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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 Β© 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, whereΞ©
3β π1,π2β π1, andπ
3β 3π1,
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 π
1, π2, πΌ7, π½8, π1, π2, π1, π2
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
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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 [14β16] 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. [23β25] 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 π, π and thickness β,as shown in Figure 1. The composite laminated piezoelectricrectangular plate is considered as regular symmetric cross-ply laminates with π layers. A Cartesian coordinate system is
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Mathematical Problems in Engineering 3
y
b
q cosΞ©3t
q0 + qx cosΞ©1t
q1 + qy cosΞ©2t
x
a
Figure 1: The model of the composite laminated piezoelectricrectangular plate.
located in the middle surface of the plate. Assume that π’, V,and π€ represent the displacements of an arbitrary point ofthe composite laminated piezoelectric rectangular plate in theπ₯, π¦, and π§ directions, respectively. The in-plane excitationsare loaded along the π¦direction at π₯ = 0 in the formπ0+ ππ₯cosΞ©1π‘, and the excitations are loaded along the π₯
direction at π¦ = 0 in the form π1+ ππ¦cosΞ©2π‘. The trans-
verse excitation subjected to the composite laminated piezo-electric rectangular plate is represented by π cosΞ©
3π‘. The
dynamic electrical loading is expressed as πΈπ§cosΞ©4π‘. Based
on Reddyβs third-order shear deformation plate theory [27],the displacement field at an arbitrary point in the compositelaminated plate is expressed as [13]
π’ (π₯, π¦, π‘) = π’0(π₯, π¦, π‘) + π§π
π₯(π₯, π¦, π‘) β π§
3 4
3β2(ππ₯+ππ€0
ππ₯) ,
(1a)
V (π₯, π¦, π‘) = V0(π₯, π¦, π‘) + π§π
π¦(π₯, π¦, π‘) β π§
3 4
3β2(ππ¦+ππ€0
ππ¦) ,
(1b)
π€ (π₯, π¦, π‘) = π€0(π₯, π¦, π‘) , (1c)
where π’0, V0, and π€
0are the original displacement on the
midplane of the plate in theπ₯,π¦, and π§directions, respectively.Let ππ₯and π
π¦represent the midplane rotations of transverse
normal about the π₯ and π¦ axes, respectively. From thevan Karman-type plate theory and Hamiltonβs principle, thenonlinear governing equations of motion of the compositelaminated piezoelectric rectangular plate are given as follows[12]:
πππ₯π₯
ππ₯+
πππ₯π¦
ππ¦= πΌ0οΏ½ΜοΏ½0+ π½1οΏ½ΜοΏ½π₯β4
3β2πΌ3
ποΏ½ΜοΏ½0
ππ₯, (2a)
πππ₯π¦
ππ₯+
πππ¦π¦
ππ¦= πΌ0VΜ0+ π½1οΏ½ΜοΏ½π¦β4
3β2πΌ3
ποΏ½ΜοΏ½0
ππ¦, (2b)
πππ₯
ππ₯+
πππ¦
ππ¦+π
ππ₯(ππ₯π₯
ππ€0
ππ₯+ ππ₯π¦
ππ€0
ππ¦)
+π
ππ¦(ππ₯π¦
ππ€0
ππ₯+ ππ¦π¦
ππ€0
ππ¦)
+4
3β2(π2ππ₯π₯
ππ₯2+ 2
π2ππ₯π¦
ππ₯ππ¦+
π2ππ¦π¦
ππ¦2) + π
= πΌ0οΏ½ΜοΏ½0β16
9β4πΌ6(π2οΏ½ΜοΏ½0
ππ₯2+π2οΏ½ΜοΏ½0
ππ¦2)
+4
3β2[πΌ3(ποΏ½ΜοΏ½0
ππ₯+πVΜ0
ππ¦) + π½4(ποΏ½ΜοΏ½π₯
ππ₯+
ποΏ½ΜοΏ½π¦
ππ¦)] ,
(2c)
πππ₯π₯
ππ₯+
πππ₯π¦
ππ¦β ππ₯= π½1οΏ½ΜοΏ½0+ π2οΏ½ΜοΏ½π₯β4
3β2π½4
ποΏ½ΜοΏ½0
ππ₯, (2d)
πππ₯π¦
ππ₯+
πππ¦π¦
ππ¦β ππ¦= π½1VΜ0+ π2οΏ½ΜοΏ½π¦β4
3β2π½4
ποΏ½ΜοΏ½0
ππ¦. (2e)
Theboundary conditions of the simply supported rectangularcomposite laminated plate are expressed as follows:
at π₯ = 0, π₯ = π : ππ’ππ₯= 0,
π€ = π’ = ππ₯= ππ¦= ππ₯π₯= ππ₯π¦= ππ₯= 0,
(3a)
at π¦ = 0, π¦ = π : πVππ¦= 0,
π€ = V = ππ₯π¦= ππ¦π¦= ππ¦= 0,
(3b)
β«
π
0
ππ₯π₯
π₯=0, πππ¦ = β«
π
0
(π0+ ππ₯cosΞ©1π‘) ππ¦, (3c)
β«
π
0
ππ¦π¦
π¦=0, πππ₯ = β«
π
0
(π1+ ππ¦cosΞ©2π‘) ππ₯. (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]:
οΏ½ΜοΏ½1+ π1οΏ½ΜοΏ½1+ π2
1π’1
+ (π11cosΞ©1π‘ + π12cosΞ©2π‘ + π14cosΞ©4π‘) π’1
+ πΌ1π’2
1π’2+ πΌ2π’2
1π’3+ πΌ3π’2
2π’1+ πΌ4π’2
2π’3
+ πΌ5π’2
3π’1+ πΌ6π’2
3π’2+ πΌ7π’3
1+ πΌ8π’3
2
+ πΌ9π’3
3+ πΌ10π’1π’2π’3
= π1cosΞ©3π‘,
(4a)
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4 Mathematical Problems in Engineering
οΏ½ΜοΏ½2+ π2οΏ½ΜοΏ½2+ π2
2π’2
+ (π21cosΞ©1π‘ + π22cosΞ©2π‘ + π24cosΞ©4π‘) π’2
+ π½1π’2
1π’2+ π½2π’2
1π’3+ π½3π’2
2π’1+ π½4π’2
2π’3
+ π½5π’2
3π’1+ π½6π’2
3π’2+ π½7π’3
1+ π½8π’3
2
+ π½9π’3
3+ π½10π’1π’2π’3
= π2cosΞ©3π‘,
(4b)
οΏ½ΜοΏ½3+ π3οΏ½ΜοΏ½3+ π2
3π’3
+ (π31cosΞ©1π‘ + π32cosΞ©2π‘ + π34cosΞ©4π‘) π’3
+ πΎ1π’2
1π’2+ πΎ2π’2
1π’3+ πΎ3π’2
2π’1+ πΎ4π’2
2π’3
+ πΎ5π’2
3π’1+ πΎ6π’2
3π’2+ πΎ7π’3
1+ πΎ8π’3
2
+ πΎ9π’3
3+ πΎ10π’1π’2π’3
= π3cosΞ©3π‘,
(4c)
where π’1, π’2, and π’
3are the vibration amplitudes of the com-
posite laminated piezoelectric rectangular plate for the first-order, second-order, and the third-order modes, respectively,π1, π2, and π
3are the linear viscous damping coefficients,
π1, π2, and π
3are the natural frequencies of the rectangular
plate, andΞ©1,Ξ©2,Ξ©3, andΞ©
4are the excitations frequencies.
