problems.
5.Piece-Wise Linear Approximation
Chapters 3 and 4 are devoted to solving linear differential equations of motion. In reality, many loading factors that are included in these equations are actually non-linear. However, the non-linearity of these factors is often not essential; it is then justifiable to consider them as being linear. There are no currently established methodologies for solving non-linear differential equations in general terms.
Many specific non-linear differential equations can be solved using particular mathematical investigations, and there are catalogs where these solutions can be found. However, these solutions have a very limited applicability to non-linear differential equations of motion.
In a significant number of real life problems, the non-linearity of the loading factors cannot be ignored. Neglecting the strong non-linearity of these factors results in essential quantitative errors; yet some important qualitative characteristics of the process could be misunderstood or not revealed at all.
The method of piece-wise linear approximation allows us — with an appropriate accuracy — to investigate problems that include non-linear loading factors. The characteristics of these factors can be represented by corresponding graphs whose curvatures reflect the extent of the factors’ non-linearity.
Piece-wise linear approximation consists of replacing the curve by a broken line. For instance, if the curve is replaced by a broken line including three straight segments, the process of motion can be divided into three intervals. For each interval, a linear differential equation will be composed with the initial conditions of motion equal to the conditions of motion at the end of the previous interval. The shorter the length of the segments, the more accurate the results of the solution will be. A reasonable compromise will decide the number and values of the replacement increments that would satisfy the goal of the investigation. The application of the piece-wise linear approximation to the solutions of real-life problems comprising non-linear loading factors is presented in a detailed way in this chapter.
6.Dynamics of Two-Degree-of-Freedom Systems
Numerous mechanical engineering systems are made up by several separate masses connected among themselves by specific links. These links allow for motion of these masses relative to each other. Each motion is described by its mass’s differential equation. The amount of these masses defines the number of degrees-of-freedom of the system. Of the actual multiple-degree-of-freedom mechanical systems, the majority have just two masses — therefore, this text is limited to considering two-degree-of-freedom structures. Two types of links allow relative motion of the connected masses: the elastic link (spring) and the hydraulic link (dashpot). The masses could be connected by a hydraulic or elastic link, or by both links acting in parallel. A simultaneous system of two differential equations of motion should be assembled in order to describe the motion of the two masses.
Chapter 6 contains a detailed discussion of the structures of the differential equations of motion and also of the considerations for composing these equations. It also includes typical examples that demonstrate the methods for investigating two-degree-of-freedom systems.
A General Note
These chapter descriptions indicate that the analysis of an actual mechanical system is a complex process engaging an interaction among several sciences.
During the first steps of the analysis, we should pay particular attention to the characteristics of the damping and stiffness resisting forces. In the majority of practical cases, these forces could be linear or non-linear whereas the rest of the forces are usually linear. Information regarding the characteristics of the actual damping and stiffness forces for a specific case should be based on the results of the investigations; these results are usually presented in graphs or can be found in corresponding sources.
Normally, our analysis of the solutions of the differential equation of motion provides the information needed to make appropriate engineering decisions. This text includes all the steps necessary for a complete analysis of actual problems in mechanical engineering dynamics.
Numerous software programs are available for computing the parameters of motion of mechanical engineering systems. These programs can be used when the differential equations of motion are already available. When investigating real life problems, the first steps are associated with composing the differential equations of motion. This text is intended to help you assemble these equations. In many practical situations, you may need to analyze the working process of a mechanical engineering system in order to estimate the influence of the parameters on each other and to reveal their specific roles. For these cases, we present the analysis in general terms without any use of related numerical data. This book will also be useful for performing this kind of analyses.
DIFFERENTIAL EQUATIONS OF MOTION
A mechanical system’s equation of motion, also called the law of motion, represents the system’s displacement as a function of running time. Analyzing the equation of motion provides comprehensive information needed for the development, design, and improvement of the system. The equation of motion is the solution of the differential equation of motion for the system performing a certain working process.
The accuracy of the analysis can be evaluated by appropriate experiments. Results that disagree with the experiments tell us the differential equation does not closely enough reflect the actual conditions of the process of motion. In such cases, we revise the differential equation. We may need to carry out a few iterations to achieve the acceptable accuracy; however, in many practical cases, our first iteration should be enough. The considerations presented below may help us develop these equations.
1.1The Left Side of Differential Equations of Motion (Sum of Resisting Loading Factors Equals Zero)
From Dynamics, we know that the differential equation of motion is a second order differential equation. As it turns out, a second order differential equation also describes the processes in electrical circuits. The structure of such an equation is predetermined by principles of mathematics without any regard to either the characteristics of motion of a mechanical system or the characteristics of processes in electrical circuits. An ordinary linear second order differential equation reads:
 | (1.0) |
where x is the function, t is the argument, c1, c2, and c3 are constant coefficients, P is a constant value, and f(t) is a certain known function of t.
Let’s examine the left side of this equation. The first term is the product of multiplying a constant coefficient by the second derivative. The second term is the product of multiplying another constant coefficient by the first derivative. The third term is the product of multiplying one more constant coefficient by the function, and finally the last term is a constant value. This constant value can be considered a product of multiplying a constant coefficient by the function (or argument) to the zeroth power.
