scholarly journals Analytical study of the natural bending oscillations of a concave beam with parabolic change in thickness

2021 ◽  
Vol 3 (7 (111)) ◽  
pp. 15-23
Author(s):  
Kirill Trapezon ◽  
Alexandr Trapezon
Keyword(s):  
Analytical Solution ◽  
Analytical Study ◽  
Variable Coefficients ◽  
Rigid Fixation ◽  
Natural Oscillations ◽  
Parabolic Profile ◽  
Rigidity Parameter ◽  
Calculation Results ◽  
Maximum Stresses ◽  
Free Beam

The synthesis of factorization and symmetry methods produced a general analytical solution to the fourth-order differential equation with variable coefficients. The form and structure of the variable coefficients correspond, in this case, to the problem of the oscillations of a concave beam of variable thickness. The solution to this equation makes it possible to study in detail the oscillations of such and similar, for example convex, beams at the different fixation of their ends' sections. A practical confirmation has been obtained that the beam whose thickness changes in line with the concave parabola law H=a2x2+1, where a is the concave factor, demonstrates an increase in the natural frequencies of its free oscillations with an increase in its rigidity. As an example, the object's maximum deflection dependence on the beam rigidity factor has been established. The nature of this dependence confirmed the obvious conclusion that the deflections had decreased while the rigidity had increased. The evidence from the calculation results can be a testament to the correctness of the reported procedure of problem-solving. The considered problem and the analytical solution to it could serve as a practical guide to the optimal design of beam structures. In this case, it is very important to take into consideration the place and nature of the distribution of cyclical extreme operating stresses. The resulting ratios to solve the problem make it possible to simulate the required normal stresses in both the fixation and central zones when the rigidity parameter is changed. Designers could predict such a parabolic profile of the beam, which would ensure the required reduction of maximum stresses in the place of fixing the beam. The considered example of solving the problem of the natural oscillations of the beam with rigid fixation of the ends illustrates the effectiveness of the factoring and symmetry methods used. The developed solution algorithm could be extended to study the natural bending oscillations of the beam at other fixing techniques, not excluding a variant of a completely free beam

An analytical solution of multi-dimensional space fractional diffusion equations with variable coefficients

2020 ◽  
pp. 2150006
Author(s):  
Pratibha Verma ◽  
Manoj Kumar
Keyword(s):  
Analytical Solution ◽  
Diffusion Coefficients ◽  
High Efficiency ◽  
Dimensional Space ◽  
Adomian Decomposition Method ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Adomian Decomposition

In this paper, we have considered the multi-dimensional space fractional diffusion equations with variable coefficients. The fractional operators (derivative/integral) are used based on the Caputo definition. This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method (TSADM). Moreover, new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem. We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions. Finally, the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods. The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM. There are no restrictions imposed on the problems for diffusion coefficients, and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.

Numerical Study of a 3-Axis Stage for Application of Fast Tool Servo

2014 ◽  
Vol 625 ◽  
pp. 219-223 ◽  
Author(s):  
Yung Tien Liu ◽  
Bo Jheng Li
Keyword(s):  
Numerical Study ◽  
Fast Response ◽  
Maximum Deviation ◽  
Stress And Strain ◽  
Fast Tool Servo ◽  
The Finite Element Method ◽  
Resonant Frequencies ◽  
Calculation Results ◽  
Flexural Hinges ◽  
Maximum Stresses

In this paper, a 3-axis stage consisted of a XY stage and Z-axis feeding tool holder is proposed for the application of fast tool servo (FTS). The XY stage actuated by six piezoelectric (PZT) actuators is designed with symmetric flexural hinges featuring low interference motions, high stiffness, and fast response. Numerical design using the finite element method (FEM) was conducted to investigate the steady characteristics (displacement, stiffness, stress, and strain) and dynamic characteristic of resonance frequency. According to calculation results, the major characteristics obtained along XYZ axes are as follows: displacements induced are 10.06, 10.28, and 20.31 μm due to the applied voltage being 50 V; stiffness are 112.84, 110.31, and 223.34 kN/mm; the maximum stresses at the hinges are 9.78, 10.9, and 100.56 N/mm2, which are lower than the allowable stress of aluminum used; and the resonant frequencies are 1.0, 0.64, and 0.4 kHz, respectively. Experimental examinations regarding to the resonant frequencies were performed with a maximum deviation of 16% along the Z-axis compared to the simulation result. As a result of the investigation, it is expected that the 3-axis stage can be effectively applied to implement a FTS.

A Series Solution for Wave Scattering by a Circular Island on a Shoal Based on the Mild-Slope Equation

2012 ◽  
Vol 28 (1) ◽  
pp. 41-51 ◽  
Author(s):  
Y.-M. Cheng ◽  
C.-T. Chen ◽  
L.-F. Tu ◽  
J.-F. Lee
Keyword(s):  
Analytical Solution ◽  
Wave Scattering ◽  
Present Method ◽  
Series Solution ◽  
Shallow Water Equation ◽  
Variable Coefficients ◽  
Fourier Cosine ◽  
Mild Slope Equation ◽  
Scattering Field ◽  
Run Up

ABSTRACTA series solution based on the mild-slope equation is produced in this study of wave scattering produced by a circular cylindrical island mounted on an axi-symmetrical shoal. The solution is presumed to be a Fourier cosine expansion with variable coefficients in the radial direction on account of the symmetric scattering field, which translates the original 2-D boundary-value problem to a 1-D one in which an ordinary differential equation is in effect treated. Approximations to the coefficients of the governing equation with the Taylor expansions enable the use of the Frobenius method, and consequently the solution is obtained in a combined Fourier and power series. For verification, the present method is mainly compared with Zhu and Zhang's [1] analytical solution of the linearised shallow water equation for a conical shoal, and with a different analytical solution of the mild-slope equation developed by Liu et al. [2] for a paraboloidal shoal. Fine agreements are achieved. The present method is then used to investigate the variation pattern of the wave run-up when the shoal profile varies from conical to paraboloidal, and some interesting phenomena are observed.

