Research Article Vibration, Stability, and Resonance of Angle ...In this paper, the perturbation method and stability of the composite laminated piezoelectric rectangular plate under

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 418374, 26 pageshttp://dx.doi.org/10.1155/2013/418374

Research ArticleVibration, Stability, and Resonance of Angle-Ply CompositeLaminated Rectangular Thin Plate under Multiexcitations

M. Sayed1,2 and A. A. Mousa1,3

1 Department of Mathematics and Statistics, Faculty of Science, Taif University, P.O. Box 888, Al-Taif, Saudi Arabia2Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt3 Department of Basic Engineering Sciences, Faculty of Engineering, Menoufia University, Shibin El-Kom, Egypt

Correspondence should be addressed to M. Sayed; moh 6 [email protected]

Received 10 November 2012; Revised 15 April 2013; Accepted 1 May 2013

Academic Editor: Dane Quinn

Copyright © 2013 M. Sayed and A. A. Mousa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

An analytical investigation of the nonlinear vibration of a symmetric cross-ply composite laminated piezoelectric rectangular plateunder parametric and external excitations is presented.Themethod of multiple time scale perturbation is applied to solve the non-linear differential equations describing the systemup to and including the second-order approximation. All possible resonance casesare extracted at this approximation order.The case of 1 : 1 : 3 primary and internal resonance, whereΩ

3≅ 𝜔1,𝜔2≅ 𝜔1, and𝜔

3≅ 3𝜔1,

is considered. The stability of the system is investigated using both phase-plane method and frequency response curves. The influ-ences of the cubic terms onnonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate are studied.The analytical results given by the method of multiple time scale is verified by comparison with results from numerical integrationof themodal equations. Reliability of the obtained results is verified by comparison between the finite differencemethod (FDM) andRunge-Kutta method (RKM). It is quite clear that some of the simultaneous resonance cases are undesirable in the design of suchsystem. Such cases should be avoided as working conditions for the system. Variation of the parameters 𝜇

1, 𝜇2, 𝛼7, 𝛽8, 𝜔1, 𝜔2, 𝑓1, 𝑓2

leads to multivalued amplitudes and hence to jump phenomena. Some recommendations regarding the different parameters of thesystem are reported. Comparison with the available published work is reported.

1. Introduction

Composite laminated plates that are widely used in severalengineering fields such as machinery, shipbuilding, aircraft,automobiles, robot arm, watercraft-hydropower, and wingsof helicopters are made of the angle-ply composite laminatedplates. Several researchers have focused their attention onstudying the nonlinear dynamics, bifurcations, and chaos ofthe composite laminated plates.

Internal resonance has been found in many engineeringproblems in which the natural frequencies of the system arecommensurable. Ye et al. [1] investigated the local and globalnonlinear dynamics of a parametrically excited symmetriccross-ply composite laminated rectangular thin plate underparametric excitation. The study is focused on the case of 1 : 1internal resonance and primary parametric resonance. Zhang[2] dealt with the global bifurcations and chaotic dynamics of

a parametrically excited, simply supported rectangular thinplate. The method of multiple scales is used to obtain theaveraged equations. The case of 1 : 1 internal resonance andprimary parametric resonance is considered. Guo et al.[3] studied the nonlinear dynamics of a four-edge simplysupported angle-ply composite laminated rectangular thinplate excited by both the in-plane and transverse loads. Theasymptotic perturbation method is used to derive the fouraveraged equations under 1 : 1 internal resonance. Zhang et al.[4] investigated the local and global bifurcations of a para-metrically and externally excited simply supported rectan-gular thin plate under simultaneous transversal and in-planeexcitations.The studies are focused on the case of 1 : 1 internalresonance and primary parametric resonance. Tien et al. [5]applied the averaging method and Melnikov technique tostudy local, global bifurcations and chaos of a two-degrees-of-freedom shallow arch subjected to simple harmonic

2 Mathematical Problems in Engineering

excitation for case of 1 : 2 internal resonances. Sayed andMousa [6] investigated the influence of the quadratic andcubic terms on nonlinear dynamic characteristics of theangle-ply composite laminated rectangular plate with para-metric and external excitations. Two cases of the subhar-monic resonances cases in the presence of 1 : 2 internal reso-nances are considered.Themethod ofmultiple time scale per-turbation is applied to solve the nonlinear differential equa-tions describing the system up to and including the second-order approximation. Zhang et al. [7] gave further studies onthe nonlinear oscillations and chaotic dynamics of a para-metrically excited simply supported symmetric cross-plylaminated composite rectangular thin plate with the geo-metric nonlinearity and nonlinear damping. Zhang et al. [8]dealt with the nonlinear vibrations and chaotic dynamics ofa simply supported orthotropic functionally graded mate-rial (FGM) rectangular plate subjected to the in-plane andtransverse excitations together with thermal loading in thepresence of 1 : 2 : 4 internal resonance, primary parametricresonance, and subharmonic resonance of order 1/2. Zhanget al. [9] investigated the bifurcations and chaotic dynamics ofa simply supported symmetric cross-ply composite laminatedpiezoelectric rectangular plate subject to the transverse, in-plane excitations and the excitation loaded by piezoelectriclayers. Zhang and Li [10] analyzed the resonant chaoticmotions of a simply supported rectangular thin plate withparametrically and externally excitations using exponentialdichotomies and an averaging procedure. Zhang et al. [11]analyzed the chaotic dynamics of a six-dimensional nonlinearsystem which represents the averaged equation of a compos-ite laminated piezoelectric rectangular plate subjected to thetransverse, in-plane excitations and the excitation loaded bypiezoelectric layers. The case of 1 : 2 : 4 internal resonances isconsidered. Zhang and Hao [12] studied the global bifurca-tions and multipulse chaotic dynamics of the composite lam-inated piezoelectric rectangular plate by using the improvedextended Melnikov method. The multipulse chaotic motionsof the system are found by using numerical simulation, whichfurther verifies the result of theoretical analysis. Guo andZhang [13] studied the nonlinear oscillations and chaoticdynamics for a simply supported symmetric cross-ply com-posite laminated rectangular thin plate with parametric andforcing excitations. The case of 1 : 2 : 3 internal resonance isconsidered. The method of multiple scales is employed toobtain the six-dimensional averaged equation.The numericalmethod is used to investigate the periodic and chaoticmotions of the composite laminated rectangular thin plate.Eissa and Sayed [14–16] and Sayed [17] investigated the effectsof different active controllers on simple and spring pendulumat the primary resonance via negative velocity feedback orits square or cubic. Sayed and Kamel [18, 19] investigated theeffects of different controllers on the vibrating system and thesaturation control to reduce vibrations due to rotor blade flap-ping motion. The stability of the system is investigated usingboth phase-plane method and frequency response curves.Sayed and Hamed [20] studied the response of a two-degree-of-freedom systemwith quadratic coupling under parametricand harmonic excitations. The method of multiple scaleperturbation technique is applied to solve the nonlinear

differential equations and obtain approximate solutions up toand including the second-order approximations. Amer et al.[21] investigated the dynamical system of a twin-tail aircraft,which is described by two coupled second-order nonlin-ear differential equations having both quadratic and cubicnonlinearities under different controllers. Best active controlof the system has been achieved via negative accelera-tion feedback. The stability of the system is investigatedapplying both frequency response equations and phase-plane method. Hamed et al. [22] studied the nonlineardynamic behavior of a string-beam coupled system subjectedto external, parametric, and tuned excitations that are pre-sented.The case of 1 : 1 internal resonance between themodesof the beam and string, and the primary and combined reson-ance for the beam is considered. The method of multiplescales is applied to obtain approximate solutions up to andincluding the second-order approximations. All resonancecases are extracted and investigated. Stability of the systemis studied using frequency response equations and the phase-plane method. Awrejcewicz et al. [23–25] studied the chaoticdynamics of continuous mechanical systems such as flexibleplates and shallow shells.The considered problems are solvedby the Bubnov-Galerkin, Ritz method with higher approxi-mations, and finite difference method. Convergence and val-idation of those methods are studied. Awrejcewicz et al. [26]investigated the chaotic vibrations of flexible nonlinear Euler-Bernoulli beams subjected to harmonic load andwith variousboundary conditions. Reliability of the obtained results isverified by the finite difference method and finite elementmethodwith the Bubnov-Galerkin approximation for variousboundary conditions and various dynamic regimes.

In this paper, the perturbation method and stabilityof the composite laminated piezoelectric rectangular plateunder simultaneous transverse and in-plane excitations areinvestigated. The method of multiple scales are applied toobtain the second-order uniform asymptotic solutions. Allpossible resonance cases are extracted at this approximationorder. The study is focused on the case of 1 : 1 : 3 internalresonance and primary resonance.The stability of the systemand the effects of different parameters on system behaviorhave been studied using frequency response curves. Stabilityis performed of figures by solid and dotted lines. Theanalytical results given by the method of multiple timescale is verified by comparison with results from numericalintegration of the modal equations. It is quite clear thatsome of the simultaneous resonance cases are undesirable inthe design of such system. Such cases should be avoided asworking conditions for the system. Some recommendationsregarding the different parameters of the system are reported.Comparison with the available published work is reported.

2. Mathematical Analysis

Consider a simply supported four edges composite laminatedpiezoelectric rectangular plate of lengths 𝑎, 𝑏 and thickness ℎ,as shown in Figure 1. The composite laminated piezoelectricrectangular plate is considered as regular symmetric cross-ply laminates with 𝑛 layers. A Cartesian coordinate system is

Mathematical Problems in Engineering 3

y

b

q cosΩ3t

q0 + qx cosΩ1t

q1 + qy cosΩ2t

x

a

Figure 1: The model of the composite laminated piezoelectricrectangular plate.

located in the middle surface of the plate. Assume that 𝑢, V,and 𝑤 represent the displacements of an arbitrary point ofthe composite laminated piezoelectric rectangular plate in the𝑥, 𝑦, and 𝑧 directions, respectively. The in-plane excitationsare loaded along the 𝑦direction at 𝑥 = 0 in the form𝑞0+ 𝑞𝑥cosΩ1𝑡, and the excitations are loaded along the 𝑥

direction at 𝑦 = 0 in the form 𝑞1+ 𝑞𝑦cosΩ2𝑡. The trans-

verse excitation subjected to the composite laminated piezo-electric rectangular plate is represented by 𝑞 cosΩ

3𝑡. The

dynamic electrical loading is expressed as 𝐸𝑧cosΩ4𝑡. Based

on Reddy’s third-order shear deformation plate theory [27],the displacement field at an arbitrary point in the compositelaminated plate is expressed as [13]

𝑢 (𝑥, 𝑦, 𝑡) = 𝑢0(𝑥, 𝑦, 𝑡) + 𝑧𝜑

𝑥(𝑥, 𝑦, 𝑡) − 𝑧

3 4

3ℎ2(𝜑𝑥+𝜕𝑤0

𝜕𝑥) ,

(1a)

V (𝑥, 𝑦, 𝑡) = V0(𝑥, 𝑦, 𝑡) + 𝑧𝜑

𝑦(𝑥, 𝑦, 𝑡) − 𝑧

3 4

3ℎ2(𝜑𝑦+𝜕𝑤0

𝜕𝑦) ,

(1b)

𝑤 (𝑥, 𝑦, 𝑡) = 𝑤0(𝑥, 𝑦, 𝑡) , (1c)

where 𝑢0, V0, and 𝑤

0are the original displacement on the

midplane of the plate in the𝑥,𝑦, and 𝑧directions, respectively.Let 𝜑𝑥and 𝜑

𝑦represent the midplane rotations of transverse

normal about the 𝑥 and 𝑦 axes, respectively. From thevan Karman-type plate theory and Hamilton’s principle, thenonlinear governing equations of motion of the compositelaminated piezoelectric rectangular plate are given as follows[12]:

