Ronald J. Anderson - Introduction to Mechanical Vibrations

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

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.

Both the simple pendulum with the EOM (Equation of Motion)

(1.49)

and the bead on a rotating wire with the EOM

(1.50)

have geometric nonlinearities arising from the sine and cosine terms. They will both be addressed in the following, starting with the simple pendulum.

The nonlinear differential equation of motion for the pendulum ( Equation 1.44) is valid for any range of motion. The difficulty is that we can't solve nonlinear differential equations without resorting to numerical methods. We get around this problem by linearizing the differential equation because we have had courses on how to solve linear differential equations.

To do this, we consider small motions near the stable equilibrium state. Let in Equation 1.44be replaced by where is a very small angle and is the equilibrium value of . That is,

and, differentiating with the knowledge that is constant, we find

and then

Substituting into Equation 1.44yields

(1.51)

We can use the trigonometric identity for the sine of the sum of two angles 7 to write

(1.52)

We now consider rewriting Equation 1.52under the condition where is a very small angle. The small angle conditions on the sine and cosine are derived from their Maclaurin's Series expansions. The expansions are

and

If is very small, then higher powers of are much smaller and are negligible 8 in the series. We therefore approximate the sine and cosine with

and

Equation 1.52can then be written as