Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique

Theoretical and experimental large deflection and stress analysis of a master leaf spring considering stress concentration effect of clamping is reported. The non-uniformly curved master leaf spring under three point bending subjected to moving boundaries is modeled. Geometrically nonlinear strain-displacement relations, as necessary for the theoretical analysis, are derived through visualization of physics behind the large deformation problem. An embedded curvilinear coordinate system is considered, to study the combined effects of non-uniform curvature, bending, stretching and shear deformation including cross-sectional warping. Governing equation of the non-uniformly curved beam system is derived in variational form using energy method, based on linear material constitutive relations and the derived nonlinear kinematic relations. An iterative solution scheme through successive geometry updation is developed and executed in MATLAB® software to solve the governing equation involving strong geometric nonlinearity together with complicating moving boundary effect. Experimental deflection profiles under static loading are obtained through manual image processing technique using AutoCAD® software. Whereas, strain measurements are performed using strain gauges with data acquisition system (HBM-MX840B). Comparison between the theoretical and experimental results lead towards observation on stress concentration effect due to presence of geometric discontinuity in form of a small hole in the physical system. A modified formulation is proposed using domain decomposition method incorporating effect of geometric discontinuity through an equivalent curved beam geometry of variable cross-section. The modified theoretical model is validated successfully with the experimental results, and observations on stress characteristics and effect of hole diameter to beam width ratio are made.

Exact Three-Dimensional Stability Analysis of Plate Using an Direct Variational Energy Method

In this paper, direct variational calculus was put into practical use to analyses the three dimensional (3D) stability of rectangular thick plate which was simply supported at all the four edges (SSSS) under uniformly distributed compressive load. In the analysis, both trigonometric and polynomial displacement functions were used. This was done by formulating the equation of total potential energy for a thick plate using the 3D constitutive relations, from then on, the equation of compatibility was obtained to determine the relationship between the rotations and deflection. In the same way, governing equation was obtained through minimization of the total potential energy functional with respect to deflection. The solution of the governing equation is the function for deflection. Functions for rotations were obtained from deflection function using the solution of compatibility equations. These functions, deflection and rotations were substituted back into the energy functional, from where, through minimizations with respect to displacement coefficients, formulas for analysis were obtained. In the result, the critical buckling loads from the present study are higher than those of refined plate theories with 7.70%, signifying the coarseness of the refined plate theories. This amount of difference cannot be overlooked. However, it is shown that, all the recorded average percentage differences between trigonometric and polynomial approaches used in this work and those of 3D exact elasticity theory is lower than 1.0%, confirming the exactness of the present theory. Thus, the exact 3D plate theory obtained, provides a good solution for the stability analysis of plate and, can be recommended for analysis of any type of rectangular plates under the same loading and boundary condition. Doi: 10.28991/CEJ-2022-08-01-05 Full Text: PDF

Elastic and Thermoelastic Responses of Orthotropic Half-Planes

A unified approach is presented for constructing explicit solutions to the plane elasticity and thermoelasticity problems for orthotropic half-planes. The solutions are constructed in forms which decrease the distance from the loaded segments of the boundary for any feasible relationship between the elastic moduli of orthotropic materials. For the construction, the direct integration method was employed to reduce the formulated problems to a governing equation for a key function. In turn, the governing equation was reduced to an integral equation of the second kind, whose explicit analytical solution was constructed by using the resolvent-kernel algorithm.

Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

Bending Analysis of Isotropic Rectangular Kirchhoff Plates Subjected to Non-Uniform Heating Using the Fourier Transform Method

The object of this paper is the bending analysis of isotropic rectangular Kirchhoff plates subjected to non-uniform heating (NUH) using the Fourier transform method. The bottom and top surfaces of the plate are assumed to have different changes in temperature, whereas the change in temperature of the mid-surface is zero. According to classical plate theory, the governing equation of the plate contains second derivatives of the NUH; these derivatives are zero by constant value of the NUH, which leads to its absence in the governing equation. This paper presented an approach by which Fourier sine transform was utilized to describe the NUH, while the double trigonometric series of Navier and the simple trigonometric series of Lévy were utilized to describe the deflection curve. Thus, the NUH appeared in the governing equation, which simplified the analysis. Rectangular plates simply supported along all edges were analyzed, bending moments, twisting moments, and deflections being determined. In addition, rectangular plates simply supported along two opposite edges were analyzed; the other edges having various support conditions (free, simply supported, and fixed).

Numerical modeling of groundwater flow based on explicit and fully implicit schemes of finite volume method

This paper documents a novel numerical model for calculating the behavior of unsteady, one-dimensional groundwater flow by using the finite volume method. The developed model determined water table fluctuations for different scenarios as follows: Drainage and recession from an unconfined aquifer, and water table fluctuations above an inclined leaky layer due to ditch recharge with a constant and variable upper boundary condition. The Boussinesq equation, which is the governing equation in this domain, is linearized and solved numerically in both of the explicit and fully implicit conditions. Meanwhile, the upwind scheme is employed to discretize the governing equation. The computed outcomes of both the explicit and implicit approaches agreed well with the results of analytical solution and laboratory experiments. The main reason is that in the first half of simulation process explicit scheme obtains slightly better results and in the second half of the simulation process fully implicit scheme predicts more reliable outcomes that are hidden in the neighbor node points. As a final point, the numerical outcomes confirm that the developed model is capable of calculating satisfactory outcomes in engineering and science applications.

Integrate-and-Differentiate Approach to Nonlinear System Identification

In this paper, we consider a problem of parametric identification of a piece-wise linear mechanical system described by ordinary differential equations. We reconstruct the phase space of the investigated system from accelerometer data and perform parameter identification using iteratively reweighted least squares. Two key features of our study are as follows. First, we use a differentiated governing equation containing acceleration and velocity as the main independent variables instead of the conventional governing equation in velocity and position. Second, we modify the iteratively reweighted least squares method by including an auxiliary reclassification step into it. The application of this method allows us to improve the identification accuracy through the elimination of classification errors needed for parameter estimation of piece-wise linear differential equations. Simulation of the Duffing-like chaotic mechanical system and experimental study of an aluminum beam with asymmetric joint show that the proposed approach is more accurate than state-of-the-art solutions.

Attempt to quantify the impact of seasonal air density variation on operating tip-speed ratio of small wind turbines

This study presents the impact of seasonal variation in air density on the operating tip-speed ratio of small wind turbines. The air density, which varies depending on the temperature, atmospheric pressure, and relative humidity, has an annual amplitude of about 5% in Tokyo, Japan. This study quantified this impact using the rotational speed equation of motion in a small wind turbine informed by previous work. This governing equation has been simplified by expanding the aerodynamic torque coefficient profile for a wind turbine rotor to the tip-speed ratio. Furthermore, this governing equation is simplified by using nondimensional forms of the air density, inflow wind velocity, and rotational speed with their characteristic values. In this study, the generator’s load is set to be constant based on a previous analysis of a small wind turbine. By considering the equilibrium between the aerodynamic torque and the load torque of the governing equation at the optimum tip-speed ratio, the impact of the variation in the air density on the operating tip-speed ratio was expressed using a simple mathematical form. As shown in this derived form, the operating tip-speed ratio was found to be less sensitive to a variation in air density than that in inflow wind velocity.