Sound quality analysis of cymbals
Toshiki NAKANISHIa)
Graduate School of Mechanical Engineering, Hosei University
3-7-2, Kajinocho, Koganei-shi, Tokyo, 184-8584, Japan
Tatsuhito AIHARAb)
Mitsuo IWAHARc)
Department of Mechanical Engineering, Hosei University
3-7-2, Kajinocho, Koganei-shi, Tokyo, 184-8584, Japan
Tetsuya SAKAId)
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University
1-1-1, Higasi-Mita, Tama-ku, Kawasaki, Kanagawa 241-8571 Japan
Gaku MINORIKAWAe)
Department of Mechanical Engineering, Hosei University
3-7-2, Kajinocho, Koganei-shi, Tokyo, 184-8584, Japan
ABSTRACT
Many researches on sound quality improvement have been conducted in various musicalinstruments focusing on human audibility. However, current researches indicate that ultrahigh-frequency over 20 kHz called human audibility limit affects sound quality. In addition,clarifying the mechanism of generating sound from the view point of manufacturing
process is necessary to improve the sound quality of musical instrument. Therefore, themain purpose of this research is to clarify the effect of the cymbal shape on the soundquality by using numerical simulations and experiments including over human audibility
range. The numerical simulations are carried out to calculate the sound pressure of various
cymbals using the Finite Element Method and the velocity potential method by varying thelength of the center part of the cymbal called cup, which is well known as important
portion to generate the high frequency sound in manufacturer’s experience. Recording
experiments are also conducted with actual cymbals equivalent to the simulation models.
As a result, the simulation results are found to correspond with the experimental ones.
a)email: 10x0078@stu.hosei.ac.jp
email: tatsuhito.aihara@hosei.ac.jpc)email: iwahara@hosei.ac.jpd)email: tetsuya_sakai@mth.biglobe.ne.jpe)email: minori@hosei.ac.jpb)Moreover, the numerical simulations are performed to clarify the effects of the design
parameters such as the height of the center on the sound quality.
1

INTRODUCTION

Recently, lower the noise level and reducing the vibration is important matter in various
industrial products. Meanwhile, sound quality improvement is essential for musical instruments
in order that generating sound becomes performance. Because of this, many researches on soundquality improvement have been conducted in various musical instruments. However, theseresearches have some problems.
1. Conventional researches1,2 on sound quality improvement is performed within an audible
frequency range below 20 kHz. However, current researches3,4,5 indicate that ultra-highfrequency over 20 kHz called human audibility limit affects sound quality. Therefore,
considering ultra-high-frequency over 20 kHz is necessary to improve sound quality.
2. Clarifying the mechanism of generating sound from the viewpoint of manufacturing process
is necessary to improve the sound quality of actual musical instrument.
Mentioned above, the subject of this study is cymbals which generate ultra-high-frequency over
20 kHz called human audibility limit and focus on the shape which is one of the elements toconstitute cymbal in order to contribute to manufacturing process. Furthermore, the effect for thecymbal shape on the sound quality is clarified by using numerical simulations and experiments
including over human audibility range.
2

NUMERICAL SIMULATION

2.1 Flow of numerical simulation method
The numerical simulation method6,7 for calculating the sound pressure of cymbal is
explained in this chapter. First, three dimensional of finite element models of cymbals are
created and natural frequencies and natural modes are calculated by eigenvalue analysis.
Secondly, natural frequencies, natural modes and modal damping ratio are inputted into theradiation sound forecast program. Modal damping ratio is identified by experimental modal
analysis because it cannot be found by theoretical calculation. Based on these data, the vibration
velocity of all nodes of the finite element model surface is calculated by theory modal analysis.
The sound is calculated using the velocity potential method from that vibration velocity and thegeometric shape of model. The surface vibration velocity and the sound are calculated in afrequency domain and frequency response function is calculated. Figure 1 shows this flowchartof numerical simulation.
2.2 Calculation of the sound by velocity potential method
The radiation sound forecast program calculates the direct sound, reflected sound anddiffracted sound. Here, influence of the reflected sound and the diffracted sound are little in thiscase. Each calculation methods are described as follows.
2.2.1 Calculation of the direct soundEquation (1) shows the velocity potential of the sound field in the half-space of the wall
surface forward in case point sound source exists on the surface of rigid wall.


where

A0 j (t kr )
e
2r

(1)

 : Velocity potential of the sound fieldr : Distance from point sound source
A0 : Strength of point sound sourcek : Wavenumber
j : Imaginary number

Expanding eqn. (1), following equation can be obtained.

