The 21st International Congress on Sound and Vibration
13-17 July, 2014, Beijing/China
A STUDY ON CHIRPING SOUND OF COLLIDING STEEL
BALLS
Yongkyun Lee
Korea Minjok Leadership Academy, Gangwondo, Korea 223-823
e-mail: [email protected]
This paper is about an unusual sound that is created when two steel balls collide. When two
hard steel balls, or similar, are brought into contact with each other, a chirping sound may be
produced. Based on the basic observation, we established a theoretical model about an inelastic,
vertical collision of two steel balls. Then, experiments were conducted in two steps. First, the
preliminary experiments were to define the chirping sound, which can be differentiated from
normal impact sound. Second, the main experiments were to find the validity of the theoretical
model. While the chirping sound can be heard by human ears, the analysis of measured sound
waves provided a clear understanding of the fundamental characteristics of the chirping sound.
Through the experiments, we defined the chirping sound as the sound produced due to the
series of consecutive collisions during which the time interval between collisions decreases.
Also, by controlling some possible variables, we verified that our theoretical model can explain
the mechanism and nature of the chirping sound.
1.
Introduction
In many games and computer generated motion films, the collisions of hard materials are quite
usual. At the moment of every collision the software should generate sounds to improve the sense of
realism. To synthesize the sound on computer simulated environments, we need a theoretical model
with variable parameters.
Research on the vibration and sound of collision is not new. Nishimura and Takahashi explained
the impact sound of steel ball collision both in theoretical perspective and experimental perspective1.
In their research the collision is assumed to be elastic. They first investigated the mechanism of how
the colliding sound is produced, then carried out experiments with balls of diverse diameters. Based
on Hertzian contact stress theory, which explains the pressure between two different surfaces like the
surfaces of spheres, the paper states that the maximum value of sound pulse is proportional to the
(6/5)th power of relative collision velocity. They also found that the free vibration of the steel is very
short, causing the sound intensity of free vibration to be negligible compared to the impact sound of
the collision. Endo, et al. analysed sound radiation on collision of cylindrical objects2. Sound radiated
from a freely supported cylindrical object elastically collided by a sphere has been classified into the
single impulsive sound due to the rigid body motion of the solids and the periodic one due to the free
vibrational motion. Another research on the sound of cylindrical object collision was done by Yufang,
et al. In their paper, the Hertzian impact theory and the acoustic theory are used to study the sound
radiated from the impact of two cylinders side-to-side, and the theoretical result was confirmed experimentally3. Mascarenhas, et al. investigated the collision of rods with square and circular cross
sections4. By analysing the spectrum of the sound to observe the longitudinal and flexural modes of
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
the ringing rod and with mathematical models, they can tell the shapes of rods. Mehraby, et al. found
that when the hard steel balls collide, 1.1 percent of the impact energy is stored in the spheres as
vibrational energy and the vibrations do not have any role in creation of audible sound5. For the
softened balls, then found that 0.15 percent of the impact energy is stored, but the vibration have more
significant effect in the sound generation than rigid body motion.
While most of the previous researches have focused on the usual impact sound of collision, this
paper looks into the unusual “chirping” sound. The chirping sound is generated when two hard steel
balls, or similar, are brought into contact with each other. In this paper we understand and define the
chirping by first building a simple theoretical model of collision then conducting experiments to better
understand the model. These two steps would demonstrate that our research can explain the mechanism and nature of the chirping sound.
2.
What is the chirping sound?
To define the chirping sound, basic experiments were conducted. Through them, it was possible
to examine the mechanism of how the chirping sound is produced. After various collision experiments,
we found that the vertical collision, where the lower steel ball is on an elastic surface or spring, always
produces the chirping sound. Though, the chirping sound is clearly different from the normal impact
sound to human ears, we recorded the collision sound for later analysis. Figure 1 (a) shows the recorded sound wave generated by horizontal collision and Figure 1 (b) shows the sound wave with
chirping sound. As shown in Figure 1 (b), the chirping sound can be defined as the sound generated
by repeated collision of two steel balls or any hard materials, where the interval of the collision is
steadily decreasing.
