Identification of Nano-Beams Rigidity Coefficient: A Numerical Analysis Using the Landweber Method

Due to their supporting function, beams are one of the main elements in structural projects. With the intense technological development in the field of nanotechnology, beams at micro- and nanoscales have become objects of intense study and research interest, see for example [8]. In this approach, we analyze numerically the inverse problem of identifying the stiffness coefficient in micro-nano-beams as a function that implicitly depends on the fractal media map for the continuum from strain measurements. Such a problem is unstable with respect to noise in strain measurements, which is inherent in practical problems. We introduce the equations that compose Landweber's iterative regularization method as a strategy to obtain a stable and convergent approximate solution with respect to the noise level in the measurements. We show some scenarios with simulated data for identifying the stiffness coefficient for different noise levels in measurements and for different coefficient of transformation of fractal medium. The results found numerically show that Landweber's method is a regularization strategy for the problem of identifying the stiffness coefficient in micro/nano-beams

Methods for the identification of material parameters in distributed models for flexible structures

Theoretical and numerical results are presented for inverse problems involving estimation of spatially varying parameters such as stiffness and damping in distributed models for elastic structures such as Euler-Bernoulli beams. An outline of algorithms used and a summary of computational experiences are presented

Computational methods for the identification of spatially varying stiffness and damping in beams

A numerical approximation scheme for the estimation of functional parameters in Euler-Bernoulli models for the transverse vibration of flexible beams with tip bodies is developed. The method permits the identification of spatially varying flexural stiffness and Voigt-Kelvin viscoelastic damping coefficients which appear in the hybrid system of ordinary and partial differential equations and boundary conditions describing the dynamics of such structures. An inverse problem is formulated as a least squares fit to data subject to constraints in the form of a vector system of abstract first order evolution equations. Spline-based finite element approximations are used to finite dimensionalize the problem. Theoretical convergence results are given and numerical studies carried out on both conventional (serial) and vector computers are discussed

Inverse problems in the modeling of vibrations of flexible beams

The formulation and solution of inverse problems for the estimation of parameters which describe damping and other dynamic properties in distributed models for the vibration of flexible structures is considered. Motivated by a slewing beam experiment, the identification of a nonlinear velocity dependent term which models air drag damping in the Euler-Bernoulli equation is investigated. Galerkin techniques are used to generate finite dimensional approximations. Convergence estimates and numerical results are given. The modeling of, and related inverse problems for the dynamics of a high pressure hose line feeding a gas thruster actuator at the tip of a cantilevered beam are then considered. Approximation and convergence are discussed and numerical results involving experimental data are presented

Identification of an unknown spatial load distribution in a vibrating beam or plate from the final state

The theoretical and numerical determination of a space-dependent load distribution in a simply supported non-homogeneous Euler-Bernoulli beam and Kirchhoff-Love plate is investigated. The uniqueness of a solution to this inverse source problem is proved, whilst counter examples are constructed to discuss the conditions under which uniqueness holds. A convergent and stable iterative algorithm is proposed for the recovery of the unknown load source and a stopping criterion is also given. Several one-dimensional numerical experiments are considered to investigate the properties of the proposed iterative procedure

The identification of a distributed parameter model for a flexible structure

A computational method is developed for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the RPL experiment) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. The Euler-Bernoulli theory is used to model the vibration of the beam and treat the hose-thruster assembly as a lumped or point mass-dashpot-spring system at the tip. Using measurements of linear acceleration at the tip, it is estimated that the parameters (mass, stiffness, damping) and a Voight-Kelvin viscoelastic structural damping parameter for the beam using a least squares fit to the data. Spline based approximations to the hybrid (coupled ordinary and partial differential equations) system are considered; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are presented and discussed

On damping mechanisms in beams

A partial differential equation model of a cantilevered beam with a tip mass at its free end is used to study damping in a composite. Four separate damping mechanisms consisting of air damping, strain rate damping, spatial hysteresis and time hysteresis are considered experimentally. Dynamic tests were performed to produce time histories. The time history data is then used along with an approximate model to form a sequence of least squares problems. The solution of the least squares problem yields the estimated damping coefficients. The resulting experimentally determined analytical model is compared with the time histories via numerical simulation of the dynamic response. The procedure suggested here is compared with a standard modal damping ratio model commonly used in experimental modal analysis

A unified framework for approximation in inverse problems for distributed parameter systems

A theoretical framework is presented that can be used to treat approximation techniques for very general classes of parameter estimation problems involving distributed systems that are either first or second order in time. Using the approach developed, one can obtain both convergence and stability (continuous dependence of parameter estimates with respect to the observations) under very weak regularity and compactness assumptions on the set of admissible parameters. This unified theory can be used for many problems found in the recent literature and in many cases offers significant improvements to existing results