Page 483 - Dynamics of Mechanical Systems

0593_C13_fm Page 464 Monday, May 6, 2002 3:21 PM

464 Dynamics of Mechanical Systems

where A is an amplitude to be determined and where ψ is defined as:

D
ψ = pt + φ (13.10.3)
where φ is a phase angle to be determined.
Next, let dA/dt and dφ/dt be given by:

2π
dA =− ∫ Fcos ψψ = α A () (13.10.4)
d
dt 2π p
0
and

2 π
dφ = ∫ ψψ = β()
dt 2 π pA Fsin d A (13.10.5)
0

If Eqs. (13.10.4) and (13.10.5) can be solved for A and φ, then an approximate solution
to Eq. (13.10.1) may be expressed in the form:

x = Asin ( pt + ) φ (13.10.6)

˙ x
with the derivative given by:
(
˙ x = Ap cos pt + ) φ (13.10.7)

To illustrate the procedure, consider again the nonlinear pendulum equation:

˙˙
θ +( ) l sinθ= 0 (13.10.8)
g
Suppose we approximate sinθ as:

≈−
3
sinθθ θ 6 (13.10.9)
Then, Eq. (13.10.8) becomes:

˙˙ θ ) 3 +(g l θ ) = 0 (13.10.10)
θ −(g 6l

Hence, by comparison with Eq. (13.10.1), we can identify , f, and p as:

= g 6l (13.10.11)

f θθ ( ) =− θ 3 (13.10.12)
˙
,