Introduction to Mechanical Vibrations. Ronald J. Anderson

is a small angle. Since is a constant, we differentiate to find that and . Substituting into Equation 1.58 gives

(1.59)

      We use the trigonometric identities

      along with the linearizing approximations for the small angle

      to get the linearized forms

      Then, the term which appears in Equation 1.59 can be approximated by the product of these two expressions

      (1.60)

      We then say that the nonlinear term can be neglected as being negligibly small compared to the linear term since is a small angle. This is at the heart of the linearization process. The linearized EOM becomes

      (1.61)

      Now we rearrange this to separate the constant terms from the terms with and its derivatives. The result is

      (1.62)

where the constant term is exactly the same as the equilibrium condition stated in Equation 1.47 and it is identically equal to zero by definition so it can be removed from the equation of motion. This will always be the case for vibrational systems. As we progress through the following chapters, we will often say something to the effect of “the gravitational forces are canceled out by preloads in the springs so they can be ignored” where we actually mean that our equations of motion are written for small motions away from an equilibrium state and that we need only be concerned with how the forces in elements change as the system moves slightly away from equilibrium. There will be forces holding the system in equilibrium and they may be large but they always add up to zero. The equilibrium condition for the bead is actually a statement that the moment that the gravitational force exerts about point is exactly equal and opposite to the moment about due to the centripetal acceleration.

      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 . That is, consider the case where

      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

(1.63)

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 by the local tangent to the curve.

      The equilibrium solution to the nonlinear equation of motion will place the system in equilibrium at (the point labeled operating point on Figure 1.6). Once there, we consider motions away from the operating point and use a Taylor's Series expansion for the nonlinear function

      (1.64)

      For small values of , we can neglect the higher order terms and write

      (1.65)

Figure 1.6 Nonlinear structural element – Linearization and effective stiffness.