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0593_C13_fm Page 465 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 465

and

p = g l (13.10.13)

Hence, from Eq. (13.11.2), F(A, ψ) is:

(
FA,ψ) =− A sin ψ (13.10.14)
3
3
Then, Eqs. (13.10.4) and (13.10.5) become:

 2π 
dA =−( g )1 2l π g l ∫ − A sin 3 ψ cos ψ ψ
d 
3
6
dt    
0
(13.10.15)
g l 3 sin 4 ψ 2π
= A | = 0
12π 4 0
and

2 π
dφ =−( g )(1 2l π g l A) ∫ 3 4 ψ dψ
dt 6 A sin
0
g l A 2  3 sin 2 ψ sin 4 ψ  π
=−  ψ − + | (13.10.16)
12 π  8 4 32  0
=− g l A 16
2

Upon integration of Eqs. (13.10.15) and (13.10.16) we obtain:

A = A (a constant ) (13.10.17)
0

and

2
φ =−(116 ) g l A t + φ 0 (13.10.18)
0
where φ is a constant.
0
Therefore, from Eq. (13.10.6), the approximate solution to Eq. (13.10.8) is:
{ [ )] }
θ = A sin g l 1 −(A 16 t − φ (13.10.19)
0 0 0

Comparing Eq. (13.10.19) with the linear equation solution, A sinωt, we see that ω is:
0
π
ω = 2 T = g l [ 1 −(A 0 16 )] (13.10.20)