Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 20

Devices called “dampers” are common in mechanical systems. These are elements that dissipate energy and they are modeled as producing forces that are proportional to their rate of change of length. The rate of change of length is the relative velocity across the damper. “Proportional” implies linearity and a force proportional to speed implies laminar, viscous flow. As a result, these elements are often referred to as “linear viscous dampers”.

Figure 1.4 shows a system where a body is attached to ground by a damper. The body is moving to the right with speed and the damping coefficient (constant of proportionality) is . The physical connection of the damper to both the ground and the body dictates that the rate of change of length of the damper is equal to the speed . The force in the damper will therefore be . The direction of the force will be such that it causes the damper to increase in length as shown in the lower part of Figure 1.4. By Newton's 3rd Law, the force on the body must be equal and opposite to the force acting on the damper. The force therefore acts to the left on the body. In other words, the damping force opposes the velocity of the body.

Consider now the more general case of a particle where the velocity of the body is given by

(1.35)

Given this velocity, the force that the damper applies to the particle will be

(1.36)

The components of can be substituted into Equation 1.26 to get the following expression for the generalized force arising from the damper

(1.37)

Figure 1.4 A linear viscous damper.

where we can write⁴

(1.38)

which can be substituted into Equation 1.37 to yield the following expression for the generalized force

(1.39)

The generalized force, as expressed in Equation 1.39, can be derived from a scalar function called Rayleigh's Dissipation Function which is defined as

(1.40)

A simple differentiation with respect to yields

(1.41)

Lagrange's Equation can then be written as

(1.42)

where now represents the generalized force corresponding to all externally applied forces that are neither conservative nor linear viscous in nature. Finally, we can transfer the Rayleigh Dissipation term to the left‐hand side and write Lagrange's Equation with dissipation as

(1.43)