COST Action FP1302 WoodMusiCK Short Term Scientific Mission Report Cello mobility and radiation study Timothy Wofford The bridge mobility and acoustic radiation transfer functions describe how a cello responds to a musician’s gestures. Mobility measurements may be modeled by a series of coupled damped harmonic oscillators, with individual modes providing frequency-dependent contributions having an a priori known functional form. On the other hand, radiation transfer functions have no a priori known functional form due to frequency-dependent directivity and room acoustics. Using indepen- dent component analysis to examine simultaneous measurements of mobility and radiation transfer functions, we attempted to identify the contribution of individual modes to the radiation transfer function. The modes of a vibrating system form a complete basis from which we can describe any excitation as a superposition. Tapping an arbitrary point on a cello with a small hammer results in a superposi- tion of all modes. Tapping different points on the cello results in different superpositions. By placing an accelerometer in a fixed position on the cello, or a microphone in a fixed position in the room, we can make measurements of these various superpositions. The contribution of an individual mode to a superposition is a mode-dependent function of frequency. The contributions of a mode to each superposition are a scalar multiples of the same function, which means that they are correlated. I hypothesize that the independent components of the measurements are simply related to the modes of the cello. We can test this hypothesis by comparing the mobility and radiation reconstruc- tion coefficients of suitably normalized independent components. George Stoppani, a violin maker based in Manchester, has made extensive measurements of this kind on violins and to a lesser extent on cellos. He has developed his own suite of modal analysis software for this purpose. The data necessary for the study described above is also useful for determining mode shapes. The mode shapes describe how the cello deforms when driven at modal frequencies and helps to identify the physical origins of features in the mobility functions. The rest of this report is structured chronologically as the work which was carried out each day is described.
Transcript
COST Action FP1302WoodMusiCK
Short Term Scientific Mission Report
Cello mobility and radiation studyTimothy Wofford
The bridge mobility and acoustic radiation transfer functions describe how a cello responds to a musician’s gestures. Mobility measurements may be modeled by a series of coupled damped harmonic oscillators, with individual modes providing frequency-dependent contributions having an a priori known functional form. On the other hand, radiation transfer functions have no a priori known functional form due to frequency-dependent directivity and room acoustics. Using indepen-dent component analysis to examine simultaneous measurements of mobility and radiation transfer functions, we attempted to identify the contribution of individual modes to the radiation transfer function.
The modes of a vibrating system form a complete basis from which we can describe any excitation as a superposition. Tapping an arbitrary point on a cello with a small hammer results in a superposi-tion of all modes. Tapping different points on the cello results in different superpositions. By placing an accelerometer in a fixed position on the cello, or a microphone in a fixed position in the room, we can make measurements of these various superpositions. The contribution of an individual mode to a superposition is a mode-dependent function of frequency. The contributions of a mode to each superposition are a scalar multiples of the same function, which means that they are correlated. I hypothesize that the independent components of the measurements are simply related to the modes of the cello. We can test this hypothesis by comparing the mobility and radiation reconstruc-tion coefficients of suitably normalized independent components.
George Stoppani, a violin maker based in Manchester, has made extensive measurements of this kind on violins and to a lesser extent on cellos. He has developed his own suite of modal analysis software for this purpose. The data necessary for the study described above is also useful for determining mode shapes. The mode shapes describe how the cello deforms when driven at modal frequencies and helps to identify the physical origins of features in the mobility functions.
The rest of this report is structured chronologically as the work which was carried out each day is described.
Monday, October 26, 2015On Monday we reviewed some of George Stoppani’s previous work with cellos and discussed which measurements would be interesting for our respective projects. We identified two cellos which we would like to measure: LAM1, a cello I brought from our lab, and GS2001, a cello built by George Stoppani in 2001. We decided to excite the LAM1 cello using a small impact hammer at 236 impact points spread over the front, back, ribs, neck, tailpiece, and bridge while the cello is supported in a vertical position by rubber bands underneath and a string around the scroll. The strings would be damped using cardstock and sponges. These measurements would allow us to compare bridge mobility and radiation transfer functions, to visualize the mode shapes below about 1500 Hz, and to visualize the motion of the bridge. A uni-axial accelerometer would be placed on the c-string corner of the bridge and an omnidirectional microphone with flat response would be placed at a position corresponding to the player’s left ear. The signal from the impact hammer would be split for record-ing by two computers. One computer would record the signal from the accelerometer and calculate the mobility. The other computer would record the signal from the microphone and calculate the radiation transfer function.
