Finite Element Modeling of a Slewing Non-linear Flexible Beam for Active Vibration Control with Arrays of Sensors and Actuators

Abstract

In this article, a new finite element model (FEM) of an EulerBernoulli beam, developed through an absolute nodal coordinate formulation (ANCF), is presented for simulation and analysis of the performance of surface-bonded piezoelectric actuators in suppressing non-linear transverse vibrations that are induced by very fast slewing. The elastic deformations experienced are an order of magnitude larger than cases considered to date, and the model employs a unique cubic spline approximation to the beam's deformed elastic line that is in terms of node positions and curvatures. To ensure relevant commentary on the vibration suppression properties of the distributed piezoelectric actuators, a material damping model was introduced in the continuum equations to capture the non-linear damping of the very slender beam that is observed in experiments. Following the ANCF methodology, the constitutive damping moment is formulated in terms of the absolute nodal coordinates with care taken to ensure the calculation is singularity free. Galerkin's method of weighted residuals is applied to discretize the revised equations of motion derived for the beam continuum. The FE beam model exploits a synergy between the twisted spline geometry and the lumped mass approximation to halve the size of the matrix equations that must be solved on each time step. However, this condensation of the matrix equations requires the use of interelement boundaries at the edges of the surface-bonded piezos. Using a single-link flexible manipulator as an example, a number of static and dynamic simulation examples that illustrate the validity of our FEM are presented, including comparisons to theoretical and other existing numerical solutions in literature. In addition, active vibration control examples are presented using proportional-and derivative-based hub motion and piezoelectric actuator controls in suppressing dramatic vibrations induced by fast slewing.

Static and dynamic finite rotation FE-analysis of thin-walled structures with piezoelectric sensor and actuator patches or layers

“…The variation of d is highly nonlinear and the scalar multiplication of an arbitrary vector  ∈ k 3 with the linearisation Δδ d can be expressed as…”

“…These so-called smart functional materials have been widely used as sensors and actuators in various industrial fields, such as aerospace, mechanical and civil engineering, as well as micro-and nano-engineering. Due to the inherent combination of sensing and actuation capabilities and high electro-mechanical coupling coefficients, piezoelectric layers or patches embedded or mounted on the host surface yield several applications, such as noise, vibration, shape and position control, non-destructive testing and health-monitoring systems (see [1][2][3]) among others.…”

On the methodology for evaluation of dynamic effects of a surface bonded piezoelectric patch for active vibration control

“…The nomenclature developed by Sorrentino et al (2003) is maintained throughout this work andmðxÞ, cðxÞ,kðxÞ are the mass, damping and stiffness per unit length, respectively, and are considered to be piecewise constant across the global domain. The need for an internal energy dissipation term,ĉðxÞ, in the model was clarified in Gilardi et al(2009) for the case of large deflections and is included in the linear analysis for completeness.…”

Analysis of electromechanical systems based on the absolute nodal coordinate formulation

“…As shown in the literature, the ANCF has already been successfully used for the coupled electromechanical analysis of some micro-electromechanical systems. ANCF beam elements within the framework of the Euler-Bernoulli beam theory assumption were used to simulate the performance of surface-bonded piezoelectric actuators [10]. Later, [28] successfully extended the ANCF approach to simulate piezoelastic laminated plates.…”