Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 23

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)

We look for solutions where the angle remains constant. To find these solutions, we let and set so that the angle can never change. This results in the equilibrium condition

(1.47)

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)

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.

 . This is satisfied ifThis is an equilibrium value of where the gravitational pull and the centripetal effects exactly balance each other. It corresponds to an angle between and because the positive values of , , and force the cosine to be positive. We will be interested in the behavior of the bead for small motions about this equilibrium state.⁶