𝜕𝑁𝑥𝑥

𝜕𝑥+

𝜕𝑁𝑥𝑦

𝜕𝑦= 𝐼0�̈�0+ 𝐽1�̈�𝑥−4

3ℎ2𝐼3

𝜕�̈�0

𝜕𝑥, (2a)

𝜕𝑁𝑥𝑦

𝜕𝑥+

𝜕𝑁𝑦𝑦

𝜕𝑦= 𝐼0V̈0+ 𝐽1�̈�𝑦−4

3ℎ2𝐼3

𝜕�̈�0

𝜕𝑦, (2b)

𝜕𝑄𝑥

𝜕𝑥+

𝜕𝑄𝑦

𝜕𝑦+𝜕

𝜕𝑥(𝑁𝑥𝑥

𝜕𝑤0

𝜕𝑥+ 𝑁𝑥𝑦

𝜕𝑤0

𝜕𝑦)

+𝜕

𝜕𝑦(𝑁𝑥𝑦

𝜕𝑤0

𝜕𝑥+ 𝑁𝑦𝑦

𝜕𝑤0

𝜕𝑦)

+4

3ℎ2(𝜕2𝑃𝑥𝑥

𝜕𝑥2+ 2

𝜕2𝑃𝑥𝑦

𝜕𝑥𝜕𝑦+

𝜕2𝑃𝑦𝑦

𝜕𝑦2) + 𝑞

= 𝐼0�̈�0−16

9ℎ4𝐼6(𝜕2�̈�0

𝜕𝑥2+𝜕2�̈�0

𝜕𝑦2)

+4

3ℎ2[𝐼3(𝜕�̈�0

𝜕𝑥+𝜕V̈0

𝜕𝑦) + 𝐽4(𝜕�̈�𝑥

𝜕𝑥+

𝜕�̈�𝑦

𝜕𝑦)] ,

(2c)

𝜕𝑀𝑥𝑥

𝜕𝑥+

𝜕𝑀𝑥𝑦

𝜕𝑦− 𝑄𝑥= 𝐽1�̈�0+ 𝑘2�̈�𝑥−4

3ℎ2𝐽4

𝜕�̈�0

𝜕𝑥, (2d)

𝜕𝑀𝑥𝑦

𝜕𝑥+

𝜕𝑀𝑦𝑦

𝜕𝑦− 𝑄𝑦= 𝐽1V̈0+ 𝑘2�̈�𝑦−4

3ℎ2𝐽4

𝜕�̈�0

𝜕𝑦. (2e)

Theboundary conditions of the simply supported rectangularcomposite laminated plate are expressed as follows:

at 𝑥 = 0, 𝑥 = 𝑎 : 𝜕𝑢𝜕𝑥= 0,

𝑤 = 𝑢 = 𝜑𝑥= 𝜑𝑦= 𝑀𝑥𝑥= 𝑁𝑥𝑦= 𝑄𝑥= 0,

(3a)

at 𝑦 = 0, 𝑦 = 𝑏 : 𝜕V𝜕𝑦= 0,

𝑤 = V = 𝑁𝑥𝑦= 𝑀𝑦𝑦= 𝑄𝑦= 0,

(3b)

∫

𝑏

0

𝑁𝑥𝑥

𝑥=0, 𝑎𝑑𝑦 = ∫

𝑏

0

(𝑞0+ 𝑞𝑥cosΩ1𝑡) 𝑑𝑦, (3c)

∫

𝑎

0

𝑁𝑦𝑦

𝑦=0, 𝑏𝑑𝑥 = ∫

𝑎

0

(𝑞1+ 𝑞𝑦cosΩ2𝑡) 𝑑𝑥. (3d)

Applying the Galerkin procedure, we obtain that the dimen-sionless differential equations of motion for the simplysupported symmetric cross-ply rectangular thin plate areshown as follows [11, 28]:

�̈�1+ 𝜇1�̇�1+ 𝜔2

1𝑢1

+ (𝑓11cosΩ1𝑡 + 𝑓12cosΩ2𝑡 + 𝑓14cosΩ4𝑡) 𝑢1

+ 𝛼1𝑢2

1𝑢2+ 𝛼2𝑢2

1𝑢3+ 𝛼3𝑢2

2𝑢1+ 𝛼4𝑢2

2𝑢3

+ 𝛼5𝑢2

3𝑢1+ 𝛼6𝑢2

3𝑢2+ 𝛼7𝑢3

1+ 𝛼8𝑢3

2

+ 𝛼9𝑢3

3+ 𝛼10𝑢1𝑢2𝑢3

= 𝑓1cosΩ3𝑡,

(4a)


�̈�2+ 𝜇2�̇�2+ 𝜔2

2𝑢2


+ 𝛽1𝑢2

1𝑢2+ 𝛽2𝑢2

1𝑢3+ 𝛽3𝑢2

2𝑢1+ 𝛽4𝑢2

2𝑢3

+ 𝛽5𝑢2

3𝑢1+ 𝛽6𝑢2

3𝑢2+ 𝛽7𝑢3

1+ 𝛽8𝑢3

2

+ 𝛽9𝑢3

3+ 𝛽10𝑢1𝑢2𝑢3

= 𝑓2cosΩ3𝑡,

(4b)

�̈�3+ 𝜇3�̇�3+ 𝜔2

3𝑢3


+ 𝛾1𝑢2

1𝑢2+ 𝛾2𝑢2

1𝑢3+ 𝛾3𝑢2

2𝑢1+ 𝛾4𝑢2

2𝑢3

+ 𝛾5𝑢2

3𝑢1+ 𝛾6𝑢2

3𝑢2+ 𝛾7𝑢3

1+ 𝛾8𝑢3

2

+ 𝛾9𝑢3

3+ 𝛾10𝑢1𝑢2𝑢3

= 𝑓3cosΩ3𝑡,

(4c)

where 𝑢1, 𝑢2, and 𝑢

3are the vibration amplitudes of the com-

posite laminated piezoelectric rectangular plate for the first-order, second-order, and the third-order modes, respectively,𝜇1, 𝜇2, and 𝜇

3are the linear viscous damping coefficients,

𝜔1, 𝜔2, and 𝜔

3are the natural frequencies of the rectangular

plate, andΩ1,Ω2,Ω3, andΩ

4are the excitations frequencies.

𝑓𝑛1, 𝑓𝑛2, 𝑓𝑛3, and 𝑓

𝑛(𝑛 = 1, 2, 3) are the amplitudes of

parametric and external excitation forces corresponding tothe three nonlinear modes, and 𝛼

𝑖, 𝛽𝑖, and 𝛾

𝑖(𝑖 = 1, 2, . . . , 10)

are the nonlinear coefficients.The linear viscous damping andexciting forces are assumed to be

𝜇𝑛= 𝜀𝜇𝑛, 𝑓𝑛1= 𝜀𝑓𝑛1, 𝑓

𝑛2= 𝜀𝑓𝑛2,

𝑓𝑛3= 𝜀𝑓𝑛3, 𝑓𝑛= 𝜀2𝑓𝑛, 𝑛 = 1, 2, 3,

(5)

where 𝜀 is a small perturbation parameter and 0 < 𝜀 ≪ 1.The external excitation forces 𝑓

𝑛are of the order 2, and

the linear viscous damping 𝜇𝑛, parametric exciting forces𝑓

𝑛1,

𝑓𝑛2, and 𝑓

𝑛3are of the order 1.

To consider the influence of the cubic terms on non-linear dynamic characteristics of the composite laminatedpiezoelectric rectangular plate, we need to obtain the second-order approximate solution of (4a), (4b), and (4c). Methodof multiple scales [29–31] is applied to obtain a second-order approximation for the system. For the second-orderapproximation, we introduce three time scales defined by

𝑇0= 𝑡, 𝑇

1= 𝜀𝑡, 𝑇

2= 𝜀2𝑡. (6)

In terms of these scales, the time derivatives become

𝑑

𝑑𝑡= 𝐷0+ 𝜀𝐷1+ 𝜀2𝐷2, (7a)

𝑑2

𝑑𝑡2= 𝐷2

0+ 2𝜀𝐷

0𝐷1+ 𝜀2(𝐷2

1+ 2𝐷0𝐷2) , (7b)

where 𝐷𝑛= 𝜕/𝜕𝑇

𝑛, 𝑛 = 0, 1, 2. We seek a uniform approxi-

mation to the solution of (4a), (4b), and (4c) in the form:

𝑢𝑛 (𝑡, 𝜀) = 𝜀𝑢𝑛1 (𝑇0, 𝑇1, 𝑇2) + 𝜀

2𝑢𝑛2(𝑇0, 𝑇1, 𝑇2)

+ 𝜀3𝑢𝑛3(𝑇0, 𝑇1, 𝑇2) + 𝑂 (𝜀

4) , 𝑛 = 1, 2, 3.

(8)

Terms of 𝑂(𝜀4) and higher orders are neglected. Substituting(5) and (7a)–(8) into (4a), (4b), and (4c) and equating thecoefficients of like powers of 𝜀, we obtain the following.

Order 𝜀

(𝐷2

0+ 𝜔2

1) 𝑢11= 0, (9a)

(𝐷2

0+ 𝜔2

2) 𝑢21= 0, (9b)

(𝐷2

0+ 𝜔2

3) 𝑢31= 0. (9c)

Order 𝜀2

(𝐷2

0+ 𝜔2

1) 𝑢12= −2𝐷

0𝐷1𝑢11− 𝜇1𝐷0𝑢11

− (𝑓11cosΩ1𝑇0+ 𝑓12cosΩ2𝑇0

+𝑓14cosΩ4𝑇0) 𝑢11+ 𝑓1cosΩ3𝑇0,

(10a)

(𝐷2

0+ 𝜔2

2) 𝑢22= −2𝐷

0𝐷1𝑢21− 𝜇2𝐷0𝑢21


+𝑓24cosΩ4𝑇0) 𝑢21+ 𝑓2cosΩ3𝑇0,

(10b)

(𝐷2

0+ 𝜔2

3) 𝑢32= −2𝐷

0𝐷1𝑢31− 𝜇3𝐷0𝑢31


+𝑓34cosΩ4𝑇0) 𝑢31+ 𝑓3cosΩ3𝑇0.

(10c)

Order 𝜀3

(𝐷2

0+ 𝜔2

1) 𝑢13= −𝐷2

1𝑢11− 2𝐷0𝐷2𝑢11− 2𝐷0𝐷1𝑢12

− 𝜇1(𝐷0𝑢12+ 𝐷1𝑢11)


+ 𝑓14cosΩ4𝑇0) 𝑢12

− 𝛼1𝑢2

11𝑢21− 𝛼2𝑢2

11𝑢31− 𝛼3𝑢2

21𝑢11

− 𝛼4𝑢2

21𝑢31− 𝛼5𝑢2

31𝑢11− 𝛼6𝑢2

31𝑢21

− 𝛼7𝑢3

11− 𝛼8𝑢3

21− 𝛼9𝑢3

31− 𝛼10𝑢11𝑢21𝑢31,

(11a)


(𝐷2

0+ 𝜔2

2) 𝑢23= −𝐷2

1𝑢21− 2𝐷0𝐷2𝑢21− 2𝐷0𝐷1𝑢22

− 𝜇2(𝐷0𝑢22+ 𝐷1𝑢21)


+𝑓24cosΩ4𝑇0) 𝑢22

− 𝛽1𝑢2

11𝑢21− 𝛽2𝑢2

11𝑢31− 𝛽3𝑢2

21𝑢11

− 𝛽4𝑢2

21𝑢31− 𝛽5𝑢2

31𝑢11− 𝛽6𝑢2

31𝑢21

− 𝛽7𝑢3

11− 𝛽8𝑢3

21− 𝛽9𝑢3

31− 𝛽10𝑢11𝑢21𝑢31,

(11b)

(𝐷2

0+ 𝜔2

3) 𝑢33= −𝐷2

1𝑢31− 2𝐷0𝐷2𝑢31− 2𝐷0𝐷1𝑢32

− 𝜇3(𝐷0𝑢32+ 𝐷1𝑢31)


+𝑓34cosΩ4𝑇0) 𝑢32

− 𝛾1𝑢2

11𝑢21− 𝛾2𝑢2

11𝑢31− 𝛾3𝑢2

21𝑢11

− 𝛾4𝑢2

21𝑢31− 𝛾5𝑢2

31𝑢11− 𝛾6𝑢2

31𝑢21

− 𝛾7𝑢3

11− 𝛾8𝑢3

21− 𝛾9𝑢3

31− 𝛾10𝑢11𝑢21𝑢31.