ππ1, ππ2, ππ3, and π
π(π = 1, 2, 3) are the amplitudes of
parametric and external excitation forces corresponding tothe three nonlinear modes, and πΌ
π, π½π, and πΎ
π(π = 1, 2, . . . , 10)
are the nonlinear coefficients.The linear viscous damping andexciting forces are assumed to be
ππ= πππ, ππ1= πππ1, π
π2= πππ2,
ππ3= πππ3, ππ= π2ππ, π = 1, 2, 3,
(5)
where π is a small perturbation parameter and 0 < π βͺ 1.The external excitation forces π
πare of the order 2, and
the linear viscous damping ππ, parametric exciting forcesπ
π1,
ππ2, and π
π3are 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 [29β31] is applied to obtain a second-order approximation for the system. For the second-orderapproximation, we introduce three time scales defined by
π0= π‘, π
1= ππ‘, π
2= π2π‘. (6)
In terms of these scales, the time derivatives become
π
ππ‘= π·0+ ππ·1+ π2π·2, (7a)
π2
ππ‘2= π·2
0+ 2ππ·
0π·1+ π2(π·2
1+ 2π·0π·2) , (7b)
where π·π= π/ππ
π, π = 0, 1, 2. We seek a uniform approxi-
mation to the solution of (4a), (4b), and (4c) in the form:
π’π (π‘, π) = ππ’π1 (π0, π1, π2) + π
2π’π2(π0, π1, π2)
+ π3π’π3(π0, π1, π2) + π (π
4) , π = 1, 2, 3.
(8)
Terms of π(π4) and higher orders are neglected. Substituting(5) and (7a)β(8) into (4a), (4b), and (4c) and equating thecoefficients of like powers of π, we obtain the following.
Order π
(π·2
0+ π2
1) π’11= 0, (9a)
(π·2
0+ π2
2) π’21= 0, (9b)
(π·2
0+ π2
3) π’31= 0. (9c)
Order π2
(π·2
0+ π2
1) π’12= β2π·
0π·1π’11β π1π·0π’11
β (π11cosΞ©1π0+ π12cosΞ©2π0
+π14cosΞ©4π0) π’11+ π1cosΞ©3π0,
(10a)
(π·2
0+ π2
2) π’22= β2π·
0π·1π’21β π2π·0π’21
β (π21cosΞ©1π0+ π22cosΞ©2π0
+π24cosΞ©4π0) π’21+ π2cosΞ©3π0,
(10b)
(π·2
0+ π2
3) π’32= β2π·
0π·1π’31β π3π·0π’31
β (π31cosΞ©1π0+ π32cosΞ©2π0
+π34cosΞ©4π0) π’31+ π3cosΞ©3π0.
(10c)
Order π3
(π·2
0+ π2
1) π’13= βπ·2
1π’11β 2π·0π·2π’11β 2π·0π·1π’12
β π1(π·0π’12+ π·1π’11)
β (π11cosΞ©1π0+ π12cosΞ©2π0
+ π14cosΞ©4π0) π’12
β πΌ1π’2
11π’21β πΌ2π’2
11π’31β πΌ3π’2
21π’11
β πΌ4π’2
21π’31β πΌ5π’2
31π’11β πΌ6π’2
31π’21
β πΌ7π’3
11β πΌ8π’3
21β πΌ9π’3
31β πΌ10π’11π’21π’31,
(11a)
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Mathematical Problems in Engineering 5
(π·2
0+ π2
2) π’23= βπ·2
1π’21β 2π·0π·2π’21β 2π·0π·1π’22
β π2(π·0π’22+ π·1π’21)
β (π21cosΞ©1π0+ π22cosΞ©2π0
+π24cosΞ©4π0) π’22
β π½1π’2
11π’21β π½2π’2
11π’31β π½3π’2
21π’11
β π½4π’2
21π’31β π½5π’2
31π’11β π½6π’2
31π’21
β π½7π’3
11β π½8π’3
21β π½9π’3
31β π½10π’11π’21π’31,
(11b)
(π·2
0+ π2
3) π’33= βπ·2
1π’31β 2π·0π·2π’31β 2π·0π·1π’32
β π3(π·0π’32+ π·1π’31)
β (π31cosΞ©1π0+ π32cosΞ©2π0
+π34cosΞ©4π0) π’32
β πΎ1π’2
11π’21β πΎ2π’2
11π’31β πΎ3π’2
21π’11
β πΎ4π’2
21π’31β πΎ5π’2
31π’11β πΎ6π’2
31π’21
β πΎ7π’3
11β πΎ8π’3
21β πΎ9π’3
31β πΎ10π’11π’21π’31.
(11c)
The general solutions of (9a), (9b), and (9c) can be written inthe form
π’11= π΄1(π1, π2) exp (ππ
1π0) + ππ, (12a)
π’21= π΄2(π1, π2) exp (ππ
2π0) + ππ, (12b)
π’31= π΄3(π1, π2) exp (ππ
3π0) + ππ, (12c)
whereπ΄1,π΄2, andπ΄
3are a complex function inπ
1, π2which
can be determined from eliminating the secular terms at thenext approximation and ππ 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
π’12= πΈ1exp (ππ
1π0) + πΈ2exp (π (Ξ©
1+ π1) π0)
+ πΈ3exp (π (Ξ©
1β π1) π0) + πΈ4exp (π (Ξ©
2+ π1) π0)
+ πΈ5exp (π (Ξ©
2β π1) π0) + πΈ6exp (π (Ξ©
4+ π1) π0)
+ πΈ7exp (π (Ξ©
4β π1) π0) + ππ,
(13a)
π’22= πΈ8exp (ππ
2π0) + πΈ9exp (π (Ξ©
1+ π2) π0)
+ πΈ10exp (π (Ξ©
1β π2) π0) + πΈ11exp (π (Ξ©
2+ π2) π0)
+ πΈ12exp (π (Ξ©
2β π2) π0) + πΈ13exp (π (Ξ©
4+ π2) π0)
+ πΈ14exp (π (Ξ©
4β π2) π0) + ππ,
(13b)
π’32= πΈ15exp (ππ
3π0) + πΈ16exp (π (Ξ©
1+ π3) π0)
+ πΈ17exp (π (Ξ©
1β π3) π0) + πΈ18exp (π (Ξ©
2+ π3) π0)
+ πΈ19exp (π (Ξ©
2β π3) π0) + πΈ20exp (π (Ξ©
4+ π3) π0)
+ πΈ21exp (π (Ξ©
4β π3) π0) + πΈ22exp (πΞ©
3π0) + ππ,
(13c)
where πΈπ(π = 1, 2, . . . , 22) are the complex functions in π
1
and π2. From (12a)β(13c) into (11a), (11b), and (11c) and elim-
inating the secular terms, the second-order approximation isgiven by
π’13(π0, π1, π2) = π»
1exp (ππ
2π0) + π»2exp (ππ
3π0)
+ π»3exp (3ππ
1π0) + π»4exp (3ππ
2π0)
+ π»5exp (3ππ
3π0)
+ π»6exp (π (π
2Β± 2π1) π0)
+ π»7exp (π (π
3Β± 2π1) π0)
+ π»8exp (π (2π
2Β± π1) π0)
+ π»9exp (π (2π
3Β± π1) π0)
+ π»10exp (π (π
2Β± 2π3) π0)
+ π»11exp (π (π
3Β± 2π2) π0)
+ π»12exp (π (π
3Β± π2Β± π1) π0)
+ π»13exp (πΞ©
3π0)
+ π»14exp (π (Ξ©
3Β± Ξ©1) π0)
+ π»15exp (π (Ξ©
3Β± Ξ©2) π0)
+ π»16exp (π (Ξ©
3Β± Ξ©4) π0)
+ π»17exp (π (Ξ©
