Nonlinear dynamics of axially moving beam with coupled longitudinal– transversal vibrations Xiao-Dong Yang & Wei Zhang
Nonlinear Dynamics An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems ISSN 0924-090X Volume 78 Number 4 Nonlinear Dyn (2014) 78:2547-2556 DOI 10.1007/s11071-014-1609-5
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Author's personal copy Nonlinear Dyn (2014) 78:2547–2556 DOI 10.1007/s11071-014-1609-5
ORIGINAL PAPER
Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations Xiao-Dong Yang · Wei Zhang
Received: 23 February 2014 / Accepted: 15 July 2014 / Published online: 2 August 2014 © Springer Science+Business Media Dordrecht 2014
Abstract In this study, the nonlinear vibrations of an axially moving beam are investigated by considering the coupling of the longitudinal and transversal motion. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. By detuning the axially velocity, the exact parameters with which the system may turn to internal resonance are detected. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. The saturation and jump phenomena of such system have been reported by investigating the nonlinear amplitude–response curves with respect to external excitation, internal, and external detuning parameters. The longitudinal external excitation may trigger only longitudinal response when excitation amplitude is weak. However, beyond the critical excitation amplitude, the response energy will be transferred from the longitudinal motion to the transversal motion even the excitation is employed on the longitudinal direction. Such energy transfer due to saturation has the potential to be used in the vibration suppression.
X.-D. Yang (B) · W. Zhang College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China e-mail:
[email protected]
Keywords Nonlinear vibrations · Axially moving beam · Coupled longitudinal–transversal vibrations · Multiple-scale method · Saturation phenomenon
1 Introduction The axially moving structures are involved in many engineering devices, such as belt drives, high-speed magnetic tapes, band saws, fiber winding, and paper sheets. The axial traveling speed plays an important role on the transverse vibrations of the structure and their dynamical stability. Study of the transverse vibrations becomes key to avoid possible resulting fatigue, failure, and low quality. String model and beam model are usually used to investigate such structures. Recent developments on axially moving continua have been reviewed in [1–3]. The linear and nonlinear transverse dynamics of systems with axially moving speed have been investigated extensively in the literature. Wickert [4] examined the sub- and super-critical nonlinear vibrations of axially moving tensioned beams employing the linear integropartial differential equation of motion. Marynowski et al. [5,6] considered several energy dissipation mechanisms in the mathematical model of axially moving systems and investigated the bifurcation and chaos due to the parametric excitation. The stability and vibration characteristics of an axially moving plate were investigated by Lin [7] by linear model. The dynamics of an axially moving beam beyond the first bifur-
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cation were examined by Pellicano and Vestroni [8]. The sub-critical vibrations of an axially moving beam were investigated by Sze et al. [9] and Huang et al. [10] using the incremental harmonic balance method. Özhan and Pakdemirli [11] examined the primary resonance of an axially moving beam by developing a general solution procedure for this class of systems. In a series of papers by Ghayesh et al. [12–20], the sub- and super-critical dynamics and stability of an axially moving beam were investigated analytically and numerically. Recently, nonlinear natural frequencies of a high-speed moving beam were obtained by Ding and Chen [21] using the Galerkin method. It can be concluded from the above references that even for the lower speed case the internal resonance may induce complex dynamics for the axially moving beam in sub-critical regime. Huang et al. [10] studied the 3:1 internal resonance for the axially moving beam. Özhan and Pakdemirli [11] discussed the nonlinear dynamics of systems with cubic nonlinearities in the internal resonance case. Riedel and Tan [22] examined the coupled forced dynamics of an axially moving beam possessing a three-to-one internal resonance. Tang and Chen [23] investigated the nonlinear response of in-plane moving plate with the case of 3:1 and 1:1 internal resonances. Ghayesh [24] studied the complex nonlinear dynamics, such as bifurcations and chaos, of axially moving beam for the 3:1 internal resonance case. Wang et al. [25] considered the nonlinear dynamics of the moving laminated circular cylindrical shells. The one-to-one internal resonance phenomenon was discussed by the method of harmonic balance. Most of the references studying internal resonances considered only the transversal vibrations of the axially moving material. Based on the Galerkin truncation, the first two or more orders have been retained. When the second transversal mode natural frequency approaches the three times of the first transversal mode natural frequency by detuning the axial speed, the 3:1 internal resonance phenomenon could be detected for the system with cubic nonlinearities. The coupled longitudinal–transversal vibrations of axially moving systems with constant axial speed, on the other hand, have not received considerable attention in the literature. Ghayesh et al. [26–28] studied the longitudinal– transversal vibrations for the axially moving beam. They discussed the contribution of external excitations to the nonlinear responses and also the 3:1 internal resonances. However, the 2:1 internal resonances have not
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been studied in the literature, which may yield interesting nonlinear dynamical phenomena. In this study, the partial differential equations governing the coupled longitudinal–transversal vibrations are studied. The Galerkin method is used to truncate the nonlinear partial differential equations into a set of ordinary differential equations. By detuning the axially moving speed, the internal resonance relations can be found. An example is proposed to illustrate the existence of jump associated with saturation in the current study for the axially moving beam.