(11c)

The general solutions of (9a), (9b), and (9c) can be written inthe form

𝑢11= 𝐴1(𝑇1, 𝑇2) exp (𝑖𝜔

1𝑇0) + 𝑐𝑐, (12a)

𝑢21= 𝐴2(𝑇1, 𝑇2) exp (𝑖𝜔

2𝑇0) + 𝑐𝑐, (12b)

𝑢31= 𝐴3(𝑇1, 𝑇2) exp (𝑖𝜔

3𝑇0) + 𝑐𝑐, (12c)

where𝐴1,𝐴2, and𝐴

3are a complex function in𝑇

1, 𝑇2which

can be determined from eliminating the secular terms at thenext approximation and 𝑐𝑐 stands for the complex conjugateof the preceding terms. Substituting (12a), (12b), and (12c)into (10a), (10b), and (10c) and eliminating the secular terms,then the first-order approximations are given by

𝑢12= 𝐸1exp (𝑖𝜔

1𝑇0) + 𝐸2exp (𝑖 (Ω

1+ 𝜔1) 𝑇0)

+ 𝐸3exp (𝑖 (Ω

1− 𝜔1) 𝑇0) + 𝐸4exp (𝑖 (Ω

2+ 𝜔1) 𝑇0)


2− 𝜔1) 𝑇0) + 𝐸6exp (𝑖 (Ω

4+ 𝜔1) 𝑇0)


4− 𝜔1) 𝑇0) + 𝑐𝑐,

(13a)


2𝑇0) + 𝐸9exp (𝑖 (Ω

1+ 𝜔2) 𝑇0)


1− 𝜔2) 𝑇0) + 𝐸11exp (𝑖 (Ω

2+ 𝜔2) 𝑇0)


2− 𝜔2) 𝑇0) + 𝐸13exp (𝑖 (Ω

4+ 𝜔2) 𝑇0)


4− 𝜔2) 𝑇0) + 𝑐𝑐,

(13b)


3𝑇0) + 𝐸16exp (𝑖 (Ω

1+ 𝜔3) 𝑇0)


1− 𝜔3) 𝑇0) + 𝐸18exp (𝑖 (Ω

2+ 𝜔3) 𝑇0)


2− 𝜔3) 𝑇0) + 𝐸20exp (𝑖 (Ω

4+ 𝜔3) 𝑇0)


4− 𝜔3) 𝑇0) + 𝐸22exp (𝑖Ω

3𝑇0) + 𝑐𝑐,

(13c)

where 𝐸𝑖(𝑖 = 1, 2, . . . , 22) are the complex functions in 𝑇

1

and 𝑇2. From (12a)–(13c) into (11a), (11b), and (11c) and elim-

inating the secular terms, the second-order approximation isgiven by

𝑢13(𝑇0, 𝑇1, 𝑇2) = 𝐻

1exp (𝑖𝜔

2𝑇0) + 𝐻2exp (𝑖𝜔

3𝑇0)

+ 𝐻3exp (3𝑖𝜔

1𝑇0) + 𝐻4exp (3𝑖𝜔

2𝑇0)


3𝑇0)

+ 𝐻6exp (𝑖 (𝜔

2± 2𝜔1) 𝑇0)


3± 2𝜔1) 𝑇0)

+ 𝐻8exp (𝑖 (2𝜔

2± 𝜔1) 𝑇0)


3± 𝜔1) 𝑇0)


2± 2𝜔3) 𝑇0)


3± 2𝜔2) 𝑇0)


3± 𝜔2± 𝜔1) 𝑇0)

+ 𝐻13exp (𝑖Ω

3𝑇0)

+ 𝐻14exp (𝑖 (Ω

3± Ω1) 𝑇0)


3± Ω2) 𝑇0)


3± Ω4) 𝑇0)


1± 𝜔1) 𝑇0)


2± 𝜔1) 𝑇0)


4± 𝜔1) 𝑇0)

+ 𝐻20exp (𝑖 (2Ω

1± 𝜔1) 𝑇0)


2± 𝜔1) 𝑇0)


4± 𝜔1) 𝑇0)


2± Ω1± 𝜔1) 𝑇0)


4± Ω1± 𝜔1) 𝑇0)


4± Ω2± 𝜔1) 𝑇0) + 𝑐𝑐,

(14a)𝑢23(𝑇0, 𝑇1, 𝑇2) = 𝐻

26exp (𝑖𝜔


3𝑇0)


1𝑇0) + 𝐻29exp (3𝑖𝜔

2𝑇0)



3𝑇0)


2± 2𝜔1) 𝑇0)


2± 𝜔1) 𝑇0)


3± 2𝜔1) 𝑇0)


3± 𝜔1) 𝑇0)


2± 2𝜔3) 𝑇0)


2± 𝜔3) 𝑇0)


3± 𝜔2± 𝜔1) 𝑇0)

+ 𝐻38exp (𝑖Ω

3𝑇0)


3± Ω1) 𝑇0)


3± Ω2) 𝑇0)


4± Ω3) 𝑇0)


1± 𝜔2) 𝑇0)


2± 𝜔2) 𝑇0)


4± 𝜔2) 𝑇0)


1± 𝜔2) 𝑇0)


2± 𝜔2) 𝑇0)


4± 𝜔2) 𝑇0)


2± Ω1± 𝜔2) 𝑇0)


4± Ω1± 𝜔2) 𝑇0)


4± Ω2± 𝜔2) 𝑇0) + 𝑐𝑐,

(14b)𝑢33(𝑇0, 𝑇1, 𝑇2) = 𝐻

51exp (3𝑖𝜔


1𝑇0)



2𝑇0)


2𝑇0)


2± 2𝜔1) 𝑇0)


2± 𝜔1) 𝑇0)


3± 2𝜔1) 𝑇0)


3± 𝜔1) 𝑇0)


2± 2𝜔3) 𝑇0)


2± 𝜔3) 𝑇0)


3± 𝜔2± 𝜔1) 𝑇0)

+ 𝐻63exp (𝑖Ω

3𝑇0)


3± Ω1) 𝑇0)


3± Ω2) 𝑇0)


3± Ω4) 𝑇0)


1± 𝜔3) 𝑇0)


2± 𝜔3) 𝑇0)


4± 𝜔3) 𝑇0)


1± 𝜔3) 𝑇0)


2± 𝜔3) 𝑇0)


4± 𝜔3) 𝑇0)


2± Ω1± 𝜔3) 𝑇0)


4± Ω1± 𝜔3) 𝑇0)


4± Ω2± 𝜔3) 𝑇0) + 𝑐𝑐,

(14c)

where 𝐻𝑖(𝑖 = 1, 2, . . . , 75) are the complex functions in 𝑇

1

and 𝑇2. From the above derived solutions, the reported

resonance cases are the following.

(i) Primary resonance:Ω1≅ 𝜔𝑛,Ω2≅ 𝜔𝑛,Ω3≅ 𝜔𝑛,Ω4≅

𝜔𝑛, and 𝑛 = 1, 2, 3.

(ii) Subharmonic resonance: Ω1≅ 2𝜔𝑛, Ω2≅ 2𝜔𝑛, Ω4≅

2𝜔𝑛, and 𝑛 = 1, 2, 3.

(iii) Internal or secondary resonance: 𝜔1≅ 𝜔2, 𝜔2≅ 𝜔3,

𝜔3≅ 𝜔1, 𝜔1≅ 3𝜔𝑠, 𝜔2≅ 3𝜔𝑟, 𝜔3≅ 3𝜔𝑚, 𝑠 = 2, 3,

𝑟 = 1, 3, and𝑚 = 1, 2.(iv) Combined resonance: 𝜔

3± 𝜔2≅ 2𝜔1, 𝜔3± 𝜔1≅ 2𝜔2,

𝜔1±𝜔2≅ 2𝜔3,𝜔2±2𝜔3≅ 𝜔1,𝜔3±2𝜔2≅ 𝜔1,𝜔3±2𝜔1≅

𝜔2, 𝜔1± 2𝜔3≅ 𝜔2, 𝜔2± 2𝜔1≅ 𝜔3, 𝜔1± 2𝜔2≅ 𝜔3,

Ω3± Ω𝑡≅ 𝜔𝑛, Ω4± Ω𝑚≅ 2𝜔𝑛, Ω2± Ω1≅ 2𝜔𝑛,

𝑡 = 1, 2, 4, 𝑚 = 1, 2, and 𝑛 = 1, 2, 3.(v) Simultaneous or incident resonance.

Any combination of the previous resonance cases is con-sidered as simultaneous resonance.

3. Stability Analysis

Thebehavior of such a system can be very complex, especiallywhen the natural frequencies and the forcing frequency sat-isfy certain internal and external resonance conditions. Thestudy is focused on the case of 1 : 1 : 3 primary resonance andinternal resonance, where Ω

3≅ 𝜔1, 𝜔2≅ 𝜔1, and 𝜔

3≅ 3𝜔1.

To describe how close the frequencies are to the resonanceconditions, we introduce detuning parameters as follows:

Ω3= 𝜔1+ 𝜎1= 𝜔1+ 𝜀�̂�1,

𝜔2= 𝜔1+ 𝜎2= 𝜔1+ 𝜀�̂�2,

𝜔3= 3𝜔1+ 𝜎3= 3𝜔1+ 𝜀�̂�3,

(15)


where𝜎1and𝜎2,𝜎3are called the external and internal detun-

ing parameters, respectively. Eliminating the secular termsleads to solvability conditions for the first- and second-orderexpansions as follows:

2𝑖𝜔1𝐷1𝐴1= −𝑖𝜇

1𝜔1𝐴1+𝑓1

2exp (𝑖�̂�

1𝑇1) , (16a)


2𝜔2𝐴2+𝑓2

2exp (𝑖 (�̂�

1− �̂�2) 𝑇1) , (16b)


3𝜔3𝐴3, (16c)

2𝑖𝜔1𝐷2𝐴1

= −𝐷2

1𝐴1− 𝜇1𝐷1𝐴1

− {𝑓2

11

2 (Ω2

1− 4𝜔2

1)+

𝑓2

12

2 (Ω2

2− 4𝜔2

1)+

𝑓2

14

2 (Ω2

4− 4𝜔2

1)}𝐴1

− {2𝛼1𝐴1𝐴1+ 2𝛼6𝐴3𝐴3+ 3𝛼8𝐴2𝐴2}𝐴2exp (𝑖�̂�

2𝑇1)

− 𝛼1𝐴2

1𝐴2exp (−𝑖�̂�

2𝑇1) − 𝛼2𝐴2

1𝐴3exp (𝑖�̂�

3𝑇1)

− {2𝛼3𝐴2𝐴2+ 2𝛼5𝐴3𝐴3+ 3𝛼7𝐴1𝐴1}𝐴1

− 𝛼4𝐴2

2𝐴3exp (𝑖 (�̂�

3− 2�̂�2) 𝑇1)

