Smart Composite String with Integrated Piezoelectric Control: Nonlinear Timoshenko Beam Formulation

This paper develops a comprehensive nonlinear model for a smart composite string integrating ultra-thin piezoelectric sensors and actuators. The modeling framework is based on the geometrically nonlinear Timoshenko beam theory, incorporating von Karman strain terms to account for large-amplitude vibrations. The formulation couples mechanical and electro-mechanical fields to describe active control of the string’s vibrational response, enabling tuning of sustain and suppression of undesired overtones. Governing equations are derived through Hamilton’s principle, leading to coupled partial differential equations for transverse and axial motions. The proposed model provides a theoretical foundation for developing next-generation actively tunable string instruments. The model is implemented to evaluate the effects of pluck amplitude, pluck position, piezoelectric patch geometry, and electrical control parameters on tonal and dynamic performance. The tonal performance index (TPI), piezoelectric performance index (PPI), and acoustic output (SPL) are computed to quantify tonal richness, energy conversion efficiency, and control responsiveness. The results reveal that geometric nonlinearity leads to a measurable increase in both TPI (up to 4.5%) and PPI (up to 1.2%), while simultaneously reducing higher-harmonic energy by approximately 7%, indicating a non-trivial trade-off between tonal richness and harmonic complexity. Under active control, optimal performance is achieved for intermediate pluck positions (≈40% of string length) and moderate pluck amplitudes (≈10 mm), producing the highest TPI (~0.9) and PPI (~0.79) with sub-millisecond (ms) settling times.