2 Governing equations Now, we consider the nonlinear dynamics of the longitudinal–transversal coupling governing equations for the axially moving uniform beam with speed v on two simple supports with distance l. Assuming that plane sections remain plane and assuming a linear stress–strain law, one can derive the following equations of planar motion for beams [29] 2 2 ∂u ∂ 2u ∂ 2u ∂ u 2∂ u + 2v + c + v − E A ρA 1 ∂t 2 ∂ x∂t ∂x2 ∂t ∂x2 ∂w ∂ 2 w = (E A − P) + F1 cos (1 t) ∂x ∂x2 2 2 ∂ 2w ∂ w 2∂ w + v + 2v ρA ∂t 2 ∂ x∂t ∂x2
(1)
∂ 2w ∂w ∂ 4w − P 2 + EI 4 ∂t ∂x ∂x 2 ∂ u ∂w ∂u ∂ 2 w + F2 cos (2 t) + = (E A − P) ∂x2 ∂x ∂x ∂x2 (2) + c2
where u is the longitudinal displacement; w is the transversal displacement; ρ is the beam density; A and I are, respectively, the area and moment of inertia of the beam cross-section; E is Young’s modulus; P is the prescribed axial load; c1 and c2 are the constants related to the structural damping; and F1 and F2 are external excitation amplitudes. Equations (1) and (2) without the traveling effect have been discussed by Nayfeh and Mook [29] in detail. In the current version, the higher terms more than two have been omitted. Equations (1) and (2) are coupled via the quadratic nonlinear terms. The similar equations governing the motions of beam without traveling have been investigated in Nayfeh and Mook [29] for the case of u =
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O(w 2 ) and the case of u = O(w). If the radius of gyration of the cross-section be extremely small, the longitudinal inertia is small compared with the restoring force, and it follows that u = O(w 2 ); if the radius of gyration be small, but not extremely small, it follows that u = O(w). Ghayesh et al. [26–28] studied the coupled vibrations of axially moving material for the case of u = O(w 2 ). In the next section, it can be verified that by detuning the axially moving velocity we can locate the vicinity where the transversal and longitudinal motions have the same order, u = O(w), even the beam is a slender one. 3 Galerkin truncation Modal analysis is one of the powerful tools to investigate the vibration engineering problems. According to the Galerkin truncation method, the time–spatial response function of the system can be expanded in terms of linear undamped natural modes of the corresponding time-independent system by neglecting the effects of the damping, the axially moving velocity, and the excitation. The response of our continuous system in terms of the linear free-vibration modes for the current pinned–pinned ends conditions can be assumed as follows: ∞ nπ x (3) pn (t) sin u(x, t) = l n=1
w(x, t) =
∞
qn (t) sin
n=1
nπ x l
(4)
Substituting (3) and (4) into (1) and (2) and multiplying both sides of the results by sin (mπ x/l) and integrate over the interval [0, l] lead to p¨n + λ2n pn = ε −μ1 p˙ n − 4v −κ
∞ m=1
∞
nm (−1)m+n − 1 p˙ m 2 −n
m2
mn (m + n) qm qm+n
m=1
+
n
m (n − m) qm qn−m 2
m=1
(−1)n f 1 cos (1 t) − n
(5) q¨n + ωn2 qn
= ε −μ2 q˙n − 4v
∞ m=1
nm (−1)m+n − 1 q˙m m 2 − n2
−κ
∞
mn (m + n) (qm pm+n + pm qn+m )
m=1
+
n
m (n − m) ( pm qn−m + qm pn−m ) 2
m=1
−
(−1)n ε f 2 cos (2 t) n
(6)
where EA− P c1 c2 , μ2 = , κ = π3 , ρ Al ρ Al 2ρ Al 5 4F1l 4F2 l f1 = , f2 = ρ Aπ ρ Aπ
μ1 =
(7)
In this study, the amplitudes of excitation and responses are considered small but finite, the structure damping weak. The longitudinal and transversal natural frequencies λn and ωn can be obtained by solving for the roots of the following two algebraic equation, respectively,
det −λ2 I + K1 + iλG = 0
(8) det −ω2 I + K2 + iωG = 0 In Eq. (8), I is the unit matrix, G is the gyroscopic matrix, and K1 andK2 are diagonal stiffness matrices. The elements of G, K1 , and K2 can be obtained as nm (−1)m+n − 1 , (G)nm = 4v 2 2 m −n 2 n π2 E 2 −v , (K1 )nn = 2 l ρ (K2 )nn =
