Longitudinal–Flexural–Torsional Dynamic Behavior of Liquid-Filled Pipelines: An Analytic Solution

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Dimitrios G. Pavlou
Keyword(s):  
Dynamic Behavior ◽  
Analytic Solution ◽  
Integral Transform ◽  
Equation System ◽  
Time And Space ◽  
Partial Differential ◽  
Coupled Equations ◽  
Contraction Ratio ◽  
The Time Domain ◽  
Poisson Contraction

Abstract The 14 partial differential equation system describing the longitudinal–flexural–torsional dynamic behavior of liquid-filled pipelines contains coupled equations due to mutual boundary conditions and Poisson contraction ratio terms. Solutions of the above system are available in the frequency-domain or in the time-domain (method of characteristics (MOC)). In this paper, an analytic solution in the domain of time and space is achieved. Double integral transform, namely, finite sine Fourier transform (FSFT) and Laplace transform, is applied to the derived system of the 14 fourth-order partial differential equations, yielding an algebraic system in terms of the transformed variables. The inversion of the FSFT is an easy task, but the analytic inversion of the Laplace transforms is very challenging. Both integral transform inversions of the 14 transformed variables are successfully performed, and an analytic matrix formula in the domain of time and space along with numerical results is obtained.

Analytical/Numerical Hybrid Solution for One-Dimensional Ablation Problem

2003 ◽  
Author(s):  
Fa´bio Yukio Kurokawa ◽  
Antonio Joa˜o Diniz ◽  
Joa˜o Batista Campos-Silva
Keyword(s):  
Differential Equations ◽  
Integral Transform ◽  
Thermal Protection ◽  
Linear Partial Differential Equation ◽  
Equation System ◽  
Partial Differential ◽  
One Dimensional ◽  
Plane Plate ◽  
Non Linear ◽  
Non Linear System

Ablation is a thermal protection process with several applications in engineering, mainly in the field of airspace industry. The use of conventional materials must be quite restricted, because they would suffer catastrophic flaws due to thermal degradation of their structures. However, the same materials can be quite suitable once being protected by well-known ablative materials. The process that involves the ablative phenomena is complex, could involve the whole or partial loss of material that is sacrificed for absorption of energy. The analysis of the ablative process in a blunt body with revolution geometry will be made on the stagnation point area that can be simplified as a one-dimensional plane plate problem. In this work the Generalized Integral Transform Technique (GITT) is employed for the solution of the non-linear system of coupled partial differential equations that model the phenomena. The solution of the problem is obtained by transforming the non-linear partial differential equation system to a system of coupled first order ordinary differential equations and then solving it by using well-established numerical routines. The results of interest such as the temperature field, the depth and the rate of removal of the ablative material are presented and compared with those ones available in the open literature.

Boundary control design based on partial differential equation observer for vibration suppression and attitude control of flexible satellites with multi-section solar panels

2021 ◽  
pp. 107754632199015
Author(s):  
Mohammad Mahdi Ataei ◽  
Hassan Salarieh ◽  
Hossein Nejat Pishkenari ◽  
Hadi Jalili
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Attitude Control ◽  
Vibration Suppression ◽  
Equation System ◽  
Solar Panels ◽  
Attitude Dynamics ◽  
Partial Differential ◽  
Satellite Attitude ◽  
Element Technique

A novel partial differential equation observer is proposed to be used in boundary attitude and vibration control of flexible satellites. Solar panels’ vibrations and attitude dynamics form a coupled partial differential equation–ordinary differential equation system which is controlled directly without discretization. Few feedback signals from boundaries are required which are estimated via a partial differential equation observer. Consequently, just satellite attitude and angular velocity should be measured and still the control system benefits information from continuous part vibrations. The closed-loop system is proved to be asymptotically stable. Simulations with a finite element technique illustrate good performance of this observer-based boundary controller.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Theory of a Time Domain Boundary Element Development for the Dynamic Analysis of Coupled Multiphase Porous Media

2017 ◽  
Vol 08 (03n04) ◽  
pp. 1750007
Author(s):  
Pooneh Maghoul ◽  
Behrouz Gatmiri
Keyword(s):  
Boundary Element ◽  
Time Domain ◽  
Unsaturated Soils ◽  
Water Pressure ◽  
Domain Boundary ◽  
Fundamental Solutions ◽  
Element Formulation ◽  
Inertia Effects ◽  
Coupled Equations ◽  
The Time Domain

