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and
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
As we did for the simple pendulum, we let
We use the trigonometric identities
along with the linearizing approximations for the small angle
to get the linearized forms
Then, the term
(1.60)
We then say that the nonlinear
(1.61)
Now we rearrange this to separate the constant terms from the terms with
(1.62)
where the constant term
It remains to pick a suitable equilibrium state from the three we found in Subsection 1.2.2. Of these, the most interesting is the one where the bead is not below or above point
With a little effort and use of one trigonometric identity, you can show that the linear equation of motion about this state can be written as
As you will see later, this is the standard form for an undamped vibrational equation of motion. If the bead is in equilibrium and is disturbed away from equilibrium by a small angle, it will begin to oscillate with a frequency that can be determined directly from Equation 1.63.9
1.3.2 Nonlinear Structural Elements
Making linear approximations to trigonometric functions is not the only consideration we have in creating linear differential equations of motion. Many times the physical properties of structural elements in the system are nonlinear. A rubber suspension element is a good example. Depending on how it is designed, it can be made to get softer or harder as it deflects. Note that “softer” and “harder” are non‐technical words relating to the stiffness of the element. Figure 1.6 shows the characteristics of a “hardening” spring where the element gets stiffer as it deflects. The stiffness is measured
