Introduction to Mechanical Vibrations. Ronald J. Anderson

Geometric Nonlinearities

      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.

      1.3.1.1 Linear EOM for a 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 1.44 be 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 1.44 yields

(1.51)

We can use the trigonometric identity for the sine of the sum of two angles⁷ to write

(1.52)

We now consider rewriting Equation 1.52 under 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 negligible⁸ in the series. We therefore approximate the sine and cosine with

      and

      Equation 1.52 can then be written as

      (1.53)

      and the equation of motion (Equation 1.51) becomes

(1.54)

      Note the constant term that appears in Equation 1.54. This is the term that was set to zero to determine the equilibrium state (see Equation 1.45) and it is still equal to zero so it can be removed. The equilibrium condition is always a set of constant terms that must be zero for the system not to move and that set of constant terms always reappears in the linearized equation of motion. After removing the equilibrium condition, the linearized equation of motion for the pendulum becomes

      (1.55)

      This equation is valid for motion about either of the two equilibrium states we found (i.e. with and with ). We write

(1.56)

      and

(1.57)

      Equation 1.56 will yield oscillating solutions (i.e. vibrations – more to come later) and Equation 1.57 will yield growing exponential solutions, showing that the system really doesn't want to stay in the unstable upright position.

      1.3.1.2 Linear EOM for the Bead on the Wire

The EOM for the bead on the rotating wire is

(1.58)

      As we did for the simple pendulum, we let