1. Near-field Acoustic Holography: The Frame Drum Grégoire Tronel Dr Steven Errede REU 2010

2. Overview 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

3. Data Acquisition (DAQ) System Lock-in Amplifiers: U scn P scn U mon P mon Thompson rods: X-Y translational stages for scan microphones Microphones: Above 2 (P,U) scan Below 2 (P,U) monitor Computer Processing the data

4. Eigen-Modes of a Vibrating Surface The vibration modes of a drum are mathematically described by the Bessel functions of the 1 st kind (J mn ). 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 J mn eigen-modes of a vibrating drum head – The number below each mode is the frequency ratio of a resonance mode normalized to J 01.

5. 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. 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 eigen- frequency. Expected modes: J 01 J 02 J 03

6. 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

7. 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) 426 Hz: J(0,2)? 662 Hz: J(0,3) Well defined J(2,1) nearby J(0,2) Coupled with J(2.2)

8. Mode-locked eigen-frequency and reference phase Drifting resonant frequency (left) for J 03, and its reference phase with negative parity (right), both in function of the measurement number (32x32). 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.

9. 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 non- linear mixing of nearby eigen-modes. Frequency dependent phase-shift effects due to LIA (<< 1 o ). Particle velocity microphone has frequency-dependent phase shift, but is relatively small (< 10 o ) 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: ma x + bv x + kx = F drive (F drive 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.

10. 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 J mn eigen-mode and its corresponding eigen-frequency. Finite stiffness, non-uniform tension across the drumhead, membrane- to-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.

11. 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.

12. Understanding the Apparatus 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. LIA: The signal from each one of these 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. 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

13. Microphone Response Calibration To extract physical quantities from the microphones, each microphones were absolutely calibrated in a L p = 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 L p = 94.0 dB sound field @ NTP: |p| = 1.0 Pa (RMS), |u| = 2.42 mm/s (RMS) EXTRA

14. Microphone + LIA Phase Calibrations EXTRA