Study on the application of optical interferometry in the measurement of the coating thickness of color-coated plates

The principle and application of optical interferometry to measure the coating thickness of color-coated plates were introduced in this paper.Additionally,several factors affecting the test results,including coating refractive index,wavelength range,and film thickness range setting,were analyzed.Among these,the refractive index of the color coating,which cannot be measured directly,was identified as the key factor.A solution to this problem was proposed.Finally,the optical interference method and the current detection methods,including the micrometer method and the magnetic eddy current method,were analyzed and compared.The results show that optical interferometry has better repeatability and reproducibility than the current methods and show no significant difference from the current methods through statistical tests.Therefore,the method can be applied to the detection of the coating thickness of color-coated plates.

Exact thickness-shear resonance frequency of electroded piezoelectric crystal plates

The determination of the precise thickness-shear frequency of electroded crystal plates has practical importance in quartz crystal resonator design and fabricatiom especially when the high fundamental thickness-shear frequency has reduced the crystal plate thickness to such a degree that proper consideration of the effect of electrodes is very important. The electrodes effect as mass loading in the estimation of the resonance frequency has to be modified to consider the stiffness of electrodes, as the relative strength is increasingly noticeable. By following a known procedure in the determination of the thickness-shear frequency of an infinite AT-cut crystal plate, frequency equations of crystal plate without and with piezoelectric effect are obtained in terms of elastic constants and the electrode material density. After solving these equations for the usual design parameters of crystal resonators, the design process can be optimized to pinpoint the precise configuration to avoid time-consuming trial and reduction steps. Since these equations and solutions are presented for widely used materials and parameters, they can be easily integrated into the existing crystal resonator design and manufacturing processes.

Cubic Spline Solutions of Nonlinear Bending and Buckling of Circular Plates with Arbitrarily Variable Thickness

The cubic B-splines taken as trial function, the large deflection of a circular plate with arbitrarily variable thickness,as well as the buckling load, have been calculated by the method of point collocation. The support can be elastic. Loads imposed can be polynomial distributed loads, uniformly distributed radial forces or moments along the edge respectively or their combinations. Convergent solutions can still be obtained by this method under the load whose value is in great excess of normal one. Under the action of the uniformly distributed loads, linear solutions of circular plates with linearly or quadratically variable thickness are compared with those obtained by the parameter method. Buckling of a circular plate with identical thickness beyond critical thrust is compared with those obtained by the power series method.

NONLINEAR VIBRATION FOR MODERATE THICKNESS RECTANGULAR CRACKED PLATES INCLUDING COUPLED EFFECT OF ELASTIC FOUNDATION

Based on Reissner plate theory and Hamilton variational principle, the nonlinear equations of motion were derived for the moderate thickness rectangular plates with transverse surface penetrating crack on the two-parameter foundation. Under the condition of free boundary, a set of trial functions satisfying all boundary conditions and crack＇s continuous conditions were proposed. By employing the Galerkin method and the harmonic balance method, the nonlinear vibration equations were solved and the nonlinear vibration behaviors of the plate were analyzed. In numerical computation, the effects of the different location and depth of crack, the different structural parameters of plates and the different physical parameters of foundation on the nonlinear amplitude frequency response curves of the plate were discussed.

Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

LARGE DEFLECTION PROBLEM OF CIRCULAR PLATES WITH VARIABLE THICKNESS UNDER THE ACTION OF COMBINED LOADS

By using the modified iteration method of large deflection theory of plates with variable thichness[1], we solve the problem of circular plates with variable thickness subjected to combined loads under the boundary conditions of the clamped edges and get comparatively more accurate second-order approximate analytical solution. If the results of this paper are degraded into the special cases, the results coinciding with those of papers [1,2] can be obtained. In this paper, the characteristic curves are plotted and some comparisons are made. The results of this paper are satisfactory.

Dynamic Behaviors of Axially Moving Viscoelastic Plate with Varying Thickness

Structural components of varying thickness draw increasing attention these days due to economy and light-weight considerations. In view of the absence of research in vibration analysis of viscoelastic plate with varying thickness, this study devotes to investigate the dynamic behaviors of axially moving viscoelastic plate with varying thickness. Based on the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of motion of the axially moving viscoelastic rectangular plate is derived, the plate constituted by Kelvin-Voigt model has linearly varying thickness in the y-direction. The dimensionless complex frequencies of axially moving viscoelastic plate with four edges simply supported are calculated by the differential quadrature method, curves of real parts and imaginary parts of the first three-order dimensionless complex frequencies versus dimensionless moving speed are obtained, the effects of the aspect ratio, thickness ratio, the dimensionless moving speed and delay time on the dynamic behaviors of the axially moving viscoelastic rectangular plate with varying thickness are analyzed. When other parameters keep constant, with the decrease of thickness ratio, the real parts of the first three-order natural frequencies decrease, and the critical divergence speeds of various modes decrease too, moreover, whether the delay time is large or small, the frequencies are all complex numbers.

