliptical modes typically have a very high quality factor and thus could not explain broadband acoustic affects. (Chatziioannou and Kausel, n.d.) .

Furthermore, elliptical modes also result in destructive interference of the sound waves and hence do not significantly contribute to coupling with the air column. (Morrison and

Hoekje, 1997). For these reasons, my study is devoted to axisymmetric modes only, not elliptical modes.

PIEZOELECTRIC SENSORS

A piezoelectric sensor is a device that uses the piezoelectric effect to measure a change in acceleration, pressure, strain, force, or temperature by converting it into an electrical charge. The piezoelectric effect occurs when a piezoelectric material is placed under mechanical stress. This causes a shifting of the positive and negative charge centers in the material and results in an external electric field. This electric field can be measured to determine how much strain or movement the sensor is exhibiting. The greater the motion that the piezoelectric sensor is undergoing, the greater the electric field.(“The

Piezoelectric Effect - Piezoelectric Motors & Motion Systems,” n.d.). Figure 5 illustrates how a piezoelectric sensor works. If a pressure causes the sensing element, which is made out of the piezoelectric material, to move, then it converts this pressure into a voltage by the piezoelectric effect. (“Piezoelectric Sensor Technology,” 2012).

Fig 5. A diagram of how the piezoelectric sensor works. Upon application of pressure to the sensing element, the pressure is converted to a voltage by the piezoelectric effect.

(“Piezoelectric Sensor Technology,” 2012).

For example, if a piezoelectric sensor was placed on a vibrating cell phone, an electric field could be detected and measured whenever the phone was vibrating because the phone vibration would cause the piezoelectric sensor to move. This movement would cause the piezoelectric sensor to exhibit the piezoelectric effect and produce an electric field. Whenever the phone is not vibrating, no electric field would be detected because the piezoelectric sensor would not be under any strain/motion so no electric field could be produced by the piezoelectric effect. Figure 6 shows what a piezoelectric sensor looks like and how its size compares to a quarter. The white plastic part of the sensor attaches

15 to the material that is moving and the two connectors of the sensor attach to a voltmeter or oscilloscope to measure the resultant voltage.

Fig 6. A piezoelectric sensor held up to a quarter to show its size. (“Piezo Vibration

Sensor”).

Piezoelectric sensors could be used to find vibrational modes of a material by placing the sensors on the material and varying the frequency of vibration. Where the material is vibrating intensely, the piezoelectric sensors will output a large voltage and at places of less intense vibration, the sensors will output a small voltage. By mapping the piezoelectric sensor behavior at different frequencies, the voltages can be compared to find the vibrational modes of the vibrating material.

SPECKLE INTERFEROMETRY

Speckle interferometry is a technique that uses laser light, together with video detection, to visualize static and dynamic displacements of surfaces. Speckle interferometry is the most sensitive and accurate optical method for full three- dimensional displacement assessment. Laser light is shined on the surface under investigation and the image formed by the light is a subjective speckle pattern. The light arriving at a point in the speckled image is scattered from a finite area of the object and its phase, amplitude, and intensity give us information about how the surface is moving.

A second beam, known as the reference beam, is superimposed on the video camera image. The two light fields interfere and create a speckle pattern. If the object is displaced, the distance between the object and the image will change and cause the phase of the image speckle pattern to change. When the object has been displaced, the new image is subtracted point by point from the first image and a speckle pattern, with fringes representing movement, is created. These speckle patterns can be used as maps to observe where the object is moving and with what intensity. (Moore, 2004).

Speckle interferometry has been used to study the vibration of musical instruments before. Richardson used speckle interferometry to examine the vibrational modes of the violin and the guitar. (Richardson, 2011). When the frequency of vibration of the instrument was equal to the resonant frequency, stationary and clear fringe patterns were observed. Figure 7 shows a speckle pattern of a violin that Richardson produced in his study. The fringes map out contours of equal vibration amplitude with adjacent bright

(or dark) fringes representing a further amplitude. Speckle interferometry is a useful

17 technique to locate the normal modes of a vibrating surface and determine at which frequencies they occur because the nodal lines stand out as very intense fringes.

Fig 7. A speckle pattern of a violin produced by Richardson. The nodal lines stand out as very intense fringes. Used with permission from Richardson. (Richardson, 2011).

CHLADNI PATTERNS

Chladni patterns are another useful technique to visualize the vibration of a moving surface and to find its vibrational modes. Chladni, for whom the technique is named, was a German acoustic physicist who found the vibrational modes of a square plate using this method. A plate (or any surface) is sprinkled with sand, salt, or any fine powder and then sinusoidally driven. The object’s surface is constantly being displaced

18 which causes the sand to move on the plate. Figure 8 shows a Chladni pattern of a metal plate.

Fig 8. A Chladni Pattern of a metal plate. The sand represents nodal lines (spots in the plate where no movement is occurring). (“Chladni Plates Kit - Arbor Scientific,” n.d.).

At certain frequencies, a normal mode is produced and can be detected by the standing wave pattern the sand makes on the plate. The sand rests upon the nodal positions where no movement of the surface is occurring.(“The Physics of Sound and

Music,” n.d.). Referring back to the standing wave section, the sand gathers at nodes.

Areas on the plate where there is no sand is where the plate is vibrating back and forth.

Figure 9 shows a diagram of how a standing wave aligns with the Chladni pattern formed on the surface of the vibrating object, in this case a circular brass plate.

Fig 9. A Chladni pattern of a circular brass plate with a standing wave overlaid on top of it. The orange arrows point to nodes, points of no vibration, where the sand gathers. The yellow depicts areas of vibration on the plate.

Chladni patterns provide a simple and useful way to detect vibrational modes.

Richardson used Chladni patterns to determine the vibrational modes of a violin. One of his patterns is shown in Figure 10.(Richardson, 2011). Figure 11 shows a Chladni pattern of a circular brass disk driven mechanically at 2631Hz that I produced.

Fig 10. A Chladni pattern of a violin produced by Richardson. The black powder fringes represent nodal lines. Used with permission from Richardson (Richardson, 2011).