p j
t

(2)

where
p : Sound Pressure : Density of medium

From eqn. (2), sound pressure can be calculated. The vibration surface is divided into micro area
elements in case vibration surface emits the sound. Direct sound by its surface is solved byconsidering the each micro area element emitting sound as new point sound source. Figure 2
shows calculation method of the direct sound.
2.2.2 Calculation of the reflected sound and the diffracted soundFigure 3 and 4 show each calculation method of sound. Reflected sound and diffracted
sound are calculated by using Huygens’ principle. In this calculation method, the sound fromsound source is caused to diffuse reflection with reflection surface. Wave front expanded fromsound source to the edge is divided into micro area elements, and reflected sound by thevibration of the surface is solved by considering the each micro area element emitting sound as
new point sound source. The calculation method of diffracted sound is also performed similarly,
too
3 EXPERIMENT
3.1 Vibration examination
The modal damping ratio is necessary in numerical simulation. However, the formulation of
the damping in theory is impossible to interact with various causes such as friction. Therefore,
value of damping is identified by analyzing the experimental data which was measured in thevibration examination. The method to clarify vibration characteristic for object by analyzing theexperimental data is called as experimental modal analysis. However, identifying modal
damping ratio by the nonlinear optimization method is difficult in this case because the cymbals

have much number of the natural mode. Therefore, modal damping ratio is manipulated tocorrespond with experiment in this study.
3.1.1 Experimental method
The cymbal is vibrated with the impulse hammer to measure the surface vibration velocity
in the Laser doppler vibrometer which installed 0.15m from the cymbal. The measurement withlaser is possible to obtain high reliability because it is not affect to the cymbal to be unnecessary
to touch it. Figure 5 shows experiment environment.
3.1.2 Identifying modal damping ratio
Figure 6 shows comparison result of the surface vibration velocity between experimental
result and numerical simulation one. As shown in Fig.6, the result of the calculation using
manipulated modal damping ratio is a tendency similar to the experimental results. However, it isdifficult to identify the modal damping ratio because the cymbals have natural frequency morethan 3500 in the frequency domain to 40kHz. Therefore, the modal damping ratio is not adjusted
at each natural frequency, but it is manipulated that the graph tendency of the calculation in all
frequency band is similar to experimental result shown in Fig.7 in this study.
3.2 Sound recording
Sound recording is conducted in actual cymbals equivalent to analytical models to comparewith numerical analytical results and experimental ones. Figure 8 and Table 1 show thedimensions of each cymbal. As shown in Table 1 and Fig.8, the length of the center part of thecymbal called as cup are mainly different. This difference may affect the frequency response
characteristics including the ultra-high-frequency sound. Figure 9 shows recording position andbeating point. The recording conditions are as follows. Analysis frequency is 50kHz which islimit of the sound level meter unit and measurement time is 0.64 sec due to be early damping of
the ultra-high-frequency. In addition, the frequency response function was calculated by thesound recording results and the measurement of the exciting force results to make a condition
same as the numerical results.
4 RESULTS
4.1 Experimental results
Figure 10 show the spectrogram of each cymbal in experiment. The horizontal axis of thespectrogram is time and the vertical axis is frequency. As shown in Fig.10, ultra-high-frequency
over human audibility damps instantly. Figure 11 shows the frequency analysis result ofexperiment. However, Fig.10 and Fig.11 are difficult to detect the difference of the highfrequency sound. Therefore, 1/1 octave band analysis are conducted to clearly compare thesound pressure. This analysis is most suitable for the sound evaluation in considering acoustic
sense because frequency response characteristic to feel in human ear are like equal ratio. In
addition, the calculation of the sound pressure level for measured sound in each band is possible
through a band filter determined in a standard.

4.2 Comparing experimental results and numerical calculation ones
It is possible to detect the difference of the frequency response characteristics by using
octave band analysis. Therefore, comparison of experimental results and numerical calculation
ones are conducted to verify the analysis method. Figures 12(a) and (b) show the numericalresults and the experimental ones. As shown in Fig.12, the tendency of the graph and relative
merit of the sound pressure level in two cymbals of numerical calculation results correspond withexperimental ones and the effectiveness of the numerical analysis is clarified. In addition, thelength of the center mainly affects the high frequency band.
4.3 Effect of the design parameter
The influence on high frequency sound by the other design parameters is examined bynumerical calculation. Here, influence of the height of the cup is confirmed by using radiation
sound forecast program. Table 2 shows that the model dimensions changed only height on thebasis of Model 2. As sown in Table 2, the height of the cup is fluctuated from Model 2 by 20%.
Figure 13 shows the results of numerical analysis of three models. As shown in Fig.13, Model B
generates sound higher than Model A in all frequencies band. However, Model C generates
sound lower than Model B in the high frequency band.
5

CONCLUSIONS

The main purpose of this research is to clarify the effect of the cymbal shape on the soundquality by using numerical simulations and experiments including over human audibility range.
This study can be summarized as follows:
1.
2.

3.

4.

6

Simulation method for calculating the sound pressure of cymbals by using the Finite
Element Method and the velocity potential method is proposed.
The experiments are carried out using the actual variant cymbals. The experimental resultshows the difference in the shape of the cymbal affects the frequency response
characteristics including the ultra-high-frequency sound.
The effectiveness of the numerical analysis is clarified by comparing the experimental
results with the numerical ones. As a result, it is possible to connect to the development of
products improvement by clarifying the mechanism of generating sound from the view pointof manufacturing process.
The numerical simulations are performed to clarify the effects of the height of the center on
the sound quality. The height of the center is proved to affect in the sound pressure level inall frequency band.
ACKNOWLEDGEMENTS

We would like to express our gratitude to Koide company cooperated to make cymbal
samples

7

REFERENCES

1.

