Introduction to Mechanical Vibrations. Ronald J. Anderson

the vector expression of the absolute position of the point of application of the forceand then writes the generalized force as(1.27) Equation 1.27 can be written for each of applied forces and the resulting scalar generalized forces can be added together to give the total generalized force for generalized coordinate as(1.28)

      2 The intuitive approachLet there be generalized coordinates specifying the position of a force acting on a dynamic system in Cartesian Coordinates. The force will be acting at the point where the coordinates , and are functions of the generalized coordinates through and of time, , as follows(1.29) Variations in the position of the force as the generalized coordinates are varied while time is held constant can be written as(1.30) If we are trying to find the generalized force corresponding to only one of the generalized coordinates, say , we rewrite Equation 1.30 with and , giving(1.31) Now consider Equation 1.26 with each side multiplied by (1.32) The terms from Equation 1.31 can be substituted into the right‐hand side of Equation 1.32 to yield(1.33) The right‐hand side of Equation 1.33 can be seen to be the work done by the applied force as its position varies due to changes in the generalized coordinate while all other generalized coordinates and time are held constant.Using the intuitive approach to finding generalized forces, the analyst will consider, in sequence, the variation of individual generalized coordinates and will write expressions for the total work done during each variation. The generalized force associated with each generalized coordinate will be the work done during the variation of that coordinate, , divided by the variation in the coordinate. That is,(1.34)

      

      1.1.3.4 Dampers – Rayleigh's Dissipation Function

      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)

1.2 Equilibrium Solutions

      Equilibrium solutions of the equations of motion are those where the degrees of freedom assume values which cause their first and second derivatives to go to zero. Under these conditions, there will be no tendency for the values of the degrees of freedom to change and the system will be in an equilibrium state.

      Some equilibrium states are stable and some are unstable and, inevitably, systems are either in a stable equilibrium state or trying to get to one. The study of small motions around a stable equilibrium state is called Vibrations.

      1.2.1 Equilibrium of a Simple Pendulum

We start by considering the simple pendulum⁵ shown in Figure 1.5. Using the angle