Fig 11. A Chladni pattern of a circular brass disk when driven at 2361 Hz.

PREVIOUS VIBRATIONAL ANALYSIS STUDIES

Khare and Mittal conducted a vibrational analysis of both circular and annular plates with varying geometric boundary conditions. They used three different boundary conditions when studying the circular plate: free, clamped, and simply supported. Their results for the circular plate are shown in Figure 12. They studied the annular disk for a set of 9 different boundary conditions. Figure 13 shows the different boundary conditions they studied. Their results for the annular plate for three of the nine different boundary conditions are shown in Figure 14. The three boundary conditions shown are free-clamped, free-free, and free-supported. The reason I present only these three is because they are the most similar to the mellophone bell and the brass plates I was studying. (Khare and Mittal, 2015). Their results are important to my research because I will be able to directly compare my results with theirs when studying brass plates. Also, when studying the mellophone bell, I compared my results to those in Figure 14.

Fig 12. Khare and Mittal’s results for the circular plate with clamped, free, and simply supported boundary conditions. The red end of the spectrum correlates to movement and the blue end of the spectrum corresponds to no movement. Modes are listed as (j,i).

Reproduced with permission from Materials Today. (Khare and Mittal, 2015).

Fig 13. These are radial cuts of the circular annular plate and the corresponding nine

boundary conditions that Mittal and Khare studied. The dashed lines correspond to

clamping and the triangle with dashed lines corresponds to simply supported.

Reproduced with permission from Materials Today. (Khare and Mittal, 2015).

Fig 14. Khare and Mittal’s results for the annular plate for three boundary conditions: free-clamped, free-free, and free-supported. The red end of the spectrum correlates to movement and the blue end of the spectrum corresponds to no movement. Modes are listed as (j,i). Reproduced with permission from Materials Today. (Khare and Mittal,

2015).

Escaler and De La Torre conducted vibrational analysis on an aluminum circular plate using the Chladni technique. The plate was harmonically excited at its center through an extension bar and its outer edge was left free. This is a very similar experimental setup to my experiment. Pictures of their Chladni patterns showing the axisymmetric modes are shown in Figure 15. Figure 16 shows top views of the simulated mode shapes in air for the axisymmetric modes of vibration with nodal radii from 1-6 produced by Escaler and De La Torre. The color patterns in Figure 16 depict the vibrational contours and behavior the plate is experiencing during these axisymmetric

24 modes. Even though Escaler and De La Torre used a circular aluminum plate and I will be using a circular brass plate, it is important to look at their Chladni patterns and note the similarities.

Fig 15 . Chladni patterns of a circular aluminum plate found by Escaler and De La Torre.

Air refers to the fact the experiment was done in air and not vacuum and s refers to the number of nodal diameters. Reproduced with permission from Journal of Fluids and

Structures. (Escaler and De La Torre, 2018).

Fig 16. Top views of the simulated mode shapes in air for the axisymmetric modes of vibration with nodal radii from 1-6 produced by Escaler and De La Torre. Blue color indicates zero deformation and red color indicates maximum deformation. s refers to the number of nodal diameters. Reproduced with permission from Journal of Fluids and

Structures. (Escaler and De La Torre, 2018).

Ni calculated the natural frequencies of shells to study the effect of boundary parameters on the dynamic characteristics of shells. Ni studied a wide geometry of shells but the one that is most relevant to my research is his calculation of natural frequencies of a pseudosphere. A pseudosphere is a kind of surface with constant negative curvature. It resembles a bell of a brass instrument. With the parameters Ni chose, his pseudosphere resembles that of a trumpet bell. Figure 17 shows the calculated natural frequencies and mode shapes of the pseudospherical shell that Ni calculated. Notice that modes 14 and 11 is axisymmetric and exactly the kind of modes I will be studying. (Ni et al., 2019).

Fig 17. Natural frequencies and mode shapes of the pseudospherical shell produced by

Ni. Blue color indicates zero deformation and red color indicates maximum deformation.

The material properties for this shell: E = 210 GPa, ν = .3, ρ = 7800 kg/m3. Reproduced with permission from International Journal of Solids and Structures. (Ni et al., 2019)

Vibrational analysis has not only been done on plates and shells but on actual brass instruments themselves. Moore used speckle interferometry to show the deflection patterns associated with the modes of vibration of the trumpet bell. Figure 18 shows his results. In his experiments, Moore used a Silver Flair model trumpet made by the King

Instrument Company. It should be noted that Moore only detected elliptical modes and not axisymmetric modes. His findings are still important though because the first detectable resonance and the Bb4 of the air column are nearly coincident. A Bb4 is around 466Hz and the first resonance Moore detected was at 467Hz. This is an elliptical mode though so we know that its acoustic effects are insignificant.

Fig 18. Electronic speckle pattern interferograms showing the deflection patterns associated with the modes of vibration of the trumpet bell produced by Moore. The white areas indicate antinodes (movement) and the black areas indicate nodes (no movement).

The view is from the front of the trumpet, looking directly into the bell orifice.

Reproduced with permission from Journal of Sound and Vibration. (Moore et al., 2005).

Gilbert conducted a vibrational analysis of a trombone (Courtois model 149).

Gilbert created a set of artificial lips to play the trombone and measured the vibration of the trombone using a laser doppler vibrometer. A laser doppler vibrometer is a scientific

27 instrument that is used to make non-contact vibration measurements of a surface. The laser beam from the laser beam vibrometer is directed at the surface of interest and the vibration amplitude and frequency are extracted from the Doppler shift of the reflected laser beam frequency due to the motion of the surface. Table 4 shows the parameters

(frequency, quality factor, and magnitude) of the first eight resonances of the trombone found by Gilbert. (Gilbert et al., 1998). Even though these results are for a trombone and not a mellophone it is useful to compare the results because a trombone has a larger and more flared bell, like a mellophone, than a trumpet does.

Table 4. Resonance frequency (Hz) , quality factor (dimensionless), magnitude (dB) of the first eight resonances of the trombone (Courtois model 149) found by Gilbert using a vibrometer. (Gilbert et al., 1998).

