scholarly journals ON EXACT ANALYTICAL SOLUTIONS OF THE TIMOSHENKO BEAM MODEL UNDER UNIFORM AND VARIABLE LOADS

2021 ◽  
Vol 16 (5) ◽  
Author(s):  
Kamran Malik
Keyword(s):  
Timoshenko Beam ◽  
Analytical Solutions ◽  
Research Work ◽  
Finite Volume Methods ◽  
Variable Load ◽  
Beam Model ◽  
Structure Model ◽  
Timoshenko Beam Model ◽  
Exact Analytical Solutions ◽  
The Mathematical Model

In this research work, we consider the mathematical model of the Timoshenko beam (TB) problem in the form of a boundary-value problem of a system of ordinary differential equations. Instead of numerical solution using finite difference and finite volume methods, an attempt is made to derive the exact analytical solutions of the model with boundary feedback for a better and explicit description of the rotation and displacement parameters of the TB structure model. The explicit analytical solutions have been successfully found for the uniform and real-time variable load cases. The rotation and displacement profiles obtained through the analytical solutions accurately picture the structure of the beam under uniform and variable loads.

scholarly journals AN EFFICIENT FINITE DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION OF TIMOSHENKO BEAM MODEL

2021 ◽  
Vol 16 (5) ◽  
Author(s):  
Kamran Malik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Timoshenko Beam ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Variable Load ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Step Size

We propose and implement a finite difference scheme for the numerical solution of the Timoshenko beam model without locking phenomenon. The averaging concept is used in approximating the function, and thus developing the scheme for elements. Finally, the system is discretized into the algebraic system using the proposed scheme and the numerical solution is attained. The numerical solutions are attained for a constant load and a variable load comprising linear and exponential functions. The mathematical model of the Timoshenko beam (TB) problem in the form of a boundary-value problem has been solved successfully for the rotation and displacement parameters. The results agree with other schemes in the literature for various values of the parameter and step size.

scholarly journals A Mathematical Model of Epidemics—A Tutorial for Students

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1174 ◽  
Author(s):  
Yutaka Okabe ◽  
Akira Shudo
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Analytical Solutions ◽  
Sir Model ◽  
Exact Analytical Solutions ◽  
The Mathematical Model ◽  
Epidemic Diseases

This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. Additionally, we treat the generalization of the SIR model including births and natural deaths.

Investigating vibrations of viscoelastic fluid-conveying carbon nanotubes resting on viscoelastic foundation using a nonlocal fractional Timoshenko beam model

2020 ◽  
pp. 239779142093170
Author(s):  
M Faraji Oskouie ◽  
R Ansari ◽  
H Rouhi
Keyword(s):  
Carbon Nanotubes ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Time Domain ◽  
Beam Model ◽  
Viscoelastic Foundation ◽  
Timoshenko Beam Model ◽  
Governing Equations ◽  
The Time Domain

On the basis of fractional viscoelasticity, the size-dependent free-vibration response of viscoelastic carbon nanotubes conveying fluid and resting on viscoelastic foundation is studied in this article. To this end, a nonlocal Timoshenko beam model is developed in the context of fractional calculus. Hamilton’s principle is applied in order to obtain the fractional governing equations including nanoscale effects. The Kelvin–Voigt viscoelastic model is also used for the constitutive equations. The free-vibration problem is solved using two methods. In the first method, which is limited to the simply supported boundary conditions, the Galerkin technique is employed for discretizing the spatial variables and reducing the governing equations to a set of ordinary differential equations on the time domain. Then, the Duffing-type time-dependent equations including fractional derivatives are solved via fractional integrator transfer functions. In the second method, which can be utilized for carbon nanotubes with different types of boundary conditions, the generalized differential quadrature technique is used for discretizing the governing equations on spatial grids, whereas the finite difference technique is used on the time domain. In the results, the influences of nonlocality, geometrical parameters, fractional derivative orders, viscoelastic foundation, and fluid flow velocity on the time responses of carbon nanotubes are analyzed.

Precise Frequency Calculation Model on BTA Boring Bar

2014 ◽  
Vol 532 ◽  
pp. 398-401
Author(s):  
Wu Zhao ◽  
Wei Tao Jia ◽  
Quan Bin Zhang ◽  
Zhan Qi Hu
Keyword(s):  
Timoshenko Beam ◽  
Calculation Model ◽  
Cutting Fluid ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Intrinsic Frequency ◽  
Boring Bar ◽  
Precise Calculation ◽  
Vibration Model ◽  
Frequency Calculation

For the purpose of precise calculation on intrinsic frequency of the deep-hole boring bar in trepanning heavy-duty processing, a new frequency calculation model is proposed, based on the synthetically investigation of the axial press effects, intermediate supported, Coriolis inertia effects induced by cutting fluid and other relevant various factors of boring bar. The boring bar can be decomposed into the two parts, corresponding to the liquid-solid coupling vibration model inside the work part and Timoshenko beam model outside the work part, respectively. Then assume the whole system as continuous equal span beam model to combine these two parts. Through nesting liquid-solid coupled vibration model (considering cutting fluid velocity) and Timoshenko beam model (containing axial pressure and lateral bending) among the continuous beam model (considering equal span), the precise calculation on intrinsic frequency of the boring system can be completed.

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