Coupled mech nical and electromagnetic modeling of eddy current sensors

To effectively monitor the vibration of blades in a rotating machine, a non-contacting method called blade tip-timing (BTT) has been used. The method is based on the analysis of the differential arrival times of the blades at sensors mounted on the stator to characterize the vibration amplitude and frequency of the blades. These sensors can also provide blade tip clearance measurement. A combination of these data can provide a robust condition monitoring approach for the early detection of blade cracks. Eddy current sensors (ECS) have shown great potential to assess the health of an engine without any need for direct access to the blade and therefore they are insensitive to the presence of any type of contaminant. Also, both tip timing and tip clearance of each blade could be measured by these sensors in real time and at high resolution. ECSs measure the magnetic field caused by eddy currents during the blade motion, and hence are a coupled mechanical and electromagnetic problem. An ECS on the casing of a machine has been modeled to fully understand how the dynamic response of the blade is measured by the sensors. Detailed 2-D and 3-D modeling and simulation of a rotating simplified bladed disk passing an ECS is presented. The effect of the variation of the rotation speed and the gap between the sensor and the blade tip on the accuracy of the measurement is investigated. Such an analysis will enable the reliable monitoring of blade damage during engine operation

Simulating eddy current sensor outputs for blade tip timing

Blade tip timing is a contactless method used to monitor the vibration of blades in rotating machinery. Blade vibration and clearance are important diagnostic features for condition monitoring, including the detection of blade cracks. Eddy current sensors are a practical choice for blade tip timing and have been used extensively. As the data requirements from the timing measurement become more stringent and the systems become more complicated, including the use of multiple sensors, the ability to fully understand and optimize the measurement system becomes more important. This requires detailed modeling of eddy current sensors in the blade tip timing application; the current approaches often rely on experimental trials. Existing simulations for eddy current sensors have not considered the particular case of a blade rotating past the sensor. Hence, the novel aspect of this article is the development of a detailed quasi-static finite element model of the electro-magnetic field to simulate the integrated measured output of the sensor. This model is demonstrated by simulating the effect of tip clearance, blade geometry, and blade velocity on the output of the eddy current sensor. This allows an understanding of the sources of error in the blade time of arrival estimate and hence insight into the accuracy of the blade vibration measurement

Non-Local Behavior of a Mixed-Mode Crack in a Functionally Graded Piezoelectric Medium

In this paper, the problem of a mixed-Mode crack embedded in an infinite medium made of a functionally graded piezoelectric material (FGPM) with crack surfaces subjected to electro-mechanical loadings is investigated. Eringen’s non-local theory of elasticity is adopted to formulate the governing electro-elastic equations. The properties of the piezoelectric material are assumed to vary exponentially along a perpendicular plane to the crack. Using Fourier transform, three integral equations are obtained in which the unknown variables are the jumps of mechanical displacements and electric potentials across the crack surfaces. To solve the integral equations, the unknowns are directly expanded as a series of Jacobi polynomials, and the resulting equations solved using the Schmidt method. In contrast to the classical solutions based on the local theory, it is found that no mechanical stress and electric displacement singularities are present at the crack tips when nonlocal theory is employed to investigate the problem. A direct benefit is the ability to use the calculated maximum stress as a fracture criterion. The primary objective of this study is to investigate the effects of crack length, material gradient parameter describing FGPMs, and lattice parameter on the mechanical stress and electric displacement field near crack tips

Nonlocal modeling of a Carbon Nanotube actuated by an electrostatic force

Carbon nanotubes (CNTs) are promising mechanical structures at the nano-scale which have attracted increasing attention due to their amazing mechanical, chemical, thermal, and electrical properties. To take into account size dependence of such small sized structures, the use of nonlocal continuum theory is proposed where intrinsic length scales is taken into account. Based on the Eringen theory, a nonlinear nonlocal model of a clamped-clamped CNT is developed in this study. Static and free vibration responses are simulated and analyzed. The main objective of this work is to study the influence of CNT size and length scale parameter on the static and free vibration response to better understand their effect on the general behavior of the CNT. It has been found that the nonlocal effect can largely influence the performance of the CNT and change qualitatively its nonlinear response

