2 The equation of motion
The problem of giving the longitudinal vibration of a harmonic external forced beam is given by
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where
w
ˆ
(
x
ˆ
,
t
ˆ
)
is the transverse displacements of the beam and ε is a small dimensionless parameter; m denotes the mass and
η
ˆ
is the damping coefficient;
F
ˆ
is the external excitation amplitude,
Ω
ˆ
is the external excitation frequencies, and
D
α
denotes the fractional derivative of order α. Here, also, the dot denotes partial differentiation with respect to time
t
ˆ
, and prime denotes the derivative with respect to spatial
x
ˆ
. On the other hand, it is assumed that the tension T is characterized as a small periodic perturbation ε
T
1
cos
Ω
on the steady-state tension
T
0
, i.e.,
T
=
T
0
+
ε
T
1
cos
Ω
,
where Ω is the frequency of a beam 12. Introducing the dimensionless parameters as
w
=
w
ˆ
L
,
x
=
x
ˆ
L
,
t
=
t
ˆ
L
T
0
ρ
A
,
we have the new dimensionless parameters
a
=
T
1
T
0
,
η
¯
=
η
ˆ
L
α
−
2
T
0
α
−
2
m
α
(
η
¯
=
ε
η
)
,
Ω
1
=
Ω
L
m
T
0
,
Ω
2
=
Ω
ˆ
L
m
T
0
,
f
(
x
)
=
L
T
0
F
ˆ
(
x
)
,
where ρ is density, A is the cross-sectional area, and L is the length of the beam. Thus, the equation in the non-dimensional form is presented as
w
¨
−
(
1
+
ε
a
cos
Ω
1
t
)
w
″
+
ε
η
D
α
w
=
f
(
x
)
cos
Ω
2
t
,
where
η
¯
equals εη. For simply supported beams, non-dimensional boundary conditions are
w
(
0
,
t
)
=
w
(
1
,
t
)
=
0
.
3 The method of multiple scales
In this section, an approximate solution will be searched by using the method of multiple scales. This method is known as the direct-perturbation method which can be applied directly to the partial differential equation. In higher-order schemes and for finite mode truncations, the method yields better approximations to the real problem 13. Let us consider the expansion
w
(
x
,
t
;
ε
)
=
w
0
(
x
,
T
0
,
T
1
)
+
ε
w
1
(
x
,
T
0
,
T
1
)
+
⋯
,
where
T
0
=
t
is the usual fast-time scale and
T
1
=
ε
t
is the slow-time scales. Now, the time derivatives are given by
![]()
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where
D
i
=
∂
/
∂
T
i
. Here,
D
+
α
e
λ
t
=
D
0
α
e
λ
t
=
λ
α
e
λ
t
can be used for calculating the fractional derivative of the exponential function, where
D
+
α
,
D
+
α
−
1
,
D
+
α
−
2
,
…
are the Riemann-Liouville fractional derivatives 1. Substituting Eqs. (8)-(11) into Eqs. (6) and (7) and separating into terms at each order of ε, we have the following:
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At order one, the solution is obtained as
w
0
(
x
,
t
)
=
[
A
n
(
T
1
)
e
i
ω
n
T
0
+
A
¯
n
(
T
1
)
e
−
i
ω
n
T
0
]
X
n
(
x
)
+
[
e
i
Ω
2
T
0
+
e
−
i
Ω
2
T
0
]
Y
(
x
)
,
where
ω
n
represents the natural frequency,
A
n
and
A
¯
n
are complex amplitudes and their conjugates, respectively. Now, substituting (16) into Eq. (12), we obtain the boundary value problems
![]()
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and
![]()
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Thus, the solutions of Eqs. (17) and (19) are
![]()
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where ψ
(
x
)
is a particular solution for Eq. (19). Here, the particular solution of Eq. (19) changes with respect to the selection of the function f
(
x
)
. Let us substitute (16) into Eq. (14) for the solution of order ε, then
D
0
2
w
1
+
L
0
[
w
1
]
=
[
−
2
i
ω
n
D
1
A
n
e
i
ω
n
T
0
+
2
i
ω
n
D
1
A
¯
n
e
−
i
ω
n
T
0
]
X
n
(
x
)
+
a
2
X
n
″
[
A
n
(
e
i
(
ω
n
+
Ω
1
)
T
0
+
e
i
(
ω
n
−
Ω
1
)
T
0
)
+
A
¯
n
(
e
−
i
(
ω
n
−
Ω
1
)
T
0
+
e
−
i
(
ω
n
+
Ω
1
)
T
0
)
]
+
a
2
Y
″
[
e
i
(
Ω
1
+
Ω
2
)
T
0
+
e
−
i
(
Ω
1
+
Ω
2
)
T
0
+
e
i
(
Ω
1
−
Ω
2
)
T
0
+
e
−
i
(
Ω
1
−
Ω
2
)
T
0
]
−
η
X
n
[
(
i
ω
n
)
α
A
n
e
i
ω
n
T
0
+
(
−
i
ω
n
)
α
A
¯
n
e
−
i
ω
n
T
0
]
−
η
Y
[
(
i
Ω
2
)
α
e
i
Ω
2
T
0
+
(
−
i
Ω
2
)
α
e
−
i
Ω
2
T
0
]
.
