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.
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