Proceedings of the Third International Conference on Mathematics and Natural Sciences (ICMNS 2010)
SIMPLE EXPERIMENT APPARATUS IN MEASURING SPEAKER VIBRATION AMPLITUDE USING LASER BEAM AND MIRROR Sitti Balkis1, Wahyu Srigutomo2, and Sparisoma Viridi3 1Magister
Program in Physics Teaching of Complex System Research Division 3Nuclear Physics and Biophysics Research Division Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung Jalan Ganesha 10, Bandung 40132, Indonesia 2Physics
3Email:
[email protected]
Abstract. Vibration amplitude of a speaker, on which a small mirror is attached on, can be measured by observing length of beam projection line. The accuracy of the experiments is increased by sampling more data for different tilt angle of the screen. The formulation is developed using light beam property, trigono-metry, and law of reflection. It is also reported that the calculated vibration amplitude depends on the voltage fed to the speaker. Keywords: experiments, vibration, amplitude, laser beam.
1
Introduction
Vibration measurements using laser beam are a common method in measuring vibration, such as that operates according to the principle of heterodyne interferometer [1] and uses optical fiber with laser diode interferometer [2]. Beside a normal, linear vibration, other type of vibration such as torsional and bending vibration can also be measured using laser beam [3]. An experiment using a speaker as source of vibration, a mirror as reflecting surface (attached on the speaker), and a screen on which the beam is projected, has been performed. Since reflected beams in different time are always parallel, variation of screen position can not be used. Then, the screen tilt angle θ is varied to obtain more data, and hopefully can increase the accuracy of measured vibration amplitude A. The formulation to obtain A from several measurements of length of beam projection line l for each θ, is derived using light beam property, trigonometry, and law of reflection [4].
2
Experiments
Experiments are conducted using several equipments as shown in Figure 1. A laser pointer is used as source of laser beam. As source of vibration, a speaker (18 W, 8 Ω, 15 cm) together with a signal generator is used. In order to have more adjustable vibration amplitude an amplifier is used to amplify the signal before it is fed to the speaker. Laser beam will be reflected by a mirror attached on the speaker and then captured by a tiltable screen.
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Simple Experiment Apparatus in Measuring Speaker
Figure 1 Speaker vibration amplitude measurement system: (1) tiltable screen, (2) amplifier, (3) oscilloscope, (4) signal generator, (5) laser pointer, (6) speaker, and (7) mirror. By varying screen tilt angle θ different lengths of beam projection line l are obtained. Illustration of beam projection line can be seen in Figure 2.
Figure 2 Beam projection line as captured by the tiltable angle. A graph paper is used as background to help the line length measurement.
3
Formulation
Vibration amplitude A can not be measured directly in our experiment but must be calculated from measured length of beam projection line l, which is illustrated in
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Balkis, S., Srigutomo W., and Viridi, S.
Figure 2. Angle of incident and reflected light is labeled with α which is measured from vertical direction. Angle of tiltable screen are labeled with θ which is also measured from vertical direction.
2
Figure 3 Beam projection line as captured by the tiltable angle. A graph paper is used as background to help the line length measurement. From Figure 3 it can be obtained that
b = c sin α
(1)
2 A = c cosα .
(2)
and
Substitution of Equation (2) into Equation (3) will yield
b=
2A sin α = 2 A tan α . cos α
(3)
We can also find the relation
2b cosα = d
(4)
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Simple Experiment Apparatus in Measuring Speaker
Substitution of Equation (3) into Equation (4) will lead to
(4 A tan α ) cos α = d ⇒ 4 A sin α = d .
(5)
With the tilt angle of the screen θ the beam projection line will be l. Then using sine rule in a triangle
l m = sin δ sin(2π − δ − α ) ⇒
l sin(π − α )
⇒m=
=
m m = sin[2π − (π − α ) − θ ] sin(α − θ )
l sin(α − θ ) sin α
(6)
can be obtained. Next step is to project m to d
⎛π ⎞ m cos⎜ − α ⎟ = d 2 ⎝ ⎠
⇒ m sin α = d .