Nonlocal modeling of a Carbon Nanotube actuated by an electrostatic force

Carbon nanotubes (CNTs) are promising mechanical structures at the nano-scale which have attracted increasing attention due to their amazing mechanical, chemical, thermal, and electrical properties. To take into account size dependence of such small sized structures, the use of nonlocal continuum theory is proposed where intrinsic length scales is taken into account. Based on the Eringen theory, a nonlinear nonlocal model of a clamped-clamped CNT is developed in this study. Static and free vibration responses are simulated and analyzed. The main objective of this work is to study the influence of CNT size and length scale parameter on the static and free vibration response to better understand their effect on the general behavior of the CNT. It has been found that the nonlocal effect can largely influence the performance of the CNT and change qualitatively its nonlinear response

Metamaterial beam with embedded nonlinear vibration absorbers

In this work the multi-mode vibration absorption capability of a nonlinear metamaterial beam is investigated. A Euler–Bernoulli beam is coupled to a distributed array of nonlinear spring–mass subsystems acting as local resonators/vibration absorbers. The dynamic behavior of the metamaterial beam is first investigated via the classical approach employed for periodic structures by which the frequency stop bands of the single cell are determined. Subsequently, the frequency response is obtained for the metamaterial beam to study a multi-frequency stop band system by adding an array of embedded nonlinear local resonators. A path following technique coupled with a differential evolutionary optimization algorithm is adopted to obtain the optimal frequency-response curves of the metamaterial beam in the nonlinear regime. The use of the local absorbers, via a proper tuning of their constitutive parameters, allows a significant reduction of the metamaterial beam oscillations associated with the lowest three vibration modes

Coupled mech nical and electromagnetic modeling of eddy current sensors

To effectively monitor the vibration of blades in a rotating machine, a non-contacting method called blade tip-timing (BTT) has been used. The method is based on the analysis of the differential arrival times of the blades at sensors mounted on the stator to characterize the vibration amplitude and frequency of the blades. These sensors can also provide blade tip clearance measurement. A combination of these data can provide a robust condition monitoring approach for the early detection of blade cracks. Eddy current sensors (ECS) have shown great potential to assess the health of an engine without any need for direct access to the blade and therefore they are insensitive to the presence of any type of contaminant. Also, both tip timing and tip clearance of each blade could be measured by these sensors in real time and at high resolution. ECSs measure the magnetic field caused by eddy currents during the blade motion, and hence are a coupled mechanical and electromagnetic problem. An ECS on the casing of a machine has been modeled to fully understand how the dynamic response of the blade is measured by the sensors. Detailed 2-D and 3-D modeling and simulation of a rotating simplified bladed disk passing an ECS is presented. The effect of the variation of the rotation speed and the gap between the sensor and the blade tip on the accuracy of the measurement is investigated. Such an analysis will enable the reliable monitoring of blade damage during engine operation

Finite-element analysis of errors on stress and strain measurements in dynamic tensile testing of low-ductile materials

International audienceTensile dynamic tests are essential experiments to develop and validate constitutive equations. In this paper, we studied the errors on stress and strain measurement in dynamic tensile tests by using finite-element analysis. Two strain and one stress measures were discussed. Mainly, we considered the influence of the material and multiple geometrical and testing parameters. We were limited to the case of elastic behaviour of the material. We observed that the errors on the strain measures are essentially influenced by the geometrical parameters. On the other hand, the error on the stress measure are highly correlated to stress field homogeneity

A surface crack in a graded coating bonded to a homogeneous substrate under general loading conditions

The elastostatic problem of a surface crack in a graded coating bonded to a homogeneous substrate under general loading conditions is considered. The coating is graded along the thickness direction and modeled as a nonhomogeneous medium with an isotropic stress-strain law. The problem is solved under the assumption of plane strain or generalized plane stress conditions. The crack surfaces are subjected to arbitrary loadings which give rise to mixed fracture modes which can be uncoupled due to the fact that the crack axis is parallel to the material gradient. Therefore, the opening and sliding mode problems may be formulated separately. For each problem, the solution of the composite medium may be determined by obtaining the solution of the homogeneous substrate and that of the graded layer. The latter solution may be expressed as the sum of two solutions, namely an infinite graded medium with a crack and a graded strip without a crack. The resulting mixed-boundary value problem is reduced to a set of two uncoupled singular integral equations which are solved numerically using Jacobi polynomials. The main objective of the paper is to study the effect of the layer thickness and nonhomogeneity parameter on the crack tip mixed-mode stress intensity factors for the purpose of gaining better understanding on the behavior of graded coatings