Solving Engineering Problems in Dynamics. Michael Spektor

href="#fb3_img_img_415b0048-2d84-5992-8c79-b0e84f591c5c.jpg" alt="img"/>(1.6)

      where R is a constant active force; A is the amplitude of a sinusoidal force; ω₁ is the frequency of the sinusoidal force; Q is the constant value of an active force at the beginning of the motion; τ is the time that the motion can last; and μ is a constant dimensionless coefficient, for τ > 0, μ > 0, and t ≤ τ; and finally C₁ and K₁ are respectively the damping and stiffness coefficients of the active damping and stiffness forces.

      The initial conditions for this case are arbitrary, which means that the parameters of motion at the beginning of the process could be equal to zero or could be different from zero. In general, the initial conditions of motion for equation (1.6) may be also presented by the expression (1.2).

      The left side of equations (1.1) and (1.6) are identical, as expected. Thus, equation (1.6) represents the structure of the most general differential equation of a rigid body’s rectilinear motion. The complexity of the differential equations of motion in the actual situations is significantly lower than in equation (1.6).

      The characteristics of a differential equation of motion’s forces determine the linearity or non-linearity of the equation.

      It is very important for the overall analysis to clarify the peculiarities of the characteristics of the forces that should be included in the equation. There are two main concerns in determining the characteristics of the forces.

      1.The first is associated with obtaining the most credible data about the particular forces for the particular real conditions.

      2.The second is related to interpreting these characteristics in terms of their linearity or non-linearity.

      Adequate information is obtainable from a comprehensive search of the relevant sources. The decision to categorize these characteristics as linear or non-linear depends on both the actual experimental data and the level of compromise that is justifiable in each particular case. The following analysis of these forces addresses these two concerns.

      These considerations are all related to those differential equations of motion in the horizontal direction that are not affected by vertical forces such as gravity. In cases of vertical motion or an incline, the force of gravity plays a role. If a mechanical system is moving up vertically or on an incline, the force of gravity or its component represents a resistance and should be included in the left side of the differential equation of motion. When the system moves down, these forces represent active or external forces and should be included in the right side of the equation. According to equation (1.6), the right side of the differential equation of motion generally includes five active forces:

      1.Constant force R

      2.Sinusoidal force A sin ω₁t

      3.Force depending on time

      4.Force depending on velocity

      5.Force depending on displacement K₁x

      Detailed descriptions of the characteristic of forces included in the differential equations of motion (1.1) and (1.6) are presented in Chapter 2.

      Now let us compose a differential equation of motion for a system that rotates around its horizontal axis and is subjected to all possible resisting and active moments. This equation is completely similar to equation (1.6) and is presented in the following shape:

(1.6a)

      where M is a constant active moment; MA is the amplitude of a sinusoidal active moment; ω₁ is the frequency of the sinusoidal moment; MQ is the constant value of an active moment at the beginning of the motion; and C₁ and K₁ are respectively the damping and stiffness coefficients; μ is a constant dimensionless coefficient, and τ > 0, μ > 0, and t ≤ τ.

      The initial conditions of motion for equation (1.6a) are presented in expression (1.2a).

      Because of the strong similarity between the characteristics of forces and moments, as well as between the appropriate differential equations of motion, we will consider only forces in this text while keeping in mind that all considerations related to forces are also applicable to moments.

      It is important to realize that all components in the differential equation of motion should be functions of time. (For constant terms, time is to the zeroth power.) In certain real situations, the movable mechanical systems are subjected to forces that may depend on temperature. Changes of fluid temperature will cause changes of the damping force. The influence of temperature on forces cannot be directly incorporated into the differential equations of motion. If we could also incorporate forces that depend on temperature into these equations, we would obtain equations with two independent variables: time and temperature. There are no differential equations of motion with multiple arguments; there can be just one independent variable — and it must be the running time. Therefore, any forces that depend on temperature or other factors, except time, cannot be included in any differential equations of motion.

      However, there is a way to account for the change of the forces due to temperature change. Consider the change of fluid viscosity due to temperature. As mentioned above, this change will cause the change of the damping force. In other words, temperature change will result in the change of the damping coefficient. The differential equation of motion and its solution are the same for different values of damping coefficients. These values should be determined for different temperatures and then used during the quantitative analysis of the parameters of motion. The results of this analysis will reveal the influence of temperature on the process of the system’s motion.