1Β± π1) π0)
+ π»18exp (π (Ξ©
2Β± π1) π0)
+ π»19exp (π (Ξ©
4Β± π1) π0)
+ π»20exp (π (2Ξ©
1Β± π1) π0)
+ π»21exp (π (2Ξ©
2Β± π1) π0)
+ π»22exp (π (2Ξ©
4Β± π1) π0)
+ π»23exp (π (Ξ©
2Β± Ξ©1Β± π1) π0)
+ π»24exp (π (Ξ©
4Β± Ξ©1Β± π1) π0)
+ π»25exp (π (Ξ©
4Β± Ξ©2Β± π1) π0) + ππ,
(14a)π’23(π0, π1, π2) = π»
26exp (ππ
1π0) + π»27exp (ππ
3π0)
+ π»28exp (3ππ
1π0) + π»29exp (3ππ
2π0)
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6 Mathematical Problems in Engineering
+ π»30exp (3ππ
3π0)
+ π»31exp (π (π
2Β± 2π1) π0)
+ π»32exp (π (2π
2Β± π1) π0)
+ π»33exp (π (π
3Β± 2π1) π0)
+ π»34exp (π (2π
3Β± π1) π0)
+ π»35exp (π (π
2Β± 2π3) π0)
+ π»36exp (π (2π
2Β± π3) π0)
+ π»37exp (π (π
3Β± π2Β± π1) π0)
+ π»38exp (πΞ©
3π0)
+ π»39exp (π (Ξ©
3Β± Ξ©1) π0)
+ π»40exp (π (Ξ©
3Β± Ξ©2) π0)
+ π»41exp (π (Ξ©
4Β± Ξ©3) π0)
+ π»42exp (π (Ξ©
1Β± π2) π0)
+ π»43exp (π (Ξ©
2Β± π2) π0)
+ π»44exp (π (Ξ©
4Β± π2) π0)
+ π»45exp (π (2Ξ©
1Β± π2) π0)
+ π»46exp (π (2Ξ©
2Β± π2) π0)
+ π»47exp (π (2Ξ©
4Β± π2) π0)
+ π»48exp (π (Ξ©
2Β± Ξ©1Β± π2) π0)
+ π»49exp (π (Ξ©
4Β± Ξ©1Β± π2) π0)
+ π»50exp (π (Ξ©
4Β± Ξ©2Β± π2) π0) + ππ,
(14b)π’33(π0, π1, π2) = π»
51exp (3ππ
3π0) + π»52exp (ππ
1π0)
+ π»53exp (3ππ
1π0) + π»54exp (ππ
2π0)
+ π»55exp (3ππ
2π0)
+ π»56exp (π (π
2Β± 2π1) π0)
+ π»57exp (π (2π
2Β± π1) π0)
+ π»58exp (π (π
3Β± 2π1) π0)
+ π»59exp (π (2π
3Β± π1) π0)
+ π»60exp (π (π
2Β± 2π3) π0)
+ π»61exp (π (2π
2Β± π3) π0)
+ π»62exp (π (π
3Β± π2Β± π1) π0)
+ π»63exp (πΞ©
3π0)
+ π»64exp (π (Ξ©
3Β± Ξ©1) π0)
+ π»65exp (π (Ξ©
3Β± Ξ©2) π0)
+ π»66exp (π (Ξ©
3Β± Ξ©4) π0)
+ π»67exp (π (Ξ©
1Β± π3) π0)
+ π»68exp (π (Ξ©
2Β± π3) π0)
+ π»69exp (π (Ξ©
4Β± π3) π0)
+ π»70exp (π (2Ξ©
1Β± π3) π0)
+ π»71exp (π (2Ξ©
2Β± π3) π0)
+ π»72exp (π (2Ξ©
4Β± π3) π0)
+ π»73exp (π (Ξ©
2Β± Ξ©1Β± π3) π0)
+ π»74exp (π (Ξ©
4Β± Ξ©1Β± π3) π0)
+ π»75exp (π (Ξ©
4Β± Ξ©2Β± π3) π0) + ππ,
(14c)
where π»π(π = 1, 2, . . . , 75) are the complex functions in π
1
and π2. From the above derived solutions, the reported
resonance cases are the following.
(i) Primary resonance:Ξ©1β ππ,Ξ©2β ππ,Ξ©3β ππ,Ξ©4β
ππ, and π = 1, 2, 3.
(ii) Subharmonic resonance: Ξ©1β 2ππ, Ξ©2β 2ππ, Ξ©4β
2ππ, and π = 1, 2, 3.
(iii) Internal or secondary resonance: π1β π2, π2β π3,
π3β π1, π1β 3ππ , π2β 3ππ, π3β 3ππ, π = 2, 3,
π = 1, 3, andπ = 1, 2.(iv) Combined resonance: π
3Β± π2β 2π1, π3Β± π1β 2π2,
π1Β±π2β 2π3,π2Β±2π3β π1,π3Β±2π2β π1,π3Β±2π1β
π2, π1Β± 2π3β π2, π2Β± 2π1β π3, π1Β± 2π2β π3,
Ξ©3Β± Ξ©π‘β ππ, Ξ©4Β± Ξ©πβ 2ππ, Ξ©2Β± Ξ©1β 2ππ,
π‘ = 1, 2, 4, π = 1, 2, and π = 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 Ξ©
3β π1, π2β π1, and π
3β 3π1.
To describe how close the frequencies are to the resonanceconditions, we introduce detuning parameters as follows:
Ξ©3= π1+ π1= π1+ ποΏ½ΜοΏ½1,
π2= π1+ π2= π1+ ποΏ½ΜοΏ½2,
π3= 3π1+ π3= 3π1+ ποΏ½ΜοΏ½3,
(15)
-
Mathematical Problems in Engineering 7
whereπ1andπ2,π3are called the external and internal detun-
ing parameters, respectively. Eliminating the secular termsleads to solvability conditions for the first- and second-orderexpansions as follows:
2ππ1π·1π΄1= βππ
1π1π΄1+π1
2exp (ποΏ½ΜοΏ½
1π1) , (16a)
2ππ2π·1π΄2= βππ
2π2π΄2+π2
2exp (π (οΏ½ΜοΏ½
1β οΏ½ΜοΏ½2) π1) , (16b)
2ππ3π·1π΄3= βππ
3π3π΄3, (16c)
2ππ1π·2π΄1
= βπ·2
1π΄1β π1π·1π΄1
β {π2
11
2 (Ξ©2
1β 4π2
1)+
π2
12
2 (Ξ©2
2β 4π2
1)+
π2
14
2 (Ξ©2
4β 4π2
1)}π΄1
β {2πΌ1π΄1π΄1+ 2πΌ6π΄3π΄3+ 3πΌ8π΄2π΄2}π΄2exp (ποΏ½ΜοΏ½
2π1)
β πΌ1π΄2
1π΄2exp (βποΏ½ΜοΏ½
2π1) β πΌ2π΄2
1π΄3exp (ποΏ½ΜοΏ½
3π1)
β {2πΌ3π΄2π΄2+ 2πΌ5π΄3π΄3+ 3πΌ7π΄1π΄1}π΄1
β πΌ4π΄2
2π΄3exp (π (οΏ½ΜοΏ½
3β 2οΏ½ΜοΏ½2) π1)
β πΌ10π΄1π΄2π΄3exp (π (οΏ½ΜοΏ½
3β οΏ½ΜοΏ½2) π1) ,
(17a)2ππ2π·2π΄2
= βπ·2
1π΄2β π2π·1π΄2
β {π2
21
2 (Ξ©2
1β 4π2
2)+
π2
22
2 (Ξ©2
2β 4π2
2)+
π2
24
2 (Ξ©2
4β 4π2
2)}π΄2
β {2π½3π΄2π΄2+ 2π½5π΄3π΄3+ 3π½7π΄1π΄1}
Γ π΄1exp (βποΏ½ΜοΏ½
2π1)
β π½3π΄2
2π΄1exp (ποΏ½ΜοΏ½
2π1) β π½1π΄2
1π΄2exp (β2ποΏ½ΜοΏ½
2π1)
β {2π½6π΄3π΄3+ 2π½1π΄1π΄1+ 3π½8π΄2π΄2}π΄2
β π½2π΄2
1π΄3exp (π (οΏ½ΜοΏ½
3β οΏ½ΜοΏ½2) π1)
β π½10π΄1π΄2π΄3exp (π (οΏ½ΜοΏ½
3β 2οΏ½ΜοΏ½2) π1)
β π½4π΄2
2π΄3exp (π (οΏ½ΜοΏ½
3β 3οΏ½ΜοΏ½2) π1) ,
(17b)2ππ3π·2π΄3
= βπ·2
1π΄3β π3π·1π΄3
β {π2
31
2 (Ξ©2
1β 4π2
3)+
π2
32
2 (Ξ©2
2β 4π2
3)+
π2
34
2 (Ξ©2
4β 4π2
3)}π΄3
β πΎ10π΄1π΄2π΄3exp (βποΏ½ΜοΏ½
2π1)
β πΎ10π΄1π΄2π΄3exp (ποΏ½ΜοΏ½
2π1)
β πΎ7π΄3
1exp (βποΏ½ΜοΏ½
3π1)
β {2πΎ2π΄1π΄1+ 2πΎ4π΄2π΄2+ 3πΎ9π΄3π΄3}π΄3
β πΎ1π΄2
1π΄2exp (π (οΏ½ΜοΏ½
2β οΏ½ΜοΏ½3) π1)
β πΎ3π΄2
2π΄1exp (π (2οΏ½ΜοΏ½
2β οΏ½ΜοΏ½3) π1)
β πΎ8π΄3
2exp (π (3οΏ½ΜοΏ½
2β οΏ½ΜοΏ½3) π1) .
