Solving Engineering Problems in Dynamics. Michael Spektor
conditions, the system possesses the potential energy of the deformed medium — the deformation is proportional to the initial displacement s0. The system also possesses the kinetic energy that is proportional to the initial velocity v0. In this case, the system’s motion is caused by the combined action of potential and kinetic energy.
Equation (1.1) describes the motion in cases where the system possesses just the potential energy (for
The left side of equation (1.1) has five components or, in the general case of the differential equation of motion, these five resisting forces:
1.Force of inertia
2.Damping force
3.Stiffness force Kx
4.Constant force P
5.Dry friction force F
These forces represent the reaction of all possible factors to the system’s motion. Their characteristics depend on the structure of the mechanical system and on the nature of the environment in which the system is moving.
The structure of the left side of the linear differential equation of motion (1.1) corresponds to the structure of the second order linear differential equation. Not all resisting forces are present in each actual problem; in reality, the left side of the equation may include any number of components from one to five. However, the force of inertia associated with the second order derivative is always present in the differential equation of motion. Thus, the shortest and simplest expression of a differential equation of motion is that the force of inertia equals zero, and the motion is caused by the initial velocity. In this case, the body moves by inertia with a constant velocity (the acceleration equals zero). This case is discussed further in Chapter 3.
Now let’s compose a similar differential equation of motion for a body in rotation around its horizontal axis. We apply the same procedures as before:
 | (1.1a) |
The initial conditions are:
For  | (1.2a) |
where J is the moment of inertia of the system, C and K are respectively the damping and stiffness coefficients in rotation, MP is a resisting constant moment, MF is a constant moment associated with dry friction, θ is the angular displacement, t is the running time, and Ω0 and θ0 are the initial angular velocity and initial angular displacement respectively. As indicated above, equations (1.1) and (1.1a) are absolutely similar as are expressions (1.2) and (1.2a). The solutions of these equations with their initial conditions are also absolutely similar.
The left side of the differential equation of motion comprises the resisting forces or moments whereas the right side consists of active or external forces or moments. These two parts are equal to each other.
Now let’s compose the differential equation of the rectilinear motion for a body that is subjected to all possible resisting and active forces. The left side of this equation is the same as in equation (1.1). Thus, we must compose the right side so that it includes all possible active or external forces. According to the structure of the second order differential equation, its components could represent
1.Constant values
2.Variables depending on the argument
3.Variables 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