Small Deflection Theory Research Articles

Investigation of MEMS Single Turn Meander-Shaped Silicon Carbide Piezoresistive Pressure Sensor on a Clamped Circular Diaphragm for High Pressure Harsh Environment Applications

Silicon-based Microelectromechanical System (MEMS) pressure sensors are extensively used and have the advantages of high accuracy and miniaturization. Despite this, due to inherent material limitations, they are not easily able to withstand high temperatures greater than 150°C. To overcome this disadvantage, Silicon Carbide (SiC) is the preferred material because it has excellent thermomechanical properties and operates above 600°C. Piezoresistive pressure sensors made of Silicon Carbide are ideally able to detect pressures beyond 600°C. Our work presents MEMS Single turn meander-shaped piezoresistive pressure sensor on a circular SiC diaphragm for high pressure applications of pressure range 0–40[Formula: see text]MPa in harsh environment. This work models and analyses the piezoresistive pressure sensor characteristics using an analytical modeling and simulation method to choose the best design. To ascertain its sensitivity, expressions are computationally simulated with MATLAB software using the thin plate and small deflection theory. To evaluate the viability of the model, COMSOL Multiphysics simulation is used. When compared to recent studies, our proposed sensor provides a sensitivity of near about 5.4[Formula: see text]mV/V/MPa across a pressure range of 0–40[Formula: see text]MPa.

An equivalent modelling approach for vibroacoustic characteristics analysis of railway aluminium extrusion

For railway carbody structures with irregular cross-sectional shapes and great longitudinal lengths, it is difficult to obtain their vibroacoustic characteristics using traditional computational methods. Moreover, traditional methods are time intensive and not conducive to the optimisation design of such complex and large structures. This study introduces an equivalent modelling approach for vibroacoustic analysis, i.e., sound transmission loss and sound radiation analysis, of railway aluminium extrusion. The aluminium extrusion is equivalent to a homogeneous orthotropic plate, and the equivalent mechanical properties of the plate are obtained via static analysis based on the small-deflection theory. The vibroacoustic characteristics are obtained using the hybrid method of finite element and statistical energy analysis. First, the equivalent modelling approach is validated using the results of a detailed prediction model and experimental test. Subsequently, the applicability of the equivalent approach is analysed by comparing it to the predicted results of the detailed model with different dimensions and boundary conditions. Ultimately, the minimum longitudinal length that needs to be considered for the analysis of railway aluminium extrusion is discussed using the equivalent approach and the computational efficiency is compared. The results reveal that the equivalent approach can effectively simulate the vibroacoustic performances of the aluminium extrusion in 1/3 octave bands centred below 1000 Hz and reduce the calculation time by 90% compared with the detailed prediction models of aluminium extrusion, which provided a foundation for vibroacoustic optimisation design of railway carbody structures.

Cell wall fracture mechanism in ultrasonic-assisted cutting of honeycomb materials

Revealing the ultrasonic cutting mechanism of honeycomb composite is important for determining the acoustic parameters of the ultrasonic system and selecting the parameters of the cutting process. Understanding more details of the stress on the cell wall from ultrasonic vibrating tool and the conditions for cell wall breakage is essential to study the machining mechanism. According to the evolution of contact state between the straight edge cutter and the honeycomb cell wall in a cycle, the cutting force acting on the cell wall is divided into three stages: transverse cutting load action, longitudinal cutting load action, and no cutting load action. The cell wall deflection and stress equations under transverse cutting load were established by applying elastic thin plate small deflection theory. The deformation and fracture characteristics of the honeycomb cell wall were analyzed by combining the analytical and the finite element model. The results showed that the ultrasonic vibration of the cutter greatly improved the stiffening effect of the cell wall and its fracture was caused by the deflection under the transverse cutting load, which exceeded the maximum allowable deformation after local stiffening. In addition, with only longitudinal cutting load, it was difficult to break the critical buckling state that leads to cell wall fracture.

The Mixed-Body Model: A Method for Predicting Large Deflections in Stepped Cantilever Beams

AbstractThis paper presents a method for predicting endpoint coordinates, stress, and force to deflect stepped cantilever beams under large deflections. This method, the mixed-body model or MBM, combines small deflection theory and the pseudo-rigid-body model for large deflections. To analyze the efficacy of the model, the MBM is compared to a model that assumes the first step in the beam to be rigid, to finite element analysis, and to the numerical boundary value solution over a large sample set of loading conditions, geometries, and material properties. The model was also compared to physical prototypes. In all cases, the MBM agrees well with expected values. Optimization of the MBM parameters yielded increased agreement, leading to average errors of <0.01 to 3%. The model provides a simple, quick solution with minimal error that can be particularly helpful in design.

