Generating Periodic Nonsinusoidal Vibrations with Inharmonicity of Overtones

Abstract

Damped bending vibrations of a high-Q string with fixed ends are analyzed. The model of a thin elastic homogeneous string with an electromagnetic pickup placed near the string is used to construct a second-order wave equation in partial time and coordinate derivatives that takes into account the elastic properties of string material, its length, tension and damping, the excitation intensity and location, and the pickup position with respect to the string. The solution as a function of time has the form of a series of higher harmonics (overtones) of fundamental-frequency vibration. It is shown that free oscillations have a nonsinusoidal shape slowly changing during the transient. To obtain a correct solution, several dozens of overtones having commensurable amplitudes should be taken into account. It is found that the oscillations are characterized by inharmonicity of overtones, which manifests itself in a progressive increase in the overtone eigenfrequency with the growth of its number in comparison with a value that is an integer multiple of the fundamental frequency. A nondamped process is accompanied by a periodically varying waveform of the carrier oscillation, and as the vibration decays, its initial nonsinusoidal shape approaches a harmonic waveform. To measure the process quantitative parameters, calibrated arrays of sound records of about 100 strings of a professional grand piano were used in the range of the fundamental tone frequencies exceeding five octaves. It is found that the equivalent Q factor of such oscillating systems varies from a few hundred to several thousand. To determine the inharmonicity parameter value, a modified cepstral transformation method is used. The basic idea of the modified method lies in finding a calibrated pre-distortion of the frequency axis after the first spectral transformation of the process time history samples at which the cepstral response is best located in the region of repiods. Measurements of the inharmonicity parameter for the above-mentioned array of records showed that its values are from a few hundredths to several percent, depending on the combination of the vibrating system parameters. It is shown that the presence of tens of higher inharmonic overtones propagating along the vibration system with different velocities leads to noticeable periodic variations in the process shape and magnitude, also in the case of undamped oscillation. The obtained results can be used for improving the quality of electronic synthesizers of quasi-periodic signals through bringing the musical synthesizer vibration structure and sound closer to those obtained in expensive natural instruments. A training laboratory work has been developed and introduced into the education course, in which the above-mentioned quasi periodic oscillation of a distributed high-Q vibration system with inharmonicity of overtones is synthesized, and a cepstral analysis of the obtained signals is carried out.