[Paper Review] Boundary Vibration Control of Strain Gradient Timoshenko Micro-Cantilevers Using Piezoelectric Actuators

Amin Mehrvarz, Hasan Salarieh+2 more|arXiv (Cornell University)|Dec 4, 2018

Nonlocal and gradient elasticity in micro/nano structuresMaterials Science37 references3 citations

TL;DR

This paper proposes a boundary vibration control strategy for strain gradient Timoshenko micro-cantilevers using piezoelectric actuators. By applying a feedback control law derived via Lyapunov stability theory and the Lasalle invariant set theorem—based on spatial derivatives of boundary states—the controller effectively suppresses vibrations, as validated through finite element simulations using a strain gradient Timoshenko beam element.

ABSTRACT

In this paper the problem of boundary control of vibration in a clamped-free strain gradient Timoshenko micro-cantilever is studied. For getting systems closer to reality, the force or moment exertion conditions should be modeled. To this end, a piezoelectric layer is laminated on one side of the beam and the controlling actuation is applied through the piezoelectric voltage. The beam and piezoelectric layer are coupled and modeled at the same time and the dynamic equations and boundary conditions of the system are achieved using the Hamilton principle. To achieve the purpose of eliminating vibration of the system, the control law is obtained from a Lyapunov function using Lasalle invariant set theorem. The control law has a form of feedback from the spatial derivatives of boundary states of the beam. The finite element method using the strain gradient Timoshenko beam element has been used and then the simulation is performed to illustrate the impact of the proposed controller to the micro beam.

Motivation & Objective

• To address the challenge of vibration control in micro-scale Timoshenko beams accounting for strain gradient effects.
• To model the coupled dynamics of a Timoshenko micro-beam and a piezoelectric actuator layer using Hamilton's principle.
• To design a boundary feedback control law that ensures asymptotic stability and vibration suppression.
• To validate the controller’s effectiveness through finite element simulation of the strain gradient Timoshenko beam element.

Proposed method

• The coupled system of the Timoshenko micro-beam and piezoelectric layer is modeled using Hamilton's principle to derive dynamic equations and boundary conditions.
• A Lyapunov function is constructed to analyze system stability, and the control law is derived using the Lasalle invariant set theorem.
• The control input is formulated as feedback from the spatial derivatives of the boundary states (e.g., transverse deflection and rotation angle).
• The finite element method is employed using a strain gradient Timoshenko beam element to discretize the system for numerical simulation.
• The controller is implemented in a simulation environment to evaluate its performance in suppressing vibrations.

Experimental results

Research questions

• RQ1 How can vibration be effectively suppressed in a strain gradient Timoshenko micro-cantilever using boundary actuation?
• RQ2 What control law ensures asymptotic stability for a micro-scale beam with non-classical material behavior?
• RQ3 How does the inclusion of strain gradient effects influence the dynamic response and control design?
• RQ4 What is the impact of spatial derivative feedback on vibration suppression performance?

Key findings

• The proposed control law successfully achieves asymptotic stability and effective vibration suppression in the micro-cantilever beam.
• The controller relies on feedback from spatial derivatives of boundary states, which enhances sensitivity to local dynamic behavior.
• Finite element simulations confirm the controller’s robustness and effectiveness in reducing vibration amplitudes over time.
• The inclusion of strain gradient effects in the beam model significantly alters the dynamic characteristics, necessitating advanced control strategies.