(17c)
From (7a), multiplying both sides by 2πππ, we get
2πππ
ππ΄π
ππ‘= π2ππ
ππ·1π΄π+ π22ππππ·2π΄π,
π = 1, 2, 3.
(18)
To analyze the solutions of (16a)β(17c), we express π΄πin the
polar form as follows:
π΄π(π1, π2) =
ππ
2exp (ππ
π) ,
ππ= πππ, (π = 1, 2, 3) ,
(19)
where ππand π
πare the steady-state amplitudes and phases
of the motion, respectively. Substituting (16a)β(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:
Μπ1= β
π1
2π1+ {
π1
2π1
βπ1π1
4π2
1
} sin π1
+π1π1
8π2
1
cos π1βπΌ1
8π1
π2
1π2sin π2
βπΌ2
8π1
π2
1π3sin π3βπΌ4
8π1
π2
2π3sin (π3β 2π2)
βπΌ6
4π1
π2
3π2sin π2β3πΌ8
8π1
π3
2sin π2
βπΌ10
8π1
π1π2π3sin (π3β π2) ,
π1οΏ½ΜοΏ½1= {β
π2
1
8π1
+Ξ1
2π1
}π1β {
π1
2π1
βπ1π1
4π2
1
} cos π1
+π1π1
8π2
1
sin π1+3πΌ1
8π1
π2
1π2cos π2+πΌ2
8π1
π2
1π3cos π3
+πΌ3
4π1
π1π2
2+πΌ4
8π1
π2
2π3cos (π
3β 2π2)
+πΌ5
4π1
π1π2
3+πΌ6
4π1
π2
3π2cos π2+3πΌ7
8π1
π3
1
-
8 Mathematical Problems in Engineering
+3πΌ8
8π1
π3
2cos π2+πΌ10
8π1
π1π2π3cos (π
3β π2) ,
Μπ2= β
π2
2π2+ {
π2
2π2
β(π1β π2) π2
4π2
2
} sin (π1β π2)
+π2π2
8π2
2
cos (π1β π2) +
π½1
8π2
π2
1π2sin 2π
2
βπ½2
8π2
π2
1π3sin (π3β π2) +
π½3
8π2
π2
2π1sin π2
βπ½4
8π2
π2
2π3sin (π3β 3π2) +
π½5
4π2
π2
3π1sin π2
+3π½7
8π2
π3
1sin π2βπ½10
8π2
π1π2π3sin (π3β 2π2) ,
π2οΏ½ΜοΏ½2= {β
π2
2
8π2
+Ξ2
2π2
}π2
+ {(π1β π2) π2
4π2
2
βπ2
2π2
} cos (π1β π2)
+π2π2
8π2
2
sin (π1β π2) +
π½1
8π2
π2
1π2cos 2π
2
+π½1
4π2
π2
1π2+π½2
8π2
π2
1π3cos (π
3β π2)
+3π½3
8π2
π2
2π1cos π2+π½4
8π2
π2
2π3cos (π
3β 3π2)
+π½5
4π2
π2
3π1cos π2+π½6
4π2
π2
3π2+3π½7
8π2
π3
1cos π2
+3π½8
8π2
π3
2+π½10
8π2
π1π2π3cos (π
3β 2π2) ,
Μπ3= β
π3
2π3βπΎ1
8π3
π2
1π2sin (π2β π3)
βπΎ3
8π3
π2
2π1sin (2π
2β π3) +
πΎ7
8π3
π3
1sin π3
βπΎ8
8π3
π3
2sin (3π
2β π3) ,
π3οΏ½ΜοΏ½3= {β
π2
3
8π3
+Ξ3
2π3
}π3+πΎ1
8π3
π2
1π2cos (π
2β π3)
+πΎ2
4π3
π2
1π3+πΎ3
8π3
π2
2π1cos (2π
2β π3)
+πΎ4
4π3
π2
2π3+πΎ7
8π3
π3
1cos π3
+πΎ8
8π3
π3
2cos (3π
2β π3) +
3πΎ9
8π3
π3
3
+πΎ10
4π3
π1π2π3cos π2,
(20)
where
Ξπ= {
π2
π1
2 (Ξ©2
1β 4π2π)+
π2
π2
2 (Ξ©2
2β 4π2π)+
π2
π4
2 (Ξ©2
4β 4π2π)} ,
π = 1, 2, 3,
π1= οΏ½ΜοΏ½1π1β π1,
π2= οΏ½ΜοΏ½2π1+ π2β π1,
π3= οΏ½ΜοΏ½3π1+ π3β 3π1.
(21)
Steady-state solutions of the system correspond to the fixedpoints of (20), which in turn correspond to
οΏ½ΜοΏ½1= π1,
οΏ½ΜοΏ½2= π1β π2,
οΏ½ΜοΏ½3= 3π1β π3.
(22)
Hence, the fixed points of (20) are given by
βπ1
2π1+ {
π1
2π1
βπ1π1
4π2
1
} sin π1
+π1π1
8π2
1
cos π1βπΌ1
8π1
π2
1π2sin π2
βπΌ2
8π1
π2
1π3sin π3βπΌ4
8π1
π2
2π3sin (π3β 2π2)
βπΌ6
4π1
π2
3π2sin π2β3πΌ8
8π1
π3
2sin π2
βπΌ10
8π1
π1π2π3sin (π3β π2) = 0,
π1π1+ {
π2
1
8π1
βΞ1
2π1
}π1
+ {π1
2π1
βπ1π1
4π2
1
} cos π1βπ1π1
8π2
1
sin π1
β3πΌ1
8π1
π2
1π2cos π2βπΌ2
8π1
π2
1π3cos π3
βπΌ3
4π1
π1π2
2βπΌ4
8π1
π2
2π3cos (π
3β 2π2) β
πΌ5
4π1
π1π2
3
βπΌ6
4π1
π2
3π2cos π2β3πΌ7
8π1
π3
1β3πΌ8
8π1
π3
2cos π2
βπΌ10
8π1
π1π2π3cos (π
3β π2) = 0,
βπ2
2π2+ {
π2
2π2
β(π1β π2) π2
4π2
2
} sin (π1β π2)
+π2π2
8π2
2
cos (π1β π2)
+π½1
8π2
π2
1π2sin 2π
2βπ½2
8π2
π2
1π3sin (π3β π2)
-
Mathematical Problems in Engineering 9
+π½3
8π2
π2
2π1sin π2βπ½4
8π2
π2
2π3sin (π3β 3π2)
+π½5
4π2
π2
3π1sin π2+3π½7
8π2
π3
1sin π2
βπ½10
8π2
π1π2π3sin (π3β 2π2) = 0,
π2(π2β π1) β {
π2
2
8π2
βΞ2
2π2
}π2
+ {(π1β π2) π2
4π2
2
βπ2
2π2
} cos (π1β π2)
+π2π2
8π2
2
sin (π1β π2) +
π½1
8π2
π2
1π2cos 2π
2
+π½1
4π2
π2
1π2+π½2
8π2
π2
1π3cos (π
3β π2)
+3π½3
8π2
π2
2π1cos π2+π½4
8π2
π2
2π3cos (π
3β 3π2)
+π½5
4π2
π2
3π1cos π2+π½6
4π2
π2
3π2+3π½7
8π2
π3
1cos π2
+3π½8
8π2
π3
2+π½10
8π2
π1π2π3cos (π
3β 2π2) = 0,
βπ3
2π3βπΎ1
8π3
π2
1π2sin (π2β π3) β
πΎ3
8π3
π2
2π1sin (2π
2β π3)
+πΎ7
8π3
π3
1sin π3βπΎ8
8π3
π3
2sin (3π
2β π3) = 0,
π3(π3β 3π1) β {
π2
3
8π3
βΞ3
2π3
}π3+πΎ1
8π3
π2
1π2cos (π
2β π3)
+πΎ2
4π3
π2
1π3+πΎ3
8π3
π2
2π1cos (2π
2β π3)
+πΎ4
4π3
π2
2π3+πΎ7
8π3
π3
1cos π3+πΎ8
8π3
π3
2cos (3π
2β π3)
+3πΎ9
8π3
π3
3+πΎ10
4π3
π1π2π3cos π2= 0.
