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 is considered to have a constant value. In this case, the differential equation of motion is linear — assuming, of course, that all other components of the equation are linear. If instead this coefficient has a variable value, the equation becomes non-linear.
There are no readily available formulas to calculate the damping coefficient. For each case, the characteristics and the value of the damping coefficient should be determined on the basis of experimental data. Note that in some cases a damping force becomes a part of the external loading factors (see Chapter 4).
The forces that depend on the function (the displacement) are exerted by the elastic media in the process of interacting with a movable object. By its nature, this force is the reaction of the medium to its deformation by a movable body. This force represents a resisting force and is called the stiffness force. The function’s coefficient is the stiffness coefficient; it depends on the type and condition of the elastic medium, the shape and dimensions of the body, and the peculiarities of the deformation.
For some elastic media, the stiffness coefficient depends on displacement of the movable object (deformer). If this dependence is negligible, the stiffness coefficient is considered to have a constant value. The corresponding differential equation of motion is linear. If, however, this dependence is significant, the stiffness coefficient is characterized by a variable value. The corresponding equation of motion is then non-linear.
Mechanical engineering systems often include elastic links in the shape of springs. The stiffness coefficient for the springs can be calculated using readily available formulas. Sometimes this coefficient is called the spring constant. For deformation of elastic media, there are no readily available formulas to calculate the stiffness coefficients; appropriate data is needed to determine the values and characteristics of these coefficients. In some cases, the stiffness force is considered an external active force (see Chapter 4).
The fourth term of equation (1.0) has a constant value. In the differential equation of motion, this value may be represented by certain constant resisting forces such as the force exerted during the deformation of a plastic medium, the dry friction force, or the force of gravity in case of an upward motion.
Consider the right side of equation (1.0) with respect to the differential equation of motion. This part may comprise a force that is a certain known function of time, velocity, or displacement — or a sum of all of them, including a constant force. In a very specific case (Chapter 4), the right side of the differential equation of motion contains a force that depends on acceleration. All these considerations let us conclude that the structure of a differential equation of motion is determined by equation (1.0).
With respect to Dynamics, the terms in the left side of equation (1.0) represent forces that resist the motion of a mechanical system, whereas the right side of the equation includes terms that cause the motion. The forces that resist the motion characterize the reaction of the system to its motion. Thus, the forces that have a reactive nature are the resisting forces. The forces in the right side of the differential equation of motion are applied to the system; they may be called the external forces or the active forces.
More considerations associated with loading factors (forces and moments) and with the structure of the differential equations of motion are discussed below.
Like any equation, a differential equation of motion consists of two equal parts. The components of the equation represent forces or moments applied to the mechanical system. Forces are used in equations of a particle’s rectilinear motion or a rigid body’s rectilinear translation, whereas moments are used for equations to describe the rotation of a body around its axis. The forces or moments can be classified into two groups:
1.Active forces and moments causing motion
2.Reactive forces and moments resisting the motion
It is justifiable to place all the resisting loading factors into the left side of the differential equation, and the active loading factors into the right side.
Based on all these considerations, it is possible to assemble the most general left side for an actual mechanical engineering system’s differential equation of motion. Let’s start with a system in the rectilinear motion. Assume that no external forces are applied to the system, which is moving in a horizontal direction. In this case, the right side of the equation equals zero. In the absence of external forces, the motion occurs due to the energy that the system possesses (kinetic, potential, or both). The initial conditions of motion contain information regarding the energy the system possesses at the beginning of the analysis. Based on these considerations and applying equation (1.0), we may write the differential equation of motion of a mechanical system, as seen in equation (1.1):
 | (1.1) |