Optimization of Holding Temperature and Holding Thickness for Controlled Rolling on Plate Mill

Holding temperature and holding thickness are main parameters for two-phase controlled rolling on plate mill. The optimization of holding temperature and holding thickness for pass schedule calculation of two-phase controlled rolling on plate mill was presented and its feature is as follows： （1） Determination of holding thickness can be automatically obtained based on the influence of mill safety limits, tracking zone length and holding time on holding thickness; （2） Determination of holding temperature can be automatically obtained and the holding time can be reduced as much as possible; （3） Algorithm can modify the holding temperature and thickness depending on slab size and product size.

Characteristics of Eddy Current Attenuation and Thickness Measurement of Metallic Plate

In eddy current testing, the law of attenuation of eddy current(EC) is of great concern. In conductive half space under the excitation of uniform magnetic field, the EC density decreases exponentially in the depth direction. However, in conductor with finite thickness tested by coil, the distribution of EC in the depth direction is more complicated. This paper studies the characteristics of EC attenuation in metallic plate of finite thickness. Simulation results show that there is an EC reflection at the bottom of plate, which changes the law of EC attenuation. A new concept, namely the equivalent attenuation coefficient, is proposed to quantify the speed of EC attenuation. The characteristics of EC attenuation are utilized to explain the nonmonotonic relation between coil voltage and plate thickness. Procedure of selecting frequency is discussed. Thereafter, measurement of plate thickness is carried out and accurate result is obtained.

Vibration of Visco-Elastic Parallelogram Plate with Parabolic Thickness Variation

The main objective of the present investigation is to study the vibration of visco-elastic parallelogram plate whose thickness varies parabolically. It is assumed that the plate is clamped on all the four edges and that the thickness varies parabolically in one direction i.e. along length of the plate. Rayleigh-Ritz technique has been used to determine the frequency equation. A two terms deflection function has been used as a solution. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The assumption of small deflection and linear visco-elastic properties of “Kelvin” type are taken. We have calculated time period and deflection at various points for different values of skew angles, aspect ratio and taper constant, for the first two modes of vibration. Results are supported by tables. Alloy “Duralumin” is considered for all the material constants used in numerical

Effect of Non-Homogeneity on Thermally Induced Vibration of Orthotropic Visco-Elastic Rectangular Plate of Linearly Varying Thickness

The analysis presented here is to study the effect of non-homogeneity on thermally induced vibration of orthotropic visco-elastic rectangular plate of linearly varying thickness. Thermal vibrational behavior of non-homogeneous rectangular plates of variable thickness having clamped boundary conditions on all the four edges is studied. For non–homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the method of separation of variables, the governing differential equation is solved. An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Time period and deflection at different points for the first two modes of vibration are calculated for various values of temperature gradients, non- homogeneity constant, taper constant and aspect ratio. Comparison studies have been carried out with non-homogeneous visco-elastic rectangular plate to establish the accuracy and versatility.

Design and application of thickness measurement calibration system based on laser displacement sensor

This study aims to improve the accuracy and safety of steel plate thickness calibration.A differential noncontact thickness measurement calibration system based on laser displacement sensors was designed to address the problems of low precision of traditional contact thickness gauges and radiation risks of radiation-based thickness gauges.First,the measurement method and measurement structure of the thickness calibration system were introduced.Then,the hardware circuit of the thickness system was established based on the STM32 core chip.Finally,the system software was designed to implement system control to filter algorithms and human-computer interaction.Experiments have proven the excellent performance of the differential noncontact thickness measurement calibration system based on laser displacement sensors,which not only considerably improves measurement accuracy but also effectively reduces safety risks during the measurement process.The system offers guiding significance and application value in the field of steel plate production and processing.

NON-LINEAR DYNAMIC BEHAVIOR OF THERMOELASTIC CIRCULAR PLATE WITH VARYING THICKNESS SUBJECTED TO NONCONSERVATIVE LOADING

The non-linear dynamic behaviors of thermoelastic circular plate with varying thickness subjected to radially uniformly distributed follower forces are considered. Two coupled non-linear differential equations of motion for this problem are derived in terms of the transverse deflection and radial displacement component of the mid-plane of the plate. Using the Kantorovich averaging method, the differential equation of mode shape of the plate is derived, and the eigenvalue problem is solved by using shooting method. The eigencurves for frequencies and critical loads of the circular plate with unmovable simply supported edge and clamped edge are obtained. The effects of the variation of thickness and temperature on the frequencies and critical loads of the thermoelastic circular plate subjected to radially uniformly distributed follower forces are then discussed.