High-Precision Large-Range Optical Fiber Interferometric Piezometer and Its Wideband Interferometry for Water Pressure Measurement

A high-precision large-range fiber-optic interferometric piezometer is designed, packaged, and demonstrated for pore water pressure measurement. The piezometer is an optical fiber extrinsic Fabry–Perot interferometer (EFPI), which comprises a fiber ferrule connector/flat contact (FC/FC) connector and an aluminum foil pasted on a thin metal plate. The EFPI forms a sealed chamber with only the outer plane of the aluminum alloy plate being subjected to the water pressure through another hydraulic chamber with a standard filter stone. The deformation of the metal plate is analyzed and simulated for parameter optimization by using the small deflection theory. By adopting the conventional wavelength tracking method, the theoretical simulation and experimental results show a high measurement accuracy of the cm level but a limited measurement range. To resolve the problem, several wideband interferometry (WBI) methods, including free spectral range (FSR)-based, spatial frequency, interference order, and fitted FSR, are employed to analyze the spectrum of the EFPI for comparison. The fitted FSR method has demonstrated its superiority in applications that require simultaneous high resolution and large measurement range. A gas pressure test verifies that the proposed piezometer can work in a wide range of up to 0.5 MPa that is equivalent to a water depth of 50 m. In addition, the proposed piezometer offers several advantages, such as cost-effectiveness, robustness, repeatability, and stability, which further benefits its practical application.

Elastic buckling of composite webs in back-to-back cold-formed steel built-up columns-Part II: Design formula

Elastic buckling of the composite webs in cold-formed steel (CFS) back-to-back built-up columns was investigated in this paper. The built-up members were fabricated from two identical C-section components and were assembled with self-drilling screws on their webs. The new buckling modes of screw composite plates with simply supported on four sides (SS) have been determined by previous experiments and numerical simulations. In this paper, the instability mechanism was analyzed based on the small deflection theory to study the elastic buckling behavior of SS screw composite plates. Immediately, a calculation model of SS screw composite plates was established considering the shear slip and the screw restraint. The analytic formula of elastic critical stresses was deduced based on the calculation model. However, the critical stress is affected by the ratio of the web length to the depth of the web, the screw diameter and the screw spacing. The proposed formula was modified using previous results obtained by extensive finite element parametric analyzes to increase the applicability for the proposed analytical formula in SS screw composite plates. Eventually, the applicability and accuracy of the design formula were validated using the previous experimental results. Therefore, a new stability calculation theory and a design formula considering the influence of shear slip were proposed to solve the defects in solving the buckling problem of screw composite plates using the classical stability theory of the single plate.

Behavioural Study of Thin Plate Under Large Deflection Theory

For a thin plate, if the deformation is on the order of the thickness and stay elastic, linear theory might not turn out correct results because it does not predict the in plane movement of the member. Therefore, to account for the inconsistencies of geometric nonlinearity, large deflection theory is required [1]. This report pertains to the analytical study dispensed to check the behavior of thin plate under fixed and pinned edge conditions, and for diverse thicknesses, under the small and large deflection theories. The deformation is additionally studied, supported by Von-Karman equations. Non linear analysis has been performed on FE model using the ANSYS software. The consequences of geometric nonlinearities are mentioned. Outline on conclusion of the theoretical and experimental results obtained, are compared so as to review the similarity of the modeling and theory.