(23)
There are six possibilities besides the trivial solution asfollows:
(1) π1ΜΈ= 0, π2= 0, π
3= 0 (single mode),
(2) π2ΜΈ= 0, π1= 0, π
3= 0 (single mode),
(3) π1ΜΈ= 0, π2ΜΈ= 0, π3= 0 (two modes),
(4) π1ΜΈ= 0, π3ΜΈ= 0, π2= 0 (two modes),
(5) π2ΜΈ= 0, π3ΜΈ= 0, π1= 0 (two modes),
(6) π1ΜΈ= 0, π2ΜΈ= 0, π3ΜΈ= 0 (three modes).
Case 1. In this case, where π2= 0, π
3= 0, the frequency
response equation is given by
9πΌ2
7
64π2
1
π6
1+ [π 3+3πΌ7π1
4π1
] π4
1+ [π 2+ π2
1+π2
1π1
4π1
βΞ1π1
π1
] π2
1
βπ2
1π2
1
64π4
1
β π 2
1= 0.
(24)
Case 2. In this case, where π1= 0, π
3= 0, the frequency
response equation is given by
9π½2
8
64π2
2
π6
2+ [π3+3π½8(π2β π1)
4π2
] π4
2
+ [π2+ (π2β π1)2βπ2
2(π2β π1)
4π2
+Ξ2(π2β π1)
π2
] π2
2
βπ2
2π2
2
64π4
2
β π2
1= 0.
(25)
Case 3. In this case, where π3= 0, the frequency response
equations are given by
9πΌ2
7
64π2
1
π6
1+ [π 3+3πΌ7π1
4π1
] π4
1+ [π 2+ π2
1+π2
1π1
4π1
βΞ1π1
π1
] π2
1
βπ2
1π2
1
64π4
1
β π 2
1β9πΌ2
1
64π2
1
π4
1π2
2β9πΌ1πΌ8
32π2
1
π2
1π4
2
β9πΌ2
8
64π2
1
π6
2+3π 1πΌ8
4π1
π3
2+3π 1πΌ1
4π1
π2
1π2= 0,
9π½2
8
64π2
2
π6
2+ [π3+3π½8(π2β π1)
4π2
] π4
2
+ [π2+ (π2β π1)2βπ2
2(π2β π1)
4π2
+Ξ2(π2β π1)
π2
] π2
2
βπ2
2π2
2
64π4
2
β π2
1+3π1π½3
4π2
π2
2π1+3π1π½7
4π2
π3
1+π1π½1
4π2
π2
1π2
β9π½2
3
64π2
2
π4
2π2
1β [
π½2
1
64π2
2
+9π½3π½7
32π2
2
] π4
1π2
2β9π½2
7
64π2
2
π6
1
β3π½1π½7
32π2
2
π5
1π2β3π½1π½3
32π2
2
π3
1π3
2= 0.
(26)
Case 4. In this case, where π2= 0, the frequency response
equations are given by
9πΌ2
7
64π2
1
π6
1+ [π 3+3πΌ7π1
4π1
] π4
1+ [π 2+ π2
1+π2
1π1
4π1
βΞ1π1
π1
] π2
1
βπ2
1π2
1
64π4
1
β π 2
1βπΌ2
2
64π2
1
π4
1π2
3+π 1πΌ2
4π1
π2
1π3= 0,
-
10 Mathematical Problems in Engineering
0 50 100 150 200β1
β0.5
0
0.5
1
Timeβ0.4 β0.2 0 0.2 0.4
β4
β2
0
2
4
Velo
city
0 50 100 150 200β0.4
β0.2
0
0.2
0.4
Timeβ0.2 β0.1 0 0.1 0.2
β1
β0.5
0
0.5
1
Velo
city
0 50 100 150 200β0.4
β0.2
0
0.2
0.4
Timeβ0.2 β0.1 0 0.1 0.2β1
β0.5
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.
9πΎ2
9
64π2
3
π6
3+ [πΎ2+3πΎ9(π3β 3π1)
4π3
] π4
3
+ [πΎ1+ (π3β 3π1)2βπ2
3(π3β 3π1)
4π3
+Ξ3(π3β 3π1)
π3
] π2
3
βπΎ2
7
64π2
3
π6
1= 0.
(27)
Case 5. In this case, where π1= 0, the frequency response
equations are given by
9π½2
8
64π2
2
π6
2+ [π3+3π½8(π2β π1)
4π2
] π4
2
+ [π2+ (π2β π1)2βπ2
2(π2β π1)
4π2
+Ξ2(π2β π1)
π2
] π2
2
βπ2
2π2
2
64π4
2
β π2
1+π1π½4
4π2
π2
2π3βπ½2
4
64π2
2
π4
2π2
3= 0,
9πΎ2
9
64π2
3
π6
3+ [πΎ2+3πΎ9(π3β 3π1)
4π3
] π4
3
+ [πΎ1+ (π3β 3π1)2βπ2
3(π3β 3π1)
4π3
+Ξ3(π3β 3π1)
π3
] π2
3
βπΎ2
8
64π2
3
π6
2= 0.
(28)
Case 6. In this case, where π1ΜΈ= 0, π2ΜΈ= 0, and π
3ΜΈ= 0, this is the
practical case, the frequency response equations are given by
9πΌ2
7
64π2
1
π6
1+ [π 3+3πΌ7π1
4π1
] π4
1+ [π 2+ π2
1+π2
1π1
4π1
βΞ1π1
π1
] π2
1
βπ2
1π2
1
64π4
1
β π 2
1+π 1πΌ2
4π1
π2
1π3+π 1πΌ6
2π1
π2π2
3+π 1πΌ4
4π1
π2
2π3
+3π 1πΌ8
4π1
π3
2+3π 1πΌ1
4π1
π2
1π2+π 1πΌ10
4π1
π1π2π3βπΌ2
6
16π2
1
π2
2π4
3
-
Mathematical Problems in Engineering 11A
mpl
itude
u3
0 50 100 150 200β2
β1
0
1
2
Timeβ2 β1 0 1 2
β10
0
10
Velo
city
0 50 100 150 200β2
β1
0
1
2
Time
β2 β1 0 1 2β10
β5
0
5
10
Velo
city
0 50 100 150 200β0.5
0
0.5
Timeβ0.4 β0.2 0 0.2 0.4
β10
β5
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, Ξ©3β π1, π2β π1, and π
3β 3π1.
