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 to these initial 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