− 𝛼10𝐴1𝐴2𝐴3exp (𝑖 (�̂�

3− �̂�2) 𝑇1) ,

(17a)2𝑖𝜔2𝐷2𝐴2

= −𝐷2

1𝐴2− 𝜇2𝐷1𝐴2

− {𝑓2

21

2 (Ω2

1− 4𝜔2

2)+

𝑓2

22

2 (Ω2

2− 4𝜔2

2)+

𝑓2

24

2 (Ω2

4− 4𝜔2

2)}𝐴2

− {2𝛽3𝐴2𝐴2+ 2𝛽5𝐴3𝐴3+ 3𝛽7𝐴1𝐴1}

× 𝐴1exp (−𝑖�̂�

2𝑇1)

− 𝛽3𝐴2

2𝐴1exp (𝑖�̂�

2𝑇1) − 𝛽1𝐴2

1𝐴2exp (−2𝑖�̂�

2𝑇1)

− {2𝛽6𝐴3𝐴3+ 2𝛽1𝐴1𝐴1+ 3𝛽8𝐴2𝐴2}𝐴2

− 𝛽2𝐴2

1𝐴3exp (𝑖 (�̂�

3− �̂�2) 𝑇1)

− 𝛽10𝐴1𝐴2𝐴3exp (𝑖 (�̂�

3− 2�̂�2) 𝑇1)

− 𝛽4𝐴2

2𝐴3exp (𝑖 (�̂�

3− 3�̂�2) 𝑇1) ,

(17b)2𝑖𝜔3𝐷2𝐴3

= −𝐷2

1𝐴3− 𝜇3𝐷1𝐴3

− {𝑓2

31

2 (Ω2

1− 4𝜔2

3)+

𝑓2

32

2 (Ω2

2− 4𝜔2

3)+

𝑓2

34

2 (Ω2

4− 4𝜔2

3)}𝐴3

− 𝛾10𝐴1𝐴2𝐴3exp (−𝑖�̂�

2𝑇1)

− 𝛾10𝐴1𝐴2𝐴3exp (𝑖�̂�

2𝑇1)

− 𝛾7𝐴3

1exp (−𝑖�̂�

3𝑇1)

− {2𝛾2𝐴1𝐴1+ 2𝛾4𝐴2𝐴2+ 3𝛾9𝐴3𝐴3}𝐴3

− 𝛾1𝐴2

1𝐴2exp (𝑖 (�̂�

2− �̂�3) 𝑇1)

− 𝛾3𝐴2

2𝐴1exp (𝑖 (2�̂�

2− �̂�3) 𝑇1)

− 𝛾8𝐴3

2exp (𝑖 (3�̂�

2− �̂�3) 𝑇1) .

(17c)

From (7a), multiplying both sides by 2𝑖𝜔𝑛, we get

2𝑖𝜔𝑛

𝑑𝐴𝑛

𝑑𝑡= 𝜀2𝑖𝜔

𝑛𝐷1𝐴𝑛+ 𝜀22𝑖𝜔𝑛𝐷2𝐴𝑛,

𝑛 = 1, 2, 3.

(18)

To analyze the solutions of (16a)–(17c), we express 𝐴𝑛in the

polar form as follows:

𝐴𝑛(𝑇1, 𝑇2) =

𝑎𝑛

2exp (𝑖𝜑

𝑛) ,

𝑎𝑛= 𝜀𝑎𝑛, (𝑛 = 1, 2, 3) ,

(19)

where 𝑎𝑛and 𝜑

𝑛are the steady-state amplitudes and phases

of the motion, respectively. Substituting (16a)–(17c) and (19)into (18) and equating the real and imaginary parts, we obtainthe following equations describing the modulation of theamplitudes and phases of the response:

̇𝑎1= −

𝜇1

2𝑎1+ {

𝑓1

2𝜔1

−𝜎1𝑓1

4𝜔2

1

} sin 𝜃1

+𝜇1𝑓1

8𝜔2

1

cos 𝜃1−𝛼1

8𝜔1

𝑎2

1𝑎2sin 𝜃2

−𝛼2

8𝜔1

𝑎2

1𝑎3sin 𝜃3−𝛼4

8𝜔1

𝑎2

2𝑎3sin (𝜃3− 2𝜃2)

−𝛼6

4𝜔1

𝑎2

3𝑎2sin 𝜃2−3𝛼8

8𝜔1

𝑎3

2sin 𝜃2

−𝛼10

8𝜔1

𝑎1𝑎2𝑎3sin (𝜃3− 𝜃2) ,

𝑎1�̇�1= {−

𝜇2

1

8𝜔1

+Γ1

2𝜔1

}𝑎1− {

𝑓1

2𝜔1

−𝜎1𝑓1

4𝜔2

1

} cos 𝜃1

+𝜇1𝑓1

8𝜔2

1

sin 𝜃1+3𝛼1

8𝜔1

𝑎2

1𝑎2cos 𝜃2+𝛼2

8𝜔1

𝑎2

1𝑎3cos 𝜃3

+𝛼3

4𝜔1

𝑎1𝑎2

2+𝛼4

8𝜔1

𝑎2

2𝑎3cos (𝜃

3− 2𝜃2)

+𝛼5

4𝜔1

𝑎1𝑎2

3+𝛼6

4𝜔1

𝑎2

3𝑎2cos 𝜃2+3𝛼7

8𝜔1

𝑎3

1


+3𝛼8

8𝜔1

𝑎3

2cos 𝜃2+𝛼10

8𝜔1

𝑎1𝑎2𝑎3cos (𝜃

3− 𝜃2) ,

̇𝑎2= −

𝜇2

2𝑎2+ {

𝑓2

2𝜔2

−(𝜎1− 𝜎2) 𝑓2

4𝜔2

2

} sin (𝜃1− 𝜃2)

+𝜇2𝑓2

8𝜔2

2

cos (𝜃1− 𝜃2) +

𝛽1

8𝜔2

𝑎2

1𝑎2sin 2𝜃

2

−𝛽2

8𝜔2

𝑎2

1𝑎3sin (𝜃3− 𝜃2) +

𝛽3

8𝜔2

𝑎2

2𝑎1sin 𝜃2

−𝛽4

8𝜔2

𝑎2

2𝑎3sin (𝜃3− 3𝜃2) +

𝛽5

4𝜔2

𝑎2

3𝑎1sin 𝜃2

+3𝛽7

8𝜔2

𝑎3

1sin 𝜃2−𝛽10

8𝜔2

𝑎1𝑎2𝑎3sin (𝜃3− 2𝜃2) ,

𝑎2�̇�2= {−

𝜇2

2

8𝜔2

+Γ2

2𝜔2

}𝑎2

+ {(𝜎1− 𝜎2) 𝑓2

4𝜔2

2

−𝑓2

2𝜔2

} cos (𝜃1− 𝜃2)

+𝜇2𝑓2

8𝜔2

2

sin (𝜃1− 𝜃2) +

𝛽1

8𝜔2

𝑎2

1𝑎2cos 2𝜃

2

+𝛽1

4𝜔2

𝑎2

1𝑎2+𝛽2

8𝜔2

𝑎2

1𝑎3cos (𝜃

3− 𝜃2)

+3𝛽3

8𝜔2

𝑎2

2𝑎1cos 𝜃2+𝛽4

8𝜔2

𝑎2

2𝑎3cos (𝜃

3− 3𝜃2)

+𝛽5

4𝜔2

𝑎2


4𝜔2

𝑎2

3𝑎2+3𝛽7

8𝜔2

𝑎3

1cos 𝜃2

+3𝛽8

8𝜔2

𝑎3

2+𝛽10

8𝜔2


3− 2𝜃2) ,

̇𝑎3= −

𝜇3

2𝑎3−𝛾1

8𝜔3

𝑎2

1𝑎2sin (𝜃2− 𝜃3)

−𝛾3

8𝜔3

𝑎2

2𝑎1sin (2𝜃

2− 𝜃3) +

𝛾7

8𝜔3

𝑎3

1sin 𝜃3

−𝛾8

8𝜔3

𝑎3

2sin (3𝜃

2− 𝜃3) ,

𝑎3�̇�3= {−

𝜇2

3

8𝜔3

+Γ3

2𝜔3

}𝑎3+𝛾1

8𝜔3

𝑎2

1𝑎2cos (𝜃

2− 𝜃3)

+𝛾2

4𝜔3

𝑎2

1𝑎3+𝛾3

8𝜔3

𝑎2

2𝑎1cos (2𝜃

2− 𝜃3)

+𝛾4

4𝜔3

𝑎2

2𝑎3+𝛾7

8𝜔3

𝑎3

1cos 𝜃3

+𝛾8

8𝜔3

𝑎3

2cos (3𝜃

2− 𝜃3) +

3𝛾9

8𝜔3

𝑎3

3

+𝛾10

4𝜔3

𝑎1𝑎2𝑎3cos 𝜃2,

(20)

where

Γ𝑛= {

𝑓2

𝑛1

2 (Ω2

1− 4𝜔2𝑛)+

𝑓2

𝑛2

2 (Ω2

2− 4𝜔2𝑛)+

𝑓2

𝑛4

2 (Ω2

4− 4𝜔2𝑛)} ,

𝑛 = 1, 2, 3,

𝜃1= �̂�1𝑇1− 𝜑1,

𝜃2= �̂�2𝑇1+ 𝜑2− 𝜑1,

𝜃3= �̂�3𝑇1+ 𝜑3− 3𝜑1.

(21)

Steady-state solutions of the system correspond to the fixedpoints of (20), which in turn correspond to

�̇�1= 𝜎1,

�̇�2= 𝜎1− 𝜎2,

�̇�3= 3𝜎1− 𝜎3.

(22)

Hence, the fixed points of (20) are given by

−𝜇1

2𝑎1+ {

𝑓1

2𝜔1

−𝜎1𝑓1

4𝜔2

1

} sin 𝜃1

+𝜇1𝑓1

8𝜔2

1

cos 𝜃1−𝛼1

8𝜔1

𝑎2

1𝑎2sin 𝜃2

−𝛼2

8𝜔1

𝑎2

1𝑎3sin 𝜃3−𝛼4

8𝜔1

𝑎2

2𝑎3sin (𝜃3− 2𝜃2)

−𝛼6

4𝜔1

𝑎2

3𝑎2sin 𝜃2−3𝛼8

8𝜔1

𝑎3

2sin 𝜃2

−𝛼10

8𝜔1

𝑎1𝑎2𝑎3sin (𝜃3− 𝜃2) = 0,

𝑎1𝜎1+ {

𝜇2

1

8𝜔1

−Γ1

2𝜔1

}𝑎1

+ {𝑓1

2𝜔1

−𝜎1𝑓1

4𝜔2

1

} cos 𝜃1−𝜇1𝑓1

8𝜔2

1

sin 𝜃1

−3𝛼1

8𝜔1

𝑎2

1𝑎2cos 𝜃2−𝛼2

8𝜔1

𝑎2

1𝑎3cos 𝜃3

−𝛼3

4𝜔1

𝑎1𝑎2

2−𝛼4

8𝜔1

𝑎2

2𝑎3cos (𝜃

3− 2𝜃2) −

𝛼5

4𝜔1

𝑎1𝑎2

3

−𝛼6

4𝜔1

𝑎2

3𝑎2cos 𝜃2−3𝛼7

8𝜔1

𝑎3

1−3𝛼8

8𝜔1

𝑎3

2cos 𝜃2

−𝛼10

8𝜔1


3− 𝜃2) = 0,

−𝜇2

2𝑎2+ {

𝑓2

2𝜔2

−(𝜎1− 𝜎2) 𝑓2

4𝜔2

2

} sin (𝜃1− 𝜃2)

+𝜇2𝑓2

8𝜔2

2

cos (𝜃1− 𝜃2)

+𝛽1

8𝜔2

𝑎2

1𝑎2sin 2𝜃

2−𝛽2

8𝜔2

𝑎2

1𝑎3sin (𝜃3− 𝜃2)


+𝛽3

8𝜔2

𝑎2

2𝑎1sin 𝜃2−𝛽4

8𝜔2

𝑎2

2𝑎3sin (𝜃3− 3𝜃2)