Thus, different cases arise depending on the numerical value of variation frequency. These cases will be treated in the following sections.
4 Case studies
In this section, we assume that one dominant mode of vibrations exists. As a result of the previous studies in the literature, it is seen that the results are the same in the finite mode analysis and in the infinite mode analysis 314. Therefore, we consider one dominant mode of vibration in this study.
4.1
Ω
1
close to 0,
Ω
2
away from ωn (
Ω
1
≅
0
,
Ω
2
≠
ω
n
)
For this case, we consider the case of the nearness of
Ω
1
to zero is expressed as
Ω
1
≅
ε
σ
n
,
where
σ
n
is a detuning parameter. Then, Eq. (23) becomes
D
0
2
w
1
−
w
1
″
=
[
−
2
i
ω
n
D
1
A
n
X
n
+
a
2
A
n
X
n
″
(
e
i
ε
σ
n
T
0
+
e
−
i
ε
σ
n
T
0
)
−
η
(
i
ω
n
)
α
A
n
X
n
]
e
i
ω
n
T
0
+
cc
+
NST
where cc and NST denote complex conjugates and non-secular terms, respectively. Thus, the solution of Eq. (25) is
w
1
=
φ
n
(
x
,
T
1
)
e
i
ω
n
T
0
+
φ
¯
n
(
x
,
T
1
)
e
−
i
ω
n
T
0
+
W
(
x
,
T
0
,
T
1
)
,
where the first term is related to the secular terms and the second term is related to the non-secular terms. Now, substituting Eq. (26) into Eq. (25), we obtain the equation
φ
n
″
+
ω
n
2
φ
n
=
2
i
ω
n
D
1
A
n
X
n
−
a
A
n
X
n
″
cos
σ
n
T
1
+
η
(
i
ω
n
)
α
A
n
X
n
with the boundary conditions
φ
n
(
0
)
=
0
,
φ
n
(
1
)
=
0
.
Using the solvability condition 15, we then find
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Thus, by the normalization given as
∫
0
1
X
n
2
(
x
)
d
x
=
1
, then Eq. (29) turns into
2
i
ω
n
D
1
A
n
+
A
n
[
a
cos
(
σ
n
T
1
)
d
1
+
(
i
ω
n
)
α
]
=
0
,
where
d
1
=
∫
0
1
X
n
(
x
)
X
n
″
(
x
)
d
x
.
Then, the amplitude solution for the first order of the problem is as follows:
A
n
(
T
1
)
=
A
0
exp
[
−
η
2
ω
n
α
−
1
T
1
sin
(
π
2
α
)
+
i
(
−
a
2
ω
n
σ
n
sin
(
σ
n
T
1
)
+
η
2
ω
n
α
−
1
cos
(
π
2
α
)
T
1
)
]
and the displacement is also obtained:
w
(
x
,
t
)
≅
A
0
exp
(
−
η
2
ω
n
α
−
1
ε
t
sin
(
π
2
α
)
)
{
exp
[
−
i
a
2
ω
n
σ
n
sin
(
σ
n
ε
t
)
∫
0
1
X
n
X
n
″
d
x
+
i
η
2
ε
t
ω
n
α
−
1
cos
(
π
2
α
)
+
i
ω
n
t
]
+
cc
}
sin
ω
n
x
+
2
cos
Ω
¯
2
t
×
[
sin
Ω
¯
2
x
sin
Ω
¯
2
[
ψ
(
0
)
cos
Ω
¯
2
−
ψ
(
1
)
]
−
ψ
(
0
)
cos
Ω
¯
2
x
+
ψ
(
x
)
]
,
where
A
0
is a constant (determined by enforcing initial conditions). Additionally, the supplementary natural frequency from the fractional derivative is also given by
ω
n
a
=
−
a
2
ω
n
σ
n
sin
(
σ
n
ε
t
)
∫
0
1
X
n
X
n
″
d
x
+
η
2
ε
t
ω
n
α
−
1
cos
(
π
2
α
)
.
