Mathematical Analysis of Vibration Suppression in Smart Structures Using Piezoelectric Materials

Abstract

This paper presents a comprehensive mathematical–simulation hybrid analysis of vibration suppression in smart structures using piezoelectric materials. The research combines Euler–Bernoulli beam theory, linear piezoelectric constitutive relations, finite element-based modal reduction, and optimal control strategies including PID, LQR, and H_? control. A detailed electromechanical model of a piezo-actuated cantilever beam is developed, and a worked numerical example with assumed material constants illustrates the induced strain, curvature, modal frequencies, and closed-loop vibration reduction. A simulation-style LQR analysis demonstrates a 76.4% reduction in peak vibration amplitude for the first mode. This study highlights the importance of intelligent material integration, optimal control design, and electromechanical coupling for modern vibration suppression and discusses future directions including nonlinear modelling and adaptive learning-based controllers.

KEYWORDS: Mathematical Analysis, Vibration, Beam, Piezoelectric, LQR.