Wavelet-based homotopy method for analysis of nonlinear bending of variable-thickness plate on elastic foundations

A unified bran-new wavelet algorithm has been proposed for solving large bending of clamped or simply supported variable-thickness plates resting on nonlinear triparametric elastic foundations. The strongly nonlinear variable-coefficient governing equations for plates with arbitrary distributed thickness have been successfully transformed and solved under the formulation of thickness distributed function meeting circled homogeneous Neumann boundary. The highly accurate wavelet solutions of dimensionless deflection and bending stresses are finally obtained by wavelet homotopy approach, which are difficult and rarely obtained by other numerical methods. Nonlinear bending of characteristic variable-thickness plates of prismatic, tapered, pitted and raised types are further investigated with different parameters of thickness profiles and elastic foundations. Nonlinear effect of inhomogeneous thickness turns to be more evident on changing deflection and stresses distributions of rectangular tapered plates, but only affect extreme values for square tapered one. Resisting bending capability of prismatic plate will be enhanced by improvement of tapered slope but weakened adding polynomial orders of thickness profile. Difference of deflection and bending stresses between raised and pitted plates are enlarged by small deflection theory but gradually narrowed by large deformation theory, which reveals thickness discrepancy has little effect at large bending stage.

Tool path design system to enhance accuracy during double sided incremental forming: An analytical model to predict compensations for small/large components

Abstract Double sided incremental forming (DSIF) has potential to form complex three-dimensional sheet metal components without using component specific tooling. Forming tool deflection and sheet spring-back are significant factors contributing to the geometrical inaccuracy of DSIF components. Numerical prediction and experimental measurement of sheet spring-back is time consuming. In addition, available analytical methods to predict and compensate sheet spring-back uses theory of small deflections by neglecting the membrane effects. With increase in sheet deflection beyond its thickness, membrane forces experienced by the middle plane of sheet due to stretching significantly resists the applied transverse load. In the present work, combination of small deflection and membrane theories are used to predict and compensate sheet deflections, so that a single methodology can be used for small as well as large components. Proposed methodology is validated using experimental and numerical predictions and they are in very good agreement. Two geometries (axisymmetric, free form components) with different component openings are formed to validate the proposed predictive methodology. Results indicate there is significant improvement (maximum error is less than 800 μ m) in accuracy of components formed using compensated tool paths developed using proposed model. In addition, support tool maintained contact with component throughout forming (maximum force on the support tool is less than 60 N).

Effect of geometrical imperfection on the thermomechanical behavior of functionally graded material rotating disk

The thermoelastic analysis of axisymmetric bending of rotating functionally graded material (FGM) disks, with and without imperfection, is presented. Material properties are assumed to be temperature dependent and graded in the thickness direction following a grading index power-law distribution. Theories of von Karman and first-order shear deformation were implemented to calculate the stress and displacement fields. Numerical results are given for evaluating the effects of temperature, material properties and imperfection on displacement and stress fields in a disk with roller-supported boundary conditions. Small and large deflection theories are considered to obtain the stress field and displacement field for the perfect and imperfect in FG disks and also in full metal (or full ceramic) disks. A series-form solution is used to solve the large-deflection nonlinear equations. The results are compared and validated with the results from the finite element method. It is observed that applying the initial geometric imperfection to a rotating solid disk would increase the value of radial stress, circumferential stress and vertical displacement.

An exact analytical solution for fractional-order thermoelasticity in a multi-stacked elliptic plate

Thermoelastic analysis of an isotropic homogeneous multi-stacked elliptical plate has been considered in this research. For which multi-layered plate is taken into consideration on a plane-parallel elliptic geometry perpendicular to the z-direction. The governing equations are considered in the context of time-fractional derivative of the order α with temperature distribution in each s layer of the stacked plate with time-dependent sectional heat supply on the lower and upper face. The multi-stacked profile consists of s discrete plates each of a different material with perfect thermal contact at each of its s-1 interface. The general solution, which perfectly satisfies the fundamental equation of heat conduction, is obtained using an integral transformation technique. It is solved using a type of quasi-orthogonality relationship by modifying Vodicka’s method and the Laplace transformation. The analysis is based on the small-deflection theory corresponding to the fundamental solutions for the fractional-order heat conduction equation. In addition to this, the intensities of bending moments, forces, maximum normal stresses and its associated stresses are formulated involving the Mathieu functions. As a special case, a multi-stacked circular plate is also discussed in detail as a limiting case. Numerical calculations are also performed, and the results are graphically illustrated.

