Particular attention is paid in Chapter 1 to explaining the structure of differential equations of motion and assembling them.
2.Analysis of Forces
The structure of the differential equation of motion is absolutely similar for rectilinear and rotational motion. So too is the process of composing the equation. To avoid redundant explanations, our analysis of loading factors focuses just on forces. However, the same characteristics and considerations are completely applicable to moments.
The left side of the differential equation of motion consists of resisting forces, whereas the right side consists of active forces. The resisting forces are variables (inertia, damping, and stiffness) and constants (e.g., dry friction, gravity, and plastic deformation). The force of inertia is present in all differential equations of motion. The resisting forces should be identified depending on the functionality and on the structure of the mechanical system as well as on the nature of the environment in which the motion occurs.
As variables, the force of inertia depends on acceleration, the damping force on velocity, and the stiffness force on displacement. These resisting forces can be linear or non-linear and their characteristics are determined by their coefficients. The coefficient of the inertia force is the mass, which is usually a constant value; consequently, the inertia force is linear. Non-linear inertia forces are not considered in this book. The damping and stiffness coefficients can also be constant or variable. If constant, the differential equation of motion is linear. If even one of these coefficients is a variable, the differential equation of motion is non-linear.
In certain mechanical systems, resisting forces could appear that represent some functions of time. However the majority of conventional mechanical systems do not have any obvious factors pointing to the existence of time-depending resisting forces which, therefore, are not discussed in this book.
In the majority of cases, the characteristics of active forces are predetermined. For conventional mechanical systems, these active forces include: constant forces, sinusoidal forces exerted by vibrators, and forces depending on time, velocity, or displacement. These last three can be linear or non-linear.
Chapter 2 looks closely at the characteristics and peculiarities of the resisting and active forces.
3.Solving Differential Equations of Motion Using Laplace Transforms
In solving the differential equations of motion, our goal is to obtain an expression for displacement as a function of time. This expression is also called the law of motion. Finding the best method for solving various linear differential equations can be challenging. However, the Laplace Transform represents a straightforward universal method for solving all linear differential equations.
The Laplace Transform lets us convert differential equations into algebraic equations whose solutions can be achieved by conventional algebraic procedures. We can apply the Laplace Transform without addressing the mathematical principles on which it is built. It provides a straightforward methodology regardless of the characteristics of the equation’s components or its initial conditions.
Chapter 3 reviews the steps of this methodology; they are identical for each differential equation. First, we convert the differential equation of motion from the time domain form into the Laplace domain form, working with a table of Laplace Transform conversion pairs compiled for this text. The second step of the methodology deals with the Laplace domain solution of the differential equation of motion. This step, based on ordinary algebraic procedures, results in an algebraic equation that represents the dependant variable (e.g., displacement) as a function of the independent variable (e.g., running time). Both variables are in the same Laplace domain. The Laplace Transform eliminates the need of calculus to solve the differential equation of motion. Therefore, we obtain an algebraic equation with the dependent variable in the left side of the equation, and a sum of algebraic expressions (proper fractions) on the right.
In the last step, we invert all the terms of the solution from the Laplace domain into the time domain form. This inversion represents the solution of the differential equation of motion. All three steps of this methodology are demonstrated in the text by solving numerous examples.
In some cases, there will be terms in the right side of the Laplace domain solution that do not have representations in this text’s table, or even in other, more comprehensive tables. For these cases, Chapter 3 discusses a method of decomposition used to resolving these expressions.
The examples in this chapter begin with a solution of a very simple differential equation. The complexity of the solutions gradually increases; ultimately, the examples include a range of diversified differential equations of motion of actual mechanical systems.
4.Analysis of Typical Mechanical Engineering Systems
Assembling the different equation of motion is a very important step when investigating the dynamics of a mechanical system. The differential equation should reflect the peculiarities of the real working process. This chapter discusses the considerations that are relevant to the process of assembling differential equations of motion. These considerations are associated with real-life problems of typical mechanical systems. We start with composing the appropriate differential equation of motion. The following step focuses on this equation’s solution. In the last step, our analysis of the solution reveals the system’s performance characteristics: energy consumption, required power, acting forces, and others. The complexity of the examples increases from example to example, and can be very helpful in solving actual 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