Near-field Acoustic Holography:
The Frame Drum
Grégoire Tronel
Dr Steven Errede
REU 2010
Overview
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•
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The use of Near-field Acoustic Holography (NAH) over a vibrating
system enables the extraction of fundamental acoustic quantities such
as the complex acoustic pressure (P) and particle velocity (U).
From these can be derived many important properties of a sound field
such as the complex acoustic impedance (Z), the sound intensity (I),
the sound power, the energy density, as well as structural wave
number indications and phase information.
The NAH setup ultimately allows to image the vibration modes of a
drum membrane to significant accuracy.
Frame drum
Data Acquisition (DAQ) System
Computer
Processing
the data
Thompson rods:
X-Y translational
stages for scan
microphones
Microphones:
Above
2 (P,U) scan
Lock-in Amplifiers:
Uscn
Pscn
Umon
Pmon
Below
2 (P,U) monitor
Eigen-Modes of a Vibrating Surface
• The vibration modes of a drum are mathematically described by the
Bessel functions of the 1st kind (Jmn).
• By correct use of the phase-sensitive P/U microphones, one is able
to visualize the normal modes of vibration.
• Two adjacent regions oscillate 180° out-of-phase to each other.
First 12 Jmn eigen-modes of a vibrating drum head – The number below
each mode is the frequency ratio of a resonance mode normalized to J01.
Finding the Resonance Peaks:
Frequency Scan
• A characteristic mode of vibration can only be observable if the
membrane vibrates at a certain frequency called eigen-frequency.
• The purpose is to maintain the scan mics at constant position above the
magnets. Then sweeping the frequency over a suitable range will reveal
the modes.
Expected
modes:
J01
J02
J03
Left:
Pressure spectrum taken
with the driving force at
the center.
For an ideal drum, only
the (0,n) modes should
be accessible.
Each positive peak
represents an eigenfrequency.
Imaging the Modes of Vibration:
Spatial Scan
• Once the eigen-frequencies are known, one can stabilize the
magnet vibrations to a desired frequency, exciting the drumhead at
its corresponding mode of resonance.
• Then the translational stages supporting the P/U mics will scan the
drumhead allowing 3D representation of the vibrating membrane.
J(0,1)
J(0,2)
J(0,3)
First three ideal modes of resonance with magnets placed at the center of the drum
Obtained modes of vibration
• In order to maximize to study of the drum harmonic spectrum, we
conducted our experiment at three different positions of the driving force
along the membrane radius (center, half-radius, edge).
• Careful measurements of the eigen-frequencies revealed the presence of
few resonance modes between 175 Hz and 685 Hz (3 at center, 5 at edge).
• Nearby resonant frequencies generate coupling of eigen-modes.
Focusing on the three modes previously mentioned:
176 Hz:
J(0,1)
Well defined
426 Hz:
J(0,2)?
J(2,1) nearby J(0,2)
662 Hz:
J(0,3)
Coupled with J(2.2)
Mode-locked eigen-frequency and
reference phase
Correction of the drifting phase and frequency (primarily due to changes
in ambient temperature) is processed throughout the entire scan period
by the mode-locking monitor microphones.
Drifting resonant frequency (left) for J03, and its reference phase with negative parity (right),
both in function of the measurement number (32x32).
Causes of Phase Shift
• Propagation time effect – mics placed a height z above/below the
drumhead which gives rise to frequency-dependent phase shift  = -kz.
• Phase-shift effects of nearby/overlapping resonances, and possible nonlinear mixing of nearby eigen-modes.
• Frequency dependent phase-shift effects due to LIA (<< 1o).
• Particle velocity microphone has frequency-dependent phase shift, but
is relatively small (< 10o) over the frequency range 60 Hz < f < 4 KHz.
• The time-delayed response of the drumhead from its driven force? No,
because of the 1-d mechanical equation of motion: max + bvx + kx = Fdrive
(Fdrive is from the coil+magnets, the coil is driven by constant current NIC
which eliminates phase shifts due to inductance of the coil).
• However, all the known phase-shift effects are corrected in the offline
data analysis upon acquirement of the raw data.
Summary
• The Data acquisition system describes a phase sensitive setup for
acoustic holography.
• By first finding the eigen-frequencies, one can then scan the drum
surface by mode-locking to a desired resonant frequency.
• Many relevant physical quantities may be derived from measurements
of the complex pressure and particle velocity.
• Ultimately, a 3D image representation of the vibrating membrane
reveals the correlation between a Jmn eigen-mode and its corresponding
eigen-frequency.
• Finite stiffness, non-uniform tension across the drumhead, membraneto-shell coupling, asymmetric clamping of the drum shell, spatial
instability of the excited system, standing waves and interference near
the setup, constant drifts of the room conditions, all these may cause
the observed model to diverge from the theoretical model.
Acknowledgments
• I would like to thank Professor Steven Errede for
introducing me to the domain of physical acoustics,
and for his devotion and assistance throughout the
research.
• I also wish to thank Adam Watts for his help, Tony
Pitts as the REU coordinator, and Katie Butler for
sharing the workspace.
EXTRA
•
Understanding the Apparatus
microphones is sent to a Lock-in Amplifier
(LIA). Each LIA measures the amplitudes of
the real/in-phase component and the
imaginary/90°out-of-phase component of
a complex harmonic (i.e periodic) signal,
both relative to a stable sine wave of
reference.
Magnets: Two super-magnets are placed
above and below the membrane. A coil
situated beneath the membrane and aligned
with the magnets is driven by a sinusoidal
current. It induces an oscillatory magnetic
field which causes the magnets to move up
and down for any input frequency.
•
•
Microphones: Two pressure microphones
and two particle velocity microphones. One
of each kind is placed above the membrane.
These “scan mics” will scan the entire drum
surface by doing 32x32 measurements. The •
two remaining P and U microphones are
placed beneath the membrane. The
“monitor mics” control the frequency by
keeping track of the resonance.
Thompson rods: Motorized and
computerized translational stages to which
the P/U scan microphones are attached Accurate to a micrometer.
Data processing: Finally, all the information
concerning the near sound field of the drum
is processed by a computer through data
acquisition.
EXTRA
Microphone Response Calibration
• To extract physical quantities from the microphones, each
microphones were absolutely calibrated in a Lp = 94.0 dB
sound field at f = 1 KHz, using a NIST-certified Extech 40774
calibrator. Upon calibration their output voltage could then be
related to either pressure or particle velocity (expressed in
RMS Pa or mm/s, rather than arbitrary RMS volts).
• In a Lp = 94.0 dB sound field @ NTP:
|p| = 1.0 Pa (RMS), |u| = 2.42 mm/s (RMS)
EXTRA
Microphone + LIA Phase Calibrations