Solving Engineering Problems in Dynamics. Michael Spektor
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) |
where x is the displacement (the function), t is the running time (the argument), m is the mass of the system, C is the damping coefficient, K is the stiffness coefficient, P is the constant resisting force of any nature except friction, and F is the constant dry friction force. The force P could represent the force of the system’s gravity in cases of upward motion, the resisting force exerted during a medium’s plastic deformation, or other. The gravity force becomes an active force when it acts in the direction of motion.
The first term of equation (1.1) is the force of inertia; it represents the product of multiplying mass m by acceleration
The third term of this equation is the stiffness resisting force that depends on the system’s displacement. This force equals the product of multiplying the stiffness coefficient by the displacement x. The nature of this force is the reaction of an elastic medium to its deformation by a movable body. By its nature, the resisting constant force associated with a medium’s plastic deformation and the dry friction constant force are also reactive forces. More details about the force of inertia, the damping and stiffness forces, and their coefficients are presented below.
From a mathematical point of view, assume that the left side of equation (1.1) could include a resisting force dependant on time. As a matter of fact, all forces in the differential equation of motion are functions of time, including the constant forces that are actually coefficients at the time that is to the zeroth power.
Let’s analyze a hypothetical case where a resisting force depends directly on time. Imagine a device programmed to increase a pressure force dependant on time. This device, which is attached to a movable system, applies a resisting force that is increasing in time. But this force is not reactive by nature. Instead, it is an external or active force; it should be included in the right side of the differential equation. Thus, a force that depends directly on time should not be included in the left side of a differential equation of motion. All this makes it clear that equation (1.1) represents the most general left side of a differential equation of motion that includes all possible resisting forces; it describes the rectilinear motion of a hypothetical mechanical system in the absence of external forces.
In order to solve the differential equation of motion for this system, first determine the initial conditions of motion:
For  | (1.2) |
where s0 and v0 are the initial displacement and velocity respectively.
According