+𝛽5

4𝜔2

𝑎2

3𝑎1sin 𝜃2+3𝛽7

8𝜔2

𝑎3

1sin 𝜃2

−𝛽10

8𝜔2

𝑎1𝑎2𝑎3sin (𝜃3− 2𝜃2) = 0,

𝑎2(𝜎2− 𝜎1) − {

𝜇2

2

8𝜔2

−Γ2

2𝜔2

}𝑎2

+ {(𝜎1− 𝜎2) 𝑓2

4𝜔2

2

−𝑓2

2𝜔2

} cos (𝜃1− 𝜃2)

+𝜇2𝑓2

8𝜔2

2

sin (𝜃1− 𝜃2) +

𝛽1

8𝜔2

𝑎2

1𝑎2cos 2𝜃

2

+𝛽1

4𝜔2

𝑎2

1𝑎2+𝛽2

8𝜔2

𝑎2

1𝑎3cos (𝜃

3− 𝜃2)

+3𝛽3

8𝜔2

𝑎2


8𝜔2

𝑎2

2𝑎3cos (𝜃

3− 3𝜃2)

+𝛽5

4𝜔2

𝑎2


4𝜔2

𝑎2

3𝑎2+3𝛽7

8𝜔2

𝑎3

1cos 𝜃2

+3𝛽8

8𝜔2

𝑎3

2+𝛽10

8𝜔2


3− 2𝜃2) = 0,

−𝜇3

2𝑎3−𝛾1

8𝜔3

𝑎2

1𝑎2sin (𝜃2− 𝜃3) −

𝛾3

8𝜔3

𝑎2

2𝑎1sin (2𝜃

2− 𝜃3)

+𝛾7

8𝜔3

𝑎3

1sin 𝜃3−𝛾8

8𝜔3

𝑎3

2sin (3𝜃

2− 𝜃3) = 0,

𝑎3(𝜎3− 3𝜎1) − {

𝜇2

3

8𝜔3

−Γ3

2𝜔3

}𝑎3+𝛾1

8𝜔3

𝑎2

1𝑎2cos (𝜃

2− 𝜃3)

+𝛾2

4𝜔3

𝑎2

1𝑎3+𝛾3

8𝜔3

𝑎2

2𝑎1cos (2𝜃

2− 𝜃3)

+𝛾4

4𝜔3

𝑎2

2𝑎3+𝛾7

8𝜔3

𝑎3

1cos 𝜃3+𝛾8

8𝜔3

𝑎3

2cos (3𝜃

2− 𝜃3)

+3𝛾9

8𝜔3

𝑎3

3+𝛾10

4𝜔3

𝑎1𝑎2𝑎3cos 𝜃2= 0.

(23)

There are six possibilities besides the trivial solution asfollows:

(1) 𝑎1̸= 0, 𝑎2= 0, 𝑎

3= 0 (single mode),

(2) 𝑎2̸= 0, 𝑎1= 0, 𝑎

3= 0 (single mode),

(3) 𝑎1̸= 0, 𝑎2̸= 0, 𝑎3= 0 (two modes),

(4) 𝑎1̸= 0, 𝑎3̸= 0, 𝑎2= 0 (two modes),

(5) 𝑎2̸= 0, 𝑎3̸= 0, 𝑎1= 0 (two modes),

(6) 𝑎1̸= 0, 𝑎2̸= 0, 𝑎3̸= 0 (three modes).

Case 1. In this case, where 𝑎2= 0, 𝑎

3= 0, the frequency

response equation is given by

9𝛼2

7

64𝜔2

1

𝑎6

1+ [𝑅3+3𝛼7𝜎1

4𝜔1

] 𝑎4

1+ [𝑅2+ 𝜎2

1+𝜇2

1𝜎1

4𝜔1

−Γ1𝜎1

𝜔1

] 𝑎2

1

−𝜇2

1𝑓2

1

64𝜔4

1

− 𝑅2

1= 0.

(24)

Case 2. In this case, where 𝑎1= 0, 𝑎

3= 0, the frequency

response equation is given by

9𝛽2

8

64𝜔2

2

𝑎6

2+ [𝑄3+3𝛽8(𝜎2− 𝜎1)

4𝜔2

] 𝑎4

2

+ [𝑄2+ (𝜎2− 𝜎1)2−𝜇2

2(𝜎2− 𝜎1)

4𝜔2

+Γ2(𝜎2− 𝜎1)

𝜔2

] 𝑎2

2

−𝜇2

2𝑓2

2

64𝜔4

2

− 𝑄2

1= 0.

(25)

Case 3. In this case, where 𝑎3= 0, the frequency response

equations are given by

9𝛼2

7

64𝜔2

1

𝑎6

1+ [𝑅3+3𝛼7𝜎1

4𝜔1

] 𝑎4

1+ [𝑅2+ 𝜎2

1+𝜇2

1𝜎1

4𝜔1

−Γ1𝜎1

𝜔1

] 𝑎2

1

−𝜇2

1𝑓2

1

64𝜔4

1

− 𝑅2

1−9𝛼2

1

64𝜔2

1

𝑎4

1𝑎2

2−9𝛼1𝛼8

32𝜔2

1

𝑎2

1𝑎4

2

−9𝛼2

8

64𝜔2

1

𝑎6

2+3𝑅1𝛼8

4𝜔1

𝑎3

2+3𝑅1𝛼1

4𝜔1

𝑎2

1𝑎2= 0,

9𝛽2

8

64𝜔2

2

𝑎6

2+ [𝑄3+3𝛽8(𝜎2− 𝜎1)

4𝜔2

] 𝑎4

2

+ [𝑄2+ (𝜎2− 𝜎1)2−𝜇2

2(𝜎2− 𝜎1)

4𝜔2

+Γ2(𝜎2− 𝜎1)

𝜔2

] 𝑎2

2

−𝜇2

2𝑓2

2

64𝜔4

2

− 𝑄2

1+3𝑄1𝛽3

4𝜔2

𝑎2

2𝑎1+3𝑄1𝛽7

4𝜔2

𝑎3

1+𝑄1𝛽1

4𝜔2

𝑎2

1𝑎2

−9𝛽2

3

64𝜔2

2

𝑎4

2𝑎2

1− [

𝛽2

1

64𝜔2

2

+9𝛽3𝛽7

32𝜔2

2

] 𝑎4

1𝑎2

2−9𝛽2

7

64𝜔2

2

𝑎6

1

−3𝛽1𝛽7

32𝜔2

2

𝑎5

1𝑎2−3𝛽1𝛽3

32𝜔2

2

𝑎3

1𝑎3

2= 0.

(26)



9𝛼2

7

64𝜔2

1

𝑎6

1+ [𝑅3+3𝛼7𝜎1

4𝜔1

] 𝑎4

1+ [𝑅2+ 𝜎2

1+𝜇2

1𝜎1

4𝜔1

−Γ1𝜎1

𝜔1

] 𝑎2

1

−𝜇2

1𝑓2

1

64𝜔4

1

− 𝑅2

1−𝛼2

2

64𝜔2

1

𝑎4

1𝑎2

3+𝑅1𝛼2

4𝜔1

𝑎2

1𝑎3= 0,


0 50 100 150 200−1

−0.5

0

0.5

1

Time−0.4 −0.2 0 0.2 0.4

−4

−2

0

2

4

Velo

city

0 50 100 150 200−0.4

−0.2

0

0.2

0.4

Time−0.2 −0.1 0 0.1 0.2

−1

−0.5

0

0.5

1

Velo

city

0 50 100 150 200−0.4

−0.2

0

0.2

0.4

Time−0.2 −0.1 0 0.1 0.2−1

−0.5

0

0.5

1Ve

loci

ty

Am

plitu

deu1

Am

plitu

deu2

Am

plitu

deu3

u1

u2

u3

Figure 2: Time response and phase-plane diagrams of the system for nonresonance case.

9𝛾2

9

64𝜔2

3

𝑎6

3+ [𝐾2+3𝛾9(𝜎3− 3𝜎1)

4𝜔3

] 𝑎4

3

+ [𝐾1+ (𝜎3− 3𝜎1)2−𝜇2

3(𝜎3− 3𝜎1)

4𝜔3

+Γ3(𝜎3− 3𝜎1)

𝜔3

] 𝑎2

3

−𝛾2

7

64𝜔2

3

𝑎6

1= 0.

(27)



9𝛽2

8

64𝜔2

2

𝑎6

2+ [𝑄3+3𝛽8(𝜎2− 𝜎1)

4𝜔2

] 𝑎4

2

+ [𝑄2+ (𝜎2− 𝜎1)2−𝜇2

2(𝜎2− 𝜎1)

4𝜔2

+Γ2(𝜎2− 𝜎1)

𝜔2

] 𝑎2

2

−𝜇2

2𝑓2

2

64𝜔4

2

− 𝑄2

1+𝑄1𝛽4

4𝜔2

𝑎2

2𝑎3−𝛽2

4

64𝜔2

2

𝑎4

2𝑎2

3= 0,

9𝛾2

9

64𝜔2

3

𝑎6

3+ [𝐾2+3𝛾9(𝜎3− 3𝜎1)

4𝜔3

] 𝑎4

3

+ [𝐾1+ (𝜎3− 3𝜎1)2−𝜇2

3(𝜎3− 3𝜎1)

4𝜔3

+Γ3(𝜎3− 3𝜎1)

𝜔3

] 𝑎2

3

−𝛾2

8

64𝜔2

3

𝑎6

2= 0.

(28)

Case 6. In this case, where 𝑎1̸= 0, 𝑎2̸= 0, and 𝑎

3̸= 0, this is the

practical case, the frequency response equations are given by

9𝛼2

7

64𝜔2

1

𝑎6

1+ [𝑅3+3𝛼7𝜎1

4𝜔1

] 𝑎4

1+ [𝑅2+ 𝜎2

1+𝜇2

1𝜎1

4𝜔1

−Γ1𝜎1

𝜔1

] 𝑎2

1

−𝜇2

1𝑓2

1

64𝜔4

1

− 𝑅2

1+𝑅1𝛼2

4𝜔1

𝑎2

1𝑎3+𝑅1𝛼6

2𝜔1

𝑎2𝑎2

3+𝑅1𝛼4

4𝜔1

𝑎2

2𝑎3

+3𝑅1𝛼8

4𝜔1

𝑎3

2+3𝑅1𝛼1

4𝜔1

𝑎2

1𝑎2+𝑅1𝛼10

4𝜔1

𝑎1𝑎2𝑎3−𝛼2

6

16𝜔2

1

𝑎2

2𝑎4

3

Mathematical Problems in Engineering 11A

mpl

itude

u3

0 50 100 150 200−2

−1

0

1

2

Time−2 −1 0 1 2

−10

0

10

Velo

city

0 50 100 150 200−2

−1

0

1

2

Time

−2 −1 0 1 2−10

−5

0

5

10

Velo

city

0 50 100 150 200−0.5

0

0.5

Time−0.4 −0.2 0 0.2 0.4

−10

−5

0

5

10Ve

loci

ty

Am

plitu

deu1

Am

plitu

deu2

u1

u2

u3

Figure 3: Time response and phase-plane diagrams of the system at simultaneous resonance case, Ω3≅ 𝜔1, 𝜔2≅ 𝜔1, and 𝜔

3≅ 3𝜔1.

