Solving Engineering Problems in Dynamics. Michael Spektor

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.

      The right side of this equation may include certain known variable and constant values. All these terms must either have the same units or be dimensionless. The solution of equation (1.0) represents an expression describing the dependence between the function x and the argument t.

      Now let’s consider equation (1.0) from the viewpoint of Dynamics. Displacement, velocity, and acceleration are the three basic parameters of motion of a mechanical system. All other parameters are derived from these three. Hence, the left side of a second order differential equation helps describe the motion of a system because it contains the same basic structural parameters: the second derivative (acceleration), the first derivative (velocity), the function (displacement), and a constant value. Newton’s Second Law states that a body’s motion is caused by a force. This Second Law is expressed by the following well known formula:

      F₀ = m₀a₀

      where F₀ is the force, m₀ is the mass of the body, and a₀ is the acceleration of the motion of the body (the indexes “0” are given in order to avoid possible confusion with similar parameters in the text).

      The first term of equation (1.0) contains the second derivative, which is the acceleration. According to Newton’s Law, the coefficient c₁ in the differential equation of motion should be replaced by the mass m. Thus, the first term of the differential equation of motion is actually a force; all other terms of this equation should have the same units and they should be forces. The product of multiplying the mass by the acceleration (second derivative) represents the force of inertia. Because the mass is a constant value and in general the second derivative (the acceleration) is a variable quantity, we conclude that the force of inertia depends on the acceleration. Similarly, by multiplying the second and third terms of equation (1.0) by certain specific coefficients, we obtain respectively a force that depends on the velocity and a force that depends on the displacement.

      The force that depends on the velocity is actually the reaction of a fluid medium to a movable body that interacts with this medium. This reaction represents a resisting damping force; the coefficient at the first derivative (the velocity) is called the damping coefficient.

      The damping coefficient depends on both the type and condition of the fluid and also on the shape and dimensions of the movable object. Special hydraulic links (dashpots) are used in some mechanical systems in order to absorb impulsive loading. These links exert damping forces and are characterized by damping coefficients. Very often the damping coefficient depends on the velocity of the movable body.

      When this coefficient does not depend on the velocity, or the dependence is negligible, the damping coefficient