Design and sensitivity analysis of capacitive MEMS pressure sensor for blood pressure measurement

This paper presents the design and simulation of a MEMS based clamped capacitive pressure sensor for blood pressure measurement. Normally, Blood pressure for human beings varies in the range of 0.1–0.14 MPa. The operating range of this sensor design, modified according to the dimensions, is between 0 and 0.16 MPa. The diaphragm deflects on application of pressure which triggers a change in the capacitance between diaphragm and bottom electrode. The plates are separated by a dielectric medium ( $$ Si_{3} N_{4} $$ ). The change in capacitance, capacitive sensitivity and diaphragm deflection are observed for every design model to determine its performance. To improve the sensor performance, parameters like separation gap and diaphragm thickness are varied. The length and width of square diaphragm and overlapping area (A) are constant for all the designs. Small deflection theory ( $$ w_{\hbox{max} } = {h \mathord{\left/ {\vphantom {h 4}} \right. \kern-0pt} 4} $$ ), pull-in voltage phenomena ( $$ w_{\hbox{max} } \le {g \mathord{\left/ {\vphantom {g 4}} \right. \kern-0pt} 4} $$ ), and thin plate theory ( $$ h \cong a/10 $$ ) have been considered while designing the model. With respect to pressure, the simulation of diaphragm deflection is performed in COMSOL Multiphysics and analytical simulation for Change in capacitance and Capacitive sensitivity is performed in MATLAB.

Flow-induced Vibrations of a Horizontal Elastic Band Plate Submerged in Fluid of Finite Depth

Abstract The paper deals with forced vibrations of a horizontal thin elastic plate submerged in a semi-infinite layer of fluid of constant depth. The pressure load on this plate is induced by water waves arriving at the plate. This load is accompanied by pressure resulting from the motion of the plate. The plate and fluid motions depend on boundary conditions, and, in particular, the pressure load depends on the width of the gap between the plate and the bottom. In theoretical description of the phenomenon, we deal with a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surfaces. The main attention is focused on transient solutions of the problem, which correspond to fluid (and plate) motion starting from rest. In formulation of this problem, a linear theory of small deflections of the plate is employed. In order to calculate the fluid pressure, a solution of Laplace’s equation is constructed in a doubly connected fluid domain. With respect to the initial value problem considered, we confine our attention to a finite fluid domain. For a finite elapse of time, measured from the starting point, the solution in the finite fluid area mimics a solution within an infinite domain, inherent for wave propagation problems. Because of the complicated structure of boundary conditions of the coupled problem considered, the fluid domain is divided into sub-domains of simple geometry, and the solutions of the problem equations are constructed separately in each of these domains. Numerical experiments have been conducted to illustrate the formulation developed in this paper.

End stiffness modeling for automatic horizontal dual-machine cooperative drilling and riveting system

Aircraft panel automatic drilling and riveting system is the key equipment to realize aircraft panel assembly with high efficiency and quality. In practical application, automatic drilling and riveting system should bear cyclic riveting force with large load, which means such systems must have enough stiffness to deal with this situation. The stiffness of each components in automatic drilling and riveting system reflects as the end stiffness finally; in the meanwhile, end stiffness and riveting ability of the system will be affected when the posture of the system changes. According to this situation, stiffness matrixes related to mechanics characteristic of motion axes, transmission parts, inner-side working head, and outer-side working head of automatic horizontal dual-machine cooperative drilling and riveting system are established, in which the Jacobi matrix method, point transformation matrix method, and finite element analysis are applied. Finally, the static end stiffness model is proposed based on multi-body small deflection theory. The end stiffness model has been validated through experimental measurement, and the stiffness relationship of dual-machine tool system was obtained, which applied important data for structure optimization and control system design.

A novel neural electrode with micromotion-attenuation capability based on small-deflection theory

A novel neural electrode with micromotion-attenuation capability based on small-deflection theory

Force-Fracture Characteristics of the Roof above Goaf in a Steep Coal Seam: A Case Study of Xintie Coal Mine

After coal is extracted from a working face in a steep coal seam (SCS), the immediate roof tends to cave in and refill the lower part of the goaf. Based on the geological conditions of a work area in a SCS and the characteristics of roof caving, this study proposed a formula for the width of the backfill in the goaf and analyzed the main factors influencing it. Based on the small-deflection theory for elastic thin plates, a working face model was created for the mechanical analysis of the main roof above a SCS before the roof fractures for the first time. Then, a roof deflection equation was derived for the estimation roof deformation under the action of both the load from overlying strata and the support provided by the backfill in the goaf. The theoretical analysis combined with the actual operational parameters at the Zuoqipian working face in #49 seam of Xintie Coal Mine shows that the maximum roof deflection is around 0.8 m and occurs at a location 39 m from the upper end of the working face. Fractures will first develop in the upper sections of the frontal and rear walls of the face and the middle of the upper suspended roof due to tension or shearing and ultimately form an E-shaped pattern. The measured support pressure and the roof deformation obtained by theoretical analysis show a similar distribution pattern, indirectly confirming the accuracy of the theoretical results.

