August 2016: An article on iterative numerical methods for finding the eigenvalues and eigenvectors of real (not complex) matrices. It covers the direct and inverse Power Methods, QR-Schur decomposition algorithm and Jacobi's method, plus reduction to Hessenberg form, leading to John Francis' implicitly shifted QR method. These algorithms are essential to solving vibration problems. There are several illustrative worked examples.
February 2016: An article on vibration analysis, aimed at modelling the frequency spectrum of vibrations of a cello. I explain the textbook theory of small oscillations, normal modes, forced oscillations and damping, linking these to finite element modelling. Particular attention is given to the damping of resonances of a cello. A lengthy Appendix describes my original measurements on a 'cello.
Here is another article (137 pages) on sound radiation from vibrating objects, including small revisions made in June 2015.
See my account of how we hear sounds and recognise pitch; how our ear work to amplify sounds and tell one frequency from another. I alsodiscusses the 'missing fundamental' by which we can hear the pitch of the lowest note on a viola or 'cello even though frequency spectrum analysis shows that the fundamental frequency is not there. Read it here.
Since 2012 I have been looking into the acoustics of the violin and viola. I have made several non-standard violins and violas, one shown above, from plywood as experiments.
I have a hobby interest in playing these instruments to an amateur standard. I have been intrigued and at times frustrated by the differences in playability and sound of different instruments. Moreover, there are conflicting anecdotes from experienced musicians and instrument technicians on what changes have what effect. So I decided to investigate for myself.
The whole project goes back to basics, starting with understanding the vibrational modes of a rectangular, flat wooden plate. I have used a combination of experimental measurements (tap sounds and Chladni figures) and finite element analysis. When there is good agreement between experiment and theory, I have taken the next step towards a realistic model of a violin, gradually building the wooden plates in a box, then into a violin shaped box. I make good use of finite element analysis using the LISA, Strand 7 and, more recently, the Mecway programs/
There are nine downloadable pdf files:
Article 9: Iterative numerical methods for real eigenvalues and eigenvectors of matrices. Informal introduction to some of the underlying mathematical methods for solving the matrices which arise in finite element modelling of vibrational frequencies. Lots of numerical examples.
Article 8: Forced vibrations, normal modes, measurements of resonance damping on a cello . This recounts the well-established theory of small oscillations using a simple example of a 3-mass-3-spring system. Matrix methods are used to find the normal, natural modes of vibration and their frequencies. When the system is disturbed it will oscillate in one of more of these modes. When the system is driven by a periodic force, it will follow the driving force but influenced by the natural modes. . If the driving frequency is close to a natural frequency, resonance will occur. Damping is energy loss due to friction and sound radiation. It blunts the resonances. All this theory can be combined with finite element methods and applied to violin- family instruments to predict how they will vibrate and radiate sound.
Article 7: Sound radiation from vibrating objects . This uses the physics and mathematics of acoustics to describe how sound moves from the surface of a vibrating object to a distant listener. The sound spreads out and contributions from different parts of the surface interfere. Small objects lock sound close to them in a reactive component of the wave energy, which explains why small musical instruments or loudspeakers are incapable for radiating loudly at low frequencies.
I have developed a computer program to model these effects based on the Kirchhoff and Rayleigh-Sommerfeld formulations of Huygen's principle of secondary wave sources. I show good results for radiation from a piston in a wall and radiation from vibrating spheres and spheroids to verify the accuracy of the program. Spheroids are much more complicated than spheres, so one of the appendices explains separation of Helmholtz's wave equation in spheroidal co-ordinates. The article also introduces integral equations and the boundary element method (BEM).
Article 6: How our ears hear the sound of a violin and iinterpret it as pitch and timbre. A venture into physiology
Article 5: April-July 2013, amended May 2014: Understanding the f-holes and Helmholtz A0 air resonance. At about C sharp on the D string of a violin the air in the f-holes is set into strong vibration and this is believed to give the instrument its full sound on the G and D strings. The paper shows experiments to display this resonance and determine how it is affected by the sizes of the f-holes and the violin / viola body. A simple mathematical model is described which gives a good prediction of the resonance frequency, taking account of the flexibility of the cavity walls.
In mid 2013 I added two new sections to this last paper dealing with the effects of air leaks, through cracks, into the resonant cavity, and with the amplitude of the sound radiated. It also details the effect of having the aperture at the edge of the violin instead of in the face of the top plate.
Article 4: February 2013: Building the plywood plates into a box. A step by step account of moving from a free plate to one mounted at its edges under various boundary conditions, to a complete rectangular box. I have used Strand7 to calculate the effects of varying plate thickness and size, changing wood grain orientation, and ungluing some of the joints in the box. LISA 8 is also used to model a box, with predictions compared with experiment.
Article 3 : This is a comparison between finite element theory (Strand7 program) and experiment. It uses the capability of Strand7 to model orthotropic sheet material with 2D plate-shell elements. Through a comparison with experiment I both determine the dominant elastic constants of plywood specimens and validate the finite element model. The agreement between FEA and experiment is very good. This gives confidence in the Strand7 program and also in the comparative method I propose for determining orthotropic elastic constants. Read this third paper here.
Article 2: The second article is a subsidiary study, the first step towards a systematic model of a viola or violin using finite element analysis (FEA).
I started with the vibration of a simple wooden plate, intending to compare experiment with theory. The experiments have looked at flat plates of 3 mm thick plywood, for which I determined the static elastic constants and also the resonant frequencies by spectrum analysis of tap sounds, and by exciting Chladni figures. The theory used FEA with the inexpensive Canadian program LISA-7. At the time this program had the limitation (removed in LISA 8) that it can only deal with isotropic materials, whereas wood is orthotropic (that is, it has a grain). I have asked myself whether it is possible to approximate orthotropic behaviour by some assemblage of isotropic elements which act as stiffeners along the grain direction. My moderate success is described in this downloadable pdf document.
Article 1: The earliest pdf file is a set of notes on a model non-standard viola which I built from plywood and hobby materials to start nvestigating the ergonomics, design and acoustics.
You can hear the sound of this early large plywood instrument here. I am not a good player, and I did not find it easy to play, so some notes are not truly in tune. Please just consider the tone from this plywood box, not my playing!. (Yes, I know the viola player jokes.) 1) Scale of C. 2) Arpeggio study.
Finite element model of a rectangular box where the top plate has become unglued along sections of the edge. Contours of displacement at the second lowest mode of vibration, 225 Hz.