Piano Hammer Modeling

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

String Excitation

Displacement-Wave Simulation

Nonlinear Spring Model

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The topic of linear systems theory is primarily about linear, time-invariant (LTI) filters. A linear filter is characterized by the property that its output-signal amplitude is linearly proportional to its input-signal amplitude. A time-invariant filter, or 'constant-coefficient' filter, performs the same filtering operation at all times. Only LTI filters can be characeterized by their impulse response, or their frequency response, or their poles and zeros plus a gain, or the like. Also, only LTI filters have the superposition property for audio signals, in which the filtering of an audio mix can be obtained alternatively by filtering each channel separately and adding the filtered channels together. Equivalently, only LTI filters exhibit no intermodulation distortion for audio signals. — Click for https://ccrma.stanford.edu/~jos/filters/Linear_Time_Invariant_Digital_Filters.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

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The BiQuad is a ubiquitous filter in audio signal processing, so named because its transfer function H(z)=B(z)/A(z) has a quadratic numerator polynomial B(z) and a quadratic denominator polynomial A(z). — Click for https://ccrma.stanford.edu/~jos/filters/BiQuad_Section.html

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The elementary impedance element in mechanics is the dashpot. — Click for http://ccrma.stanford.edu/~jos/pasp/Dashpot.html

The wave impedance in a propagation medium relates the force-related component of the wave motion to the displacement-related component of the motion. — Click for http://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

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The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

According to Hooke's law, the force F exerted by a spring stretched a distance x from its rest position is F=-kx. — Click for http://en.wikipedia.org/wiki/Spring_%28device%29

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Pressing a key on the piano causes a felt-tipped hammer to strike a vibrating string. — Click for http://www.speech.kth.se/music/5_lectures/

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

String Excitation

Displacement-Wave Simulation

Nonlinear Spring Model

Piano Hammer Modeling

The previous section treated an ideal point-mass striking an ideal string. This can be considered a simplified piano-hammer model. The model can be improved by adding a damped spring to the point-mass, as shown in Fig.9.22 (cf. Fig.9.12).

**Figure 9.22:** Ideal string excited by a mass and damped spring (a more realistic piano-hammer model).
$\includegraphics{eps/pianohammer}$

The impedance of this plucking system, as seen by the string, is the parallel combination of the mass impedance and the damped spring impedance $\mu+k/s$ . (The damper $\mu$ and spring are formally in series--see §7.2, for a refresher on series versus parallel connection.) Denoting the driving-point impedance of the hammer at the string contact-point by , we have

$\displaystyle R_h(s) \eqsp ms \left\Vert \left(\mu+\frac{k}{s}\right)\right. \eqsp \frac{\mu s^2 + ks}{s^2+\frac{\mu}{m}s+\frac{k}{m}}. \protect$

(10.19)

Thus, the scattering filters in the digital waveguide model are second order (biquads), while for the string struck by a mass (§9.3.1) we had first-order scattering filters. This is expected because we added another energy-storage element (a spring).

The impedance formulation of Eq.(9.19) assumes all elements are linear and time-invariant (LTI), but in practice one can normally modulate element values as a function of time and/or state-variables and obtain realistic results for low-order elements. For this we must maintain filter-coefficient formulas that are explicit functions of physical state and/or time. For best results, state variables should be chosen so that any nonlinearities remain memoryless in the digitization [364,351,560,558].

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Piano Hammer Modeling

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University