Variations of properties across plate thickness for Al alloy 7010

The variations of electrical conductivity and hardness across the thickness of an Al alloy 7010 plate under the temper condition T7651 were investigated. The electrical conductivity and hardness respond in a reciprocal manner. Cross-sectional slices of the plate subjected to re-solutionising/natural ageing and re-solutionising/artificial ageing show the similar tendencies in property changes as in the as-received raw material. This clearly suggests that the property inhomogeneity across the plate thickness is inherent of the manufacturing route. The differences in properties through the plate thickness are due to the changes in the concentrations of the strengthening alloying elements in the solid solution and the associated changes in microstructure; these are believed to be mainly due to the nature of plate solidification and prolonged high temperature during the rolling operation. The combination of electrical conductivity and hardness can be used as an integral quality property indicator for assessing inhomogeneity of thick products.

Ultimate Compressive Strength Computational Modeling for Stiffened Plate Panels with Nonuniform Thickness

The aim of this paper is to develop computational models for the ultimate compressive strength analysis of stiffened plate panels with nonuniform thickness.Modeling welding-induced initial deformations and residual stresses was presented with the measured data.Three methods,i.e.,ANSYS finite element method,ALPS/SPINE incremental Galerkin method,and ALPS/ULSAP analytical method,were employed together with existing test database obtained from a full-scale collapse testing of steel-stiffened plate structures.Sensitivity study was conducted with varying the difference in plate thickness to define a representative(equivalent)thickness for plate panels with nonuniform thickness.Guidelines are provided for structural modeling to compute the ultimate compressive strength of plate panels with variable thickness.

SCATTERING OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN MINDLIN'S THICK PLATES WITH A CUTOUT

Using the complex variable method and conformal mapping,scat- tering of flexural waves and dynamic stress concentrations in Mindlin's thick plates with a cutout have been studied.The general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of cutouts is obtained. Applying the orthogonal function expansion technique,the dynamic stress problem can be reduced into the solution of a set of infinite algebraic equations.As examples, numerical results for the dynamic stress concentration factor in Mindlin's plates with a circular,elliptic cutout are graphically presented in sequence.

Residual stress in quenched 7075 aluminum alloy thick plates

The influence of quenching water temperature, pre stretching amount and aging temperature and times on residual stress in 7075 aluminum thick plate was studied by the measurement of residual stress using drilling hole method. The results indicate that residual stress decreases by 30% with increasing quenching water temperature from 40 ℃ to 80 ℃, 20% with increasing aging temperature from 100 ℃ to 180 ℃,and 20% with increasing aging times from 5 h to 25 h. Also, residual stress decreases to zero with increasing pre stretching amount to approximately 2%. Hence, residual stress in 7075 aluminum thick plate is reduced by the control of quenching water temperature at 80 ℃ and with pre stretching amount of about 2%. An optimal aging temperature and time should be systemically investigated to obtain combination of high mechanical performances and lower residual stress for manufacturing of 7075 aluminum alloy thick plates.

GENERALIZED VARIATIONAL PRINCIPLES FOR VISCOELASTIC THIN AND THICK PLATES WITH DAMAGE

From the constitutive model with generalized force fields for a viscoelastic body with damage, the differential equations of motion for thin and thick plates with damage are derived under arbitrary boundary conditions. The convolution-type functionals for the bending of viscoelastic thin and thick plates with damage are presented, and the corresponding generalized variational principles are given. From these generalized principles, all the basic equations of the displacement and damage variables and initial and boundary conditions can be deduced. As an example, we compare the difference between the dynamical properties of plates with and without damage and consider the effect of damage on the dynamical properties of plates.

Improvement of Temperature Drop Model Parameters for 4200mm/3500mm Rolling of Baosteel Medium and Thick Plate

Based on the temperature situation during the rolling process of 4200 mm/3500 mm medium and thick plates at Baosteel,this article analyzes the influencing factors of temperature changes during the rolling process from the perspective of heat transfer theory and the temperature change law during the rolling process.The temperature loss during the four rolling processes is tracked on site,and the temperature drop model parameters during the rolling process of 4200mm/3500mm medium and thick plates at Baosteel are quantitatively provided.It is applied to actual rolling production and has achieved good results.