−𝛼4𝛼6

16𝜔2

1

𝑎3

2𝑎3

3−𝛼2

2

64𝜔2

1

𝑎4

1𝑎2

3−9𝛼2

1

64𝜔2

1

𝑎4

1𝑎2

2−9𝛼1𝛼8

32𝜔2

1

𝑎2

1𝑎4

2

−9𝛼2

8

64𝜔2

1

𝑎6

2−3𝛼4𝛼8

32𝜔2

1

𝑎5

2𝑎3−3𝛼8𝛼10

32𝜔2

1

𝑎1𝑎4

2𝑎3−𝛼2𝛼10

32𝜔2

1

𝑎3

1𝑎2𝑎2

3

−𝛼4𝛼10

32𝜔2

1

𝑎1𝑎3

2𝑎2

3−𝛼6𝛼10

16𝜔2

1

𝑎1𝑎2

2𝑎3

3−𝛼2𝛼6

16𝜔2

1

𝑎2

1𝑎2𝑎3

3

−3𝛼1𝛼10

32𝜔2

1

𝑎3

1𝑎2

2𝑎3−3𝛼1𝛼2

32𝜔2

1

𝑎4

1𝑎2𝑎3− 𝑅4𝑎2

1𝑎2

2𝑎2

3

− 𝑅5𝑎4

2𝑎2

3− 𝑅6𝑎3

2𝑎2

1𝑎3= 0,

9𝛽2

8

64𝜔2

2

𝑎6

2+ [𝑄3+3𝛽8(𝜎2− 𝜎1)

4𝜔2

] 𝑎4

2

+ [𝑄2+ (𝜎2− 𝜎1)2−𝜇2

2(𝜎2− 𝜎1)

4𝜔2

+Γ2(𝜎2− 𝜎1)

𝜔2

] 𝑎2

2

−𝜇2

2𝑓2

2

64𝜔4

2

− 𝑄2

1+3𝑄1𝛽3

4𝜔2

𝑎2

2𝑎1+3𝑄1𝛽7

4𝜔2

𝑎3

1+𝑄1𝛽1

4𝜔2

𝑎2

1𝑎2

+𝑄1𝛽4

4𝜔2

𝑎2

2𝑎3+𝑄1𝛽5

2𝜔2

𝑎1𝑎2

3+𝑄1𝛽2

4𝜔2

𝑎2

1𝑎3+𝑄1𝛽10

4𝜔2

𝑎1𝑎2𝑎3

−3𝛽1𝛽7

32𝜔2

2

𝑎5

1𝑎2−9𝛽2

3

64𝜔2

2

𝑎4

2𝑎2

1−9𝛽2

7

64𝜔2

2

𝑎6

1−𝛽2

4

64𝜔2

2

𝑎4

2𝑎2

3

−3𝛽1𝛽3

32𝜔2

2

𝑎3

1𝑎3

2−𝛽2

5

16𝜔2

2

𝑎2

1𝑎4

3−𝛽2𝛽5

16𝜔2

2

𝑎3

1𝑎3

3−3𝛽3𝛽4

32𝜔2

2

𝑎1𝑎4

2𝑎3

−3𝛽2𝛽7

32𝜔2

2

𝑎5

1𝑎3−𝛽4𝛽5

16𝜔2

2

𝑎1𝑎2

2𝑎3

3−𝛽5𝛽10

16𝜔2

2

𝑎2

1𝑎2𝑎3

2−𝛽4𝛽10

32𝜔2

2

𝑎1𝑎3

2𝑎2

3

− 𝑄4𝑎2

1𝑎2

2𝑎2

3− 𝑄5𝑎4

1𝑎2

2− 𝑄6𝑎4

1𝑎2

3− 𝑄7𝑎3

1𝑎2

2𝑎3− 𝑄8𝑎2

1𝑎3

2𝑎3

− 𝑄9𝑎3

1𝑎2𝑎2

3− 𝑄10𝑎4

1𝑎2𝑎3= 0,

9𝛾2

9

64𝜔2

3

𝑎6

3+ [𝐾2+3𝛾9(𝜎3− 3𝜎1)

4𝜔3

] 𝑎4

3

+[𝐾1+ (𝜎3−3𝜎1)2−𝜇2

3(𝜎3− 3𝜎1)

4𝜔3

+Γ3(𝜎3− 3𝜎1)

𝜔3

] 𝑎2

3


0 50 100 150 200−2

−1

0

1

2

3

Time0 50 100 150 200

−3

−2

−1

0

1

2

3

Time

0 50 100 150 200

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time

Numerical solution𝑎1(𝑡) perturbation solution



Am

plitu

deu1

Am

plitu

deu2

Am

plitu

deu3

Figure 4: Comparison between numerical solution (using RKM) and analytical solution (using perturbation method) of the system atresonance case, Ω

3≅ 𝜔1, 𝜔2≅ 𝜔1, and 𝜔

3≅ 3𝜔1.

−𝛾2

8

64𝜔2

3

𝑎6

2−

𝛾2

7

64𝜔2

3

𝑎6

1−𝛾1𝛾7

32𝜔2

3

𝑎5

1𝑎2−𝛾3𝛾8

32𝜔2

3

𝑎1𝑎5

2

− 𝐾3𝑎4

1𝑎2

2− 𝐾4𝑎2

1𝑎4

2− 𝐾5𝑎3

1𝑎2

2= 0,

(29)

where

𝑅1= [

𝑓1

2𝜔1

−𝜎1𝑓1

4𝜔2

1

] ,

𝑅2= [

𝜇2

1

4+𝜇4

1

64𝜔2

1

+Γ2

1

4𝜔2

1

−Γ1𝜇2

1

8𝜔2

1

] ,

𝑅3= [

3𝛼7𝜇2

1

32𝜔2

1

−3𝛼7Γ1

8𝜔2

1

] ,

𝑅4= [

𝛼2

10

64𝜔2

1

+3𝛼1𝛼6

16𝜔2

1

+𝛼2𝛼4

32𝜔2

1

] ,

𝑅5= [

3𝛼6𝛼8

16𝜔2

1

+𝛼2

4

64𝜔2

1

] ,

𝑅6= [

3𝛼1𝛼4

32𝜔2

1

+3𝛼2𝛼8

32𝜔2

1

] ,

𝑄1= [

𝑓2

2𝜔2

−(𝜎1− 𝜎2) 𝑓2

4𝜔2

2

] ,


200 200.5 201 201.5 202 202.5 203−0.5

0

0.5

Time200 200.5 201 201.5 202 202.5 203

−0.2

−0.1

0

0.1

0.2

Time

200 200.5 201 201.5 202 202.5 203−0.2

−0.1

0

0.1

0.2

Time

RKMFDM

RKMFDM

RKMFDM

Am

plitu

deu1

Am

plitu

deu2

Am

plitu

deu3

Figure 5: Time response of composite laminated rectangular plate using RKM and FDM.

𝑄2= [

𝜇2

2

4+𝜇4

2

64𝜔2

2

+Γ2

2

4𝜔2

2

−Γ2𝜇2

2

8𝜔2

2

] ,

𝑄3= [

3𝛽8Γ2

8𝜔2

2

−3𝛽8𝜇2

2

32𝜔2

2

] ,

𝑄4= [

𝛽2

10

64𝜔2

2

+3𝛽3𝛽5

16𝜔2

2

+𝛽2𝛽4

32𝜔2

2

] ,

𝑄5= [

𝛽2

1

64𝜔2

2

+9𝛽3𝛽7

32𝜔2

2

] ,

𝑄6= [

𝛽2

2

64𝜔2

2

+3𝛽5𝛽7

16𝜔2

2

] ,

𝑄7= [

3𝛽2𝛽3

32𝜔2

2

+3𝛽4𝛽7

32𝜔2

2

+𝛽1𝛽10

32𝜔2

2

] ,

𝑄8= [

3𝛽3𝛽10

32𝜔2

2

+𝛽1𝛽4

32𝜔2

2

] ,

𝑄9= [

𝛽2𝛽10

32𝜔2

2

+𝛽1𝛽5

16𝜔2

2

] ,

𝑄10= [

3𝛽7𝛽10

32𝜔2

2

+𝛽1𝛽2

32𝜔2

2

] ,

𝐾1= [

𝜇2

3

4+𝜇4

3

64𝜔2

3

+Γ2

3

4𝜔2

3

−Γ3𝜇2

3

8𝜔2

3

] ,

𝐾2= [

3𝛾9Γ3

8𝜔2

3

−3𝛾9𝜇2

3

32𝜔2

3

] ,

𝐾3= [

𝛾2

1

64𝜔2

3

+𝛾3𝛾7

32𝜔2

3

] ,

𝐾4= [

𝛾2

3

64𝜔2

3

+𝛾1𝛾8

32𝜔2

3

] ,

𝐾5= [

𝛾1𝛾3

32𝜔2

3

+𝛾7𝛾8

32𝜔2

3

] .

(30)


−1.5 −1 −0.5 0 0.5 1 1.5

2

4

6

8

10

Analytical solutionNumerical solution

𝜎1a 1

Figure 6: Comparison between analytical prediction using multiple time scale and numerical integration of the first mode.

−1 −0.5 0 0.5 1

5

10

15

20

0.2

0.1

0.3

a 1

𝜎1

Figure 7: Effects of the linear damping 𝜇1.

−1.5 −1 −0.5 0 0.5 1 1.5

2

4

6

8

10

12

141.7

2.3

4.3

a 1

𝜎1

Figure 8: Effects of the natural frequency 𝜔1.

−1.5 −1 −0.5 0 0.5 1 1.5

5

10

15

20

4

8

3

a 1

𝜎1

Figure 9: Effects of the external excitation 𝑓1.


−1.5 −1 −0.5 0 0.5 1 1.5

2

4

6

8

10

12

0.06

−0.06

0.01

a 1𝜎1

Figure 10: Effects of the nonlinear parameter 𝛼7.

−1.5 −1 −0.5 0 0.5 1 1.5

2

4

6

8

10

0.5510

a 1

𝜎1

Figure 11: Effects of the parametric excitation 𝑓14.

−1.5 −1 −0.5 0 0.5 1 1.5

123456

Analytical solutionNumerical solution

a 2

𝜎2

Figure 12: Comparison between analytical prediction using multiple time scale and numerical integration of the second mode.

−1.5 −1 −0.5

−0.08

0

0.2

0.5 1 1.50

2

0.01

4

6

a 2

𝜎2

Figure 13: Effects of the nonlinear parameter 𝛽8.


−1 −0.5 0 0.5

0.1

0.2

0.3

10

4

8

a 2

𝜎2

Figure 14: Effects of the linear damping 𝜇2.

−1.5 −1 −0.5 0

4

2.2

0.5 1 1.50

2

4

6

8

a 2

𝜎2

Figure 15: Effects of the natural frequency 𝜔2.

−1.5 −1 −0.5 0 0.5 1 1.50

2

4

6

8

2

4

6

a 2

𝜎2

Figure 16: Effects of the external excitation 𝑓2.

In the frequency response curves, the stable (unstable)steady-state solutions have been represented by solid (dotted)lines.

4. Numerical Results and Discussion

To study the behavior of the system, the Runge-Kutta fourth-order method (RKM) was applied to (4a), (4b), and (4c)

governing the oscillating system. A good criterion of bothstability and dynamic chaos is the phase plane trajectories.Figure 2 illustrates the response and the phase-plane for thenonresonant system at some practical values of the equationparameters: 𝜇

1= 0.05, 𝜇

2= 0.05, 𝜇

3= 0.05, 𝛼

1= 𝛽1= 𝛾1=

1.5, 𝛼2= 𝛽2= 𝛾2= 1.9, 𝛼

3= 𝛽3= 𝛾3= 0.4, 𝛼

4= 𝛽4= 𝛾4=

0.6, 𝛼5= 𝛽5= 𝛾5= 0.6, 𝛼

6= 𝛽6= 𝛾6= 0.2, 𝛼

7= 𝛽7= 𝛾7=

0.01, 𝛼8= 𝛽8= 𝛾8= 0.01, 𝛼

9= 𝛽9= 𝛾9= 0.4, 𝛼

10= 𝛽10=

𝛾10= 1.8, Ω

1= 6.4, Ω

2= 6.1, Ω

3= 5.7, Ω

4= 2, 𝜔

1= 4.7,


−2 −1 0 1 20

2

4

6

Am

plitu

des

𝜎1

a1

a1

a2a2

a3a3

Figure 17: Effects of the detuning parameter 𝜎1.