(7)
Substitution of Equation (6) into Equation (7) will give us
l sin(α − θ ) sin α = d sin α ⇒ d = l sin(α − θ ) .
(8)
And substitution of Equation (5) into Equation (8) produces
4 A sin α = l sin(α − θ )
⇒
sin(α − θ ) 1 = A 4 sin α l
(9)
We can consider a new variable y and x, which are
y=
sin(α − θ ) 4 sin α
(10)
and
1 x= . l
(11)
Using Equation (10) and (11), Equation (9) can be written as
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Balkis, S., Srigutomo W., and Viridi, S.
y = Ax
(12)
From Equation (12) we can find A as gradient of the curve. This gradient will be calculated using linear regression.
4
Results and discussion
Voltage Vpp that is fed to the speaker will determine how large the vibration amplitude of the speaker. We use three different values of Vpp as shown in the following Table 1. For every value of Vpp several values of l are obtained as the tilt angle θ is varied. In all measurement the incident beam has α = 45 °. Table 1 Length of beam projection line for several value of Vpp. θ (°)
3.6
0 15 30 45
12 15 23 120
Vpp (V) 6 l (mm) 15 20 29 165
8 18 22 35 200
Using length of beam projection line tabulated in Table 1, Equation (10)-(12), and linear regression (with intercept is force at 0), it can be obtained that for Vpp with value of 3.6, 6, and 8 V the vibration amplitude A are 2.7, 3.5, and 4.1 mm respectively. These values are considered acceptable in order of magnitude. It also shows that Vpp is proportional to A, which is A = 0.3187Vpp + 1.5637 with R2 = 0.9991. Unfortunately this function A(Vpp) is not valid for Vpp = 0 since it should give A = 0. Further measurements are needed to calibrate value of calculated A.
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Simple Experiment Apparatus in Measuring Speaker
0.28
y 3 = 4.094x R 2 = 0.9524
0.23
y
0.18
0.13
y 2 = 3.508x R 2 = 0.9548
0.08
y 1 = 2.7416x
0.03
-0.02 -0.02
R 2 = 0.9489 0.00
0.02
0.04
0.06
0.08
0.10
-1
x (m ) Figure 4 Gradient of curve y against x for different value of Vpp: 3.6 V (y1, diamond), 6 V (y2, square), and 8 V (y3, triangle).
5
Conclusion
Vibration amplitude of a speaker can be measured indirectly using laser beam and mirror attached to the speaker. Different voltage Vpp (3.6, 6, and 8 V) fed to the speaker produces different amplitude A (2.7, 3.5, and 4.1 mm). It has been observe that A is proportional to Vpp with proportional factor 0.3187.
6
Nomenclature
Meaning of some variables and symbols used in this work are as shown below: A l V
= = =
vibration amplitude length of beam projection line voltage of input signal
α
=
tilt angle laser pointer
θ
=
tilt angle of the projection screen
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Balkis, S., Srigutomo W., and Viridi, S.
7 [1] [2] [3] [4]
References Steinlechner, S., Drabarek, P., and Van Keulen, M., Laser vibrometer for vibration measurements, United States Patent 5883715, URL http://www.freepatentsonline.com/5883715.html, (9 September 2010). Suzuki, T., Okada, T., Sasaki, O., and Maruyama, T., Real-time vibration measurement using a feedback type of laser diode interferometer with an optical fiber, Optical Engineering, 36(9), 2496-2502, 1997. Miles, T. J., Lucas, M., Halliwell, N. A., and Rothberg, S. J., Torsional and Bending Vibration Measurement of Rotors using Laser Technology, Journal of Sound and Vibration, 226(3), 441-467, 1999. R. Nave, Law of Reflection, HyperPhysics, URL http://hyperphysics.phyastr.gsu.edu/hbase/phyopt/fermat.html, (30 October 2010).
Acknowledgement Authors would like to thank the Ministry of Religious Affairs (Departemen Agama) of Republic of Indonesia and the Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, for supporting this work financially.
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