n2π 2 2 n4π 4 E I P − v + 4 2 l ρA l ρA
(9)
By studying the roots of (8), we discuss the longitudinal and transversal natural frequencies effected by the axially moving velocity. Based on the parameters given by Table 1, the variation of the first two longitudinal and transversal natural frequencies with respect to the axial velocity is presented in Fig. 1. The natural frequencies decrease with the increase of the axially moving velocity, which has been discussed in many references in the vibration study of axially moving material [30–32]. The natural frequencies of the longitudinal vibration are usually higher than those of the transversal vibration when axially moving velocity is low. Although all the natural frequencies decrease with the increase of the axially moving velocity, the values
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Table 1 The values of the parameters
E
ρ
A
I
l
P
6.5 MPa
0.8 × 103 kg/m3
1.0 × 10−3 m2
8.0 × 10−6 m4
1m
500 N
natural frequencies are ω1 = 70.77 and ω2 = 28.501. In such case, the internal resonance may occur since λ1 = ω1 + ω2 .
4 Analysis by the method of multiple scales
Fig. 1 The natural frequencies versus the axially moving velocity
for longitudinal and transversal natural frequencies are varying with different rates. By detuning the axially moving velocity, we can locate some internal resonance points. For example, when velocity v = 9.46, the first longitudinal natural frequency λ1 = 355.78, and the first two transversal
Now, we use the method of multiple scales to study the governing ordinary differential Eqs. (5) and (6) coupled via the nonlinear terms. The approximate solutions for small but finite amplitudes can be assumed in the form of pn (t) = pn0 (T0 , T1 ) + εpn1 (T0 , T1 ) + O ε2 qn (t) = qn0 (T0 , T1 ) + εqn1 (T0 , T1 ) + O ε2 (10) where T0 = t and T1 = εt represent the fast and slow timescales. Substituting (10) and their derivatives into (5) and (6) and equating coefficients of like powers of ε, we obtain the following set of second-order ordinary differential equations:
D2 p + λ2 p = 0 n n0 0 n0 O ε0 : D20 qn0 + ωn2 qn0 = 0 ⎧ ∞ ⎪ nm m+n 2 p + ω2 p = −μ D p − 4v ⎪ D − 1 D0 pm0 ⎪ n1 n1 1 0 n0 2 −n 2 (−1) n 0 ⎪ m ⎪ m=1 ⎪ ∞ ⎪ n ⎪ ⎪ 2 (n − m) q q ⎪ ⎪ − 2D D p − κ mn + n) q q + m (m 0 1 n0 m0 m0 (m+n)0 (n−m)0 ⎪ ⎪ ⎪ m=1 m=1 ⎪ n ⎪ f cos t) ( ⎨ − (−1) ⎪ 1 1 n ∞ O ε1 : nm 2 2 D q + γ q = −μ D q − 4v (−1)m+n − 1 D0 qm0 ⎪ n1 n1 2 0 n0 n 0 ⎪ m 2 −n 2 ⎪ ⎪ m=1 ∞ ⎪ ⎪ ⎪ ⎪ ⎪ − 2D0 D1 qn0 − κ mn (m + n) qm0 p(m+n)0 + pm0 q(n+m)0 ⎪ ⎪ ⎪ m=1 ⎪ ⎪ n ⎪ n ⎪ 2 ⎪ + m (n − m) pm0 q(n−m)0 + qm0 p(n−m)0 − (−1) ⎩ n ε f 2 cos (2 t) m=1
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(11)
(12)
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In the above equations, the differential operator Dn denotes the derivative with respect to Tn . The solutions to (11) can be written as pn0 = An (T1 ) exp (iλn T0 ) + A¯ n (T1 ) exp (−iλn T0 ) qn0 = Bn (T1 ) exp (iωn T0 ) + B¯ n (T1 ) exp (−iωn T0 ) (13) Substituting (13) into (12) leads to D20 pn1 + λ2n pn1 = −μ1 iλn An eiλn T0 ∞ nm − 4v (−1)m+n − 1 iλm Am eiλm T0 m 2 − n2
The cc symbol denotes the complex conjugate of all the terms of the right-hand side. By considering the possible secular terms of (14), we conclude that when λn ≈ |ωn+m ± ωm | or λn ≈ |ωm ± ωn−m | extra internal resonance link connecting pn1 and qn1 may exist. The similar conclusions can be obtained by studying the possible secular term of (15). The nonresonant case and the internal resonant case of the particular solutions of (14) and (15) need to be distinguished, which will be discussed in the following subsection.
