scholarly journals Dynamic Response of a Beam Subjected to Moving Load and Moving Mass Supported by Pasternak Foundation

2012 ◽  
Vol 19 (2) ◽  
pp. 205-220 ◽  
Author(s):  
Rajib Ul Alam Uzzal ◽  
Rama B. Bhat ◽  
Waiz Ahmed
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamic Response ◽  
Moving Load ◽  
Bending Moment ◽  
Dynamic Responses ◽  
Moving Mass ◽  
Pasternak Foundation ◽  
Partial Differential ◽  
Foundation Parameters

This paper presents the dynamic response of an Euler- Bernoulli beam supported on two-parameter Pasternak foundation subjected to moving load as well as moving mass. Modal analysis along with Fourier transform technique is employed to find the analytical solution of the governing partial differential equation. Shape functions are assumed to convert the partial differential equation into a series of ordinary differential equations. The dynamic responses of the beam in terms of normalized deflection and bending moment have been investigated for different velocity ratios under moving load and moving mass conditions. The effect of moving load velocity on dynamic deflection and bending moment responses of the beam have been investigated. The effect of foundation parameters such as, stiffness and shear modulus on dynamic deflection and bending moment responses have also been investigated for both moving load and moving mass at constant speeds. Numerical results obtained from the study are presented and discussed.

Dynamic Analysis of Electrostatically Actuated Nanobeam Based on Strain Gradient Theory

2015 ◽  
Vol 15 (04) ◽  
pp. 1450059 ◽  
Author(s):  
Ehsan Maani Miandoab ◽  
Hossein Nejat Pishkenari ◽  
Aghil Yousefi-Koma
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamic Response ◽  
Strain Gradient ◽  
Strain Gradient Theory ◽  
Mode Shapes ◽  
Gradient Theory ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Electrostatically Actuated

In this study, dynamic response of a micro- and nanobeams under electrostatic actuation is investigated using strain gradient theory. To solve the governing sixth-order partial differential equation, mode shapes and natural frequencies of beam using Euler–Bernoulli and strain gradient theories are derived and then compared with classical theory. Galerkin projection is utilized to convert the partial differential equation to ordinary differential equations representing the system mode shapes. Accuracy of proposed one degree of freedom model is verified by comparing the dynamic response of the electrostatically actuated micro-beam with analogue equation and differential quadrature methods. Moreover, the static pull-in voltages of micro-beams found by one DOF model are compared with the reported data in literature. The main advantage of proposed method based on the Galerkin method is its simplicity and also its low computational cost in analyzing the dynamic and static responses of micro- and nanobeams. Additionally, effect of axial force, beam thickness and applied voltage are analyzed. The results obtained based on strain gradient theory, are compared with classical and modified couple stress theories which are the special cases of the strain gradient theory. It is shown that strain gradient theory leads to higher frequency and lower amplitude in comparison with two other theories.

scholarly journals Modal Analysis of Vibration of Euler-Bernoulli Beam Subjected to Concentrated Moving Load

2020 ◽  
pp. 2324-2334
Author(s):  
Usman M. A ◽  
Makinde T. A. ◽  
Daniel D. O.
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Methods ◽  
Modal Analysis ◽  
Series Solution ◽  
Moving Load ◽  
Bernoulli Beam ◽  
Partial Differential ◽  
Concentrated Load ◽  
Euler Bernoulli Beam

This paper investigates the modal analysis of vibration of Euler-Bernoulli beam subjected to concentrated load. The governing partial differential equation was analysed to determine the behaviour of the system under consideration. The series solution and numerical methods were used to solve the governing partial differential equation. The results revealed that the amplitude increases as the length of the beam increases. It was also found that the response amplitude increases as the foundation increases at fixed length of the beam.

scholarly journals Dynamic Response to Moving Distributed Masses of Pre-Stressed Uniform Rayleigh Beam Resting on Variable Elastic Pasternak Foundation