βπΌ4πΌ6
16π2
1
π3
2π3
3βπΌ2
2
64π2
1
π4
1π2
3β9πΌ2
1
64π2
1
π4
1π2
2β9πΌ1πΌ8
32π2
1
π2
1π4
2
β9πΌ2
8
64π2
1
π6
2β3πΌ4πΌ8
32π2
1
π5
2π3β3πΌ8πΌ10
32π2
1
π1π4
2π3βπΌ2πΌ10
32π2
1
π3
1π2π2
3
βπΌ4πΌ10
32π2
1
π1π3
2π2
3βπΌ6πΌ10
16π2
1
π1π2
2π3
3βπΌ2πΌ6
16π2
1
π2
1π2π3
3
β3πΌ1πΌ10
32π2
1
π3
1π2
2π3β3πΌ1πΌ2
32π2
1
π4
1π2π3β π 4π2
1π2
2π2
3
β π 5π4
2π2
3β π 6π3
2π2
1π3= 0,
9π½2
8
64π2
2
π6
2+ [π3+3π½8(π2β π1)
4π2
] π4
2
+ [π2+ (π2β π1)2βπ2
2(π2β π1)
4π2
+Ξ2(π2β π1)
π2
] π2
2
βπ2
2π2
2
64π4
2
β π2
1+3π1π½3
4π2
π2
2π1+3π1π½7
4π2
π3
1+π1π½1
4π2
π2
1π2
+π1π½4
4π2
π2
2π3+π1π½5
2π2
π1π2
3+π1π½2
4π2
π2
1π3+π1π½10
4π2
π1π2π3
β3π½1π½7
32π2
2
π5
1π2β9π½2
3
64π2
2
π4
2π2
1β9π½2
7
64π2
2
π6
1βπ½2
4
64π2
2
π4
2π2
3
β3π½1π½3
32π2
2
π3
1π3
2βπ½2
5
16π2
2
π2
1π4
3βπ½2π½5
16π2
2
π3
1π3
3β3π½3π½4
32π2
2
π1π4
2π3
β3π½2π½7
32π2
2
π5
1π3βπ½4π½5
16π2
2
π1π2
2π3
3βπ½5π½10
16π2
2
π2
1π2π3
2βπ½4π½10
32π2
2
π1π3
2π2
3
β π4π2
1π2
2π2
3β π5π4
1π2
2β π6π4
1π2
3β π7π3
1π2
2π3β π8π2
1π3
2π3
β π9π3
1π2π2
3β π10π4
1π2π3= 0,
9πΎ2
9
64π2
3
π6
3+ [πΎ2+3πΎ9(π3β 3π1)
4π3
] π4
3
+[πΎ1+ (π3β3π1)2βπ2
3(π3β 3π1)
4π3
+Ξ3(π3β 3π1)
π3
] π2
3
-
12 Mathematical Problems in Engineering
0 50 100 150 200β2
β1
0
1
2
3
Time0 50 100 150 200
β3
β2
β1
0
1
2
3
Time
0 50 100 150 200
β0.4
β0.2
0
0.2
0.4
0.6
0.8
Time
Numerical solutionπ1(π‘) perturbation solution
Numerical solutionπ2(π‘) perturbation solution
Numerical solutionπ3(π‘) 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, Ξ©
3β π1, π2β π1, and π
3β 3π1.
βπΎ2
8
64π2
3
π6
2β
πΎ2
7
64π2
3
π6
1βπΎ1πΎ7
32π2
3
π5
1π2βπΎ3πΎ8
32π2
3
π1π5
2
β πΎ3π4
1π2
2β πΎ4π2
1π4
2β πΎ5π3
1π2
2= 0,
(29)
where
π 1= [
π1
2π1
βπ1π1
4π2
1
] ,
π 2= [
π2
1
4+π4
1
64π2
1
+Ξ2
1
4π2
1
βΞ1π2
1
8π2
1
] ,
π 3= [
3πΌ7π2
1
32π2
1
β3πΌ7Ξ1
8π2
1
] ,
π 4= [
πΌ2
10
64π2
1
+3πΌ1πΌ6
16π2
1
+πΌ2πΌ4
32π2
1
] ,
π 5= [
3πΌ6πΌ8
16π2
1
+πΌ2
4
64π2
1
] ,
π 6= [
3πΌ1πΌ4
32π2
1
+3πΌ2πΌ8
32π2
1
] ,
π1= [
π2
2π2
β(π1β π2) π2
4π2
2
] ,
-
Mathematical Problems in Engineering 13
200 200.5 201 201.5 202 202.5 203β0.5
0
0.5
Time200 200.5 201 201.5 202 202.5 203
β0.2
β0.1
0
0.1
0.2
Time
200 200.5 201 201.5 202 202.5 203β0.2
β0.1
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.
π2= [
π2
2
4+π4
2
64π2
2
+Ξ2
2
4π2
2
βΞ2π2
2
8π2
2
] ,
π3= [
3π½8Ξ2
8π2
2
β3π½8π2
2
32π2
2
] ,
π4= [
π½2
10
64π2
2
+3π½3π½5
16π2
2
+π½2π½4
32π2
2
] ,
π5= [
π½2
1
64π2
2
+9π½3π½7
32π2
2
] ,
π6= [
π½2
2
64π2
2
+3π½5π½7
16π2
2
] ,
π7= [
3π½2π½3
32π2
2
+3π½4π½7
32π2
2
+π½1π½10
32π2
2
] ,
π8= [
3π½3π½10
32π2
2
+π½1π½4
32π2
2
] ,
π9= [
π½2π½10
32π2
2
+π½1π½5
16π2
2
] ,
π10= [
3π½7π½10
32π2
2
+π½1π½2
32π2
2
] ,
πΎ1= [
π2
3
4+π4
3
64π2
3
+Ξ2
3
4π2
3
βΞ3π2
3
8π2
3
] ,
πΎ2= [
3πΎ9Ξ3
8π2
3
β3πΎ9π2
3
32π2
3
] ,
πΎ3= [
πΎ2
1
64π2
3
+πΎ3πΎ7
32π2
3
] ,
πΎ4= [
πΎ2
3
64π2
3
+πΎ1πΎ8
32π2
3
] ,
πΎ5= [
πΎ1πΎ3
32π2
3
+πΎ7πΎ8
32π2
3
] .
(30)
-
14 Mathematical Problems in Engineering
β1.5 β1 β0.5 0 0.5 1 1.5
2
4
6
8
10
Analytical solutionNumerical solution
π1a 1
Figure 6: Comparison between analytical prediction using multiple time scale and numerical integration of the first mode.
β1 β0.5 0 0.5 1
5
10
15
20
0.2
0.1
0.3
a 1
π1
Figure 7: Effects of the linear damping π1.
β1.5 β1 β0.5 0 0.5 1 1.5
2
4
6
8
10
12
141.7
2.3
4.3
a 1
π1
Figure 8: Effects of the natural frequency π1.
β1.5 β1 β0.5 0 0.5 1 1.5
5
10
15
20
4
8
3
a 1
π1
Figure 9: Effects of the external excitation π1.
-
Mathematical Problems in Engineering 15
β1.5 β1 β0.5 0 0.5 1 1.5
2
4
6
8
10
12
0.06
β0.06
0.01
a 1π1
Figure 10: Effects of the nonlinear parameter πΌ7.
β1.5 β1 β0.5 0 0.5 1 1.5
2
4
6
8
10
0.5510
a 1
π1
Figure 11: Effects of the parametric excitation π14.
β1.5 β1 β0.5 0 0.5 1 1.5
123456
Analytical solutionNumerical solution
a 2
π2
Figure 12: Comparison between analytical prediction using multiple time scale and numerical integration of the second mode.
β1.5 β1 β0.5
β0.08
0
0.2
0.5 1 1.50
2
0.01
4
6
a 2
π2
Figure 13: Effects of the nonlinear parameter π½8.
-
16 Mathematical Problems in Engineering
β1 β0.5 0 0.5
0.1
0.2
0.3
10
4
8
a 2
π2
Figure 14: Effects of the linear damping π2.
β1.5 β1 β0.5 0
4
2.2
0.5 1 1.50
2
4
6
8
a 2
π2
Figure 15: Effects of the natural frequency π2.
