depending on the function
4.Variables depending on the derivatives of the function
In other words, the active forces include constant and variable forces. The variable forces may depend on the time, the displacement, the velocity, and the acceleration.
As an example of where the active force depends on the acceleration, consider the motion of a rear-wheel or front-wheel drive automobile. The automobile’s force of inertia is responsible for the redistribution of its force of gravity between the rear and front axels. This redistribution causes a certain increase of the rear axle’s loading, and a corresponding decrease of loading on the front axle.
For a rear-wheel drive automobile, the increase of loading on the rear axle leads to the increase of the friction force between the rear wheels and the ground. This results in the increase of the active force that causes the motion of the automobile. For a front-wheel drive automobile, the force of inertia plays an opposite role — the active force decreases. An example is provided in Chapter 4.
In this example, the value of the force of inertia from the right side of the automobile’s differential equation of motion represents just a small fraction of the total force of inertia from the left side of the equation. In the case of a four-wheel drive automobile, the redistribution of the weight between the axles does not influence the total resultant active force.
It is problematic to find other examples where an active force depends on the acceleration. Furthermore, in the rotational motion, it would probably be impossible to find a situation where the angular acceleration influences the applied active moment. Therefore, it is justifiable not to include in the right side of the equation a force that depends on acceleration.
Figure 1.1 shows a general case of a variable active force that is dependant on time. This graph approximates a random variable force. It represents the action of a random force by using a sinusoidal curve that has a maximum force Rmax, and a minimum force Rmin.
The mean force R divides the graph into two equal parts; it is calculated from the following formula:
 | (1.3) |
Figure 1.1 The variable random active force.
The amplitude A of the sinusoidal force can be determined from equation (1.4):
 | (1.4) |
The frequency ω1 of the sinusoidal force is:
 | (1.5) |
where T is the period of fluctuation of the sinusoidal force.
A variable random force can be replaced by a superposition of a constant force R and a sinusoidal force A sin ω1t. The sinusoidal force is a harmonic function of time.
Active forces can be expressed as linear or non-linear functions of time. In most practical cases, these active forces are considered as linear functions of time. Active forces can also be presented as functions that are dependent on displacement or velocity. Based on all these considerations, we assemble equation (1.6), the most general differential equation of motion of a mechanical system that moves in the horizontal direction:
 | (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