As seen in Figure 1, the fractional derivative α-order has an effect on the displacement-time curves. In Figure 2 and Figure 3, the effect of the variation of the coefficient λ is observed for the different functions on displacement-time curves.
<p>Figure 1</p>
Displacement-time curves for different values of the order of the fractional derivative for
f
(
x
)
=
x
e
−
x
(
a
=
0.8
,
ε
=
0.1
,
η
=
8
,
λ
=
1
,
x
=
0.5
).
Displacement-time curves for different values of the order of the fractional derivative for
f
(
x
)
=
x
e
−
x
(
a
=
0.8
,
ε
=
0.1
,
η
=
8
,
λ
=
1
,
x
=
0.5
).
![]()
<p>Figure 2</p>
Displacement-time graph for different values of
λ
for
f
(
x
)
=
x
e
−
x
(
a
=
0.5
,
ε
=
0.1
,
η
=
8
,
α
=
0.6
,
x
=
0.5
).
Displacement-time graph for different values of
λ
for
f
(
x
)
=
x
e
−
x
(
a
=
0.5
,
ε
=
0.1
,
η
=
8
,
α
=
0.6
,
x
=
0.5
).
![]()
<p>Figure 3</p>
Displacement-time graph for various values of
λ
for
f
(
x
)
=
x
2
e
−
x
(
a
=
0.001
,
ε
=
0.1
,
η
=
8
,
α
=
0.8
,
x
=
0.5
).
Displacement-time graph for various values of
λ
for
f
(
x
)
=
x
2
e
−
x
(
a
=
0.001
,
ε
=
0.1
,
η
=
8
,
α
=
0.8
,
x
=
0.5
).
![]()
4.2 Ω1 close to 2
ω
n
, Ω2 away from ωn (
Ω
1
≅
2
ω
n
, Ω2≠ωn)
If we consider the parametric resonance, then
Ω
1
=
2
ω
n
+
ε
σ
n
.
Hence, the solvability condition requires that
2
i
ω
n
D
1
A
n
−
a
2
d
1
A
¯
n
e
i
ε
σ
n
T
1
+
η
(
i
ω
n
)
α
A
n
=
0
,
where
d
1
is given by (31). To perform the stability analysis, one introduces the transformation
A
n
(
T
1
)
=
B
n
(
T
1
)
e
i
σ
n
T
1
/
2
,
where
B
n
can be written as
B
n
(
T
1
)
=
(
b
n
R
+
i
b
n
I
)
e
λ
T
1
.
Substituting Eq. (38) into Eq. (37) and also obtaining the result placed into Eq. (36) (and separating into real and imaginary parts), we get
[
λ
+
η
2
ω
n
α
−
1
sin
(
π
2
α
)
−
d
1
4
ω
n
+
(
η
2
ω
n
α
−
1
cos
(
π
2
α
)
−
σ
n
2
)
−
d
1
4
ω
n
−
(
η
2
ω
n
α
−
1
cos
(
π
2
α
)
−
σ
n
2
)
λ
+
η
2
ω
n
α
−
1
sin
(
π
2
α
)
]
[
b
n
R
b
n
I
]
=
[
0
0
]
.
For a non-trivial solution (
b
n
R
≠
0
,
b
n
I
≠
0
), the determinant of the coefficient matrix must be
[
λ
+
η
2
ω
n
α
−
1
sin
(
π
2
α
)
]
2
−
[
(
d
1
4
ω
n
)
2
−
(
η
2
ω
n
α
−
1
cos
(
π
2
α
)
−
σ
n
2
)
2
]
=
0
.