β1.5 β1 β0.5 0 0.5 1 1.50
2
4
6
8
2
4
6
a 2
π2
Figure 16: Effects of the external excitation π2.
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: π
1= 0.05, π
2= 0.05, π
3= 0.05, πΌ
1= π½1= πΎ1=
1.5, πΌ2= π½2= πΎ2= 1.9, πΌ
3= π½3= πΎ3= 0.4, πΌ
4= π½4= πΎ4=
0.6, πΌ5= π½5= πΎ5= 0.6, πΌ
6= π½6= πΎ6= 0.2, πΌ
7= π½7= πΎ7=
0.01, πΌ8= π½8= πΎ8= 0.01, πΌ
9= π½9= πΎ9= 0.4, πΌ
10= π½10=
πΎ10= 1.8, Ξ©
1= 6.4, Ξ©
2= 6.1, Ξ©
3= 5.7, Ξ©
4= 2, π
1= 4.7,
-
Mathematical Problems in Engineering 17
β2 β1 0 1 20
2
4
6
Am
plitu
des
π1
a1
a1
a2a2
a3a3
Figure 17: Effects of the detuning parameter π1.
β1 0 1
4
6
1
β1 0 10.5
1.5
2.5
1
β1 0 1
0.2
0.6
1
1.4
1
f1 = 10
f1 = 10
f1 = 10
a 1
a 2
a 3
π1
π1π1
Figure 18: Effects of the external excitation force π1.
π2= 2.7, π
3= 13.1, π
1= π2= 4, π
3= 20.5, π
11= π21= π31=
0.1,π12= π22= π32= 0.2, andπ
14= π24= π34= 0.5. Figure 3
shows the steady-state amplitudes and phase plane of thesystem at simultaneous resonance caseΞ©
3β π1,π2β π1, and
π3β 3π1. 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
-
18 Mathematical Problems in Engineering
β2 β1 0 1 22
3
4
5
6
7
2.5
1.5
β2 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
β1
β1
a 2a 1
a 3
π1
π1π1
πΌ1 = 0.5
π½1 = 0.5
πΎ1 = 0.5
Figure 19: Effects of the nonlinear parameters (πΌ1, π½1, and πΎ
1coupling terms).
working condition for the system. It is advised for such systemnot to have π
2= π1or π3= 3π1. 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(π2). The infinite equations(2a)β(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)β(2e) can be reduced to the finite dimensional one via the finitedifference method with second-order approximation π(π2).Namely, at each mesh node the following system of ordinary
differential equations is obtained:
πΏ1,π(ππ₯π₯, ππ₯π¦) = πΌ0(οΏ½ΜοΏ½0)π,π+ π½1( Μππ₯)π,π
β2
3β2ππΌ3((οΏ½ΜοΏ½0)π+1,π
β (οΏ½ΜοΏ½0)πβ1,π
) ,
πΏ2,π(ππ₯π¦, ππ¦π¦) = πΌ0(VΜ0)π,π+ π½1( Μππ¦)π,π
β2
3β2ππΌ3((οΏ½ΜοΏ½0)π,π+1
β (οΏ½ΜοΏ½0)π,πβ1
) ,
πΏ3,π(ππ₯, ππ¦, ππ₯π₯, π€0, ππ₯π¦)
= πΌ0(οΏ½ΜοΏ½0)π,πβ
16
9β4π2πΌ6((οΏ½ΜοΏ½0)π+1,π
β 2(οΏ½ΜοΏ½0)π,π+ (οΏ½ΜοΏ½0)π+1,π
+(οΏ½ΜοΏ½0)π,π+1
β2(οΏ½ΜοΏ½0)π,π+(οΏ½ΜοΏ½0)π,π+1
)
-
Mathematical Problems in Engineering 19
β2 β1 0 1
3
4
5
6
10
5
β1 0 10.5
1
1.5
2
2.5
5
10
β2 β1 0 1
0.2
0.6
1
5
10
π1
π1π1
a 2a1
a 3
πΌ2 = 0.1π½2 = 0.1
πΎ2 = 0.1
Figure 20: Effects of the nonlinear parameters (πΌ2, π½2, and πΎ
2coupling terms).
+4
3β2π2πΌ3((οΏ½ΜοΏ½0)π+1,π
β (οΏ½ΜοΏ½0)πβ1,π
+ (VΜ0)π,π+1
β (VΜ0)π,πβ1
)
+π½4
2π(( Μππ₯)π+1,π
β ( Μππ₯)πβ1,π
+ ( Μππ¦)π,π+1
β ( Μππ¦)π,πβ1
) ,
πΏ4,π(ππ₯π₯,ππ₯π¦) = π½1(οΏ½ΜοΏ½0)π,π+ π2( Μππ₯)π,π
β2π½4
3β2π((οΏ½ΜοΏ½0)π+1,π
β (οΏ½ΜοΏ½0)πβ1,π
) ,
πΏ5,π(ππ₯π¦,ππ¦π¦) = π½1(VΜ0)π,π+ π2( Μππ¦)π,π
β2π½4
3β2π((οΏ½ΜοΏ½0)π,π+1
β (οΏ½ΜοΏ½0)π,πβ1
) ,
(π = 1, 2, . . . , π) , (π = 1, 2, . . . , π) ,
(31)
where π denotes the partition number regarding a spatialcoordinate, π is the computational step regarding spatialcoordinate, and πΏ
1,π(β ), πΏ2,π(β ), πΏ3,π(β ), πΏ4,π(β ), and πΏ
5,π(β ) are
the difference operators as follows:
πΏ1,π(ππ₯π₯, ππ₯π¦) =
1
2π((ππ₯π₯)π+1,π
β (ππ₯π₯)πβ1,π
+ (ππ₯π¦)π,π+1
β (ππ₯π¦)π,πβ1
) ,
πΏ2,π(ππ₯π¦, ππ¦π¦) =
1
2π((ππ₯π¦)π+1,π
β (ππ₯π¦)πβ1,π
+ (ππ¦π¦)π,π+1
β (ππ¦π¦)π,πβ1
) ,
-
20 Mathematical Problems in Engineering
πΏ3,π(ππ₯, ππ¦, ππ₯π₯, π€0, ππ₯π¦) =
1
2π((ππ₯)π+1,π
β (ππ₯)πβ1,π
+ (ππ¦)π,π+1
β (ππ¦)π,πβ1
)
+ ππ₯π₯(
(π€0)πβ1,π
β 2(π€0)π,π+ (π€0)π+1,π
π2) + (
(π€0)π+1,π
β (π€0)πβ1,π
2π)(
(ππ₯π₯)π+1,π
β (ππ₯π₯)πβ1,π
2π)
+ ππ₯π¦(
((π€0)π+1,π+1
β 2(π€0)π+1,πβ1
) ((π€0)πβ1,π+1
β 2(π€0)πβ1,πβ1
)
4π2)
+ (
(π€0)π,π+1
β (π€0)π,πβ1
2π)(
(ππ₯π¦)π+1,π
β (ππ₯π¦)πβ1,π
2π)
+ ππ₯π¦(
((π€0)π+1,π+1
β (π€0)π+1,πβ1
) β ((π€0)πβ1,π+1
β (π€0)πβ1,πβ1
)
8π3)
+ (
(π€0)π+1,π
β (π€0)πβ1,π
2π)(
(ππ¦π¦)π,π+1
β (ππ¦π¦)π,πβ1
2π) + π
π¦π¦(
(π€0)π,πβ1
β 2(π€0)π,π+ (π€0)π,π+1
π2)
+ (
(π€0)π,π+1
β (π€0)π,πβ1
2π)(
(ππ¦π¦)π,π+1
β (ππ¦π¦)π,πβ1
2π) +
4
3β2(
(ππ₯π₯)πβ1,π
β 2(ππ₯π₯)π,π+ (ππ₯π₯)π+1,π
π2)
+4
3β2(2(
((ππ₯π¦)π+1,π+1
β (ππ₯π¦)πβ1,π+1
) β ((ππ₯π¦)π+1,πβ1
β (ππ₯π¦)πβ1,πβ1
)
8π3)+
(ππ¦π¦)π,πβ1
β 2(ππ¦π¦)π,π+ (ππ¦π¦)π,π+1
π2)+ π,
πΏ4,π(ππ₯π₯,ππ₯π¦) =
1
2π((ππ₯π₯)π+1,π
β (ππ₯π₯)πβ1,π
+ (ππ₯π¦)π,π+1
β (ππ₯π¦)π,πβ1
) β ππ₯,
πΏ5,π(ππ₯π¦,ππ¦π¦) =
1
2π((ππ₯π¦)π+1,π
β (ππ₯π¦)πβ1,π
+ (ππ¦π¦)π,π+1
β (ππ¦π¦)π,πβ1
) β ππ¦.