−1 0 1

4

6

1

−1 0 10.5

1.5

2.5

1

−1 0 1

0.2

0.6

1

1.4

1

f1 = 10

f1 = 10

f1 = 10

a 1

a 2

a 3

𝜎1

𝜎1𝜎1

Figure 18: Effects of the external excitation force 𝑓1.

𝜔2= 2.7, 𝜔

3= 13.1, 𝑓

1= 𝑓2= 4, 𝑓

3= 20.5, 𝑓

11= 𝑓21= 𝑓31=

0.1,𝑓12= 𝑓22= 𝑓32= 0.2, and𝑓

14= 𝑓24= 𝑓34= 0.5. Figure 3

shows the steady-state amplitudes and phase plane of thesystem at simultaneous resonance caseΩ

3≅ 𝜔1,𝜔2≅ 𝜔1, and

𝜔3≅ 3𝜔1. It is clear from Figure 3 that the steady-state ampli-

tude of the first, second, and thirdmodes is increased to about

370%, 890%, and 260%, respectively, of its value shown inFigure 2. Also, it can be seen that the time response of thesystem is tuned with multilimit cycle.

It is quite clear that such case is undesirable in thedesign of such system because it represents one of the worstbehaviors of the system. Such case should be avoided as


−2 −1 0 1 22

3

4

5

6

7

2.5

1.5

−2 0 10.5

1.5

2.5

3.5

4.5

1.5

2.5

0 1

0.2

0.6

1.2

2.5

1.5

−1

−1

a 2a 1

a 3

𝜎1

𝜎1𝜎1

𝛼1 = 0.5

𝛽1 = 0.5

𝛾1 = 0.5

Figure 19: Effects of the nonlinear parameters (𝛼1, 𝛽1, and 𝛾

1coupling terms).

working condition for the system. It is advised for such systemnot to have 𝜔

2= 𝜔1or 𝜔3= 3𝜔1. Figure 4 shows the com-

parison between numerical integration for the system equa-tion (4a), (4b), and (4c) solid lines and the amplitude-phase modulating equation (20) dashed lines. We foundthat all predictions from analytical solutions dashed linesare in good agreement with the numerical simulation solidlines.

4.1. FDM with Approximation O(𝑐2). The infinite equations(2a)–(2e) will be solved via a finite differencemethod (FDM).We briefly describe the procedure here. More details areavailable in [23, 26].The infinite dimensional equations (2a)–(2e) can be reduced to the finite dimensional one via the finitedifference method with second-order approximation 𝑂(𝑐2).Namely, at each mesh node the following system of ordinary

differential equations is obtained:

𝐿1,𝑐(𝑁𝑥𝑥, 𝑁𝑥𝑦) = 𝐼0(�̈�0)𝑖,𝑗+ 𝐽1( ̈𝜙𝑥)𝑖,𝑗

−2

3ℎ2𝑐𝐼3((�̈�0)𝑖+1,𝑗

− (�̈�0)𝑖−1,𝑗

) ,

𝐿2,𝑐(𝑁𝑥𝑦, 𝑁𝑦𝑦) = 𝐼0(V̈0)𝑖,𝑗+ 𝐽1( ̈𝜙𝑦)𝑖,𝑗

−2

3ℎ2𝑐𝐼3((�̈�0)𝑖,𝑗+1

− (�̈�0)𝑖,𝑗−1

) ,

𝐿3,𝑐(𝑄𝑥, 𝑄𝑦, 𝑁𝑥𝑥, 𝑤0, 𝑁𝑥𝑦)

= 𝐼0(�̈�0)𝑖,𝑗−

16

9ℎ4𝑐2𝐼6((�̈�0)𝑖+1,𝑗

− 2(�̈�0)𝑖,𝑗+ (�̈�0)𝑖+1,𝑗

+(�̈�0)𝑖,𝑗+1

−2(�̈�0)𝑖,𝑗+(�̈�0)𝑖,𝑗+1

)


−2 −1 0 1

3

4

5

6

10

5

−1 0 10.5

1

1.5

2

2.5

5

10

−2 −1 0 1

0.2

0.6

1

5

10

𝜎1

𝜎1𝜎1

a 2a1

a 3

𝛼2 = 0.1𝛽2 = 0.1

𝛾2 = 0.1


2coupling terms).

+4

3ℎ2𝑐2𝐼3((�̈�0)𝑖+1,𝑗

− (�̈�0)𝑖−1,𝑗

+ (V̈0)𝑖,𝑗+1

− (V̈0)𝑖,𝑗−1

)

+𝐽4

2𝑐(( ̈𝜙𝑥)𝑖+1,𝑗

− ( ̈𝜙𝑥)𝑖−1,𝑗

+ ( ̈𝜙𝑦)𝑖,𝑗+1

− ( ̈𝜙𝑦)𝑖,𝑗−1

) ,

𝐿4,𝑐(𝑀𝑥𝑥,𝑀𝑥𝑦) = 𝐽1(�̈�0)𝑖,𝑗+ 𝑘2( ̈𝜙𝑥)𝑖,𝑗

−2𝐽4

3ℎ2𝑐((�̈�0)𝑖+1,𝑗

− (�̈�0)𝑖−1,𝑗

) ,

𝐿5,𝑐(𝑀𝑥𝑦,𝑀𝑦𝑦) = 𝐽1(V̈0)𝑖,𝑗+ 𝑘2( ̈𝜙𝑦)𝑖,𝑗

−2𝐽4

3ℎ2𝑐((�̈�0)𝑖,𝑗+1

− (�̈�0)𝑖,𝑗−1

) ,

(𝑖 = 1, 2, . . . , 𝑛) , (𝑗 = 1, 2, . . . , 𝑛) ,

(31)

where 𝑛 denotes the partition number regarding a spatialcoordinate, 𝑐 is the computational step regarding spatialcoordinate, and 𝐿

1,𝑐(⋅), 𝐿2,𝑐(⋅), 𝐿3,𝑐(⋅), 𝐿4,𝑐(⋅), and 𝐿

5,𝑐(⋅) are

the difference operators as follows:

𝐿1,𝑐(𝑁𝑥𝑥, 𝑁𝑥𝑦) =

1

2𝑐((𝑁𝑥𝑥)𝑖+1,𝑗

− (𝑁𝑥𝑥)𝑖−1,𝑗

+ (𝑁𝑥𝑦)𝑖,𝑗+1

− (𝑁𝑥𝑦)𝑖,𝑗−1

) ,

𝐿2,𝑐(𝑁𝑥𝑦, 𝑁𝑦𝑦) =

1

2𝑐((𝑁𝑥𝑦)𝑖+1,𝑗

− (𝑁𝑥𝑦)𝑖−1,𝑗

+ (𝑁𝑦𝑦)𝑖,𝑗+1

− (𝑁𝑦𝑦)𝑖,𝑗−1

) ,


𝐿3,𝑐(𝑄𝑥, 𝑄𝑦, 𝑁𝑥𝑥, 𝑤0, 𝑁𝑥𝑦) =

1

2𝑐((𝑄𝑥)𝑖+1,𝑗

− (𝑄𝑥)𝑖−1,𝑗

+ (𝑄𝑦)𝑖,𝑗+1

− (𝑄𝑦)𝑖,𝑗−1

)

+ 𝑁𝑥𝑥(

(𝑤0)𝑖−1,𝑗

− 2(𝑤0)𝑖,𝑗+ (𝑤0)𝑖+1,𝑗

𝑐2) + (

(𝑤0)𝑖+1,𝑗

− (𝑤0)𝑖−1,𝑗

2𝑐)(

(𝑁𝑥𝑥)𝑖+1,𝑗

− (𝑁𝑥𝑥)𝑖−1,𝑗

2𝑐)

+ 𝑁𝑥𝑦(

((𝑤0)𝑗+1,𝑖+1

− 2(𝑤0)𝑗+1,𝑖−1

) ((𝑤0)𝑗−1,𝑖+1

− 2(𝑤0)𝑗−1,𝑖−1

)

4𝑐2)

+ (

(𝑤0)𝑖,𝑗+1

− (𝑤0)𝑖,𝑗−1

2𝑐)(

(𝑁𝑥𝑦)𝑖+1,𝑗

− (𝑁𝑥𝑦)𝑖−1,𝑗

2𝑐)

+ 𝑁𝑥𝑦(

((𝑤0)𝑖+1,𝑗+1

− (𝑤0)𝑖+1,𝑗−1

) − ((𝑤0)𝑖−1,𝑗+1

− (𝑤0)𝑖−1,𝑗−1

)

8𝑐3)

+ (

(𝑤0)𝑖+1,𝑗

− (𝑤0)𝑖−1,𝑗

2𝑐)(

(𝑁𝑦𝑦)𝑖,𝑗+1


2𝑐) + 𝑁

𝑦𝑦(

(𝑤0)𝑖,𝑗−1

− 2(𝑤0)𝑖,𝑗+ (𝑤0)𝑖,𝑗+1

𝑐2)

+ (

(𝑤0)𝑖,𝑗+1

− (𝑤0)𝑖,𝑗−1

2𝑐)(

(𝑁𝑦𝑦)𝑖,𝑗+1


2𝑐) +

4

3ℎ2(

(𝑃𝑥𝑥)𝑖−1,𝑗

− 2(𝑃𝑥𝑥)𝑖,𝑗+ (𝑃𝑥𝑥)𝑖+1,𝑗

𝑐2)

+4

3ℎ2(2(

((𝑃𝑥𝑦)𝑖+1,𝑗+1

− (𝑃𝑥𝑦)𝑖−1,𝑗+1

) − ((𝑃𝑥𝑦)𝑖+1,𝑗−1

− (𝑃𝑥𝑦)𝑖−1,𝑗−1

)

8𝑐3)+

(𝑃𝑦𝑦)𝑖,𝑗−1

− 2(𝑃𝑦𝑦)𝑖,𝑗+ (𝑃𝑦𝑦)𝑖,𝑗+1

𝑐2)+ 𝑞,

𝐿4,𝑐(𝑀𝑥𝑥,𝑀𝑥𝑦) =

1

2𝑐((𝑀𝑥𝑥)𝑖+1,𝑗

− (𝑀𝑥𝑥)𝑖−1,𝑗

+ (𝑀𝑥𝑦)𝑖,𝑗+1

− (𝑀𝑥𝑦)𝑖,𝑗−1

) − 𝑄𝑥,

𝐿5,𝑐(𝑀𝑥𝑦,𝑀𝑦𝑦) =

1

2𝑐((𝑀𝑥𝑦)𝑖+1,𝑗

− (𝑀𝑥𝑦)𝑖−1,𝑗

+ (𝑀𝑦𝑦)𝑖,𝑗+1

− (𝑀𝑦𝑦)𝑖,𝑗−1

) − 𝑄𝑦.

(32)

The obtained system of (31) with the supplemented boundaryconditions equation and the initial conditions equation issolved by the fourth-order Runge-Kutta method. Figure 5shows a comparison between the time responses of the systemequations (4a), (4b), and (4c) using the Runge-Kutta offourth-order method and the time response of the problems(31), using the finite difference method at the same values ofthe parameters shown in Figure 2.

4.2. Frequency Response Curves. When the amplitudeachieves a constant nontrivial value, a steady-state vibrationexists. Using the frequency response equations we can assessthe influence of the damping coefficients, the nonlinear para-meters, and the excitation amplitude on the steady-stateamplitudes. The frequency response equations (24)–(29) arenonlinear equations in 𝑎

1, 𝑎2, and 𝑎

3which are solved

numerically.Thenumerical results are shown in Figures 6–25,and in all figures the region of stability of the nonlinearsolutions is determined by applying the Routh-Hurwitz

criterion.The solid lines stand for the stable solution, and thedotted lines stand for the unstable solution. From the geo-metry of the figures, we observe that each curve is continuousand has stable and unstable solutions.