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Tuesday, October 27, 2015On Tuesday we prepared the instrument supports and painted markers on the LAM1 cello. Photos of the cello were taken and marker positions were recorded in the outline file for visualizing mode shapes. George set-up the two computers to take simultaneous measurements of audio or accelera-tion with the common excitation signal. George explained how he uses a small aluminum rod of a certain mass to calibrate the gain on his data acquisition system so that the largest acceptable impact signal yields the largest acceptable accelerometer signal at typical impact points. We began taking measurements, but had to stop early as I was too ill to continue.
Wednesday, October 28, 2015On Wednesday we finished measurements on the LAM1 cello and examined the data for obvious acquisition errors. I introduced George to the topics of Independent Component Analysis (ICA) and Blind Source Separation (BSS) 1 and their applications to Operational (or Output-Only) Modal Analysis (OMA) 2. The rest of the night was spent importing the data, and attempting to compare the independent components of the mobility measurements to the independent components of the radiation measurements. I had expected a clear correspondence between the independent compo-nents of the mobility measurements and those of the radiation measurements, but none was found. I was forced to conclude that the physically based vibrational modes of a cello do not correspond to the information based independent components of the mobility measurements.
Using four impact points on the front of the LAM1 cello, I calculated the independent component of the mobility functions using the FastICA algorithm, maximizing Gaussian negentropy 3. I then recon-structed the measurements from the independent components. I then rescaled the independent components so that the first impact point would have unit reconstruction coefficients. I then per-formed the same calculation on the corresponding radiation transfer functions. The results are presented below. Each row represents a measurement point, each column represents an indepen-dent component, and the entries of the table are the reconstruction coefficients. I had hoped to see common columns between the two sets of reconstruction coefficients, though not necessarily in the same order. That is I expected the column ICM1 to match one of the columns ICR1, 2, 3, or 4. This is clearly not the case.
It is possible that the analyzed impact points were poorly chosen, and that another set of impact points might give a more promising result. I plan to investigate this possibility at a later date.
1. Comon, Pierre, and Christian Jutten, eds. Handbook of Blind Source Separation: Independent component analysis and applications. Academic press, 2010.2. Poncelet, Fabien. “Experimental Modal Analysis using Blind Source Separation Techniques.” Diss. University of Liège. (2010).3. http://fr.mathworks.com/matlabcentral/fileexchange/38300-pca-and-ica-package
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Thursday, October 29, 2015On Thursday we discussed methods of extracting modal parameters. We discussed peak picking, circle fitting, complexity indicator functions, rational fractional polynomials, and global singular value decomposition. We talked about the role of the user in picking analysis windows, estimating the number of modes, and including residual terms necessary to model the data within the window. This discussion lead to ideas for automating some of this process through model stability diagrams, sliding analysis windows, and incorporating previous estimates. The picture below shows the mea-sured mobility functions from the top plate in black along with their reconstructions in red. The reconstruction is based on four modes identified through the global singular value decomposition (GSVD) method. There seem to be five modes in this analysis window, but the GSVD method with five modes does not separate the two modes at 198 and 200 Hz. The details of the picture demon-strate some of the realities of extracting modal parameters from imperfect measurements. Some resonance frequencies manifest as large movements across nearly all measurement points, as can be seen by the large peaks in nearly every mobility function around 169, 185, and 210. Some measurement points, such as those near the edges of the plates, have much lower signal-to-noise ratios, and are not fit well. These are the curves which stay near 0 dB. There are also cases where a resonance could be interpreted as noise except for a rather strong excitation at a small number of points. This is the case around 198 and 200 Hz where the tailpiece of the cello is active, causing points immediately around the bridge feet to be slightly active while the rest of the top plate hardly moves. From the figure, we can see a trend around 200 Hz across many measurements, so we might suspect a mode is present there. But it is very easy to interpret the trend as a single peak, as the two modes distinguish themselves at only a few measurement points in the top plate.
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Friday, October 30, 2015On Friday we made measurements on the GS2001 cello, repeating the data collection steps per-formed on Tuesday and Wednesday. When measuring the LAM1 cello, we concentrated on experi-mental design and taking measurements. Measuring the GS2001 cello, was an exercise in using George’s software, including setting up outline and test files, creating an organized directory struc-ture for recording data, setting data collection parameters, and extracting modal parameters. The aim was to be able to use his software by myself when I return to LAM so that we can continue to collaborate.