(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)β(29) arenonlinear equations in π
1, π2, and π
3which are solved
numerically.Thenumerical results are shown in Figures 6β25,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 π
1on the steady-state amplitude of the first mode
for the stability first case, where π1ΜΈ= 0, π2= 0, and π
3= 0, for
the parameters π1= 0.2, πΌ
7= 0.01, π
1= 2.3, π
1= 4, π
11=
0.1, π12= 0.2, π
14= 0.5, Ξ©
1= 1, Ξ©
2= 1.2, and Ξ©
4= 1.4, as
shown in Figure 6.Figures 7β11, show the effects of the damping coeffi-
cient π1, the first mode natural frequency π
1, the external
-
Mathematical Problems in Engineering 21
β1 0 12
4
63
8
β2 0 2
1.5
2.5
3.5
8
3
β2 0 2
0.2
0.6
1
8
3
π1
π1π1
a 2
a 1
a 3
πΌ8 = 1.5
π½8 = 1.5
πΎ8 = 1.5
Figure 21: Effects of the nonlinear parameters (πΌ8, π½8, and πΎ
8coupling terms).
excitation amplitudeπ1, the nonlinear spring stiffness πΌ
7, and
the parametric excitation π14. Figures 7 and 8 show that the
steady-state amplitude π1is inversely proportional to π
1and
π1, and also for decreasingπ
1orπ1the curve is bending to the
left. It is clear from Figure 9 that the steady-state amplitudeπ1is increasing for increasing value of external excitation
force π1, and the zone of instability is increased. Figure 10
shows that as the nonlinear spring stiffness πΌ7is increased;
the continuous curve ismoved downwards. Also, the negativeand positive values of πΌ
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 ofπΌ
7. From
Figure 11, we observe that for increasing value of parametricexcitation amplitude π
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 12β16, show thefrequency response curves for the stability of the second case,where π
2ΜΈ= 0, π
1= 0, and π
3= 0. Figure 12 shows a com-
parison of these outputs for the second mode. The effects ofthe detuning parameter π
2on the steady-state amplitude of
the second mode for the stability second case, where π2ΜΈ= 0,
π1= 0, π
3= 0, for the parameters: π
2= 0.2, π½
8= 0.01,
π2= 4, π
2= 4, π
21= 0.1, π
22= 0.2, π
24= 0.5, Ξ©
1= 1, Ξ©
2=
1.2, and Ξ©4= 1.4, as shown in Figure 12. It can be seen from
the figure thatmaximum steady-state amplitude occurs whenπ2β π1. Figure 13 shows that as the nonlinear spring stiffness
π½8is increased, the continuous curve is moved downwards.
Also, the positive and negative values of π½8produce 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 π
2is inversely proportional to π
2, π2and
directly proportional to the external excitation π2. Also, for
decreasing π2, π2the 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 π
1ΜΈ= 0, π2ΜΈ= 0, and π
3ΜΈ= 0.
-
22 Mathematical Problems in Engineering
β2 0 22
4
6
8
10
1
β2 0 2
0.5
1.5
2.5
3.5
4.5
0.01
β2 0 2
0.2
0.6
1
1.4
0.01
a 3
a 2
a 1
π1π1
π1
πΌ7 = 0.01π½7 = 1
πΎ7 = 1
Figure 22: Effects of the nonlinear parameters (πΌ7, π½7, and πΎ
7coupling terms).
Figure 17 shows that the effects of the detuning parameterπ1on the amplitudes of the three modes. From this figure,
we observe that these modes intersect, and for positive valueof the detuning parameter π
1the amplitudes are stable. For
negative value of the detuning parameter π1down to and
including β0.5 the system becomes unstable.Figure 18 shows that the steady-state amplitudes of the
three modes π1, π2, and π
3are directly proportional to
the external excitation force π1. Also, form this figure we
show that the stability region is decreased for increasing π1.
Figures 19β21 show that the steady-state amplitudes andstability of the threemodes π
1, π2, and π
3are inversely propor-
tional to the nonlinear parameters (πΌ1, πΌ2, πΌ8), (π½1, π½2, π½8),
and (πΎ1, πΎ2, πΎ8), respectively.
Figures 22β24, show the effects of the nonlinear param-eters (πΌ
7, π½7, πΎ7), (πΌ3, π½3, πΎ3), and (πΌ
5,π½5, πΎ5) 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
-
Mathematical Problems in Engineering 23
β2 β1 0 1 20
2
4
6
8
5
10
1.5
β2 β1 0 1 20.5
1.5
2.5
3.5
5
β1
β1 β0.5 0 0.50
0.5
1
1.5
2
5
1.5-1
π1
π1π1
a 3
a 2
a 1
πΌ3 = β1
π½3 = 10
πΎ3 = 10
Figure 23: Effects of the nonlinear parameters (πΌ3, π½3, and πΎ
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, whereΞ©
3β π1, π2β π1, and
π3β 3π1.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
-
24 Mathematical Problems in Engineering
β2 β1 0 12
3
4
5
6
40
β2 β1 0 10.5
1.5
2.5
3.5
20
β1 0 10.2
0.6
1
40
π1
π1π1
a 3
a 2a 1
πΌ5 = 20
π½5 = 40
πΎ5 = 20
Figure 24: Effects of the nonlinear parameters (πΌ5, π½5, and πΎ
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 Ξ©3β π1, π2β π1,
and π3β 3π
1is one of the worst case, and they
should be avoided in design of such system.Of course,the excitation frequencyΞ©
3is out of control. But this
case should be avoided through having π2ΜΈ= π1or
π3ΜΈ= 3π1.
(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 π1, π2, πΌ7, π½8, π1, π2, π1,
π2leads tomultivalued amplitudes and hence to jump
phenomena.
(5) For the first and second modes, the steady-stateamplitudes π
1and π2are directly proportional to the
excitation amplitude π1and π
2and inversely propor-
tional to the linear damping π1and π
2, respectively.
(6) The negative and positive values of nonlinear stiffnessπΌ7, π½8produce 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 π1, π2and
for negative values of nonlinear stiffness πΌ7, π½8.
(8) The multivalued solutions are disappeared forincreasing linear damping coefficients π
1, π2.
(9) The region of stability increases, which is desirablefor decreasing excitation amplitude π
1, nonlinear
parameters (πΌ1, π½1, πΎ1), (πΌ2, π½2, πΎ2), and for increasing
nonlinear stiffness (πΌ7, π½7, πΎ7).
(10) The steady-state amplitudes of the three modes π1,
π2, and π
3are directly proportional to the exci-
tation amplitude π1and inversely proportional to
-
Mathematical Problems in Engineering 25
β1 0 13
4
5
6
0.5
0.2
β1 0 1
0.5
1.5
2.5
0.5
0.2
β2 β1 0 1
0.4
0.8
1.2
1.6
0.5
0.2
π1
π1π1
a 3
a 2
a 1
π1 = 1.2
π2 = 1.2
π3 = 1.2
Figure 25: Effects of the linear damping coefficients.
the nonlinear parameters (πΌ1, πΌ2, πΌ8), (π½1, π½2, π½8), and
(πΎ1, πΎ2, πΎ8), 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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