Tsuji K, Kobayashi A, Ishiwata T, Iwahara M, Minorikawa G and Nagamatsu A, “Research
for the improvement of cymbal sound quality”, JSME annual meeting, vol.5, 125-126(2003)
in Japanese

2. Fujitsuka K, Kokubun K, Oe K, Iwahara M, Tanaka Y and Minorikawa G, Vibration
analyses of a Japanese traditional stringed instrument called shamisen, internoise (2012)3.

Oohashi T, Nishina E, Honda M, Yonekura Y, Fuwamoto Y, Kawai N, Maekawa T,
Nakamura S, Fukuyama H and Shibasaki H, “Inaudible high-frequency sounds affect brain
activity: hypersonic effect”, Journal of Neurophysiology, vol. 83: 3548-3558 (2000)

4.

Yagi R, Nishina E, Honda M and Oohashi T, “Modulatory effect of inaudible highfrequency sounds on human acoustic perception”, Neuroscience Letter, vol. 351: 191-195(2003)

5.

Yagi R, Nishina E and Oohashi T, “A method for behavioral evaluation of the "hypersoniceffect"”, Acoustical Science and Technology, vol. 24: 197-200 (2003) in Japanese

6.

Iwahara M, Kubota Y, Kubota K and Nagamatsu A “Accuracy Improvement of FEM Modelfor Impact Sound Prediction of Golf Clubs”, Japan Society of Sports Industry vol.23, No.1
1-10 (2013) in Japanese

7.

Saitou Y, Iwahara M, Kubota K and Nagamatsu A, “Accuracy Improvement of Batted BallSound Prediction for Golf Clubs”, Japan Society of Sports Industry vol.21, No2, 97-110(2011) in Japanese

Table 1 – Dimension of the cymbals used in experiment
Db , (mm)Dc , (mm)Namet, (mm)Model 1
1.55
144.0
58.5
Model 2
1.45
131.7
70.8

h, (mm)36.8
39.5

r, (mm)6.2
6.2

h1 , (mm)Table 2 - Dimension of the cymbals used in numerical simulation
Db , (mm)Dc , (mm)Namet, (mm)h, (mm)Model A
0.88
131.7
70.8
39.5
Model B
0.88
131.7
70.8
39.5
Model C
0.88
131.7
70.8
39.5

r, (mm)
6.2
6.2
6.2

h1 , (mm)

20.1
21.6

17.3
21.6
26.0

StartFinite element model

Eigenvalue analysis
Natural frequency・Natural mode

Modal damping ratio

Input data

Calculation of vibration caused by theoretical modal analysis（Frequency domain）
The geometryof the surface
elements

Calculation of the sound by velocity
potential （Frequency domain）

End

Fig.1 – Flowchart of the numerical simulation
Response point

r

θ

dS

Vibration surface S

Sound source

Fig.2 - Calculation method of direct soundElementary wave
Reflection surface

Direct soundr

R

Elementary wave

Direct sound

rR

θ

r

rD

θ

Sound source

Vibration surface
Vibration surface

Sound source

Response point

Response point

Fig.3 - Calculation method of reflected soundFig.4 - Calculation method of diffracted sound102

Calculation
Experiment

Velocity [m/s]

10

10

10

10

0

-2

-4

-6

0

50

100

150

200

250

300

350

400

450

500

Frequency [Hz]Fig.5 - Measure the surface vibration velocity

Fig.6 - Comparison result of the surface
vibration velocity to 500Hz102Calculation
Experiment

Velocity [m/s]

10

10

10

10

0

-2

-4

-6

0

0.5

1

1.5

2

2.5

3

3.5

Frequency [Hz]

4
x 10

4

Fig.7 - Comparison result of the surface vibration velocity to 40kHzt
CymbalFig.8 – Cymbal shapes
Recording point

Beating point
100mm

CymbalFig.9 – Sound recording point(a) Model 1
Fig.10 - Spectrogram of cymbals sound(b) Model 2

10Sound Pressure [Pa]

10
10
10
10
10
10

4

Model 1Model 2

3

2

1

0

-1

-2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency [Hz]

5
4

x 10

Fig.11 - Frequency domain Sound Pressure of two Cymbals
250

250

Model 1Model 2

150

100

150

100

50

50

0

Model 1Model 2200

Sound Pressure [Pa]Sound Pressure [Pa]

200

500

1000

2000

4000

8000

16000

0

32000

500

1000

2000

4000

8000

Center Frequency [Hz]Frequency [Hz](a) Numerical result(b) Experimental result
Fig.12 - Frequency domain Sound Pressure of two Cymbals
250
Model A
Model B
Model CSound Pressure [Pa]

200

150

100

50

0

500

1000

2000

4000

8000

Frequency [Hz]Fig.13 - Frequency domain Sound Pressure of three models

16000

32000

16000

32000