METHODOLOGY AND EXPERIMENT

BRASS PLATE SETUP

When studying the normal modes and resonant frequencies of brass plates a circular brass plate 0.15 m in diameter and 0.38 mm thick was used. The technique of

Chladni patterns was used to find the resonant frequencies and normal modes of the circular brass plate under varying boundary conditions at the center and edge of the plate.

A small hole was drilled though the center of the circular plate so that a transducer could be attached to mechanically drive the plate at its center. The transducer was hooked up to a function generator and an amplifier. The circular brass plate was sprinkled with fine sand to create the Chladni patterns. When a normal mode was reached a Chladni pattern would form and I would take a picture of it and record the resonant frequency. The setup of the experiment is shown in Figure 19. The brass plate was studied under four different types of boundary conditions: free edge, free center; free edge, clamped center; clamped edge, free center; and clamped edge, clamped center.

Fig 19. The experimental setup of the Chladni Pattern experiment of circular brass plates.

BRASS PLATE: FREE EDGE, FREE CENTER

The first boundary condition studied using Chladni patterns of the circular brass plate was free edge, free center. This meant that the plate was completely free to move at both the outer edge of the circular brass plate and at the plate’s center. A picture of a

Chladni pattern of the circular brass plate under the free edge, free center boundary condition is shown in Figure 20.

Fig 20. A picture of a Chladni pattern of a circular brass plate under the free edge, free center boundary condition.

BRASS PLATE: FREE EDGE, CLAMPED CENTER

The next boundary condition studied using Chladni patterns of the circular brass plate was free edge, clamped center. This meant that the plate was completely free to move at the outer edge of the circular brass plate but was clamped at the plate’s center, allowing no movement. Two different sized clamps were used to study this boundary condition: a 2.39 inch diameter clamp, and a 4 inch diameter clamp. Pictures of a Chladni pattern of the circular brass plate under the free edge, clamped center boundary condition using a 2.39 inch diameter clamp and a 4 inch diameter clamp is shown in Figure 21 and

Figure 22 respectively. In Figure 22 you will notice that a hook is now attached at the plate’s center. This was to lift the plate up to offset the weight of the clamp.

Fig 21. A picture of a Chladni pattern of a circular brass plate under the free edge, clamped center boundary condition. A 2.39’’ diameter clamp is used.

Fig 22. A picture of a Chladni pattern of a circular brass plate under the free edge, clamped center boundary condition. A 4’’ diameter clamp is used.

BRASS PLATE: CLAMPED EDGE, FREE CENTER

The next boundary condition studied using Chladni patterns of the circular brass plate was clamped edge, free center. This meant that the plate was completely free to move at the center of the circular brass plate but not at the outer edge. To emulate the bell wire found on most bells of brass instruments a copper wire was soldered onto the outside rim of the circular brass plate. To study the effects of the weight of the bell wire, a light lead trim, and then eventually a heavier lead tubing was attached on top of the soldered on copper wire to produce data for three different weights of bell wire. A picture of the circular brass plate under the clamped edge, free center boundary condition is shown in Figure 23. In figure 23 only the copper wire is used to clamp the edge of the

33 plate, not the lead trim or heavier lead tubing. Figure 24 shows the similarity between the bell wire of an actual mellophone and the copper bell wire used on the circular brass plate.

Fig 23. A picture of the circular brass plate under the clamped edge (copper wire), free center boundary condition.

Fig 24. On the left is a picture of the actual bell wire of a mellophone and on the right is a picture of the copper bell wire that was soldered onto the circular brass plate for this experiment. The bell wire of the mellophone is 4.8 millimeters thick. The bell wire on the plate is 2 millimeters thick.

BRASS PLATE: CLAMPED EDGE, CLAMPED CENTER

The next boundary condition studied using Chladni patterns of the circular brass plate was clamped edge, clamped center. This meant that the plate was clamped with no movement allowed at the center and outer edge of the plate. The outer edge of the plate was again clamped with the soldered-on copper bell wire. The center of the plate was clamped with both the 2.39 inch diameter and 4 inch diameter clamp. Pictures of a

Chladni pattern of the circular brass plate under the clamped edge, clamped center

35 boundary condition using a 2.39 inch diameter clamp and a 4 inch diameter clamp is shown in Figure 25 and Figure 26 respectively.

Fig 25. A picture of a Chladni pattern of a circular brass plate under the clamped edge, clamped center boundary condition. A 2.39’’ diameter clamp is used.

Fig 26. A picture of a Chladni pattern of a circular brass plate under the clamped edge, clamped center boundary condition. A 4’’ diameter clamp is used.

MELLOPHONE SETUP: ARTIFICIAL LIPS

In order for the horn to be played consistently every time, we attempted to create a set of artificial lips that would play the horn as opposed to human lips. These artificial lips would be connected to the mouthpiece of the horn and driven with a pressurized air source. The hope was that the lips would be more consistent than human lips. The artificial lips buzzed and produced an air column through the horn, however the sound would only last a short amount of time (normally less than a minute) and only played at very high frequencies. For these two reasons we could not use the artificial lips in the experiment.

The set of artificial lips we created is shown in Figure 27. The lips were made out of a spongy rubber material that resembled lip tissue. The aluminum plate and screws were designed to apply pressure to the lips so that the lips would be under enough pressure to buzz. The screws could be adjusted to alter the amount of pressure on the lips in order to change notes. The plastic cylinder had a rounded top to emulate teeth to apply back pressure to the mouthpiece and instrument.

Artificial Lips

Fig 27. The artificial lips created to drive the mellophone.

MELLOPHONE SETUP: SPECKLE INTERFEROMETRY

Speckle interferometry was also used to try to find resonances of the horn. Figure

28 shows the setup. The mellophone was hung in place with wire and mechanically driven with the red speaker that was hooked up to an amplifier and function generator. A red laser was shined on the bell and a camera recorded the moving pattern of the reflected laser light in the bell. The camera recordings could be time averaged to see if the laser light was moving or stationary as the frequency driving the horn was varied. This method

39 proved unsuccessful because the camera could not pick up on three dimensional movement. It could only detect side-to-side motions and not back-and-forth motions.

