scholarly journals Recent Advances in the Application of Differential Equations in Mechanical Engineering Problems

2018 ◽  
Vol 2018 ◽  
pp. 1-3 ◽  
Author(s):  
Rahmat Ellahi ◽  
Constantin Fetecau ◽  
Mohsen Sheikholeslami
Keyword(s):  
Differential Equations ◽  
Mechanical Engineering ◽  
Recent Advances ◽  
Engineering Problems

scholarly journals Applied Dynamics In School And Practice

2019 ◽  
Vol 9 (2) ◽  
pp. 39-50
Author(s):  
Michael Spektor ◽  
Walter W. Buchanan ◽  
Lawrence Wolf
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Mechanical Engineering ◽  
Equations Of Motion ◽  
Engineering Problem ◽  
Linear Differential Equations ◽  
Engineering System ◽  
Engineering Technology ◽  
Engineering Problems ◽  
Linear Differential

Mechanical engineering, mechanical engineering technology, and related educational programs are not addressing in a sufficient way the principles associated with applying analytical investigations in solving actual engineering problems. Because of this, graduates do not have the adequate skills required to use the methods of applied dynamics in the process of analyzing mechanical systems. These methods allow one to obtain an understanding of the role of the parameters of a system and to carry out a purposeful control of the values of these parameters with the goal to achieve the desired performance. Engineering and engineering technology programs pay very little attention to addressing these steps. It should be stressed that these programs do not offer a universal straightforward methodology of solving linear differential equations of motion that allow revealing all important interrelationships between the aspects of the engineering problem. It is difficult to formulate the reasons why there is such a low interest in applying the analytical approach in order to reveal the interrelationships between decisive aspects of the operational process of an engineering system in order to achieve the desired goal. Actually, there is almost a complete silence with regard to this issue. Hence, we assume that the first reason could be that there is no recognition of the existence of such a problem. In other words, there is no need to apply these analytical methods since these methods are not beneficial. We do not believe that the engineering community supports this reason. It is not a matter of demonstrating factual data that show how many times the theory was helpful. Without the support of the theory we cannot justifiably evaluate the results of our solutions. If we agree that there is problem, then why are there no publications that would stimulate discussions leading toward a solution of the problem? Here is the second reason. Until now, engineering programs do not present the straightforward universal theoretically sound methodologies for solving the second order linear differential equations that are vital for mechanical and electrical engineering. Without any suggestions of how to solve this problem, it did not make much sense to begin a discussion. In our opinion, this is why we have silence with the regard to this problem. However, it is well known that Laplace Transforms allow solving any linear differential equation of motion. It is justifiable to assume that the main reason why the Laplace Transform methodology is not adopted by learning environments consists in the absence of the majority of tables of Laplace Transform Pairs that are needed for solving differential equations of motion as well as differential equations describing electrical circuits. However, the situation is changed. Current publications comprise the adequate tables that are needed for solving linear differential equations of motion associated with all common mechanical engineering problems. Practicing engineers and students need assistance in acquiring the knowledge of composing differential equations of motion. They need certain training in solving these equations using Laplace Transform methodology. Several recommendations are proposed on how to expedite the implementation in academia and in industry of the methods of applied dynamics in solving common mechanical engineering problems.

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
A. M. Nagy ◽  
N. H. Sweilam ◽  
Adel A. El-Sayed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Algebraic System ◽  
Order Differential Equation ◽  
Fundamental Problem ◽  
Operational Matrix ◽  
Variable Order ◽  
Engineering Problems ◽  
Fractional Differential ◽  
Fourth Kind

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

Identification of Modal Parameters of an Elastic Rotor With Oil Film Bearings

1984 ◽  
Vol 106 (1) ◽  
pp. 107-112 ◽  
Author(s):  
Rainer Nordmann
Keyword(s):  
Modal Analysis ◽  
Dynamic Behavior ◽  
Design Process ◽  
Mechanical Engineering ◽  
Mechanical Systems ◽  
Rotating Machinery ◽  
Modal Parameters ◽  
Powerful Method ◽  
Oil Film ◽  
Engineering Problems

Investigations of the dynamic behavior of structures have become increasingly important in the design process of mechanical systems. To have a better understanding of the dynamic behavior of a structure, the knowledge of the modal parameters is very important. The powerful method of experimental modal analysis has been used to measure modal parameters in many mechanical engineering problems. But the method was mainly applied to nonrotating structures. This presentation shows improvements of the classical modal analysis for a successful application in rotating machinery with nonconservative effects. An example is given, investigating the modal parameters of an elastic rotor with oil film bearings.

scholarly journals Recent Advances in Symmetry Groups and Conservation Laws for Partial Differential Equations and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-2 ◽  
Author(s):  
Maria Gandarias ◽  
Mariano Torrisi ◽  
Maria Bruzón ◽  
Rita Tracinà ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Symmetry Groups ◽  
Partial Differential ◽  
Recent Advances

scholarly journals Data Driven Solutions and Discoveries in Mechanics Using Physics Informed Neural Network

2020 ◽  
Author(s):  
Qi Zhang ◽  
Yilin Chen ◽  
Ziyi Yang
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Data Driven ◽  
Attractive Alternative ◽  
The Finite Element Method ◽  
The Neural Network ◽  
Engineering Problems ◽  
Free Parameters

Deep learning has achieved remarkable success in diverse computer science applications, however, its use in other traditional engineering fields has emerged only recently. In this project, we solved several mechanics problems governed by differential equations, using physics informed neural networks (PINN). The PINN embeds the differential equations into the loss of the neural network using automatic differentiation. We present our developments in the context of solving two main classes of problems: data-driven solutions and data-driven discoveries, and we compare the results with either analytical solutions or numerical solutions using the finite element method. The remarkable achievements of the PINN model shown in this report suggest the bright prospect of the physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters. More broadly, this study shows that PINN provides an attractive alternative to solve traditional engineering problems.

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