The picture below summarizes the use of George’s ModeFit program. The top left corner shows the data which is being imported, in this case the mobility measurements for the top plate of the GS2001 cello. The bottom left corner visualizes this data. The user moves the red vertical lines to select an analysis window. The lower right window shows the Analysis window: measurements are in black and their reconstructions are in red. At the bottom right of the Analysis window we can read the average reconstruction error, a measure of quality of fit. Poor fits can easily have 50% or even 1500% error. In the Analysis window, we can choose between several analysis methods, notably the rational fractional polynomial method and the demonstrated global singular value decomposition method. We can collect the results of our attempts to fit the data and view them in the TableFrm window in the top right corner. Notable entries in the TableFrm window are the modal parameters (frequency and damping) and the complexity, another measure of quality of fit. Complexity values less than 20 (preferrably less than 10) indicate a good fit.
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Saturday, October 31, 2015On Saturday, having some extra time before my train, we looked at mode shapes in an attempt to name the modes we identified earlier in the week. We found that the mode shapes of the cello are close to those of the violin. In the pictures which follow, blue regions represent motion away from the viewer, red regions represent motion toward the viewer, and white lines separating blue and red regions represent stationary nodal lines. The left outline is that of the top, the right outline is that of the back. The left of each outline is the bass side and the right of each outline is the treble. Thus we are looking at the exterior side of the top plate and the interior side of the back plate.
The picture below shows the mode shape associated with the peak at 169 Hz of the LAM1 cello, which we can identify as the mode historically called the CBR mode. In the CBR mode, the bass center bouts of the top and back plate move in one direction while the treble center bouts of the top and back plate move in the opposite direction. A modal line separates the bridge feet so that the bridge rocks back and forth.
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The picture below shows the mode shape associated with the peak at 185 Hz of the LAM1 cello, which we can identify as the mode historically called the B1- mode. In this mode, the top plate and back plate are generally moving in opposite directions, meaning air is breathing in and out of the f-holes. Note that the back is not very active compared to the top (light shades of blue compared to dark shades of red), the section between the f-holes tends to move in the opposite direction from the parts outside the f-holes, and that the treble foot of the bridge is sitting on a nodal line.
The picture below shows the mode shape associated with the peak at 210 Hz of the LAM1 cello. In this mode the lower half of both plates seem to be moving in the same direction while the top halves are breathing. The same flexing shape seen in the top plate at 185Hz is seen in the bottom plate at 210Hz, indicating the lower plate is stiffer than the top plate. Again we see the region between the f-holes moving in the opposite direction of the regions outside the f-holes. However, this time the bass foot of the bridge is sitting on a nodal line. According to George, this does not happen in violins, a detail which George seemed to find interesting.
The final picture below shows the top and back mode shapes at 200 Hz. A very similar figure applies to the mode shape at 198 Hz. The dark colors in this picture are an artifact of scaling. In fact all motions are quite small compared to movements in the tailpiece (not shown). The difference between the two modes lies in the relative phase of the tailpiece motion.
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The final picture below shows the top and back mode shapes at 200 Hz. A very similar figure applies to the mode shape at 198 Hz. The dark colors in this picture are an artifact of scaling. In fact all motions are quite small compared to movements in the tailpiece (not shown). The difference between the two modes lies in the relative phase of the tailpiece motion.
ConclusionsI spent a week with violin maker George Stoppani who actively does research on violins using modal analysis, for which he has developed a suite of useful software. Together we made simultane-ous measurements of sound radiation and bridge mobility on two cellos. Through our collaboration I learned more about the realities of modal analysis and George got some ideas for improving his software by automating his expert interactions. The measurements we made were useful to George in understanding how cellos behave differently from violins. Though more analysis is needed, the data collected should provide a definitive conclusion regarding my hypothesis that independent components of radiation transfer functions can be simply related to modal parameters, with the initial findings being negative. George and I look forward to future collaborations.
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10th November 2015
To whom it may concern,
This letter is to confirm that Timothy Wofford visited me in Manchester for a Short Term
Scientific Mission between Sunday 25 to Saturday 31 October as arranged.
Best regards.
George Stoppani
Maker of Bowed String Instruments
6 Needham Avenue, Chorlton-cum-Hardy, Manchester M21 8AA UK 0161 860 7386 (workshop) e-mail [email protected] www.stoppani.co.uk