Fig 28. The setup for the speckle interferometry experiment on the mellophone. From left to right: A red laser, a camera (underneath the laser, pointing into the bell of the mellophone), a function generator hooked up to an amplifier, an oscilloscope, a mellophone hung from the table with wire, and a red speaker that mechanically drove the mellophone.

MELLOPHONE SETUP: PIEZOELECTRIC SENSORS

Another method to find the normal modes of the mellophone was using piezoelectric sensors. Figure 29 shows the setup of the piezoelectric sensors on the mellophone bell to find the normal modes. Piezoelectric sensors were secured on the bell of the mellophone and each one was hooked up to an oscilloscope in order to track the voltage output for each piezoelectric sensor. The horn was mechanically driven and swept through frequencies. As the frequencies varied, the voltage of the piezoelectric sensors was tracked to see which areas of the bell were vibrating and which were not.

Although the piezoelectric sensors did an accurate job at tracking the motion at certain points on the bell, it would have taken piezoelectric sensors on every point of the bell to fully track the motion of the bell and find the normal modes of the bell. This would have required many oscilloscopes and was thus deemed impractical. For these reasons, piezoelectric sensors were not used to find the normal modes of the mellophone.

Fig 29. Four piezoelectric sensors were secured on the bell of the mellophone and each one was connected to an oscilloscope to track the voltage of each piezoelectric sensor.

The mellophone was mechanically driven with a speaker that was hooked up to a function generator.

MELLOPHONE SETUP: CHLADNI PATTERN

The next method used to analyze the vibrations of the mellophone was Chladni patterns. Obviously, we could not sprinkle sand directly on the bell to form a Chladni pattern because the bell is sloped and would result in the sand just falling to the rim of the bell. In order to use the Chladni technique on the mellophone, we placed the bell perpendicular to the ground and drove the bell from the back with a speaker. Stick-on foam lining was attached to the sides of the bell column to hold the sand in place. Sand

42 was sprinkled on the bell column and the frequency of the speaker that was driving the bell was varied. This setup allowed the sand to move as the frequency was altered and showed where nodal points (no movement) occurred on the bell column. Figure 30 shows a picture of the setup.

Fig 30. A labeled picture of the setup of the mellophone using the Chladni pattern technique.

MELLOPHONE SETUP: DRIVING WITH PIEZOELECTRIC SENSOR

A piezoelectric sensor mounted on the mellophone bell was used to drive oscillations in the bell. The resonance of the acoustic spectrum was found using a microphone that recorded the acoustic output. In the previous experiments we were finding the resonances of the brass plate or the mellophone bell itself, which were the actual surfaces being vibrated. In this experiment we are finding the resonances of the acoustic spectrum that the instrument is producing with the air column created when the bell is being driven. These resonances are not the resonances of the bell or vibrating surface, but instead the resonances of the acoustic spectrum.

Figure 31 shows the piezoelectric sensor placed on the bell. Here, the piezoelectric sensor is being used to drive the bell. In the previous experiments, the piezoelectric sensors were being used to record the vibration of the bell. Figure 32 shows the setup for this experiment. The mellophone has a piezoelectric sensor hooked up to a function generator that is driving the mellophone bell. A microphone is placed in front of the bell and is hooked up to an dynamic signal analyzer that maps the Fourier transform of the acoustic spectrum that the microphone is recording. By varying the frequency the piezoelectric sensor is driving the bell at and observing the acoustic spectrum the microphone is detecting, the resonances of the acoustic spectrum can be found. A resonance is observed when there is a peak in the acoustic spectrum at that frequency.

Fig 31. A piezoelectric sensor is attached to the edge of the bell and hooked up to a function generator. The piezoelectric sensor drives the horn.

Fig. 32. The mellophone bell is being driven with a piezoelectric sensor that is hooked up to a function generator. A microphone is placed in front of the bell and hooked up to a dynamic signal analyzer that records the Fourier transform of the acoustic spectrum.

MELLOPHONE SETUP: PLAY TEST

To see if the acoustic spectrum, that was produced when the bell was driven by the piezoelectric sensor, matches what happens when an actual musician plays the horn, a play test was conducted. The bell was soldered back onto the valves and a musician played the mellophone. The mellophone was played at frequencies that aligned with the resonances of the acoustic spectrum that were found beforehand by driving the bell with the piezoelectric sensor. The sound output produced by the musician was recorded with the microphone that was hooked up to the dynamic signal analyzer and the acoustic spectrums were recorded. By comparing the acoustic spectrums produced by the musician playing the instrument versus the piezoelectric sensor driving the instrument, the significance of the bell vibrations were determined. Previous studies conclude that the vibration of the bell will be insignificant compared to the vibrations produced by the air column. This play test will confirm or refute those claims by comparing the acoustic spectrums of the musician versus the piezoelectric sensor driving the instrument.

DATA AND RESULTS

In the background section, I described three vibrational techniques to observe and study normal modes: piezoelectric sensors, speckle interferometry, and Chladni patterns. I tested out all three techniques but the Chladni pattern technique proved most successful so those are the results shown in this paper. Speckle interferometry proved difficult because we did not have sensitive enough equipment to detect such small vibrations using this technique.

Piezoelectric sensors, although very accurate, would have required many sensors to be placed on the vibrating surface and proved to be impractical given the time constraints.

BRASS PLATE: FREE EDGE, FREE CENTER

Figure 33 shows the Chladni patterns of normal modes of the circular brass plate under the free edge, free center boundary condition. The corresponding resonant frequency is given for each mode. Modes are listed as (i, j). Modes 3-11 are shown.

Modes 1 and 2 could not be found because they were so low and intense that the sand could not stay on the vibrating plate to form a detectable Chladni pattern.

Fig 33. Chladni patterns of normal modes of the circular brass plate under the free edge, free center boundary condition.

