Page 485 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

472 Nonlinear Vibrations Chap. 14

(i.e., we have a condition of resonance). Such terms violate the initial stipulation
that the motion is to be periodic; hence, we impose the condition
= o
Thus, «1, which we stated earlier to be some function of the amplitude A, is
evaluated to be
«1 = (14.5-8)
With the forcing term cos cot eliminated from the right side of the equation,
the general solution for is

jCj = C, sin cot + C2 cos cot + cos 3cot
(14.5-9)

By imposing the initial conditions X|(0) = ii(0) = 0, constants Cj and C2 are
Cl - 0 C, = -

Thus,
r(cos3cot —cos cot) (14.5-10)

and the solution at this point from Eq. (14.5-2) becomes
A^

X = A cos cot + JJL y (cos 3cot —cos cot)
32 c
(14.5-11)
CO—co^^| 1

COn
The solution is thus found to be periodic, and the fundamental frequency co is
found to increase with the amplitude, as expected for a hardening spring.
Mathieu equation. Consider the nonlinear equation
X co^x jjix^ = F cos cot (14.5-12)
and assume a perturbation solution

x = x ,( 0 + ^ ( 0 (14.5-13)^
Substituting Eq. (14.5-13) into (14.5-12), we obtain the following two equations:
x'l + ojI x ^ -I- = F cos cot (14.5-14)

(o)2-h = 0 (14.5-15)

^See Ref. (4], pp. 259-273.