New Orleans, Louisiana
June 26, 2016
June 26, 2016
August 28, 2016
The programs in mechanical engineering and related fields do not provide adequate training to enable graduates to conduct analytical investigations of actual problems in dynamics. An analytical approach to the improvement and development of mechanical systems allows the purposeful control of the parameters and in the same time saves time and resources. The analytical investigation in applied dynamics comprises the following three steps: 1) Composing the differential equation of motion of the system and determining the initial conditions of motion; 2) Solving the differential equation of motion for the initial conditions of motion; 3) Analyzing the solution according to the goal of the investigation.
As it is well known, the differential equation of motion consists of loading factors: forces or moments. Active loading factors cause the motion, while resisting loading factors oppose the motion. The left side of the differential equation of motion consists of the sum of resisting loading factors including the force or moment of inertia, while the right side of the equation includes the sum of active loading factors (in the absence of active factors the right side of the equation equals zero). This looks simple, however sometimes it becomes not that simple when confronting a real-life engineering problem. On the beginning, the investigator, based on the information of the problem, should figure out the characteristics of the loading factors that are applied to the system. Obviously, insignificant loading factors could be ignored. It should be emphasized that the results of the investigation depend upon the measure of accuracy that the differential equation reflects the working process of the system.
The engineering programs do not offer a straightforward universal methodology of solving linear differential equations of motion. However, as it is known, the Laplace Transform allows solving any linear differential equation of motion or a system of two linear differential equations of motion for a two-degree-of-freedom system. The Laplace Transform methodology comprises the following three steps: 1) Converting the differential equation of motion into a corresponding algebraic equation of motion based on Laplace Transform Pairs that are represented in tables. There is no need to memorize the fundamentals of Laplace Transform in order to use the tables that are available in numerous publications; 2) Solving the obtained algebraic equation of motion for the displacement in the Laplace domain. 3) Inverting the obtained solution in the second step into the time domain. This inversion is also based on the Laplace Transform Pairs and actually represents the solution of the differential equation of motion.
The solution represents the law of motion of the system or its displacement as a function of time. The analysis of the solution includes taking the first and second derivatives of the displacement in order to determine the velocity and acceleration. By applying conventional mathematical actions to the obtained parameters of motion, it is possible to determine the role of the parameters of the system in achieving the goal of the investigation.
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