BRASS PLATE: FREE EDGE, CLAMPED CENTER

Figure 34 shows the Chladni patterns of normal modes of the circular brass plate under the free edge, clamped center boundary condition. The clamp used was 2.39 inches in diameter. The corresponding resonant frequency for each mode is given. Modes 2-11 are shown. The first mode could not be found because it was so intense that the sand could not stay on to form a detectable Chladni pattern.

Fig 34. Chladni patterns of normal modes of the circular brass plate under the free edge, clamped center boundary condition. 2.39’’ diameter clamp used.

Figure 35 shows the Chladni patterns of normal modes of the circular brass plate under the free edge, clamped center boundary condition. The clamp used was 4 inches in diameter. The corresponding resonant frequency for each mode is given. Modes 1-7 are shown.

Fig 35. Chladni patterns of normal modes of the circular brass plate under the free edge, clamped center boundary condition. 4’’ diameter clamp used.

BRASS PLATE: CLAMPED EDGE, FREE CENTER

Figure 36 shows the Chladni patterns of vibrational modes of the circular brass plate under the clamped edge, free center boundary condition. The clamped edge was accomplished with the soldered on copper bell wire. The corresponding resonant frequency for each mode is given. Modes 4-8 are shown. The vibrational modes for this boundary condition were difficult to find. The patterns were not perfectly circular which made it difficult to detect the Chladni pattern. Also, in the photos dead spots occur where the sand piled up and the plate was not moving at all.

Fig 36. Chladni patterns of the vibrational modes of the circular brass plate under the clamped edge, free center boundary condition.

BRASS PLATE: CLAMPED EDGE, CLAMPED CENTER

Figure 37 shows the Chladni patterns of vibrational modes of the circular brass plate under the clamped edge, clamped center boundary condition. The clamp used was 4 inches in diameter. Figure 38 shows the Chladni patterns for this same boundary condition but with the 2.39 inch diameter clamp used. The corresponding resonant frequency for each mode is given. Just like with the clamped edge, free center boundary condition, the low modes were hard to find, the rings were not perfectly circular, and there were dead spots.

Fig 37. Chladni patterns of the vibrational modes of the circular brass plate under the clamped edge, clamped center boundary condition. 4’’ diameter clamp used.

Fig 38. Chladni patterns of the vibrational modes of the circular brass plate under the clamped edge, clamped center boundary condition. 2.39’’ diameter clamp used.

ADDING WEIGHT TO THE BELL WIRE

Figures 39 and 40 show the Chladni patterns of vibrational modes of the circular brass plate under the clamped edge, clamped center boundary condition. Modes 4-8 are shown and the corresponding resonant frequencies are given. The center is clamped with the 2.39 inch diameter clamp and in both Figures 39 and 40. In Figure 39 the edge is clamped with the soldered on copper bell wire and a thin layer of lead trim. In Figure 40 the edge is clamped with the soldered on copper bell wire and heavier lead tubing. Figure

41 shows a close-up of the layer of lead trim (blue arrow) and the heavier lead tubing

(orange arrow). The lead was added to see if the weight of the outer edge clamp has an effect on the vibrational modes and resonant frequencies.

Fig 39. Chladni patterns of vibrational modes of circular brass plate under clamped edge, clamped center boundary condition. Light lead trim and 2.39’’ diameter clamp used.

Fig 40. Chladni patterns of vibrational modes of circular brass plate under clamped edge, clamped center boundary condition. Heavy lead tubing and 2.39’’ diameter clamp used.

Fig 41. A close-up of the layer of lead trim (blue arrow) and heavier lead tubing (orange arrow).

MELLOPHONE: CHLADNI PATERN

Using the Chladni technique on the bell column as described in the methodology and experiment section, a clear Chladni pattern reoccurred on the bell column at all frequencies. Figure 42 shows a picture of this reoccurring Chladni pattern. In the picture, two nodal points are visible on the bell column where the sand is piled up. There are red marks on the bell column underneath the nodal points. These marks were made to see if the sand moved from that position when the frequency was changed. The sand did not move significantly from those two red marks and thus those nodal points were constant at all frequencies the bell was driven at.

Fig 42. A picture of the reoccurring Chladni pattern that occurred at all frequencies.

The nodal point closer to the bell tells us something very interesting about the design of the mellophone bell. Because that nodal point never shifts it means that that nodal point acts like a clamp. The designers of the bell constructed the bell with the intention that at that point the vibrations would dampen and decay. This nodal point occurs where the bell begins to slope outward more dramatically. This drastic cornering of the bell is what creates this clamping behavior.

This is a clever feature of the design of the bell because if the vibrations were not clamped and dampened at that point, the bell would vibrate vigorously with any note played. This nodal point occurs where the diameter of the bell column reaches 3 inches.

This 3’’ diameter circular nodal point is analogous to having a 3’’ diameter circular brass plate clamped at its outer edge inserted at this location in the bell column. This scenario would produce the same clamping and nodal point that is occurring in the bell column in

Figure 42. Figure 43 shows the wave pattern overlaid on the Chladni pattern of the bell column.

Fig 43. Wave pattern overlaid on the Chladni pattern of the bell column. The dashed black line represents the normal line. The red dots show the location of the nodal points where the sand piles up and no movement is occurring. The yellow curve represents the standing wave traveling along the bell and bell column. The blue circle shows the circular cross section of the nodal point that acts as a clamp. The blue dashed line shows the diameter (3 inches) of this circular cross section where the nodal point occurs.

MELLOPHONE: DRIVING WITH PIEZOELCTRIC SESNOR

By using the microphone to record the acoustic spectrum created by the bell when driven by the piezoelectric sensor, two resonances of the acoustic spectrum were found.

One resonance occurred at 1109 Hz and the other at 347 Hz. Figure 44 shows the

62 resonance at 1109 Hz and Figure 45 shows the resonance at 347 Hz. The graph on the dynamic signal analyzer has the frequency in Hertz on the x-axis and the sound output in decibels on a logarithmic scale on the y-axis.

