Introduction to Mechanical Vibrations. Ronald J. Anderson
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Название: Introduction to Mechanical Vibrations

Автор: Ronald J. Anderson

Издательство: John Wiley & Sons Limited

Жанр: Физика

Серия:

isbn: 9781119053644

isbn:

СКАЧАТЬ 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

Illustration of a simple pendulum, a massless rigid rod swinging at angle theta in a plane about the frictionless point where it is connected to the ground.

      The motion will depend on the way in which it is started. That is, if the pendulum is rotated to some arbitrary starting angle, images, and released from rest, it will swing through the position where images and will eventually return to where it started before reversing and starting the cyclic motion over again. If the pendulum is stopped and returned to images and then released, not from rest but with an initial velocity, the resulting motion will be different and the pendulum will pass through images when it returns. The motion will, however, still be cyclic.

      The question we ask now is Are there initial values of images where the pendulum can be released from rest and remain stationary? These are the equilibrium states.

      Consider Equation 1.44 under the conditions that there is an initial angle images and there is no angular velocity (i.e. images so that images does not change with time) and, further, that there is no angular acceleration (i.e. images so that images does not change with time and thus there will never be a change in images). This is an equilibrium position and Equation 1.44 becomes the equilibrium condition.

      Since images, images, and images are never zero, this can only be satisfied by:

equation

      The total range of images is images. In this range, only images (the pendulum hangs vertically downward) and images (the pendulum stands upright) satisfy the requirements. These are the two equilibrium states for the pendulum.

      There are formal methods for testing the stability of the equilibrium states but that we leave to courses on control systems. It is sufficient for us to be able to see that the state where the pendulum stands upright is unstable and the pendulum will try to get to the stable equilibrium position where images.

      The vibrations question is What will be the response of the system for small motions away from the stable equilibrium condition where images?

      1.2.2 Equilibrium of the Bead on the Wire

      We now return to our continuing example problem – the bead on a rotating semicircular wire as shown in Figure 1.1. The equation of motion (see Equation 1.23) is

      (1.46)equation

      The equilibrium condition is a group of constant terms summing up to zero that becomes an identity for us. We will see this group of terms again when we write the equation of motion for small motions around equilibrium and every time we see it, we will be able to set it equal to zero.

      With some simple factoring out of terms, we get

      (1.48)equation

      This expression will hold for two cases:

       . This is satisfied when and when . These correspond to the bead being directly below point and directly above point respectively. Being above point is, of course, physically impossible for the semicircular wire but would be possible for a complete hoop.

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