2018 ◽  
pp. 1-9
Author(s):  
AS Adeoye ◽  
TO Awodola
Keyword(s):  
Differential Equation ◽  
Dynamic Response ◽  
Moving Mass ◽  
Order Partial Differential Equation ◽  
Pasternak Foundation ◽  
Mass System ◽  
Force System ◽  
Rayleigh Beam ◽  
Moving Force ◽  
Pasternak Elastic Foundation

The dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam resting on variable elastic Pasternak foundation is examined. The equation governing this problem is a fourth order partial differential equation with variable and singular co-efficients. To solve this cumbersome equation, the method of Galerkin approach is adopted to reduce the governing differential equation to a sequence of coupled second order ordinary differential equation which is then simplified further with modified asymptotic method of Struble. The more simplified equation is solved using the Laplace transformation technique. The closed form solutions obtained are analyzed in order to show the conditions of resonance, and to show that resonance is attained earlier in moving mass system than in the moving force system. The results in plotted graphs show that as the axial force, the rotatory inertia, foundation modulus and shear modulus increase, the deflection of the elastically supported non-uniform Rayleigh beam decreases in each case. The transverse deflections of the beam on variable Pasternak elastic foundation are higher under the action of moving masses than those when only the force effects of the moving load are considered. This implies that resonance is reached faster in moving mass problem than in moving force problem.

scholarly journals Influence of a Moving Mass on the Dynamic Behaviour of Viscoelastically Connected Prismatic Double-Rayleigh Beam System Having Arbitrary End Supports

2017 ◽  
Vol 2017 ◽  
pp. 1-30 ◽  
Author(s):  
Jacob Abiodun Gbadeyan ◽  
Fatai Akangbe Hammed
Keyword(s):  
Dynamic Response ◽  
Integral Transform ◽  
Moving Load ◽  
Dynamic Responses ◽  
Solution Technique ◽  
Moving Mass ◽  
Transform Method ◽  
Rayleigh Beam ◽  
Beam System ◽  
Finite Integral

This paper deals with the lateral vibration of a finite double-Rayleigh beam system having arbitrary classical end conditions and traversed by a concentrated moving mass. The system is made up of two identical parallel uniform Rayleigh beams which are continuously joined together by a viscoelastic Winkler type layer. Of particular interest, however, is the effect of the mass of the moving load on the dynamic response of the system. To this end, a solution technique based on the generalized finite integral transform, modified Struble’s method, and differential transform method (DTM) is developed. Numerical examples are given for the purpose of demonstrating the simplicity and efficiency of the technique. The dynamic responses of the system are presented graphically and found to be in good agreement with those previously obtained in the literature for the case of a moving force. The conditions under which the system reaches a state of resonance and the corresponding critical speeds were established. The effects of variations of the ratio (γ1) of the mass of the moving load to the mass of the beam on the dynamic response are presented. The effects of other parameters on the dynamic response of the system are also examined.

Qxygen transfer in gravity flow sewers

2000 ◽  
Vol 42 (3-4) ◽  
pp. 417-422 ◽  
Author(s):  
T.Y. Pai ◽  
C.F. Ouyang ◽  
Y.C. Liao ◽  
H.G. Leu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dissolved Oxygen ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Eddy Diffusion ◽  
Partial Differential ◽  
Diffusion Term ◽  
Sewer Pipes ◽  
Gravity Sewer

Oxygen diffused to water in gravity sewer pipes was studied in a 21 m long, 0.15 m diameter model sewer. At first, the sodium sulfide was added into the clean water to deoxygenate, then the pump was started to recirculate the water and the deoxygenated water was reaerated. The dissolved oxygen microelectrode was installed to measure the dissolved oxygen concentrations varied with flow velocity, time and depth. The dissolved oxygen concentration profiles were constructed and observed. The partial differential equation diffusion model that considered Fick's law including the molecular diffusion term and eddy diffusion term were derived. The analytic solution of the partial differential equation was used to determine the diffusivities by the method of nonlinear regression. The diffusivity values for the oxygen transfer was found to be a function of molecular diffusion, eddy diffusion and flow velocity.

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