Illustration of the modal behavior of violins using "vibration maps" and screen animation. Discussion of certain results.
The results of experimental modal analysis can be presented in the form of "vibration maps" or screen animations. Modal analysis makes it possible to see the mode shapes ("modes"). This helps us to gain fundamental insight into the movements of the instrument that result from the eigenmodes of vibration and thus into its acoustic functioning. Going beyond hearing the sound and being able to "see" how it is produced boosts the violinmaker's intuitive sense of certain of the creation steps (arching, thickness graduation, etc.). The violinmaker gains a better sense of what happens to the instrument when it is played.
Here, we are dealing with the measured vibration behavior of the actual instrument and not a simulation. However, the amplitudes are greatly exaggerated and the vibration frequency is greatly slowed down. Click here to observe the eigenmodes of vibration for a violin by Antonio Stradivari. When we analyzed it, we found that this violin had an incredibly high mode density with more than 65 resonances in the frequency range up to 2000 Hz. Four of the typical corpus resonances of this instrument are shown in the following figure as a "vibration map" (left) and as a skeleton at maximum excursion with the position of rest (right) superimposed:
Typical mode shapes of the corpus resonances. Measured using experimental modal analysis (595 transfer functions).
Presentation using
"Vibration maps": Interpolation and plot of contour lines. Lines with the same amplitude (contour lines) are black and white. Nodal lines appear between the black and white contour lines. The top plate is on the left, back plate on the right, and under that the ribs on the bass bar side, lateral views and top view of the tailpiece and fingerboard, ribs on the soundpost side. View of the plates from the outside. Phase convention: Simultaneously expanding (black-gray) or simultaneously compressing (white-gray) corpus regions are considered in-phase.
Skeleton and animation: Tailpiece and fingerboard (top), top plate (center), back plate (bottom). Neutral position and maximum excursion.
Instrument: A0015 (Stradivarius 1712 "Schreiber")
Individual acoustic properties of a master or school
The modal analyses performed in the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING on various violins by Antonio Stradivari have revealed, for example, that there is clear agreement in the characteristics of essential corpus resonances. The frequency intervals and coupling between the different modes are conspicuously similar. In contrast, for example, the frequency interval between the two main corpus resonances (T1 and B1) is significantly larger in instruments by Guarneri del Gesu. Awareness of these and other similar acoustic traits of different masters (or schools) represents an innovative addition to traditional purely stylistic, construction-based description of bowed stringed instruments. Insights into the typical acoustic properties of a given master provide a good guideline for precision sound work when making new bowed stringed instruments.
B1 main corpus mode. Measurement of an Antonio Stradivari (left) and a Guarneri del Gesu (right). What we see here is the eigenmode of vibration (residue) of the B1 main corpus mode computed from the measured transfer functions (v/F). The measurement was made with the instrument in its playing state with damped strings. View of both plates from the outside.
Interpretation:
The mode shape for the B1 mode is relatively similar between the two violins. However, the Guarneri del Gesu exhibits higher corpus amplitudes. In contrast, in the Stradivarius instrument the B1 mode is coupled with a fingerboard torsion which is why the B1 mode is split into two strongly radiant modes (at 512 Hz and 523 Hz).
In both violins, the damping of the B1 mode is very similar (Stradivarius 1.53%; Guarneri 1.54%). This is an effective control quantity for the acoustic efficiency of the violin varnish that is used.
The eigenfrequency of this mode shape is fairly different in the two violins (at 512 Hz and 544 Hz). This eigenfrequency is a useful guideline for the tonal color of the instrument since the B1 mode serves as a sort of "modal barometer" during the construction process.
Technical details for this figure: Line spacing = 5(Hz*m)/(s*N). The line spacing is defined as the separation between two adjacent contour lines of the same type (the amplitudes are shown as absolute values.) Phase convention: Expanding or compressing plate regions of the corpus are considered in-phase and are shown with the same pattern (gray-white or gray-black). Labeling next to the eigenmodes of vibration: Mode #; Frequency (Hz); Damping (%); Max. neg. residue mag; Max. pos. residue mag. (Hz*m)/(s*N).
Fault diagnostics for "problem instruments":
By making measurements on many different instruments over the years, the MARTIN SCHLESKE MASTER STUDIO FOR VIOLINMAKING has compiled an extensive database of the typical acoustic and construction properties of violins, violas and celli. When combined with insight into the typical modal properties of high-quality instruments, modal analysis is a very powerful diagnostic tool for working on problem instruments. If modal abnormalities are detected (e.g. in the vibration amplitude distribution of the characteristic modes), it is possible to draw conclusions about construction- or adjustment-related defects (voice, bridge, etc.).