Figure 1. The sound wave acquired from (a) horizontal collision (b) vertical collision
Figure 2 is the FFT result of the collision. Figure 2 (a) is an FFT result of single impact sound.
Figure 2 (b) is the mix of individual FFTs of all collisions in Figure 1 (b).
Figure 2. (a) FFT of a single collision sound and (b) the mix of FFTs of all collisions
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
As shown in Figure 2 and Figure 1 (b), we can be sure that the chirping sound is generated by
repeated collisions, and the sound wave of individual collision has almost the same frequency profil
e as the whole chirping sound.
To reassure the definition of chirping sound, we observed some more vertical collision sound
waves on various bases with different elasticity. While Figure 1 (b) is the sound wave recorded when
the lower steel ball is on a spring, two other cases of different base material, styroform and hardboard,
are shown in Figure 3.
Figure 3. The sound wave acquired from (a) styroform base (b) hardboard base
From Figure 1 (b) and Figure 3, the elasticity of the base material below the lower ball changes
the shape of impact sound wave as well as the intensity of the sound. That’s because the base material
absorbed some of the impact energy in different pattern. Though the chirping sound waves in Figure
3 are not clear as in Figure 1 (b), we still can tell them from the single impact sound even by human
ears.
3.
Theoretical model of chirping sound
A chirping is a series of the same collision wave pattern. Thus, we decided to focus on the
intervals of the pattern, not the individual collisions. Figure 4 describes our model of vertical collision.
In the left hand side of Figure 4 we assumed a lower ball is on a spring with infinite spring constant,
i.e., 𝑘 = ∞. And the upper ball is falling down to the lower ball from the initial height of H. After
colliding at the speed of V, it bounds up at speed of v up to h, where e is coefficient of restitution. In
the right hand side box in Figure 4, the three vertical bars describe simplified sound waves generated
by three collisions.
Figure 4. A simple model for vertical collision of steel balls in spring
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
The interval of nth chirping collision, 𝑇𝑛 equals 2𝑡𝑛 , where tn is an one-way travel time to the
highest position hn of upper ball before nth chirping collision. Because the ball speed 𝑣𝑛 at nth collision
is,
𝑣𝑛 = e ∙ 𝑣𝑛−1 = 𝑒 𝑛 ∙ 𝑣0 = 𝑒 𝑛 ∙ √2𝑔𝐻 ,
and 𝑣𝑛 = g𝑡𝑛 , the interval 𝑇𝑛 is defined as:
𝑇𝑛 =
2∙𝑒 𝑛 ∙√2𝑔𝐻
𝑔
(1)
Our theoretical model, equation (1), defining the chirping interval only depends on the coefficient of restitution of the colliding objects and the initial height, but has no relation with other factors,
such as the mass of colliding objects. We verified the model with experiments with different parameters.
4.
Experiment Results
To verify our theoretical model, we conducted many vertical collision experiments with various
parameters. The parameters include the coefficient of restitution of ball material, the mass of ball,
and initial height which controls the speed of collision.
As the three different independent variables were chosen, it was then necessary to decide the
appropriate dependent variable. The y-axis of the experiment result graph needs to be scientifically
meaningful. The chirping sound we perceive would feel clearer as the number of collisions occurring
during a unit time is larger. In other words, when the two balls collide a given number of times over
a shorter period of time, a clearer chirping would be produced. Considering the observation that the
sound recorded by every experiment contains more than 20 collisions, we would set the standard
number of collision as 20. Therefore, we would calculate the time it takes for the balls to collide 20
times with respect to three independent variables controlled.
Figure 5 is the diagram of the experiment setting for the research.
Figure 5. Experiment Setting
In Figure 5, the steel ball (A) is dropped from onto another steel ball placed on the spring (C).
The sound is measured by a microphone ECM-8000 (E). The setting uses an electromagnet (B) as a
trigger to drop the ball. Cutting electricity from the electromagnet lets the steel ball to fall. Its falling
height is controlled by the support jack (F). A paper tube (D) is placed around the spring and the ball
on the ground to prevent the fallen ball from slipping off to the floor.