m=1
− 2D1 An iλn eiλn T0 − κ
∞
4.1 The nonresonant condition
mn (n + m)
m=1
+ B¯ m Bn+m ei(ωn+m −ωm )T0
i(ωm +ωn+m )T0
Bm Bn+m e −κ
n
m 2 (n − m) Bm Bn−m ei(ωm +ωn−m )T0
m=1
+Bm B¯ n−m ei(ωm −ωn−m )T0 −
D1 An + 21 μ1 An = 0
(−1) f 1 exp (i1 T0 ) + cc 2n n
(14)
D20 qn1 + ωn2 qn1 = −μ2 iωn Bn eiωn T0 ∞ nm − 4v (−1)m+n − 1 iωm Bm eiωm T0 m 2 − n2 m=1
− 2iωn D1 Bn eiωn T0 − κ
∞
An+m Bm ei(ωm +λn+m )T0 + An+m B¯ m ei(λn+m −ωm )T0 ∞
mn (n + m) Am Bn+m ei(λm +ωn+m )T0
m=1
+ Am B¯ n+m ei(λm −ωn+m )T0 −κ
n
m 2 (n − m) Am Bn−m ei(λm +ωn−m )T0
m=1
+ Am B¯ n−m ei(λm −ωn−m )T0 −κ
n m=1
−
(16)
Substituting the solutions of (16) into (13) pn0 = exp (−εμ1 t) an exp (iλn t) + cc qn0 = exp (−εμ2 t) bn exp (iλn t) + cc
(17)
where an and bn are complex constants. There are only zero value steady-state solutions for (17), which is the typical result of the damped free vibrations.
1 = λn + εσ1 , λn = ωn+m + ωn + εσ2
(−1)n f 2 exp (i2 T0 ) + cc n
4.2 The resonant condition In this subsection, we consider the coupled internal– external resonance conditions for the Eqs. (14) and (15) governing the longitudinal and transversal vibrations of the beam due to the longitudinal excitation. Internal resonance may occur when λn ≈ |ωn+m ± ωm | or λn ≈ |ωm ± ωn−m |, and primary external resonance could be described by 1 = λn . Now, we consider the internal resonance when λn ≈ ωn+m + ωm as a case study. We introduce the external and internal detuning parameters σ1 and σ2
m 2 (n − m) An−m Bm ei(ωm +λn−m )T0
+ A¯ n−m Bm ei(ωm −λn−m )T0
D1 Bn + 21 μ2 Bn = 0
mn (n + m)
m=1
−κ
In this case, the only terms that produce secular terms are the terms proportional to exp (±iλn T0 ) in (14) and the terms proportional to exp (±iωn T0 ) in (15). Hence, the solvability conditions become
(15)
(18)
To study the external–internal resonance better, we use the case of n = 1, m = 1, f 2 = 0 as an example.
