Modelling of sound generation in pianos

Abstract

This project aims to quantify and understand the relative importance of different physical phenomena occurring in the sound generation mechanisms in pianos, and the role of the different components involved. To achieve this, a complete physical model of the piano is developed and implemented. Different components are considered from hammer-string interaction, string vibration and the concluding sound radiation. This work examines the transmission of vibration and subsequent sound generation in pianos. A state-space time domain modal model is used to model the interactions between the hammer, the strings and the soundboard. The piano action was characterised both numerically and experimentally, showing that a power law is sufficient to represent the collision with a rigid stop, and later on with a string. The effects of hysteresis were explored, appearing to be small. The implications of using different models are addressed as well as the significance of the variables in the modelling. The effects of using a flexible hammer shank during the collision with a rigid stop and a string, on the transmitted force to the soundboard are small. Linear and non-linear string models are considered in the different directions of vibration. The string models chosen correspond to the classical stiff string models to represent the transverse motions of the string, and axial vibration in a rod to simulate longitudinal motion. The string models are extended to include different orders of non-linearity arising from the stretching of the string. The soundboard is modelled including coupling between the directions of motion at the connection point with the string on the bridge, and hence, when the string is connected to it, its directions of motion are linearly coupled. Different soundboard representations are studied, from lumped element modelling approaches to FE models including several hundred modes. Simpler soundboard representations including a limited set of modes are analysed. For unidirectional representations in the transverse direction of the string, few modes are needed to represent the dynamical behaviour of the soundboard at lower frequencies and the general trend at higher frequencies. But to include more directions a reduced order model, including a larger amount of modes, can represent the soundboard mobility response better. Once the hammer-string-soundboard model is set up, different studies are presented. The effect of different excitation phenomena is analysed including changes in note, hammer impact velocity, the influence of adding directions of motion, adding strings, and the effects of the non-linear coupling between the transverse and longitudinal motion, yielding in phantom partials. The coupling in strings is addressed in two manners, by analysing the coupling between directions of one string due to the soundboard connection, or by considering the sympathetic vibration of unison strings due to the use of the una corda pedal. The una corda pedal can provide even and uniform decays of string response. Sympathetic vibration proves to be important if strings are close to each other or if they are identical. Bidirectional non-linear models using the transverse and longitudinal motions of the string showed that including second-order non-linear terms, and cross-coupling between the directions of motion is enough to represent the string response. The implications of using soundboard models that differ in complexity are studied to obtain the main characteristics of the soundboard dynamics while keeping the computational expenses as low as possible. Including the additional second transverse response has a negligible effect on the response. Vibration across the soundboard can be obtained through the transfer frequency response functions and the multi-directional forcing at the bridge connection. Finally, simplified sound generation models are implemented to simulate the sound pressure at a point in the surrounding fluid. The Rayleigh integral method was used and validated by means of FE modelling. Comparisons with measurements showed an acceptable agreement as the modelling can represent the main characteristics of the acoustic response. Parametric studies examined the influence of soundboard design parameters on radiated sound pressure. Key parameters included soundboard thickness, the number of ribs, and bridge height. Soundboard thickness proved to be important at lower frequencies, as a decreased thickness increased the level of the response and shifted the resonances lower. At mid-frequencies the frequency response is independent from thickness, while at higher frequencies the response is higher with decreasing thickness. The influence on the amount of stiffeners on the response is important at mid-high frequencies. Bridge height can improve the longitudinal and second transverse responses, relative to the transverse response.
The results from this PhD thesis could potentially inform piano design. In general, these insights improve the understanding of piano acoustics and provide a framework for design modifications in other stringed instruments, allowing for improved quality and versatility.

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Identifiers

ORCID for Pablo Martin Miranda Valiente: orcid.org/0000-0002-5939-7554

ORCID for Giacomo Squicciarini: orcid.org/0000-0003-2437-6398

ORCID for David Thompson: orcid.org/0000-0002-7964-5906

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