Here, λ also must be zero for the steady-state condition. Thus, the stability boundaries are determined as follows:
σ
n
=
η
ω
n
α
−
1
cos
(
π
2
α
)
±
(
d
1
2
ω
n
)
2
−
(
η
ω
n
α
−
1
sin
(
π
2
α
)
)
2
.
Inserting
σ
1
into Eq. (29), we obtain
Ω
1
=
2
ω
n
+
ε
[
η
ω
n
α
−
1
cos
(
π
2
α
)
±
(
d
1
2
ω
n
)
2
−
(
η
ω
n
α
−
1
sin
(
π
2
α
)
)
2
]
for the external excitation frequency. Thus, the two different values of Ω1 denote the stability boundaries for small ε. Additionally, it is seen that the stability boundaries depend not only on natural frequency but also on α.
The variation of an unstable region for different values of λ is observed in Figure 4. Since the rigidity of the system is increased by decreasing the value of λ, the unstable region reduces expeditiously for smaller values of λ.
<p>Figure 4</p>
Stability boundaries for different values of
λ
for
f
(
x
)
=
x
e
−
x
(
α
=
0.5
,
η
=
5
).
Stability boundaries for different values of
λ
for
f
(
x
)
=
x
e
−
x
(
α
=
0.5
,
η
=
5
).
![]()
The variation of an unstable region with some different values of α for λ
=
5
and η
=
5
is shown in Figure 5. Here, it is expected that the critical value of a becomes zero for α
=
0
. This situation is clearly observed in Figure 5. On the other hand, the unstable region diminishes while α is increasing. Finally, the effect of the variation of α on the critical value of a is presented in Figure 6. Figure 7 shows that critical value a changes nonlinearly with the order of fractional derivative.
<p>Figure 5</p>
Stability boundaries for different values of
α
for
f
(
x
)
=
x
e
−
x
(
λ
=
5
,
η
=
5
).
Stability boundaries for different values of
α
for
f
(
x
)
=
x
e
−
x
(
λ
=
5
,
η
=
5
).
![]()
<p>Figure 6</p>
Critical value of
a
versus the value of
η
for various fractional orders (
λ
=
1
).
Critical value of
a
versus the value of
η
for various fractional orders (
λ
=
1
).
![]()
<p>Figure 7</p>
Critical value of
a
versus the value of
α
for various damping coefficients (
λ
=
1
).
Critical value of
a
versus the value of
α
for various damping coefficients (
λ
=
1
).
![]()
4.3 Ω1 away from 2ωn and 0, Ω2 away from ωn (
Ω
1
≠
2
ω
n
, 0,
Ω
2
≠
ω
n
)
This case corresponds to the absence of any resonances. Then, Eq. (23) turns into
D
0
2
w
1
+
L
0
[
w
1
]
=
[
−
2
i
ω
n
D
1
A
n
X
n
−
η
(
i
ω
n
)
α
A
n
X
n
]
e
i
ω
n
T
0
+
cc
+
NST
,
where cc is a complex conjugate and NST denotes non-secular terms. Substituting Eq. (26) into Eq. (43), we obtain the equation
φ
n
″
+
ω
n
2
φ
n
=
2
i
ω
n
D
1
A
n
X
n
+
η
(
i
ω
n
)
α
A
n
X
n
with the boundary conditions
φ
n
(
0
)
=
φ
n
(
1
)
=
0
.
Using the solvability condition 15, we find
2
i
ω
n
D
1
A
n
∫
0
1
X
n
2
(
x
)
d
x
+
η
(
i
ω
n
)
α
A
n
∫
0
1
X
n
2
(
x
)
d
x
=
0
.
By the normalization, then Eq. (46) becomes
2
i
ω
n
D
1
A
n
+
(
i
ω
n
)
α
d
1
A
n
=
0
.
Thus, the displacement is obtained as follows:
w
(
x
,
t
)
≅
A
0
exp
(
−
η
2
ω
n
α
−
1
sin
(
π
2
α
)
ε
t
)
×
[
exp
(
i
η
2
ε
t
ω
n
α
−
1
cos
(
π
2
α
)
+
i
ω
n
t
)
+
cc
]
sin
ω
n
x
+
2
cos
Ω
¯
2
t
[
sin
Ω
¯
2
x
sin
Ω
¯
2
[
ψ
(
0
)
cos
Ω
¯
2
−
ψ
(
1
)
]
−
ψ
(
0
)
cos
Ω
¯
2
x
+
ψ
(
x
)
]
.