This paper presents an advanced formulation of the time-domain two-dimensional (2D) boundary element method (BEM) for an elastic, homogeneous unsaturated soil subjected to dynamic loadings. Unlike the usual time-domain BEM, the present formulation applies a convolution quadrature which requires only the Laplace-domain instead of the time-domain fundamental solutions. The coupled equations governing the dynamic behavior of unsaturated soils ignoring contributions of the inertia effects of the fluids (water and air) are derived based on the poromechanics theory within the framework of a suction-based mathematical model. In this formulation, the solid skeleton displacements [Formula: see text], water pressure [Formula: see text] and air pressure [Formula: see text] are presumed to be independent variables. The fundamental solutions in Laplace transformed-domain for such a dynamic [Formula: see text] theory have been obtained previously by authors. Then, the BE formulation in time is derived after regularization by partial integrations and time and spatial discretizations. Thereafter, the BE formulation is implemented in a 2D boundary element code (PORO-BEM) for the numerical solution. To verify the accuracy of this implementation, the displacement response obtained by the boundary element formulation is verified by comparison with the elastodynamics problem.

Transient Conjugated Conduction: External Convection With Front Face Imposed Wall Heat Flux

2007 ◽  
Author(s):  
Carolina P. Naveira ◽  
Renato M. Cotta ◽  
Mohammed Lachi ◽  
Jacques Padet
Keyword(s):  
Heat Flux ◽  
Integral Transform ◽  
Heat Conduction Problem ◽  
Solid Wall ◽  
Method Of Lines ◽  
Wall Heat Flux ◽  
Front Face ◽  
Flat Plates ◽  
Partial Differential ◽  
Differential Formulation

This work presents hybrid numerical-analytical solutions for transient laminar forced convection over flat plates of non-negligible thickness, subjected to arbitrary time variations of applied wall heat flux at the interface fluid-solid wall. This conjugated conduction-convection problem is first simplified through the employment of the Coupled Integral Equations Approach (CIEA) to reformulate the heat conduction problem on the plate by averaging the related energy equation in the transversal direction. As a result, a partial differential formulation for the average wall temperature is obtained, while a third kind boundary condition is achieved for the fluid in the heat balance at the solid-fluid interface. From the available velocity distributions, the solution method is then proposed for the coupled partial differential equations, based on the Generalized Integral Transform Technique (GITT) under its partial transformation mode, combined with the method of lines implemented in the Mathematica 5.2 routine NDSolve.

scholarly journals Sinc-Galerkin method for solving hyperbolic partial differential equations

2018 ◽  
Vol 8 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Galerkin Method ◽  
Hyperbolic Equations ◽  
Approximate Solutions ◽  
Equation System ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Numerical Examples

In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

scholarly journals A separated representation involving multiple time scales within the Proper Generalized Decomposition framework

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Angelo Pasquale ◽  
Amine Ammar ◽  
Antonio Falcó ◽  
Simona Perotto ◽  
Elías Cueto ◽  
...  
Keyword(s):  
Time Scales ◽  
Time Discretization ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Proper Generalized Decomposition ◽  
Partial Differential ◽  
Time Coordinate ◽  
Separated Representation ◽  
The Time Domain ◽  
Fine Time

AbstractSolutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.

scholarly journals A New Numerical Procedure for Vibration Analysis of Beam under Impulse and Multiharmonics Piezoelectric Actuators

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Yassin Belkourchia ◽  
Lahcen Azrar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Methodological Approach ◽  
Numerical Procedure ◽  
Piezoelectric Actuators ◽  
Partial Differential ◽  
Simple Method ◽  
Regularization Procedure ◽  
Dirac Delta

The dynamic behavior of structures with piezoelectric patches is governed by partial differential equations with strong singularities. To directly deal with these equations, well adapted numerical procedures are required. In this work, the differential quadrature method (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. The DQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, the DQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to the fact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodological approach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-delta function are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can be directly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in the computation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonics piezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.

scholarly journals Invariant subspace method: A tool for solving fractional partial differential equations

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Sangita Choudhary ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Caputo Derivative ◽  
Linear Time ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Time And Space ◽  
Partial Differential ◽  
Non Linear

AbstractIn this paper invariant subspace method has been employed for solving linear and non-linear time and space fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the effectiveness and applicability of the method.

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