Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 26

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.