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The first natural frequency of the longitudinal vibration is detuned approximately equal to the sum of the first two natural frequencies of the transversal vibration. The external excitation on the longitudinal direction is focused while the excitation on the transversal direction is omitted. In such case, the solvability conditions of (14) and (15) are −iλ1 (2D1 A1 + μ1 A1 ) − 2κ B1 B2 e
−iσ2 T1
+ 21 f 1 eiσ1 T1 = 0
−iω1 (2D1 B1 + μ2 B1 ) − 2κ A1 B¯ 2 eiσ2 T1 = 0 −iω2 (2D1 B2 + μ2 B2 ) − 2κ A1 B¯ 1 eiσ2 T1 = 0.
Bn = 21 bn eiβn
(20)
where an , bn , αn , and βn are real functions of T1 . Substituting (20) into (19), separating real and imaginary parts of the resulting equations, and doing some manipulations, we obtain the following differential equations: 1 κ f1 sin γ1 a˙ 1 = − μ1 a1 + b1 b2 sin γ2 + 2 λ1 4λ1 1 κ a1 b2 sin γ2 b˙1 = − μ2 b1 − 2 ω1 1 κ a1 b1 sin γ2 b˙2 = − μ2 b2 − 2 ω2 κ b1 b2 f1 γ˙1 = σ1 − cos γ2 + cos γ1 , λ1 a 1 4λ1 a1 κ b1 b2 f1 γ˙2 = σ2 + cos γ2 − cos γ1 λ1 a 1 4λ1 a1 κ a1 b2 κ a1 b1 − cos γ2 − cos γ2 ω1 b1 ω2 b2
(21)
γ2 = σ2 T1 + α1 − β1 − β2 . (22)
To study the steady-state response, we set time derivatives to be zero and time-dependent variables to be constants in (21) and (22). The steady-state solutions can be obtained from the algebraic equations by letting the right-hand side to zero. By the set of algebraic equations, the amplitude response curve can be obtained. It can be found that there are two possible steady-state solutions: either a1 = 0 and b1 = b2 = 0, or a1 , b1
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f1 2 1/2 . 2λ1 μ1 + 4σ12
(23)
For the case of a1 , b1 , and b2 all nonzero, the solutions are 1/2 (σ1 + σ2 )2 + 41 (μ2 + μ2 )2 a1 = κ2 2 ω1 ω2 =
1 ω1 ω2 (σ1 + σ2 )2 + 2κ
1 4
(μ1 + μ2 )2
1/2
(24) 1/2 1/2 b1 = χ (ω2 /ω1 )1/2 , b2 = χ (ω1 /ω2 )1/2 . (25) In (25), the values of χ are determined by the roots of the following quadratic equation κ κ2 2 χ + a1 (μ1 sin γ2 − 2σ1 cos γ2 ) χ 2 λ1 λ1 f1 2 1 2 2 2 2 =0 + μ1 a1 + a1 σ1 − 4 4λ1
(26)
where
where the dot over any variables denotes the derivative with respect to T1 . In Eq. (21), two new variables are introduced γ1 = σ1 T1 − α1 ,
a1 =
(19)
We introduce the polar form for the amplitudes as An = 21 an eiαn ,
and b2 are all nonzero. For the first case b1 = b2 = 0, the analytical solution of a1 can be obtained:
1/2 sin γ2 = −μ2 (σ1 + σ2 )2 + μ22 1/2 cos γ2 = (σ1 + σ2 ) . (σ1 + σ2 )2 + μ22
(27)
The stability of steady-state solutions (23), (24), and (25) can be determined by the eigenvalues of the linearized coefficients matrix of the system (21) near the corresponding steady-state solution. If the real part of each eigenvalue of the coefficient matrix is not positive, then the corresponding steady-state solution is stable otherwise is unstable. In the following section, the stability of the steady-state solutions represented by amplitude responses will be checked by such criteria.