Fig 44. The resonance of the acoustic spectrum at 1109 Hz. Notice the peak at 1109 Hz on the dynamic signal analyzer. Red arrow points to 1109 Hz resonance.

Fig 45. The resonance of the acoustic spectrum at 347 Hz. Red arrow points to 347 Hz resonance.

These resonances align with notes played on the mellophone. The resonance at

1109 Hz aligns with Csharp 6 which occurs at 1109 Hz. The resonance at 347 Hz aligns with F4 which occurs at 349 Hz. The resonance that occurs at F4 is an open partial of the horn. An open partial mean that no valves on the instrument have to be depressed for the note to play. The designers of the bell created it this way so that one of the strongest resonances (347 Hz) aligns with an open partial on the horn. It is also important to note that F4 is one of the most common notes played on the mellophone. The fact that F4 aligns with a strong resonance of the acoustic spectrum explains why some musicians perceive the note to be more rich and lush (more resonant). The design of the bell itself, with the structural intent that a resonance occurs at an open partial, is what makes that note sound more resonant than others.

One of the strongest resonances of the acoustic spectrum was at 1109 Hz. To see how strong this resonance was, we drove the horn at other frequencies with the piezoelectric sensor (frequencies that were not 1109 Hz) to see if the 1109 Hz resonance showed up in the acoustic spectrum even though the horn was not being driven at 1109

Hz. The driving frequencies that produced the 1109 Hz resonance in their acoustic spectrums were exact fractions of 1109 Hz and this makes sense because it agrees with acoustic theory and the overtone series. For example, when the horn was driven at 554

Hz (exactly half of 1109 Hz) the 1109 Hz appeared in the acoustic spectrum. When the horn was driven at 221 Hz (exactly one fifth of 1109 Hz) the 1109 Hz resonance appeared in the acoustic spectrum.

Figure 46 shows the acoustic spectrum at these frequencies. The red arrow points to the 1109 Hz resonance in the acoustic spectrum. You can see that the 1109 Hz

64 resonance appears in all of the acoustic spectrums at these frequencies. It is important to note that the 1109 Hz is stronger than the actual frequency that the horn is being driven at

(note that the y-axis is on a logarithmic scale). This was not to be expected and proves that 1109 Hz is a very strong resonance when the bell is being driven. Because the horn can be driven at one frequency and still produce other resonances at other frequencies, most notably the very strong 1109 Hz resonance, shows that the resonances of the acoustic spectrum are coupled and therefore not normal modes. Normal modes require the condition that they are not coupled.

The fact that the 1109 Hz resonance can be produced (and that it is a very strong resonance at that) when the horn is driven at a frequency other than 1109 Hz is an interesting feature of the acoustics of the mellophone. It shows that vibrating the bell at one frequency can produce resonances of different frequencies in the acoustic spectrum that are even greater than the note that is driving the horn.

Fig 46. The acoustic spectrums of the horn when driven at certain frequencies. The frequencies the horn is being driven at by the piezoelectric sensor are listed underneath each graph. The red arrow points to the 1109 Hz resonance that is produced in each acoustic spectrum. The x-axis is frequency in Hertz and the y-axis is sound output in decibels and is on a logarithmic scale.

MELLOPHONE: PLAY TEST

Figure 47 shows the acoustic spectrum when the mellophone was played by an actual musician. The musician was playing a concert A3 (220 Hz). You can see in the acoustic spectrum that the overtones of A3 appear and are all very strong. One of the overtones in the spectrum is around 1109 Hz, however, this peak is now similarly sized to all the other overtones.

Fig 47. The acoustic spectrum when the mellophone is played by a musician playing A3

(220 Hz). The blue arrow points to the peak at 220 Hz and the red arrow points to the peak at 1109 Hz. The x-axis is frequency in Hertz and the y-axis sound output in decibels on a logarithmic scale.

ANALYSIS

CIRCULAR BRASS PLATE:

Table 5 lists all the resonant frequencies found from the Chladni patterns of the circular brass plate. The first column lists the mode number, this corresponds to the number of rings of each Chladni pattern. Columns 2-9 are labeled with the specific boundary conditions the plate was under. FF means free edge, free center. FC means free edge, clamped center. CF means clamped edge, free center. CC means clamped edge, clamped center. Also, in columns 3-4 and 6-9 the size of the clamp is listed in the first row. In columns 8 and 9, the type of lead used to clamp the outside edge is listed.

EFFECT OF CLAMPING AT CENTER

Looking at columns 2 and 3 of Table 5 and comparing the resonant frequencies, it is shown that modes with a clamp at the center are higher than modes that are free at the center. Graph 1 plots the resonant frequency versus the mode number for the FF boundary condition and the FC (using a 2.39 inch diameter clamp) boundary condition.

Looking at Graph 1 and comparing the FF and FC resonant frequencies, the lower numbered modes are similar for both boundary conditions. As the mode number increases, the more divergent the two data sets become. It shows that clamping at the center increases the resonant frequency compared to a free center and the magnitude of the increase in frequency between the two boundary conditions is greater for higher mode numbers.

MODE FF FC (2.39’’) FC (4’’) CF CC (2.39’’) CC (4’’) CC (2.39’’, CC (2.39’’, NUMBER (Hz) (Hz) (Hz) (Hz) (Hz) (Hz) light lead heavy lead trim) tubing) (Hz) (Hz) 1 N/A N/A 285 N/A N/A N/A N/A N/A 2 N/A 286 329 N/A N/A N/A N/A N/A 3 447 460 630 N/A N/A N/A N/A N/A 4 510 700 1033 471 695 960 620 673 5 740 1044 1556 715 932 1365 1014 950 6 1040 1450 2187 1097 1390 1905 1384 1328 7 1360 1930 2899 1283 1752 2640 1736 1748 8 1710 2474 N/A 1580 2276 3490 2266 2261 9 2140 3075 N/A N/A N/A N/A N/A N/A 10 2625 3758 N/A N/A N/A N/A N/A N/A 11 3123 4213 N/A N/A N/A N/A N/A N/A

Table 5. Lists resonant frequencies of modes of circular brass plate under the various boundary conditions . N/A means a resonant frequency for that particular mode number was not found.

