Introduction to Mechanical Vibrations. Ronald J. Anderson

the single degree of freedom, the equation of motion is

(1.44)

Figure 1.5 A simple pendulum.

Once started in motion the pendulum will swing about the point of connection to the ground. In the case of the simple pendulum there is no mechanism for removing energy from the system as it swings (i.e. no friction or other forces that do work) so the motion, once started, will persist.

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

      The question we ask now is Are there initial values of 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 and there is no angular velocity (i.e. so that does not change with time) and, further, that there is no angular acceleration (i.e. so that does not change with time and thus there will never be a change in ). This is an equilibrium position and Equation 1.44 becomes the equilibrium condition.

(1.45)

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

      The total range of is . In this range, only (the pendulum hangs vertically downward) and (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 .

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

      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)

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

1.3 Linearization

      There two types of nonlinearities that we will often be required to deal with. They are (1) geometric nonlinearities and (2) structural nonlinearities. Geometric nonlinearities arise from trigonometric functions of large angles and structural nonlinearities are due to the inherent nonlinear stiffness (i.e. force versus deflection characteristic) of materials for large deflections.

      1.3.1