An Approximate Analytical Solution of Transversal …
47
× (cos τ − cos 5τ ) +
a
24
(A 6 C 1 + A 3 C 3 + A 1 C 5 )(5 sin τ − sin 5τ )
+
a
48
(A 3 C 4 − A 6 C 3 )(cos τ − cos 7τ ) +
a
48
(A 6 C 2 + A 3 C 5 )(7 sin τ
− sin 7τ ) −
a
80
A 6 C 5 (cos τ − cos 9τ ) +
a
80
A 6 C 4 (9 sin τ − sin 9τ )
(24)
where the values of the convergence-control parameters C i , i = 1, 2,…, 5 are
optimally determined.
4 Numerical Example
We illustrate the accuracy of our procedure, considering the case corresponding to
the following values of the physical parameters involved in (11): γ = 2.5; α = 2.5;
δ = 0.8; β = 3.5; μ = −0.05; ω n = 9.83.
By means of the procedure described in [16], we obtain
C 1 = 0.145927735933; C 2 = −0.128431942959; C 3 = −0.0043147480105;
C 4 = 0.070184012124; C 5 = −0.229972288472;
a = 1.464442738889; Ω= 9.420966459723
(25)
As depicted in Fig. 3, the analytical results and numerical integration results
obtained using a fourth-order Runge–Kutta method are almost identical.
Note that our technique can be expanded to predict the beam response under
different boundary conditions and also to other nonlinear problems.
Fig. 3 Comparison between
the approximate solution
(24) and numerical solution
for γ = 2.5; α = 2.5; δ =
0.8; β = 3.5; μ = −0.05; ω n
= 9.83
_ _ _ analytical solution (OAFM), ____ numerical integration results
47
× (cos τ − cos 5τ ) +
a
24
(A 6 C 1 + A 3 C 3 + A 1 C 5 )(5 sin τ − sin 5τ )
+
a
48
(A 3 C 4 − A 6 C 3 )(cos τ − cos 7τ ) +
a
48
(A 6 C 2 + A 3 C 5 )(7 sin τ
− sin 7τ ) −
a
80
A 6 C 5 (cos τ − cos 9τ ) +
a
80
A 6 C 4 (9 sin τ − sin 9τ )
(24)
where the values of the convergence-control parameters C i , i = 1, 2,…, 5 are
optimally determined.
4 Numerical Example
We illustrate the accuracy of our procedure, considering the case corresponding to
the following values of the physical parameters involved in (11): γ = 2.5; α = 2.5;
δ = 0.8; β = 3.5; μ = −0.05; ω n = 9.83.
By means of the procedure described in [16], we obtain
C 1 = 0.145927735933; C 2 = −0.128431942959; C 3 = −0.0043147480105;
C 4 = 0.070184012124; C 5 = −0.229972288472;
a = 1.464442738889; Ω= 9.420966459723
(25)
As depicted in Fig. 3, the analytical results and numerical integration results
obtained using a fourth-order Runge–Kutta method are almost identical.
Note that our technique can be expanded to predict the beam response under
different boundary conditions and also to other nonlinear problems.
Fig. 3 Comparison between
the approximate solution
(24) and numerical solution
for γ = 2.5; α = 2.5; δ =
0.8; β = 3.5; μ = −0.05; ω n
= 9.83
_ _ _ analytical solution (OAFM), ____ numerical integration results