Graph 1. Plot of the resonant frequencies of FF and FC circular brass plate versus the mode number. Shows that clamping at the center results in higher resonant frequencies.

Comparing columns 3 and 4 of Table 5, the resonant frequencies found using the

4 inch diameter clamp are greater than the resonant frequencies using 2.39 inch diameter clamp under the FC boundary condition. Graph 2 plots the resonant frequencies of the circular brass plate under the FF boundary condition and the FC boundary condition for both the 2.39 inch diameter clamp and the 4 inch diameter clamp. This shows that using a larger clamp at the center results in higher resonant frequencies.

Graph 2. Plot of the resonant frequencies of circular brass plate under the FF boundary condition and FC boundary condition with both the 2.39 inch diameter clamp and the 4 inch diameter clamp. Shows that using a larger size clamp at the center results in a higher resonant frequencies.

COMPARING ALL BOUNDARY CONDTIONS

Graph 3 plots the resonant frequencies of the circular brass plate for all the boundary conditions studied except the added weight of the bell wire. Graph 2 corresponds to columns 2-7 of Table 5. Columns 8 and 9 are not shown because the effect of the bell wire will be discussed in the next section.

RESONANT MODES 3600 3300 3000 2700 2400 2100 1800 1500

FREQUENCY (Hz) FREQUENCY 1200 900 600 300 0 4 5 6 7 8 MODE NUMBER

Graph 3. Plot of the resonant frequencies of the circular brass plate under various boundary conditions.

Graph 3 shows that the boundary conditions in order from highest resonant frequencies to lowest are: free edge, 4’’ clamped center; clamped edge, 4’’ clamped

72 center; free edge, 2.39’’ clamped center; free edge, free center; clamped edge, free center.

Comparing the free edge and clamped edge data shows that clamping the edge lowers the resonant modes. This decrease in resonant modes must be the reason why brass instruments, like the mellophone, have bell wire attached at the edge of the bell

EFFECTS OF WEIGHT OF BELL WIRE

To determine the effects of the weight of the bell wire, the percent difference between the clamped edge with just the bell wire and the clamped edge with the light lead trim was calculated. These results are shown in Table 6. The percent difference between the clamped edge with just the bell wire and the clamped edge with the heavy lead tubing was calculated as well. These results are shown in Table 7.

Looking at last column of Table 6 and Table 7 shows the percent difference is fairly low for both cases meaning that the weight of the bell wire does not significantly change the frequency at which the resonant modes occur. It is also important to note that at the higher modes, the resonant frequencies are so high that the percent differences between the modes with and without the extra weight is below 1% and the effect of the weight of the bell wire is negligible. This result occurs at the higher resonant frequencies because the nodes move radially outward with frequency.

MODE CC (2.39’’) CC (2.39’’, light PERCENT NUMBER (Hz) lead trim) DIFFERENCE (Hz) 4 695 620 11% 5 932 1014 8.4% 6 1390 1384 0.4% 7 1752 1736 0.9% 8 2276 2266 0.4% Table 6. Lists the resonant frequencies of the clamped edge, clamped center boundary condition with and without the extra weight of the light lead trim. The last column shows the percent difference between the two sets of data.

MODE CC (2.39’’) CC (2.39’’, heavy PERCENT NUMBER (Hz) lead tubing) DIFFERENCE (Hz) 4 695 620 3.2% 5 932 1014 1.9% 6 1390 1384 4.5% 7 1752 1736 0.2% 8 2276 2266 0.6% Table 7. Lists the resonant frequencies of the clamped edge, clamped center boundary condition with and without the extra weight of the heavy lead tubing. The last column shows the percent difference between the two sets of data.

COMPARING THEORETICAL CALCULATIONS WITH EXPERIMENTAL

RESULTS

In the background section, equation 2 is the equation used for calculating resonant frequencies for circular and rectangular plates. The variables in equation 2 and equation 3 for my experiment are listed in Table 8.

VARIABLE VALUE a 0,15 m E 1x1011 N/m2 ν 0.33 h 3.8x10−4 m ρ 8500 kg/m3 ϒ 3.23 kg/m2 Table 8. Lists of variables and their corresponding values for equations 2 and 3.

Plugging in the values from Table 8 and the 휆푖푗 values from Table 3 and Table 1 into equation 2, the resonant frequencies for a completely free circular brass plate and a clamped edge, free center circular brass plate were calculated. The calculated resonant frequencies for the completely free and clamped edge, free center circular brass plate are shown in Table 9 and Table 10, respectively, alongside the experimental resonant frequencies found using Chladni patterns. The relative error percentage between the experimental results and theoretical calculations are also shown in Table 9 and Table 10.

Only modes 1-10 could be calculated for the both boundary conditions because Table 3 and Table 1 only provided the dimensionless frequency parameters for the first 10 modes.

MODE NUMBER CALCULATED EXPERIMENTAL RELATIVE

푓푖푗 푓푖푗 ERROR (Hz) (Hz) PERCENTAGE 1 25.62 N/A N/A 2 108.74 N/A N/A 3 247.67 N/A N/A 4 442.8 447 0.92% 5 693.64 510 36% 6 1000.27 740 35% 7 1263.75 1040 30% 8 1779.95 1360 31% 9 2252.72 1710 31% 10 2781.35 2140 29% 11 N/A 2625 N/A 12 N/A 3123 N/A Table 9. Lists the calculated resonant frequencies and the experimental resonant frequencies for the free edge, free center circular brass plate. N/A means the result could not be found.

MODE NUMBER CALCULATED EXPERIMENTAL REALTIVE

푓푖푗 푓푖푗 ERROR (Hz) (Hz) PERCENTAGE 1 28.82 N/A N/A 2 112.19 N/A N/A 3 251.35 N/A N/A 4 446.21 471 5.3% 5 696.76 715 2.6% 6 1003.00 1097 8.6% 7 1364.92 1283 6.4% 8 1782.53 1580 12.8% 9 2255.83 N/A N/A 10 2784.78 N/A N/A Table 10. Lists the calculated resonant frequencies and the experimental resonant frequencies for the clamped edge, free center circular brass plate. N/A means the result could not be found.