On the other hand, the amplitude is
A
n
(
T
1
)
=
A
0
exp
(
[
i
2
ω
n
α
−
1
η
cos
(
π
2
α
)
−
η
2
ω
n
α
−
1
sin
(
π
2
α
)
]
T
1
)
.
The displacement-time variation for different values of α is seen in Figure 8. Also, it is shown that the damping increases while the value of coefficient λ diminishes in Figure 9.
<p>Figure 8</p>
Displacement-time graph for the different fractional order for
f
(
x
)
=
x
e
−
x
(
ε
=
0.1
,
λ
=
1
,
η
=
5
,
x
=
0.5
).
Displacement-time graph for the different fractional order for
f
(
x
)
=
x
e
−
x
(
ε
=
0.1
,
λ
=
1
,
η
=
5
,
x
=
0.5
).
![]()
<p>Figure 9</p>
Displacement-time graph for different values of
λ
for
f
(
x
)
=
x
e
−
x
(
ε
=
0.1
,
η
=
8
,
α
=
0.5
,
x
=
0.5
).
Displacement-time graph for different values of
λ
for
f
(
x
)
=
x
e
−
x
(
ε
=
0.1
,
η
=
8
,
α
=
0.5
,
x
=
0.5
).
![]()
4.4 Ω1 away from 2ωn and Ω2 close to ωn (
Ω
1
≠
2
ω
n
≠
0
,
Ω
2
≅
ω
n
)
This case deals with the primary resonance
Ω
2
=
ω
n
+
ε
σ
n
when the frequency of the transverse loading is approximately equal to the natural frequency. Then, the steady-state solutions of amplitude-phase modulation equations and their stability can be discussed. Using the polar form
A
n
=
1
2
a
n
(
T
1
)
e
i
β
n
(
T
1
)
and substituting Eq. (50) into the equation below,
2
i
ω
n
D
1
A
n
+
η
(
i
ω
n
)
α
A
n
+
η
(
i
ω
n
)
α
d
2
e
i
σ
n
T
1
=
0
,
where
d
2
=
∫
0
1
X
n
(
x
)
Y
(
x
)
d
x
,
we then obtain
a
n
′
+
i
a
n
β
n
′
−
i
η
ω
n
α
−
1
[
cos
(
π
2
α
)
+
i
sin
(
π
2
α
)
]
(
1
2
a
n
+
d
2
e
i
(
σ
n
T
1
−
β
n
)
)
=
0
,
where
γ
n
=
σ
n
T
1
−
β
n
. Separating the equation into real and imaginary parts and also substituting the equation
a
n
′
=
γ
n
′
=
0
into Eq. (53), we find
![]()
![]()
By the same mathematical manipulation, the stability boundaries are calculated as follows:
σ
n
=
η
2
ω
n
α
−
1
cos
(
π
2
α
)
±
η
d
2
2
ω
n
2
α
−
2
a
n
2
−
(
1
2
ω
n
α
−
1
sin
(
π
2
α
)
)
2
.
4.5 Sum type of resonance (
Ω
1
+
Ω
2
≅
ω
n
+
ε
σ
n
)
In this case, we consider the sum or difference of internal and external forced frequency since
Ω
1
≠
0
,
Ω
1
≠
2
ω
n
, and Ω2≠ωn. Likewise, Eq. (23) is arranged once again; it is found that
2
i
ω
n
D
1
A
n
+
1
2
d
3
e
i
σ
n
T
1
+
η
(
i
ω
n
)
α
A
n
=
0
,
where
d
3
=
−
a
∫
0
1
X
n
(
x
)
Y
(
x
)
d
x
.
Substituting Eq. (50) into Eq. (58) and also separating the equation into real and imaginary parts, we get
![]()
![]()
Inserting Eq. (54) into Eqs. (60) and (61), then we have
![]()
![]()
Therefore, the stability boundaries are obtained as follows:
σ
n
=
η
2
ω
n
α
−
1
cos
(
π
2
α
)
±
1
2
d
3
2
ω
n
2
a
n
2
−
η
2
ω
n
2
α
−
2
sin
2
(
π
2
α
)
.