## 5pMU1. Applying root-locus techniques to the analysis of coupled modes in piano strings.

### Session: Friday Afternoon, December 6

### Time: 2:00

**Author: Timothy S. Stilson**

**Location: Ctr. for Comput. Res. in Music and Acoust., Stanford Univ., 660 Lomita Dr., Stanford, CA 94305**

**Author: Scott A. Van Duyne**

**Location: Ctr. for Comput. Res. in Music and Acoust., Stanford Univ., 660 Lomita Dr., Stanford, CA 94305**

**Abstract:**

Previous work in the study of coupled piano string behavior, e.g., by
Weinreich and Nakamura, has focused analytically on the interaction of a pair of
coupled modes, noting that the rest of the string modes couple similarly. The
coupling of three or more modes, as occurs in sets of unison piano strings, is
studied. Equivalent circuits (modeling a finite number of modes) of coupled
piano strings can be written in state-space form and their eigenvalues analyzed
at various levels of detuning and coupling. Alternatively, the transfer
functions of the system can often be written in a root-locus analysis form,
which lends a control-theory perspective to the analysis. Often, an intuitive
understanding of pole movement under variations in a single parameter, such as a
single-mode frequency or coupling coefficient, can be developed. Many effects,
for example, two-stage decay, can be understood in terms of system pole
location. Three-mode coupling is explored and interpreted, as is variation in
coupling behavior at different string harmonics.

ASA 132nd meeting - Hawaii, December 1996