4.2.1. Response Curve of Case 1. To check the accuracy ofthe analytical solution derived by the multiple time scalein predicting the amplitude of the first mode, we comparethe amplitude of the first mode obtained from frequencyresponse equation of Case 1 with values obtained fromnumerical integration of (4a). Figure 6 shows a comparisonof these outputs for the first mode.The effects of the detuningparameter 𝜎

1on the steady-state amplitude of the first mode

for the stability first case, where 𝑎1̸= 0, 𝑎2= 0, and 𝑎

3= 0, for

the parameters 𝜇1= 0.2, 𝛼

7= 0.01, 𝜔

1= 2.3, 𝑓

1= 4, 𝑓

11=

0.1, 𝑓12= 0.2, 𝑓

14= 0.5, Ω

1= 1, Ω

2= 1.2, and Ω

4= 1.4, as

shown in Figure 6.Figures 7–11, show the effects of the damping coeffi-

cient 𝜇1, the first mode natural frequency 𝜔

1, the external


−1 0 12

4

63

8

−2 0 2

1.5

2.5

3.5

8

3

−2 0 2

0.2

0.6

1

8

3

𝜎1

𝜎1𝜎1

a 2

a 1

a 3

𝛼8 = 1.5

𝛽8 = 1.5

𝛾8 = 1.5


8coupling terms).

excitation amplitude𝑓1, the nonlinear spring stiffness 𝛼

7, and

the parametric excitation 𝑓14. Figures 7 and 8 show that the

steady-state amplitude 𝑎1is inversely proportional to 𝜇

1and

𝜔1, and also for decreasing𝜇

1or𝜔1the curve is bending to the

left. It is clear from Figure 9 that the steady-state amplitude𝑎1is increasing for increasing value of external excitation

force 𝑓1, and the zone of instability is increased. Figure 10

shows that as the nonlinear spring stiffness 𝛼7is increased;

the continuous curve ismoved downwards. Also, the negativeand positive values of 𝛼

7produce either hard or soft spring,

respectively, as the curve is either bent to the right or to theleft, leading to the appearance of the jump phenomenon.Theregion of stability is increased for increasing value of𝛼

7. From

Figure 11, we observe that for increasing value of parametricexcitation amplitude 𝑓

14, the steady-state amplitude of the

first mode is increased, and the curve is shifted to the left.

4.2.2. Response Curve of Case 2. Figures 12–16, show thefrequency response curves for the stability of the second case,where 𝑎

2̸= 0, 𝑎

1= 0, and 𝑎

3= 0. Figure 12 shows a com-

parison of these outputs for the second mode. The effects ofthe detuning parameter 𝜎

2on the steady-state amplitude of

the second mode for the stability second case, where 𝑎2̸= 0,

𝑎1= 0, 𝑎

3= 0, for the parameters: 𝜇

2= 0.2, 𝛽

8= 0.01,

𝜔2= 4, 𝑓

2= 4, 𝑓

21= 0.1, 𝑓

22= 0.2, 𝑓

24= 0.5, Ω

1= 1, Ω

2=

1.2, and Ω4= 1.4, as shown in Figure 12. It can be seen from

the figure thatmaximum steady-state amplitude occurs when𝜔2≅ 𝜔1. Figure 13 shows that as the nonlinear spring stiffness

𝛽8is increased, the continuous curve is moved downwards.

Also, the positive and negative values of 𝛽8produce either

soft or hard spring, respectively, as the curve is either bent tothe left or to the right, leading to the appearance of the jumpphenomenon. Figures 14, 15, and 16 show that the steady-state amplitude 𝑎

2is inversely proportional to 𝜇

2, 𝜔2and

directly proportional to the external excitation 𝑓2. Also, for

decreasing 𝜇2, 𝜔2the curve is bending to the left.

4.2.3. Response Curve of Case 6. Figures 17, 18, 19, 20, 21,22, 23, 24, and 25 show that the frequency response curvesfor practical case stability, where 𝑎

1̸= 0, 𝑎2̸= 0, and 𝑎

3̸= 0.


−2 0 22

4

6

8

10

1

−2 0 2

0.5

1.5

2.5

3.5

4.5

0.01

−2 0 2

0.2

0.6

1

1.4

0.01

a 3

a 2

a 1

𝜎1𝜎1

𝜎1

𝛼7 = 0.01𝛽7 = 1

𝛾7 = 1


7coupling terms).

Figure 17 shows that the effects of the detuning parameter𝜎1on the amplitudes of the three modes. From this figure,

we observe that these modes intersect, and for positive valueof the detuning parameter 𝜎

1the amplitudes are stable. For

negative value of the detuning parameter 𝜎1down to and

including −0.5 the system becomes unstable.Figure 18 shows that the steady-state amplitudes of the

three modes 𝑎1, 𝑎2, and 𝑎

3are directly proportional to

the external excitation force 𝑓1. Also, form this figure we

show that the stability region is decreased for increasing 𝑓1.

Figures 19–21 show that the steady-state amplitudes andstability of the threemodes 𝑎

1, 𝑎2, and 𝑎

3are inversely propor-

tional to the nonlinear parameters (𝛼1, 𝛼2, 𝛼8), (𝛽1, 𝛽2, 𝛽8),

and (𝛾1, 𝛾2, 𝛾8), respectively.

Figures 22–24, show the effects of the nonlinear param-eters (𝛼

7, 𝛽7, 𝛾7), (𝛼3, 𝛽3, 𝛾3), and (𝛼

5,𝛽5, 𝛾5) on the steady-

state amplitudes of the three modes. It is clear that thestability regions are increased for increasing these nonlinear

parameters. For increasing value of linear viscous dampingcoefficients, we note that the steady-state amplitudes areincreasing or decreasing for the first, second, and thirdmodes, respectively, as shown in Figure 25. The region ofstability system is increased for decreasing value of dampingcoefficients.

4.3. Comparison Study. In the previous work [11], the chaoticdynamics of a six-dimensional nonlinear system whichrepresents the averaged equation of a composite laminatedpiezoelectric rectangular plate subjected to the transverse, in-plane excitations and the excitation loaded by piezoelectriclayers are analyzed. The case of 1 : 2 : 4 internal resonances isconsidered.

In our study, the nonlinear analysis and stability ofa composite laminated piezoelectric rectangular thin plateunder simultaneous external and parametric excitation forces


−2 −1 0 1 20

2

4

6

8

5

10

1.5

−2 −1 0 1 20.5

1.5

2.5

3.5

5

−1

−1 −0.5 0 0.50

0.5

1

1.5

2

5

1.5-1

𝜎1

𝜎1𝜎1

a 3

a 2

a 1

𝛼3 = −1

𝛽3 = 10

𝛾3 = 10


3coupling terms).

are investigated.The second-order approximation is obtainedto consider the influence of the cubic terms on nonlin-ear dynamic characteristics of the composite laminatedpiezoelectric rectangular plate using the multiple scalemethod. All possible resonance cases are extracted at thisapproximation order. The case of 1 : 1 : 3 internal resonanceand primary resonance is considered. The stability of thesystem and the effects of different parameters on systembehavior have been studied using phase plane and frequencyresponse curves. The analytical results given by the methodof multiple time scale are verified by comparison with resultsfrom numerical integration of the modal equations. Reliabil-ity of the obtained results is verified by comparison betweenthe finite difference method (FDM) and Runge-Kuttamethod (RKM). Variation of the some parameters leads tomultivalued amplitudes and hence to jump phenomena. It isquite clear that some of the simultaneous resonance cases areundesirable in the design of such system. Such cases shouldbe avoided as working conditions for the system.

5. Conclusions

Multiple time scale perturbation method is applied to deter-mine second-order approximate solutions for rectangularsymmetric cross-ply laminated composite thin plate sub-jected to external and parametric excitations. Second-orderapproximate solutions are obtained to study the influenceof the cubic terms on nonlinear dynamic characteristics ofthe composite laminated piezoelectric rectangular plate. Allpossible resonance cases are extracted at this approximationorder. The study is focused on the case of 1 : 1 : 3 primary res-onance and internal resonance, whereΩ

3≅ 𝜔1, 𝜔2≅ 𝜔1, and

𝜔3≅ 3𝜔1.The analytical results given by themethod ofmulti-

ple time scale are verified by comparison with results fromnumerical integration of the modal equations. Reliability ofthe obtained results is verified by comparison between thefinite difference method (FDM) and Runge-Kutta method(RKM). The stability of a composite laminated piezoelectricrectangular thin plate is investigated. The phase-plane


−2 −1 0 12

3

4

5

6

40

−2 −1 0 10.5

1.5

2.5

3.5

20

−1 0 10.2

0.6

1

40

𝜎1

𝜎1𝜎1

a 3

a 2a 1

𝛼5 = 20

𝛽5 = 40

𝛾5 = 20


5coupling terms).

method and frequency response curves are applied to studythe stability of the system. From the previous study thefollowing may be concluded.

(1) The simultaneous resonance case Ω3≅ 𝜔1, 𝜔2≅ 𝜔1,

and 𝜔3≅ 3𝜔

1is one of the worst case, and they

should be avoided in design of such system.Of course,the excitation frequencyΩ

3is out of control. But this

case should be avoided through having 𝜔2̸= 𝜔1or

𝜔3̸= 3𝜔1.

(2) A comparison between the solutions obtained numer-ically with that prediction from the multiple scalesshows an excellent agreement.

(3) Reliability of the obtained results is verified by com-parison between the finite difference method (FDM)and Runge-Kutta method (RKM).

(4) Variation of the parameters 𝜇1, 𝜇2, 𝛼7, 𝛽8, 𝜔1, 𝜔2, 𝑓1,

𝑓2leads tomultivalued amplitudes and hence to jump

phenomena.

(5) For the first and second modes, the steady-stateamplitudes 𝑎

1and 𝑎2are directly proportional to the

excitation amplitude 𝑓1and 𝑓

2and inversely propor-

tional to the linear damping 𝜇1and 𝜇

2, respectively.

(6) The negative and positive values of nonlinear stiffness𝛼7, 𝛽8produce either hard or soft spring, respectively,

as the curve is either bent to the right or to the left.(7) The region of instability increase, which is undesir-

able, for increasing excitation amplitudes 𝑓1, 𝑓2and

for negative values of nonlinear stiffness 𝛼7, 𝛽8.

(8) The multivalued solutions are disappeared forincreasing linear damping coefficients 𝜇

1, 𝜇2.

(9) The region of stability increases, which is desirablefor decreasing excitation amplitude 𝑓

1, nonlinear

parameters (𝛼1, 𝛽1, 𝛾1), (𝛼2, 𝛽2, 𝛾2), and for increasing

nonlinear stiffness (𝛼7, 𝛽7, 𝛾7).

(10) The steady-state amplitudes of the three modes 𝑎1,

𝑎2, and 𝑎

3are directly proportional to the exci-

tation amplitude 𝑓1and inversely proportional to


−1 0 13

4

5

6

0.5

0.2

−1 0 1

0.5

1.5

2.5

0.5

0.2

−2 −1 0 1

0.4

0.8

1.2

1.6

0.5

0.2

𝜎1

𝜎1𝜎1

a 3

a 2

a 1

𝜇1 = 1.2

𝜇2 = 1.2

𝜇3 = 1.2

Figure 25: Effects of the linear damping coefficients.

the nonlinear parameters (𝛼1, 𝛼2, 𝛼8), (𝛽1, 𝛽2, 𝛽8), and

(𝛾1, 𝛾2, 𝛾8), respectively.

For further work, we intend to extend our work to explorethe modeling formulation by investigating the role staticloading (in addition to the dynamic component).

Acknowledgment

The authors would like to thank the anonymous reviewers fortheir valuable comments and suggestions for improving thequality of this paper.

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Research Article Vibration, Stability, and Resonance of Angle ...In this paper, the perturbation method and stability of the composite laminated piezoelectric rectangular plate under