
373
Appendix D: Derivation of ratio
of successive amplitude reduction
values for fixed–fixed beam using
explicit and implicit approaches
In this appendix, the derivation steps of the successive amplitude reduc-
tion ratio are given for the xed–xed beam with the explicit and implicit
approaches. The ODE ψ(t) obtained by the explicit approach is given in
Equation (5.71) as
ψ
∂ψ
∂
+
∂ψ
∂
=−ωk
tt
1
n
3
3
2
2
2
(D.1)
The roots of Equation (D.1) are computed by substituting
ψ=e
st
, and
solving ++ω=ks s 0
n
32 2
in MATLAB® to get the general solution as
ψ=
ψψψ
tCeCeCe()
ttt
123
123
(D.2)
where
ψ= β+
β
−
ψ= −β −
β
−
+β−
β
ψ= −β −
β
−
−β−
β
kk
i
k
k
i