5 Numerical examples Based on steady-state solutions (23)–(25), some numerical figures are presented to show the nonlinear phenomenon found in such system. In Fig. 2, the first-order longitudinal amplitude a1 and the first- and second-order transversal amplitudes
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Fig. 2 Amplitudes of response as functions of the amplitude of the excitation; σ1 = 0.005; σ2 = 0
Fig. 3 Amplitudes of response b1 as functions of the amplitude of the excitation; σ1 = 0.005
b1 and b2 are plotted as functions of the longitudinal excitation amplitude f 1 , where stable solutions are indicated in solid lines while unstable solutions in dashed lines. The internal resonance is perfectly tuned σ2 = 0, and the external resonance is slightly detuned as σ1 = 0.005. In Fig. 2, one can clearly find the phenomenon of saturation. As longitudinal external excitation amplitude f 1 increases from zero, so does the first-order longitudinal amplitude response a1 indicated in black solid line. This natural relation is determined by the linear equation of (23). This agrees with the solution of the corresponding linear problem. Beyond a critical value, the solution of (23) loses stability (black dashed line) and another branch of solution determined by (24) dominates. It is clear that the first-order longitudinal response a1 of (24) is independent of external longitudinal excitation amplitude f 1 . Hence, the longitudinal response a1 holds the maximum value even for further increases in f 1 . The longitudinal vibration mode is saturated. Beyond the saturation point, the first and second transversal vibration amplitudes b1 and b2 (indicated in red and blue lines, respectively) jump from the zero response to a finite value. With further increase of the longitudinal excitation amplitude, the transversal vibration amplitudes keep increasing. The energy due to the longitudinal excitation is transferred to the transversal motion due to the saturation.
When there exist multiple stable solutions, a jump phenomenon associated with varying the excitation amplitude may occur. The trend of amplitude responses a1 , b1 , and b2 can be pursued from two ways, i.e., from f 1 = 0 to higher values and vice versa. The jump phenomena can be chased by tracking the cyan and magenta arrows, respectively. We now try to study the effect of internal resonance detuning parameter σ2 to the amplitude response curves. In Fig. 3, the variation of transversal amplitude response b1 has been plotted with respect to the internal resonance detuning parameter σ2 . It can be found that the increase of detuning parameter from exact internal resonance may postpone the critical excitation value where the saturation phenomenon occurs. Although the phenomenon of saturation is delayed because of the increase of the detuning parameter, the response amplitudes will go higher once the saturation triggered. The saturation has been discussed in some engineering fields, and the readers can refer to [33–35] for detailed investigation. Since energy transfer is the character of the saturation, this method can be used as vibration suppression technique as presented in Oueini and Nayfeh [36]. In Fig. 4, the effect of the external excitation frequency detuning parameter to the first-order transversal response b1 has been presented. It can be concluded that the contribution of the detuning parameter σ1 is similar to that of σ2 .
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Fig. 4 Amplitudes of response b1 as functions of the amplitude of the excitation; σ2 = 0.01
In Fig. 5, the amplitude responses of a1 , b1 , and b2 are plotted as function of internal resonance detuning parameter σ1 for σ2 = 0, σ2 > 0, and σ2 < 0, respectively. The jump phenomenon associated with varying the external excitation frequency is indicated by the arrows in Fig. 5a. The symmetric behavior of the frequency–response curve versus external resonance detuning parameter σ1 can be seen in Fig. 5a where the internal resonance is perfectly detuned, i.e., σ2 = 0 The nonzero value of σ2 will cause the unsymmetrical configurations in frequency–response curves as shown in Fig. 5b, c.
6 Conclusions The longitudinal and transversal coupling vibrations of an axially moving beam are considered in this paper. It is found that by detuning the axially velocity, the exact parameters with which the system present internal resonance are located. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. Examples are presented to show the saturation and jump phenomena when the first longitudinal natural
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Fig. 5 Frequency–response curves. a σ2 = 0, b σ2 = 0.02, c σ2 = −0.02
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frequency is equal to the sum of the first two natural frequencies of the transversal vibration. The nonlinear longitudinal and transversal amplitude–response curves with respect to external excitation amplitude and the external detuning parameters have been investigated for the case that the only longitudinal external excitation is implemented. It is concluded that firstorder longitudinal external excitation may trigger the transversal vibration response due to the saturation phenomenon. This phenomenon of saturation has the potential to be used in the vibration suppression technique. The effects of the longitudinal external excitation and internal frequency detuning parameters to the transversal response b1 have been studied. It is found that the phenomenon of saturation is delayed due to the increase of the detuning parameters, while the response amplitudes obtain higher values once the saturation triggered for higher value detuning parameters. The symmetric and unsymmetrical configurations of the amplitude responses are found as function of internal resonance detuning parameter. Acknowledgments This investigation is supported by the National Natural Science Foundation of China (Project Nos. 11322214, 11172010, and 11290152) and by the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) under Grant No. MCMS-0112G01.
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