Analytical Solution of Stress and Displacement Fields of Frozen Wall in Deep Alluvium According for Time-Space Effect

2012 ◽  
Vol 170-173 ◽  
pp. 1820-1826
Author(s):  
Mao Yan Ma ◽  
Hua Cheng ◽  
Chuan Xin Rong
Keyword(s):  
Analytical Solution ◽  
Radial Stress ◽  
Displacement Fields ◽  
Time Space ◽  
Stress And Displacement Fields ◽  
Calculation Results ◽  
Space Effect ◽  
Frozen Wall ◽  
Theory Of Viscoelasticity ◽  
Shaft Wall

Based on the theory of viscoelasticity and the principle of interaction between surrounding rock and structure in unload state, analytical solution of the stress field and displacement field of the frozen wall is obtained. Calculation results of the stress and displacement fields suggest that radial stress and displacement in the sidewall are very large within 15 days after pouring concrete of the outer shaft wall, which is proved by the fact that engineering incidents such as shaft wall rupture happen exactly at this time. The results also indicate that radial stress at different points in frozen wall all tend to the imposed loads on outer frozen wall finally, and that means the frozen wall is fluidized. This calculation theory can be used in the design of frozen wall in deep alluvium.

scholarly journals An analytical solution for a fractional heat-like equation with variable coefficients

2017 ◽  
Vol 21 (4) ◽  
pp. 1759-1764 ◽  
Author(s):  
Run-qing Cui ◽  
Tao Ban
Keyword(s):  
Analytical Solution ◽  
Power Series ◽  
Solution Process ◽  
Fractional Power ◽  
Variable Coefficients ◽  
Series Method ◽  
Power Series Method ◽  
Fractional Power Series

The fractional power series method is used to solve a fractional heat-like equations with variable coefficients. The solution process is elucidated, and the results show that the method is simple but effective.

scholarly journals A Finite-Difference Solution of Solute Transport through a Membrane Bioreactor

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
B. Godongwana ◽  
D. Solomons ◽  
M. S. Sheldon
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Membrane Bioreactor ◽  
Elliptic Partial Differential Equation ◽  
Reaction Rates ◽  
Variable Coefficients ◽  
Concentration Profiles ◽  
Algebraic Equations ◽  
Regular Perturbation ◽  
Sink Term

The current paper presents a theoretical analysis of the transport of solutes through a fixed-film membrane bioreactor (MBR), immobilised with an active biocatalyst. The dimensionless convection-diffusion equation with variable coefficients was solved analytically and numerically for concentration profiles of the solutes through the MBR. The analytical solution makes use of regular perturbation and accounts for radial convective flow as well as axial diffusion of the substrate species. The Michaelis-Menten (or Monod) rate equation was assumed for the sink term, and the perturbation was extended up to second-order. In the analytical solution only the first-order limit of the Michaelis-Menten equation was considered; hence the linearized equation was solved. In the numerical solution, however, this restriction was lifted. The solution of the nonlinear, elliptic, partial differential equation was based on an implicit finite-difference method (FDM). An upwind scheme was employed for numerical stability. The resulting algebraic equations were solved simultaneously using the multivariate Newton-Raphson iteration method. The solution allows for the evaluation of the effect on the concentration profiles of (i) the radial and axial convective velocity, (ii) the convective mass transfer rates, (iii) the reaction rates, (iv) the fraction retentate, and (v) the aspect ratio.

Scattering of SH waves around circular canyon in inhomogeneous wedge space

2020 ◽  
Vol 223 (1) ◽  
pp. 45-56
Author(s):  
Zailin Yang ◽  
Xinzhu Li ◽  
Yunqiu Song ◽  
Guanxixi Jiang ◽  
Menghan Sun ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Helmholtz Equation ◽  
Ground Motion ◽  
Governing Equation ◽  
Function Method ◽  
Complex Function ◽  
Variable Coefficients ◽  
Sh Waves ◽  
Free Condition ◽  
Complex Function Method

SUMMARY Scattering of SH waves around a circular canyon in radial inhomogeneous wedge space is investigated in this paper. Based on the complex function method, the governing equation with variable coefficients is transformed into a standard Helmholtz equation and the corresponding analytical solution to this problem is derived. The unknown coefficients in the wavefield is obtained by enforcing the stress-free condition in the circular canyon, then the incident, reflected and scattering waves in the total wavefield are all acquired. Through the calculation and analysis of the parameters that affect the ground motion, the influence of the circular canyon on the ground motion in the radial inhomogeneous wedge space is obtained. Moreover, combined with graphical results, the effects by inhomogeneous parameter on ground motion with circular canyon in wedge space are conspicuous.

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