Large and Small Deflection Analysis of a Cantilever Beam

This research focuses on the geometrically nonlinear large deflection analysis of a cantilever beam subjected to a concentrated tip load. Initially, a step-by-step development of the theoretical solution is provided and is compared with numerical analysis based on beam and shell elements. It is shown that the large deflections predicted by numerical analysis using beam elements accurately capture the theoretical results as compared to shell elements. Comparison of above deflections with theoretical and numerical approaches based on small deflection theory is also provided to show the extent of latter’s applicability. Finally, it is shown that for a linear elastic working range of common engineering metals, both small and large deflection approaches yield same results and one can adopt the simple small deflection approach for engineering design. It is highlighted that the theoretical approach of large deflection commonly available in design texts is valid only within the linear elastic strain limit and recommends a careful approach to designers. Further, the effect of parametric variation in geometry and stiffness of beam on large deflection, and resulting bending strains and tip reactions are analyzed and discussed.

An Analysis of Thermal-Bending Stresses in a Simply Supported Thin Elliptical Plate

In this paper, a transient thermal stress investigation on a simply supported thin elliptical plate during sectional heating with time-dependent temperature supply is considered. The solution of heat conduction equation with corresponding initial and boundary conditions is obtained by employing an integral transform approach. The governing equation solution for the small deflection theory is obtained and utilized to preserve the intensities of thermal bending moments, involving the Mathieu and modified functions and its derivatives. The deflection results show an approximately good agreement with the previously given results. It is also demonstrated that the temperature field in a circular solution could be resulted in a particular case of the present mathematical solution. The obtained numerical results utilizing computational implements are precise enough for practical purposes.

Analytical modeling and simulation of MEMS piezoresistive pressure sensors with a square silicon carbide diaphragm as the primary sensing element under different loading conditions

Although silicon is the preferred choice for microelectromechanical systems (MEMS) piezoresistive pressure sensors, such devices are not preferred for application in harsh environmental conditions due to the exponential increase in leakage current with temperature. To alleviate such shortcomings of silicon-based pressure sensors in extreme conditions including elevated temperature and intense vibration, this study strives to shift focus from core complementary metal–oxide–semiconductor (CMOS) materials to silicon carbide. In this work, we adopt an analytical and simulation approach to model and analyze various characteristics of such silicon carbide piezoresistive sensors and determine an optimal design. A square diaphragm is modeled using the analytical expressions for a thin plate in combination with small-deflection theory, providing quick insight for estimation of critical parameters and thus the behavior of the pressure sensor. Both clamped and freely supported edge conditions of the diaphragm are explored. Although many studies and discussions are available on the rigidly supported loading condition, the freely supported edge condition for a square diaphragm has received little attention. The deflection, stress, strain, and sensitivity of the square diaphragm under both loading conditions are reported herein then compared to understand which of the two loading conditions results in more significant outputs.

Added Mass of Fluid and Fundamental Frequencies of a Horizontal Elastic Circular Plate Vibrating in Fluid of Constant Depth

Abstract The paper deals with free vibrations of a horizontal thin elastic circular plate submerged in an infinite layer of fluid of constant depth. The motion of the plate is accompanied by the fluid motion, and thus, the pressure load on this plate results from displacements of the plate in time. The plate and fluid motions depend on boundary conditions, and, in particular, the pressure load depends on the gap between the plate and the fluid bottom. In theoretical description of this phenomenon, we deal with a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surfaces. This coupling leads to the so-called co-vibrating (added) mass of fluid, which significantly changes the fundamental frequencies (eigenfrequencies) of the plate. In formulation of the problem, a linear theory of small deflections of the plate is employed. At the same time, one assumes the potential fluid motion with the potential function satisfying Laplace’s equation within the fluid domain and appropriate boundary conditions at fluid boundaries. In order to solve the problem, the infinite fluid domain is divided into sub-domains of simple geometry, and the solution of problem equations is constructed separately for each of these domains. Numerical experiments have been conducted to illustrate the formulation developed in this paper.