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Based on the experiment setting, we first checked the chirping period with the coefficient of
restitution. We used balls with different materials, steel, ABS, and glass of which the coefficients of
restitution are 0.23, 0.58 and 0.65 respectively. We measured the time between the first collision and
the twentieth chirping collision. Initial height is 1cm, and the experiments were done three times.
Figure 6 shows the results.
Figure 6. Chirping period with respect to coefficient of restitution
In Figure 6, black dots mean the averages of three measured values. The measured time and Tn
defined by equation (1) has some difference for two main reasons. First, our model assumes the spring
constant to be infinite. Second, the measurement of the coefficient of restitution was conducted for
the first collision, with the effects of the spring neglected. The speed of the ball after collision in
reality with the spring would be relatively small compared to the assumed situation without a spring.
Hence, the real coefficient of restitution with the influence of the spring ignored must be larger than
one without consideration of the spring.
Despite the difference between the theoretical value and experimental data, we can see that the
coefficient of restitution has exponentially affects the chirping periods. If we use balls with lower
coefficient of restitution, we can observe clearer chirping sound
The second set of experiments shows the relationship of the chirping period and the mass of
balls. We used steel balls with mass of 28.3g, 44.7g, 67g, and 73.9g respectively. Figure 7 shows the
results.
Figure 7. Chirping period with repect to ball mass
As in Figure 7, experiments showed that there is no correlation between the chirping period and
ball mass. When the mass of ball is larger, the more kinetic energy is applied to the collision and then
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
we can observe higher amplitude of the sound wave. But the chirping period have no linear or nonlinear relation with the mass of balls.
The third set of experiments was about the speed of initial collision. The initial collision velocity 𝑣0 is determined by the initial height, or the vertical distance H in Figure 4. Figure 8 shows the
results.
Figure 8. Chirping period with repect to initial height
As in Figure 8, we can find very clear correlation between chirping period and initial height.
This is quite obvious result.
Through series of experiments with various parameters, we verified that our theoretical model
well describes the intervals of repeated collisions which produces the chirping sound.
5.
Conclusion
The chirping sound is an unusual audible sound when two steel balls are brought into collision.
Chirping sound is generated by series of collision. In this paper, through various experiments, we
defined the chirping sound and build a theoretical model for it. To verify our model, we conducted
colliding experiment with different parameters. The larger coefficient of restitution of the colliding
material has, and the higher the initial colliding velocity is, the longer chirping is. Long chirping is
more audible to human ears. The mass of colliding objects does not affect the intervals of repeated
collision. Our theoretical model can be used in many computer games and computer generated motion
films, where the collisions of hard materials are quite usual. At the moment of every collision the
software should generate sounds to improve the sense of realism. Our theoretical model is useful to
synthesize the collision sound.
Our model and experiments have some limitation. We assumed the spring constant of the base
material as an infinite value. In real vertical colliding situation, lower ball moves up and down, and
this phenomenon would complicate the theoretical model. We leave it as a further research to model
more realistic collision.
REFERENCES
1
2
Nishimura, G., and Takahashi, K. Impact Sound of Steel Ball Collision, Journal of the Society
for Precision Mechanics of Japan, 4(28), 220-230, (1962).
Endo, M., Nishi, S., Nakagawa, M. and Sakata, M. Sound radiation from a circular cylinder
subjected to elastic collision by a sphere, Journal of Sound and Vibration, 75(2), 285-302,
(1981).
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
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4
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Yufang, W., and Zhongfang, T. Sound radiated from the impact of two cylinders. Journal of
sound and vibration, 159(2), 295-303, (1992).
Mascarenhas, F. M., Spillmann, C. M., Lindner, J. F., and Jacobs, D. T. Hearing the shape of a
rod by the sound of its collision, American Journal of Physics, 66(8), 692-697, (1998).
Mehraby, K., Beheshti, H. K., and Poursina, M. Impact noise radiated by collision of two
spheres: Comparison between numerical simulations, experiments and analytical results, Journal of mechanical science and technology, 25(7), 1675-1685, (2011).
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