Solving Engineering Problems in Dynamics. Michael Spektor
6.2.3Solutions for a System with a Combination of a Hydraulic and an Elastic Link
6.2.4A System with a Hydraulic Link where the First Mass Is Subjected to a Constant External Force
6.2.5A Vibratory System Subjected to an External Sinusoidal Force
Index
Purposeful control and improvement of how existing mechanical systems perform is an important real-life problem, as is the development of new systems. We can obtain solutions to these problems by investigating the working processes of machines and their units and elements. These investigations should be based on fundamentals of dynamics combined with a variety of related sciences. The working processes that characterize system performance can be described by mathematical expressions that actually represent equations of motion of these systems. Analyzing these equations of motion reveals the relationship between the parameters of the system and their influence on performance and other system characteristics or elements.
This book contains comprehensive methods for analyzing the motion of engineering systems and their components. The analysis covers three basic phases: 1) composing the differential equation of motion, 2) solving the differential equation of motion, and 3) analyzing the solution. Engineering education provides the fundamental skills for completing these three phases. However, many engineers would benefit from additional training in using these fundamentals to solve real-life engineering problems. This book provides this training by describing in a step-by-step order the methods related to each of these three phases.
When assembling a differential equation of motion, it is essential to completely understand the components of this equation as well as the system’s working process. This book describes all possible components of the differential equation of motion and all possible factors of the working process. In mechanical engineering, all these components and factors represent forces and moments. The characteristics of all these loading factors and their application to particular differential equations of motion are presented in this book.
This book also introduces a straightforward universal methodology for solving differential equations of motion by using the Laplace Transform. This approach replaces calculus with conventional algebraic procedures that do not represent any difficulties for engineers. Using the Laplace methodology to solve differential equations of motion does not require memorizing the fundamentals of the Laplace Transform. Instead, this book presents an appropriate table of Laplace Transform pairs. It then explains how to use the pairs to convert differential equations into algebraic equations and then how to invert the solutions of these algebraic equations into conventional equations representing the functions of displacement of time.
Analyzing the solutions of differential equations of motion reveals the role of the system’s parameters, the influence of these parameters on each other, and how to control the performance of the system.
The motion of a mechanical system is characterized by its displacement, velocity, and acceleration. These three characteristics are the three basic parameters of the system’s motion. All other characteristics of the working processes can be determined by analyzing these three parameters. The equation of motion represents the displacement of the system as a function of time. The other two parameters — velocity and acceleration — are respectively the first and second derivatives from the displacement. Thus, the equation of motion is the basis for solving the mechanical engineering problem.
The equations of motion represent the solutions of differential equations of motion that reflect the real working processes of the systems. When we assemble these differential equations of motion, we use methodologies that are built on a close interaction between theoretical and applied sciences. Rapidly advancing technology stimulates intensive searches for more sophisticated engineering solutions. Therefore, we must be familiar with the methodologies for solving actual mechanical engineering problems.
This text can help you achieve the level of competence you need to successfully analyze real mechanical systems. An engineering educational background is sufficient to comprehend the contents of this text. We develop a comprehensive, step-by-step guide to solving mechanical engineering problems. Numerous examples demonstrate the methodologies that enable us to control the parameters of real systems. A wide range of readers can benefit from this book. Accounting for the different levels of their backgrounds, the step-by-step approach begins with the simplest examples and then gradually increases the complexity of the problems.
The text consists of six chapters. Let us consider briefly the contents of each.
1.Differential Equations of Motion
Our analysis of problems associated with dynamics is based on the laws of motion. These laws (or equations) of motion are the subject of Chapter 1. They represent displacement (the dependent variable, the function) as a function of running time (the independent variable, the argument). In general, motion has three phases: acceleration, uniform motion, and deceleration.
Displacement in uniform motion is a product of multiplying a constant velocity by the running time. This formula is known from basic physics; it is applicable to any uniformly moving object. Analysis of this formula, however, adds very limited help in understanding the working process and performance of an actual mechanical system.
Solutions that lead to performance control can be obtained from the expressions that describe acceleration and deceleration — equations representing the displacement, velocity, and acceleration as functions of time. For the plurality of real problems, there are no readily available formulas for these three parameters. Instead, mathematical expressions of these three parameters can often be obtained from solutions of corresponding differential equations of motion.
For each case, we should assemble an appropriate differential equation of motion that reflects the physical nature of the problem. As the book will show, composing differential equations of motion is not a trivial procedure.
A differential equation of motion is a second order differential equation made up of the second and first derivatives, the function, the argument, and, the constant terms. The structure of a second order differential equation is based on principles of mathematics without any dependence on laws of motion. The same approach is applicable to all mathematical rules used for practical calculations in different fields. The characteristics of motion include the second derivative (acceleration), the first derivative (velocity), the function (displacement), the argument (running time), and the constant terms. A natural linkage exists between the second order differential equation and the parameters of motion — the process of motion is described by a second order differential equation. (The second order differential equation is also applicable to electrical circuits and other physical phenomena; this text can be used for electrical engineering as well.)
In mathematics, the components of differential equations are dimensionless. In dynamics, each component of a differential equation should have the same physical units. Differential equations of motion are made up of loading factors that represent forces or moments whereas differential equations of electrical circuits include components that represent voltage.
The three basic parameters of motion are not loading factors — they have different units. These parameters cannot be directly included in a differential equation of motion. Each parameter should be multiplied by appropriate coefficients in such a way that the products have the units of loading factors, which cause the motion of objects.
Both the structure and the solution of the differential equations of motion are absolutely identical for rectilinear and rotational motions; their parameters are completely similar. Thus, the examples are presented just for rectilinear motion. Keep in mind that, if necessary, forces should be replaced by moments while the masses should be replaced by moments of inertia; the rectilinear parameters of motion should be replaced by the corresponding angular parameters. All this will not change the structure of the differential equation of motion and its solution. All considerations regarding forces