Looking at the results in Table 9 and Table 10 shows that the experimental data matches the theoretical calculations. The calculated and experimental resonant frequencies are on the same order of magnitude and the relative error between the experimental and theoretical results is low for both boundary conditions.

Looking at the values of the relative error in Tables 9 and 10 and the Chladni patterns that correspond to them, the uncertainties can be explained. By looking at the

Chladni patterns, we know that i, the number of nodal circles, is correct because we can count with accuracy how many there are on each Chladni pattern. We know that the source of error in the function generator is only 1 Hz. The most probable source of

77 uncertainty when comparing the experimental results to the calculated frequencies is that the modes are “mixed”. This means that the modes are not entirely axisymmetric and that other modes are present. You can even see in the Chladni patterns that some of the patterns have scallops and dead spots. I suspect that those are other, elliptical, modes having an effect on the mode. Brass is not an isotropic material because how it is rolled and created and explains why the other modes are present. The experimental results will not exactly match the calculated resonant frequencies for this reason.

It is important to understand how high the Q factors were in all these experiments and thus how narrow the frequency ranges are where the modes occurred. The brass plate was driven with the red speaker using the function generator and the amplitude of the vibrational motion was measured using the piezoelectric sensor attached to the edge of the plate. The sensor voltage was displayed on the oscilloscope. Three resonances and the corresponding full width at half maximum were measured to obtain values for Q, the quality factor. The Q factor for a resonance found at 931 Hz was 155.2. For a resonance at 720 Hz the Q factor was 60 and for a resonance at 438 Hz the Q factor was 36.5. This not only shows that the Q factors in these experiments were high, but shows that higher modes are more narrow than lower modes because the Q values have a direct relationship with frequency.

MELLOPHONE

By comparing the acoustic spectrums when the piezoelectric sensor was driving the horn at 221 Hz (Figure 46B) compared to when the bell was being played by a musician at 220 Hz (Figure 47) you can see how insignificant the effect of the vibrating

78 bell is. In Figure 46B when the bell is vibrated by the piezoelectric sensor, the 1109 Hz is a very strong resonance compared to any other frequency in the acoustic spectrum. Also, not all the overtones are seen in the acoustic spectrum when the piezoelectric sensor is driving the bell. On the other hand, when you look at Figure 47 when the musician is driving the horn, the peak of the 1109 Hz resonance is comparable in size to all the other resonances in the acoustic spectrum. Also, all the overtones are present when the musician plays the horn compared to when the piezoelectric sensor is driving the horn.

This proves that resonances in the air column when the musician plays overpowers any resonances produced by the vibrating bell. Thus, the resonances of the bell are negligible compared to the resonances of the air column produced by musicians. Resonances in the bell may cause the musician to perceive the note as more full and resonant but they do not actually significantly affect the acoustic spectrum because they are so minute compared to the resonances of the air column.

CONCLUSION

Piezoelectric sensors, speckle interferometry, and Chladni patterns were used to vibrationally analyze the resonant modes of a circular brass plate and mellophone. From the Chladni patterns of the circular brass plate is was found that clamping at the center of the plate increases the resonant frequencies compared to a plate free to vibrate at the center. The magnitude of this increase in resonant frequencies between the two boundary conditions is greater for the higher modes. These Chladni patterns also showed that a larger size clamp at the center of the plate corresponds to higher resonant frequencies. It was also discovered that clamping the edge of the plate lowers the resonant frequencies of the modes. This discovery explains why bell wire is added to brass instruments.

By looking at Figures 36-40, the Chladni patterns are not perfectly circular, they look scalloped. By clamping the outside edge of the plate with the bell wire, the modes become coupled. They are no longer perfectly circular. Coupling of the modes was also noticed in the resonances of the acoustic spectrum of the mellophone, proving that the modes were not normal.

Chladni patterns of the bell column of the mellophone showed that there are two nodal points that stay at the same position on the bell column no matter what frequency the mellophone was being driven at. This shows that the nodal point closer to the bell acts like a clamp that dampens vibrations to the bell.

Two of the strongest resonances of the acoustic spectrum of the mellophone occurred at 1109 Hz (Csharp 6) and 347 Hz (F4). The design of the mellophone bell produces a strong resonance at an open partial (F4) on the horn. F4 is a common note played on the mellophone and the fact that this note aligns with a strong resonance of the

80 acoustic spectrum explains why some musicians perceive the note to be more rich and lush (more resonant). However, it was found that resonances in the air column when the musician plays overpower any resonances produced by the vibrating bell. Resonances in the bell may cause the musician to perceive the note as more full and resonant but they do not actually significantly affect the acoustic spectrum.

ACKNOWLEDGMENTS

I am very grateful for this thesis project because it allowed me to combine two of my passions: music and physics. I would like to thank my thesis advisor, Dr. Martin

Kordesch, for all of his support and expertise throughout this entire project and through my undergraduate education. I would also like to thank Dr. David Drabold and Dr.

David Tees for all the advising and support over the years. I would also like to thank

Doug Shafer for making the disks and soldering on the bell wire. I would also like to thank all my music educators and physics educators who helped me get to this point. And lastly, I would like to thank the Honors Tutorial College for granting me the opportunity to complete this project.

WORKS CITED

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APPENDIX

Listed below are the copyright approvals to use images from other studies in my thesis.

Fig 48. Permission from Chatziioannou to use Figure 4 in my thesis.

Fig 49. Permission from Richardson to use Figures 7 and 10 in my thesis.

Fig 50. Permission from Materials Today to use Figures 12-14 in my thesis.

Fig 51. Permission from Journal of Fluids and Structures to use Figures 15 and 16 in my thesis.

Fig 52. Permission from International Journal of Solids and Structures to use Figure 17 in my thesis.

Fig 53. Permission from Journal of Sound and Vibration to use Figure 18.